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These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good
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4453:{\displaystyle {\begin{aligned}f_{M_{t}}(m)&=\int _{-\infty }^{m}f_{M_{t},W_{t}}(m,w)\,dw=\int _{-\infty }^{m}{\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}}\,dw\\&={\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}},\qquad m\geq 0,\end{aligned}}}
10379:
is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact â a special case of a general theorem stating that all
Brownian martingales are continuous. A Brownian martingale is, by definition, a
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2127:{\displaystyle {\begin{aligned}\operatorname {cov} (W_{s},W_{t})&=s,\\\operatorname {corr} (W_{s},W_{t})&={\frac {\operatorname {cov} (W_{s},W_{t})}{\sigma _{W_{s}}\sigma _{W_{t}}}}={\frac {s}{\sqrt {st}}}={\sqrt {\frac {s}{t}}}.\end{aligned}}}
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of the origin infinitely often) whereas it is not recurrent in dimensions three and higher (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes). Unlike the random walk, it is
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9884:
Two random processes on the time interval appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of . With no further conditioning, the process takes both positive and negative values on and is called
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5751:{\displaystyle {\begin{array}{rcl}W_{1,s}(t)&=&W(t+s)-W(s),\quad s\in \mathbb {R} \\W_{2,\sigma }(t)&=&\sigma ^{-1/2}W(\sigma t),\quad \sigma >0\\W_{3}(t)&=&tW(-1/t).\end{array}}}
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13457:
Navarro-moreno, J.; Estudillo-martinez, M.D; Fernandez-alcala, R.M.; Ruiz-molina, J.C. (2009), "Estimation of
Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory",
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behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.
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5468:
8523:{\displaystyle \limsup _{\varepsilon \to 0+}\sup _{0\leq s<t\leq 1,t-s\leq \varepsilon }{\frac {|w(s)-w(t)|}{\sqrt {2\varepsilon \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.}
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1184:
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11506:
13067:
The
Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter
10539:
10368:
given the Wiener process on the time interval (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on belongs to
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5346:
5031:{\displaystyle \,F_{M_{W_{t}}}(m)=\Pr \left(M_{W_{t}}=\max _{0\leq s\leq t}W(s)\leq m\mid W(t)=W_{t}\right)=\ 1-\ e^{-2{\frac {m(m-W_{t})}{t}}}\ \,,\,\ \ m>\max(0,W_{t})}
4649:{\displaystyle \operatorname {E} =\int _{0}^{\infty }mf_{M_{t}}(m)\,dm=\int _{0}^{\infty }m{\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}}\,dm={\sqrt {\frac {2t}{\pi }}}}
524:
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675:
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Kipnis, A., Goldsmith, A.J. and Eldar, Y.C., 2019. The distortion-rate function of sampled Wiener processes. IEEE Transactions on
Information Theory, 65(1), pp.482-499.
10110:
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4122:
11122:
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1469:
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T. Berger, "Information rates of Wiener processes," in IEEE Transactions on
Information Theory, vol. 16, no. 2, pp. 134-139, March 1970. doi: 10.1109/TIT.1970.1054423
9757:
2136:
These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that
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The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time
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3701:{\displaystyle W_{t}={\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \left(\left(n-{\frac {1}{2}}\right)\pi t\right)}{\left(n-{\frac {1}{2}}\right)\pi }}}
12054:
7071:
5117:. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically.
1568:
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In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale
7862:
12124:
8813:
8344:{\displaystyle \limsup _{\varepsilon \to 0+}{\frac {|w(\varepsilon )|}{\sqrt {2\varepsilon \log \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.}
14218:
9943:
12718:
3312:
2426:{\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left)\cdot (W_{t_{2}}-\operatorname {E} )\right]=\operatorname {E} \left.}
2845:{\displaystyle {\begin{aligned}\operatorname {E} &=\operatorname {E} \left\\&=\operatorname {E} \left+\operatorname {E} \left.\end{aligned}}}
14042:
12847:
11988:
11925:
10357:
be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and
7672:
with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely.
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A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent
14476:
11912:
1037:. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed
12257:
1733:
14160:
1787:
1049:
253:. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in
14486:
14170:
11741:
11627:
10016:
It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.
9568:{\displaystyle D_{\theta }={\frac {T_{s}}{6}}+T_{s}\int _{0}^{1}\min \left\{S(\varphi )-{\frac {1}{6}},\theta \right\}d\varphi ,}
7301:{\displaystyle a(x,t)=\left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t).}
14528:
14425:
12195:
5974:{\displaystyle W_{g}(t)=(ct+d)W\left({\frac {at+b}{ct+d}}\right)-ctW\left({\frac {a}{c}}\right)-dW\left({\frac {b}{d}}\right),}
9824:
9578:
4081:{\displaystyle f_{M_{t},W_{t}}(m,w)={\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}},\qquad m\geq 0,w\leq m.}
14715:
14705:
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14243:
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are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if
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720:
14615:
14579:
6475:
1678:
6899:{\displaystyle \left({\frac {\partial }{\partial t}}-{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t)=0}
14883:
14620:
9881:, which means that they are the only continuous LĂ©vy processes, as a consequence of the LĂ©vyâKhintchine representation.
8704:(namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density
14532:
13730:
13631:
10342:
7315:
6909:
203:
for his investigations on the mathematical properties of the one-dimensional
Brownian motion. It is often also called
14685:
13548:
13512:
13199:
12473:
7657:
5416:
207:
due to its historical connection with the physical process of the same name originally observed by
Scottish botanist
70:
48:
13320:. Cambridge series in statistical and probabilistic mathematics. Cambridge: Cambridge University Press. p. 18.
41:
14730:
14536:
14520:
14435:
14263:
14233:
13655:
2857:
1307:{\displaystyle W_{n}(t)={\frac {1}{\sqrt {n}}}\sum \limits _{1\leq k\leq \lfloor nt\rfloor }\xi _{k},\qquad t\in .}
14635:
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14569:
14564:
14000:
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10381:
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7802:
7757:
6962:
5136:
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958:
250:
14574:
14203:
14198:
14005:
13902:
13402:
10083:
8216:{\displaystyle \limsup _{t\to +\infty }{\frac {|w(t)|}{\sqrt {2t\log \log t}}}=1,\quad {\text{almost surely}}.}
1143:
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9814:
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266:
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6083:
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1004:
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12476:
Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify
12336:
11826:
11681:
11444:
8693:{\displaystyle \int _{0}^{t}f(w(s))\,\mathrm {d} s=\int _{-\infty }^{+\infty }f(x)L_{t}(x)\,\mathrm {d} x}
14273:
13857:
13802:
13718:
10497:
6255:
5988:
5208:
2931:
299:
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study
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13673:
13116:
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1375:
1105:
1038:
1033:, or other discrete-time stochastic processes with stationary independent increments. This is known as
308:
289:
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13602:
12479:
12354:
stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.
14690:
14491:
14405:
14390:
14321:
13897:
13780:
13678:
13145:
11324:{\displaystyle A={\frac {\operatorname {cov} (V_{f}(t+a),V_{f}(t))}{\operatorname {Var} (V_{f}(t))}}}
9937:
2139:
316:
9166:
7702:
7388:
5300:
3548:{\displaystyle W_{t}=\xi _{0}t+{\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \pi nt}{\pi n}}}
480:
14524:
14410:
13912:
13887:
13832:
12956:
12662:
11003:{\displaystyle \operatorname {cov} (V_{f}(t+a),V_{f}(t))=\int _{0}^{t}(f(t+a)-f(s))(f(t)-f(s))\,ds}
10213:
10141:
8807:
8723:
4463:
634:
35:
12597:
12508:
4705:, it is possible to calculate the conditional probability distribution of the maximum in interval
1478:
136:
14909:
14825:
14815:
14630:
14506:
14288:
14213:
14027:
13892:
13748:
13703:
13438:
Revuz, D., & Yor, M. (1999). Continuous martingales and
Brownian motion (Vol. 293). Springer.
8762:
outside this interval the local time evidently vanishes.) Treated as a function of two variables
7542:
6973:
4746:
324:
281:
13426:
12362:
The complex-valued Wiener process may be defined as a complex-valued random process of the form
11049:
9441:{\displaystyle R(T_{s},D_{\theta })={\frac {T_{s}}{2}}\int _{0}^{1}\log _{2}^{+}\leftd\varphi ,}
9233:
328:
14767:
14695:
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14110:
13954:
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9065:
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223:
52:
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528:
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451:
390:
220:
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10338:
9817:. The image above is of the Brownian motion on a special manifold: the surface of a sphere.
9203:
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7693:
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4681:
1873:
1504:
1472:
1317:
1034:
815:
557:
359:
340:
239:
235:
11434:{\displaystyle B^{2}=\operatorname {Var} (V_{f}(t+a))-A^{2}\operatorname {Var} (V_{f}(t))}
10183:
9759:
is the mean squared error associated only with the sampling operation (without encoding).
9295:(in estimating the continuous-time Wiener process) follows the parametric representation
7796:
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6446:
6397:
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6054:
5359:
8:
14860:
14700:
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8071:
7009:
1869:
1781:
1554:
1109:
972:
270:
254:
227:
9227:
before applying a binary code to represent these samples, the optimal trade-off between
3284:{\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left=t_{1}.}
14830:
14795:
14710:
14680:
14511:
14450:
14445:
14268:
14105:
13770:
13708:
13647:
13503:
Path
Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
13501:
13475:
13384:
13246:
13070:
12933:
12638:
12541:
11638:
Every continuous martingale (starting at the origin) is a time changed Wiener process.
10384:
adapted to the
Brownian filtration; and the Brownian filtration is, by definition, the
9890:
9278:
9102:
9041:
8995:
8085:
6426:
4661:
842:
795:
700:
680:
614:
427:
336:
196:
108:
12535:
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.
9893:. In both cases a rigorous treatment involves a limiting procedure, since the formula
6763:{\displaystyle \sigma (t)=S^{-1}(t):\quad t=\int _{0}^{\sigma (t)}|f'(W(s))|^{2}\,ds.}
5517:
4708:
3789:
3711:
14850:
14063:
13980:
13949:
13842:
13822:
13812:
13668:
13663:
13544:
13508:
13321:
13296:
13238:
13195:
12340:
9273:
8806:
of the Wiener process with respect to the squared error distance, i.e. its quadratic
8779:
8093:
8064:
7844:
3161:{\displaystyle \operatorname {E} \left=\operatorname {E} \cdot \operatorname {E} =0.}
1011:
344:
332:
320:
312:
258:
129:
14655:
14306:
10692:{\displaystyle V_{f}(t)=\int _{0}^{t}f'(s)W(s)\,ds=\int _{0}^{t}(f(t)-f(s))\,dW_{s}}
14870:
14757:
14640:
14516:
14253:
14010:
13985:
13934:
13785:
13738:
13598:
Discusses history, botany and physics of Brown's original observations, with videos
13479:
13467:
13374:
13228:
12344:
12324:
10199:
8803:
8552:
7532:{\displaystyle \left(W_{t}^{2}-t\right)^{2}-4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s}
1043:
1019:
277:
262:
13862:
14835:
14735:
14720:
14481:
14415:
14093:
14037:
14020:
13765:
13567:
Takenaka, Shigeo (1988), "On pathwise projective invariance of Brownian motion",
12656:
10191:
9886:
7029:
3411:
Wiener (1923) also gave a representation of a Brownian path in terms of a random
300:
204:
193:
14650:
13882:
13610:"Interactive Web Application: Stochastic Processes used in Quantitative Finance"
12837:{\displaystyle Z_{t}^{2}=\left(X_{t}^{2}-Y_{t}^{2}\right)+2X_{t}Y_{t}i=U_{A(t)}}
11544:
10195:
9878:
212:
14840:
14805:
14725:
14331:
14078:
13995:
13964:
13939:
13929:
13872:
13847:
13827:
13792:
13743:
13496:
13062:
10831:{\displaystyle \operatorname {Var} (V_{f}(t))=\int _{0}^{t}(f(t)-f(s))^{2}\,ds}
10202:
6080:
be a two-dimensional Wiener process, regarded as a complex-valued process with
5762:, being invariant under the generators of the group. The action of an element
3412:
1672:
293:
285:
200:
101:
13867:
13520:
11551:-times-integrated Wiener process is a zero-mean normal variable with variance
14903:
14742:
14283:
14115:
14073:
14015:
13837:
13753:
13693:
13471:
13379:
13360:
13242:
13233:
13216:
10243:
of a Brownian motion describes the time that the process spends at the point
1528:
approaches a Wiener process, which explains the ubiquity of Brownian motion.
1026:
420:
11046:
is always a zero mean normal random variable. This allows for simulation of
10494:/3), calculated using the fact that the covariance of the Wiener process is
10486:. It arises in many applications and can be shown to have the distribution
7539:
is a martingale, which shows that the quadratic variation of the martingale
1780:
These results follow immediately from the definition that increments have a
93:
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14762:
14316:
14248:
14137:
14132:
13944:
13877:
13852:
13688:
13187:
13126:
10206:
10175:
5982:
303:, the diffusion of minute particles suspended in fluid, and other types of
5045:
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14380:
14364:
14359:
14354:
14344:
14147:
14088:
14083:
14047:
13807:
13698:
13281:
Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)
13132:
Probability distribution of extreme points of a Wiener stochastic process
9767:
9193:
9059:
8954:
8096:
4739:
Probability distribution of extreme points of a Wiener stochastic process
3442:
are independent Gaussian variables with mean zero and variance one, then
1536:
1030:
274:
216:
185:
13587:
12343:) is related to the Wiener process via a combination of time change and
12115:{\displaystyle -\infty <M_{\infty }^{-}=M_{\infty }^{+}<+\infty ,}
10186:. The family of these random variables (indexed by all positive numbers
7685:
takes both (strictly) positive and (strictly) negative values on (0, Δ).
7171:{\displaystyle M_{t}=p(W_{t},t)-\int _{0}^{t}a(W_{s},s)\,\mathrm {d} s,}
1662:{\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}.}
1003:(0, 1) random variables. This representation can be obtained using the
14855:
14395:
14339:
14223:
14176:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
13616:
13541:
Probability and Random Processes with Applications to Signal Processing
13388:
13250:
12316:
Especially, a nonnegative continuous martingale has a finite limit (as
8711:
is (more exactly, can and will be chosen to be) continuous. The number
6778:
953:
An alternative characterisation of the Wiener process is the so-called
13316:
Mörters, Peter; Peres, Yuval; Schramm, Oded; Werner, Wendelin (2010).
7958:{\displaystyle \lim _{s\to t}{\frac {|w(s)-w(t)|}{|s-t|}}\to \infty .}
946:
are independent random variables, and the similar condition holds for
168:
14349:
13447:
Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.
12185:{\displaystyle -\infty =M_{\infty }^{-}<M_{\infty }^{+}=+\infty ;}
11619:{\displaystyle {\frac {t}{2n+1}}\left({\frac {t^{n}}{n!}}\right)^{2}}
9889:. Conditioned also to stay positive on (0, 1), the process is called
9783:
9228:
304:
331:
can be represented in terms of the Wiener process) and the study of
8888:{\displaystyle R(D)={\frac {2}{\pi ^{2}\ln 2D}}\approx 0.29D^{-1}.}
5759:
1723:
1125:
13592:
176:
13403:"Interview Questions VII: Integrated Brownian Motion â Quantopia"
13217:"Stochastic and Multiple Wiener Integrals for Gaussian Processes"
13121:
10007:{\displaystyle e^{\mu t-{\frac {\sigma ^{2}t}{2}}+\sigma W_{t}}.}
9877:
and infinitesimal variance Ï. These processes exhaust continuous
1540:
Five sampled processes, with expected standard deviation in gray.
1531:
957:
that says that the Wiener process is an almost surely continuous
243:
231:
5814:{\displaystyle g={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}
5758:
Thus the Wiener process is invariant under the projective group
3392:{\displaystyle W_{t_{2}}=W_{t_{1}}+{\sqrt {t_{2}-t_{1}}}\cdot Z}
12323:
All stated (in this subsection) for martingales holds also for
8770:, the local time is still continuous. Treated as a function of
12306:{\displaystyle M_{\infty }^{-}<M_{\infty }^{+}<+\infty }
7692:
is continuous everywhere but differentiable nowhere (like the
12923:{\displaystyle A(t)=4\int _{0}^{t}|Z_{s}|^{2}\,\mathrm {d} s}
12042:{\displaystyle M_{\infty }^{+}=\limsup _{t\to \infty }M_{t}.}
11979:{\displaystyle M_{\infty }^{-}=\liminf _{t\to \infty }M_{t},}
9799:
3293:
A corollary useful for simulation is that we can write, for
1018:) of a zero mean, unit variance, delta correlated ("white")
273:, the Wiener process is used to represent the integral of a
13603:"Einstein's prediction finally witnessed one century later"
13361:"A relation between Brownian bridge and Brownian excursion"
11805:{\displaystyle A(t)=4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s}
10182:> 0 by the Wiener process is a random variable with the
1135:
14156:
Autoregressive conditional heteroskedasticity (ARCH) model
13293:
Stochastic Calculus for Finance II: Continuous Time Models
11543:. All these results can be seen as direct consequences of
10326:{\displaystyle L^{x}(t)=\int _{0}^{t}\delta (x-B_{t})\,ds}
9200:
it first. When the Wiener process is sampled at intervals
8992:
and recover it with expected mean squared error less than
863:
That the process has independent increments means that if
350:
180:
A single realization of a three-dimensional Wiener process
13605: : a test to observe the velocity of Brownian motion
13315:
12247:{\displaystyle M_{\infty }^{-}=M_{\infty }^{+}=+\infty ,}
8989:
6245:{\displaystyle \tau _{D}=\inf\{t\geq 0|W(t)\not \in D\}.}
6046:
3779:{\displaystyle {\sqrt {c}}\,W\left({\frac {t}{c}}\right)}
2528:{\displaystyle W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}}
13684:
Independent and identically distributed random variables
9645:{\displaystyle S(\varphi )=(2\sin(\pi \varphi /2))^{-2}}
8543:
of a set under a Brownian motion doubles almost surely.
172:
A single realization of a one-dimensional Wiener process
7632:{\displaystyle 4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s.}
14161:
Autoregressive integrated moving average (ARIMA) model
10341:. The behaviour of the local time is characterised by
8036:+ Δ). (Local increase is a weaker condition than that
7810:
7765:
5780:
1190:
random variables with mean 0 and variance 1. For each
784:{\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).}
13073:
13032:
13005:
12959:
12936:
12850:
12721:
12665:
12641:
12600:
12564:
12544:
12511:
12482:
12451:
12424:
12368:
12260:
12198:
12127:
12057:
11991:
11928:
11829:
11744:
11684:
11557:
11514:
11447:
11336:
11216:
11130:
11094:
11052:
11016:
10843:
10731:
10705:
10550:
10500:
10402:
10253:
10144:
10118:
10092:
10025:
9946:
9827:
9730:
9658:
9581:
9453:
9301:
9281:
9236:
9206:
9169:
9111:
9068:
9044:
9018:
8998:
8963:
8901:
8816:
8580:
8362:
8238:
8124:
7865:
7805:
7760:
7731:
7705:
7583:
7545:
7445:
7391:
7318:
7188:
7074:
6976:
6912:
6804:
6635:
6539:
6478:
6449:
6429:
6400:
6367:
6331:
6296:
6258:
6184:
6157:
6129:
6086:
6057:
5991:
5827:
5768:
5515:
5476:
5419:
5391:
5362:
5303:
5211:
5139:
5055:
4782:
4755:
4711:
4684:
4664:
4472:
4141:
4096:
3907:
3836:
3792:
3746:
3714:
3561:
3448:
3421:
3315:
3176:
2988:
2934:
2860:
2541:
2441:
2182:
2142:
1908:
1882:
1790:
1736:
1681:
1571:
1507:
1481:
1436:
1378:
1347:
1320:
1200:
1146:
1052:
845:
818:
798:
723:
703:
683:
637:
617:
587:
560:
531:
483:
454:
430:
393:
362:
139:
111:
6623:{\displaystyle S(t)=\int _{0}^{t}|f'(W(s))|^{2}\,ds}
1010:
Another characterisation of a Wiener process is the
11199:{\displaystyle V_{f}(t+a)=A\cdot V_{f}(t)+B\cdot Z}
13500:
13079:
13045:
13018:
12991:
12942:
12922:
12836:
12702:
12647:
12619:
12586:
12550:
12519:
12497:
12464:
12437:
12410:
12305:
12246:
12184:
12114:
12041:
11978:
11877:
11804:
11730:
11618:
11535:
11500:
11433:
11323:
11198:
11116:
11080:
11038:
11002:
10830:
10717:
10691:
10533:
10471:{\displaystyle W^{(-1)}(t):=\int _{0}^{t}W(s)\,ds}
10470:
10325:
10163:
10130:
10104:
10074:
10006:
9865:
9751:
9716:
9644:
9567:
9440:
9287:
9264:
9219:
9181:
9155:
9093:
9050:
9030:
9004:
8981:
8945:
8887:
8692:
8522:
8343:
8215:
7957:
7832:
7787:
7746:
7717:
7663:
7631:
7569:
7531:
7431:
7378:
7300:
7170:
7068:the following stochastic process is a martingale:
7000:
6953:
6898:
6772:
6762:
6622:
6526:
6464:
6435:
6415:
6382:
6353:
6317:
6278:
6244:
6170:
6143:
6115:
6072:
6036:
5973:
5813:
5750:
5497:
5462:
5405:
5377:
5340:
5256:
5189:
5105:
5030:
4768:
4729:
4697:
4670:
4648:
4452:
4116:
4080:
3884:
3810:
3778:
3732:
3700:
3547:
3434:
3391:
3283:
3160:
2974:
2920:
2844:
2527:
2425:
2168:
2126:
1894:
1853:
1770:
1712:
1661:
1520:
1493:
1463:
1422:
1360:
1333:
1306:
1178:
1088:
851:
831:
804:
783:
709:
689:
669:
623:
602:
573:
546:
518:
469:
436:
412:
375:
155:
117:
12710:is a time-changed complex-valued Wiener process.
7976:> 0 satisfies the following for some Δ in (0,
14901:
14043:Stochastic chains with memory of variable length
12357:
12051:Then only the following two cases are possible:
12011:
11948:
10544:For the general case of the process defined by
10513:
9684:
9512:
8383:
8364:
8240:
8126:
8112:
7867:
7379:{\displaystyle p(x,t)=\left(x^{2}-t\right)^{2},}
6198:
5505:. Then the following are all Wiener processes (
5421:
5003:
4849:
4820:
3885:{\displaystyle M_{t}=\max _{0\leq s\leq t}W_{s}}
3851:
13087:while others define it for general dimensions.
5463:{\displaystyle \lim _{t\to \pm \infty }tW(t)=0}
18:Stochastic process generalizing Brownian motion
13053:are independent Wiener processes, as before).
12538:Rotation invariance: for every complex number
3830:The joint distribution of the running maximum
1771:{\displaystyle \operatorname {Var} (W_{t})=t.}
1532:Properties of a one-dimensional Wiener process
1314:This is a random step function. Increments of
1194:, define a continuous time stochastic process
384:is characterised by the following properties:
13632:
13507:(4th ed.). Singapore: World Scientific.
13214:
10391:
8539:The dimension doubling theorems say that the
8532:
7972:has no points of local increase, that is, no
5049:A demonstration of Brownian scaling, showing
2921:{\displaystyle W_{t_{1}}=W_{t_{1}}-W_{t_{0}}}
1863:
1854:{\displaystyle W_{t}=W_{t}-W_{0}\sim N(0,t).}
1096:is a Wiener process for any nonzero constant
1025:The Wiener process can be constructed as the
13531:Conformally invariant processes in the plane
13215:Huang, Steel T.; Cambanis, Stamatis (1978).
13194:(5th ed.). Cambridge University Press.
9711:
9687:
9126:
9112:
8916:
8902:
6236:
6201:
4743:cumulative probability distribution function
3403:is an independent standard normal variable.
1261:
1252:
1089:{\displaystyle \alpha ^{-1}W_{\alpha ^{2}t}}
13543:(3rd ed.). New Jersey: Prentice Hall.
12351:
9717:{\displaystyle \log ^{+}=\max\{0,\log(x)\}}
7833:{\displaystyle ({\tfrac {1}{2}}-\epsilon )}
7788:{\displaystyle ({\tfrac {1}{2}}+\epsilon )}
5190:{\displaystyle V_{t}=(1/{\sqrt {c}})W_{ct}}
5106:{\displaystyle V_{t}=(1/{\sqrt {c}})W_{ct}}
315:. It also forms the basis for the rigorous
14171:Autoregressiveâmoving-average (ARMA) model
13639:
13625:
13588:Brownian Motion for the School-Going Child
13560:Continuous martingales and Brownian motion
13538:
12950:is another complex-valued Wiener process.
12627:is another complex-valued Wiener process.
11665:is another Wiener process (different from
8107:
280:, and so is useful as a model of noise in
92:
13593:Brownian Motion, "Diverse and Undulating"
13378:
13232:
12911:
12513:
12485:
11793:
10993:
10821:
10675:
10617:
10461:
10316:
9101:distinct elements such that the expected
8681:
8617:
7675:
7617:
7520:
7156:
6750:
6613:
6272:
6137:
6109:
5596:
5470:(which holds almost surely) and as usual
5399:
4990:
4986:
4783:
4619:
4542:
4364:
4242:
4090:To get the unconditional distribution of
3754:
1179:{\displaystyle \xi _{1},\xi _{2},\ldots }
199:named in honor of American mathematician
71:Learn how and when to remove this message
13646:
13566:
13557:
13495:
10396:The time-integral of the Wiener process
9866:{\displaystyle X_{t}=\mu t+\sigma W_{t}}
9798:
9782:
9766:
8754:are the least and the greatest value of
8225:
5506:
5351:
5044:
3406:
1535:
1136:Wiener process as a limit of random walk
1128:based on Wiener measure may be called a
175:
167:
34:This article includes a list of general
13460:IEEE Transactions on Information Theory
13427:"Variance of integrated Wiener process"
13358:
13186:
11628:Cauchy formula for repeated integration
10348:
351:Characterisations of the Wiener process
14902:
14477:Doob's martingale convergence theorems
13528:
13290:
11913:Doob's martingale convergence theorems
10075:{\displaystyle X_{t}=e^{-t}W_{e^{2t}}}
9803:The generator of a Brownian motion is
8895:Therefore, it is impossible to encode
8056:).) The same holds for local decrease.
7008:is a martingale, which shows that the
6391:
6116:{\displaystyle W(0)=0\in \mathbb {C} }
6047:Conformal invariance in two dimensions
4462:the probability density function of a
1726:, using the computational formula, is
14229:Constant elasticity of variance (CEV)
14219:ChanâKarolyiâLongstaffâSanders (CKLS)
13620:
10205:of this process is given by times of
8734:on . It is strictly positive for all
7053:More generally, for every polynomial
6527:{\displaystyle Y(t)=f(W(\sigma (t)))}
6144:{\displaystyle D\subset \mathbb {C} }
4678:the Wiener process has a known value
1713:{\displaystyle \operatorname {E} =0.}
13095:
12530:
12330:
11878:{\displaystyle M_{t}-M_{0}=V_{A(t)}}
11731:{\displaystyle W_{t}^{2}-t=V_{A(t)}}
11501:{\displaystyle V_{f}(t)=W^{(-1)}(t)}
9762:
6361:is a time-changed Wiener process in
1124:, induced by the Wiener process. An
812:has almost surely continuous paths:
226:) and occurs frequently in pure and
20:
13562:(Second ed.). Springer-Verlag.
10534:{\displaystyle t\wedge s=\min(t,s)}
9192:In many cases, it is impossible to
8797:
7656:more general than polynomials, see
6279:{\displaystyle f:D\to \mathbb {C} }
6037:{\displaystyle (W_{g})_{h}=W_{gh}.}
5257:{\displaystyle V_{t}=W_{1-t}-W_{1}}
5121:
2975:{\displaystyle W_{t_{2}}-W_{t_{1}}}
1544:
1236:
554:are independent of the past values
13:
14716:Skorokhod's representation theorem
14497:Law of large numbers (weak/strong)
13539:Stark, Henry; Woods, John (2002).
12913:
12411:{\displaystyle Z_{t}=X_{t}+iY_{t}}
12300:
12284:
12266:
12238:
12222:
12204:
12176:
12160:
12142:
12131:
12106:
12090:
12072:
12061:
12021:
11997:
11958:
11934:
11795:
11210:is a standard normal variable and
9821:The stochastic process defined by
8683:
8645:
8637:
8619:
8139:
7949:
7619:
7522:
7256:
7246:
7221:
7217:
7158:
6851:
6841:
6816:
6812:
5434:
5040:
4562:
4508:
4473:
4260:
4187:
3825:
3598:
3501:
3229:
3103:
3071:
2989:
2799:
2717:
2602:
2546:
2367:
2327:
2269:
2235:
1682:
1488:
758:
677:is normally distributed with mean
40:it lacks sufficient corresponding
14:
14931:
14686:Martingale representation theorem
13581:
13558:Revuz, Daniel; Yor, Marc (1994).
13056:
10388:generated by the Wiener process.
9156:{\displaystyle \{w_{t}\}_{t\in }}
9031:{\displaystyle \varepsilon >0}
8946:{\displaystyle \{w_{t}\}_{t\in }}
7681:For every Δ > 0, the function
6290:which is not constant, such that
6151:be an open set containing 0, and
5406:{\displaystyle t\in \mathbb {R} }
5292:
1423:{\displaystyle W_{n}(t)-W_{n}(s)}
14731:Stochastic differential equation
14621:Doob's optional stopping theorem
14616:DoobâMeyer decomposition theorem
13192:Probability: Theory and Examples
12630:
12498:{\displaystyle \mathbb {R} ^{2}}
11922:be a continuous martingale, and
11823:is a continuous martingale then
9787:2D Wiener processes with drift (
7965:The same holds for local minima.
6954:{\displaystyle M_{t}=p(W_{t},t)}
6394:). More precisely, the process
5200:
25:
14601:Convergence of random variables
14487:FisherâTippettâGnedenko theorem
13450:
13441:
13432:
13419:
13395:
13352:
13265:"PĂłlya's Random Walk Constants"
10364:the conditional probability of
8786:measure on the set of zeros of
8778:is fixed), the local time is a
8511:
8332:
8204:
7664:Some properties of sample paths
7028:. It follows that the expected
6773:A class of Brownian martingales
6676:
5675:
5588:
4433:
4053:
3708:represent a Brownian motion on
2169:{\displaystyle t_{1}\leq t_{2}}
1279:
839:is almost surely continuous in
265:. It is the driving process of
14199:Binomial options pricing model
13343:
13334:
13309:
13284:
13275:
13257:
13208:
13180:
13177:N.Wiener Collected Works vol.1
13171:
12901:
12885:
12860:
12854:
12829:
12823:
12697:
12691:
12682:
12669:
12574:
12566:
12018:
11955:
11870:
11864:
11754:
11748:
11723:
11717:
11633:
11524:
11518:
11495:
11489:
11484:
11475:
11464:
11458:
11428:
11425:
11419:
11406:
11384:
11381:
11369:
11356:
11315:
11312:
11306:
11293:
11282:
11279:
11273:
11257:
11245:
11232:
11181:
11175:
11153:
11141:
11111:
11105:
11075:
11063:
11033:
11027:
10990:
10987:
10981:
10972:
10966:
10960:
10957:
10954:
10948:
10939:
10927:
10921:
10900:
10897:
10891:
10875:
10863:
10850:
10812:
10808:
10802:
10793:
10787:
10781:
10760:
10757:
10751:
10738:
10672:
10669:
10663:
10654:
10648:
10642:
10614:
10608:
10602:
10596:
10567:
10561:
10528:
10516:
10458:
10452:
10428:
10422:
10417:
10408:
10313:
10294:
10270:
10264:
9708:
9702:
9678:
9672:
9630:
9626:
9609:
9597:
9591:
9585:
9529:
9523:
9403:
9397:
9331:
9305:
9259:
9240:
9182:{\displaystyle D-\varepsilon }
9148:
9136:
9086:
9080:
8976:
8970:
8938:
8926:
8826:
8820:
8700:for a wide class of functions
8678:
8672:
8659:
8653:
8614:
8611:
8605:
8599:
8496:
8482:
8464:
8460:
8454:
8445:
8439:
8432:
8371:
8317:
8303:
8279:
8275:
8269:
8262:
8247:
8165:
8161:
8155:
8148:
8133:
7946:
7939:
7925:
7918:
7914:
7908:
7899:
7893:
7886:
7874:
7827:
7806:
7782:
7761:
7741:
7735:
7718:{\displaystyle \epsilon >0}
7432:{\displaystyle a(x,t)=4x^{2};}
7407:
7395:
7334:
7322:
7292:
7280:
7204:
7192:
7153:
7134:
7110:
7091:
6948:
6929:
6887:
6875:
6740:
6735:
6732:
6726:
6720:
6708:
6702:
6696:
6670:
6664:
6645:
6639:
6603:
6598:
6595:
6589:
6583:
6571:
6549:
6543:
6521:
6518:
6515:
6509:
6503:
6497:
6488:
6482:
6459:
6453:
6410:
6404:
6377:
6371:
6348:
6335:
6306:
6300:
6268:
6227:
6221:
6214:
6096:
6090:
6067:
6061:
6006:
5992:
5865:
5850:
5844:
5838:
5738:
5721:
5705:
5699:
5669:
5660:
5626:
5620:
5582:
5576:
5567:
5555:
5542:
5536:
5486:
5480:
5451:
5445:
5428:
5372:
5366:
5341:{\displaystyle V_{t}=tW_{1/t}}
5171:
5153:
5087:
5069:
5025:
5006:
4972:
4953:
4900:
4894:
4879:
4873:
4814:
4808:
4724:
4712:
4539:
4533:
4492:
4479:
4342:
4326:
4291:
4276:
4239:
4227:
4169:
4163:
4028:
4012:
3977:
3962:
3950:
3938:
3805:
3793:
3727:
3715:
3223:
3183:
3149:
3109:
3097:
3077:
3060:
3020:
2788:
2748:
2699:
2676:
2636:
2633:
2592:
2552:
2502:
2462:
2356:
2353:
2333:
2304:
2298:
2295:
2275:
2246:
2229:
2189:
2042:
2016:
1997:
1971:
1945:
1919:
1845:
1833:
1756:
1743:
1701:
1688:
1651:
1642:
1595:
1589:
1485:
1471:by the central limit theorem.
1458:
1440:
1417:
1411:
1395:
1389:
1298:
1286:
1217:
1211:
775:
763:
519:{\displaystyle W_{t+u}-W_{t},}
341:mathematical theory of finance
339:. It is also prominent in the
211:. It is one of the best known
1:
14666:Kolmogorov continuity theorem
14502:Law of the iterated logarithm
13489:
12992:{\displaystyle 2X_{t}+iY_{t}}
12703:{\displaystyle f(Z_{t})-f(0)}
12358:Complex-valued Wiener process
10164:{\displaystyle \sigma ^{2}=2}
9771:Wiener processes with drift (
9012:. On the other hand, for any
8546:
8232:Local modulus of continuity:
8114:Law of the iterated logarithm
8103:1/2 (therefore, uncountable).
6797:partial differential equation
1557:with mean = 0 and variance =
670:{\displaystyle W_{t+u}-W_{t}}
14671:Kolmogorov extension theorem
14350:Generalized queueing network
13858:Interacting particle systems
12620:{\displaystyle c\cdot Z_{t}}
12520:{\displaystyle \mathbb {C} }
12313:etc.) are of probability 0.
8354:Global modulus of continuity
6906:then the stochastic process
1551:probability density function
1494:{\displaystyle n\to \infty }
1341:are independent because the
156:{\displaystyle \sigma ^{2}t}
90:Probability density function
7:
13803:Continuous-time random walk
13190:(2019). "Brownian Motion".
13090:
11816:is another Wiener process.
9875:Wiener process with drift Ό
9196:the Wiener process without
7570:{\displaystyle W_{t}^{2}-t}
7001:{\displaystyle W_{t}^{2}-t}
6178:be associated Markov time:
5348:is another Wiener process.
5197:is another Wiener process.
1368:are independent. For large
10:
14936:
14811:Extreme value theory (EVT)
14611:Doob decomposition theorem
13903:OrnsteinâUhlenbeck process
13674:Chinese restaurant process
13060:
12337:continuous semimartingales
11081:{\displaystyle V_{f}(t+a)}
10480:integrated Brownian motion
10392:Integrated Brownian motion
10084:OrnsteinâUhlenbeck process
9265:{\displaystyle R(T_{s},D)}
9163:from this code is at most
8534:Dimension doubling theorem
8099:of Lebesgue measure 0 and
5356:Consider a Wiener process
1864:Covariance and correlation
219:stochastic processes with
14879:
14783:
14691:Optional stopping theorem
14588:
14550:
14492:Large deviation principle
14459:
14373:
14330:
14297:
14244:HeathâJarrowâMorton (HJM)
14189:
14181:Moving-average (MA) model
14166:Autoregressive (AR) model
14146:
14056:
13991:Hidden Markov model (HMM)
13973:
13925:SchrammâLoewner evolution
13729:
13654:
13295:. Springer. p. 114.
13291:Shreve, Steven E (2008).
13221:The Annals of Probability
10484:integrated Wiener process
10105:{\displaystyle \theta =1}
9938:geometric Brownian motion
9815:LaplaceâBeltrami operator
9094:{\displaystyle 2^{TR(D)}}
7668:The set of all functions
6171:{\displaystyle \tau _{D}}
4117:{\displaystyle f_{M_{t}}}
1784:, centered at zero. Thus
631:has Gaussian increments:
317:path integral formulation
267:SchrammâLoewner evolution
133:
128:
105:
100:
88:
14606:Doléans-Dade exponential
14436:Progressively measurable
14234:CoxâIngersollâRoss (CIR)
13519:(also available online:
13472:10.1109/TIT.2009.2018329
13164:
13140:Numerical path sampling:
11117:{\displaystyle V_{f}(t)}
11039:{\displaystyle V_{f}(t)}
10209:from closed intervals .
10082:is distributed like the
8808:rate-distortion function
8758:on , respectively. (For
6354:{\displaystyle f(W_{t})}
4464:Half-normal distribution
3786:is a Brownian motion on
3435:{\displaystyle \xi _{n}}
1464:{\displaystyle N(0,t-s)}
1361:{\displaystyle \xi _{k}}
1014:(from time zero to time
547:{\displaystyle u\geq 0,}
288:), instrument errors in
14826:Mathematical statistics
14816:Large deviations theory
14646:Infinitesimal generator
14507:Maximal ergodic theorem
14426:Piecewise-deterministic
14028:Random dynamical system
13893:Markov additive process
13117:Chernoff's distribution
11626:. This is given by the
10019:The stochastic process
9925:) does not apply when
9752:{\displaystyle T_{s}/6}
8108:Quantitative properties
7855:has a local maximum at
1895:{\displaystyle s\leq t}
996:is also a martingale).
603:{\displaystyle s<t.}
470:{\displaystyle t>0,}
413:{\displaystyle W_{0}=0}
282:electronics engineering
55:more precise citations.
14661:KarhunenâLoĂšve theorem
14596:CameronâMartin formula
14560:BurkholderâDavisâGundy
13955:Variance gamma process
13380:10.1214/aop/1176995155
13234:10.1214/aop/1176995480
13151:Walk-on-spheres method
13112:Classical Wiener space
13081:
13047:
13020:
12993:
12944:
12924:
12838:
12704:
12649:
12621:
12588:
12552:
12521:
12499:
12466:
12439:
12412:
12352:qualitative properties
12307:
12248:
12186:
12116:
12043:
11980:
11879:
11806:
11732:
11620:
11537:
11536:{\displaystyle f(t)=t}
11502:
11435:
11325:
11200:
11118:
11082:
11040:
11004:
10832:
10719:
10718:{\displaystyle a>0}
10693:
10535:
10472:
10327:
10165:
10132:
10131:{\displaystyle \mu =0}
10106:
10076:
10008:
9867:
9818:
9796:
9780:
9753:
9718:
9646:
9569:
9442:
9289:
9266:
9221:
9183:
9157:
9095:
9052:
9032:
9006:
8983:
8947:
8889:
8694:
8524:
8345:
8217:
7959:
7834:
7789:
7748:
7719:
7676:Qualitative properties
7633:
7571:
7533:
7433:
7380:
7302:
7172:
7002:
6955:
6900:
6764:
6624:
6528:
6466:
6437:
6417:
6384:
6355:
6319:
6318:{\displaystyle f(0)=0}
6280:
6246:
6172:
6145:
6117:
6074:
6038:
5975:
5815:
5752:
5499:
5498:{\displaystyle W(0)=0}
5464:
5413:, conditioned so that
5407:
5379:
5342:
5258:
5191:
5118:
5107:
5032:
4770:
4745:of the maximum value,
4731:
4699:
4672:
4650:
4454:
4118:
4082:
3886:
3820:KarhunenâLoĂšve theorem
3812:
3780:
3734:
3702:
3602:
3549:
3505:
3436:
3393:
3285:
3162:
2976:
2922:
2846:
2529:
2427:
2170:
2128:
1896:
1855:
1772:
1714:
1663:
1541:
1522:
1495:
1465:
1424:
1362:
1335:
1308:
1180:
1090:
1005:KarhunenâLoĂšve theorem
853:
833:
806:
785:
711:
691:
671:
625:
604:
575:
548:
520:
477:the future increments
471:
446:independent increments
438:
414:
377:
347:option pricing model.
224:independent increments
181:
173:
157:
119:
14791:Actuarial mathematics
14753:Uniform integrability
14748:Stratonovich integral
14676:LĂ©vyâProkhorov metric
14580:MarcinkiewiczâZygmund
14467:Central limit theorem
14069:Gaussian random field
13898:McKeanâVlasov process
13818:Dyson Brownian motion
13679:GaltonâWatson process
13529:Lawler, Greg (2005),
13366:Annals of Probability
13146:EulerâMaruyama method
13107:Abstract Wiener space
13082:
13048:
13046:{\displaystyle Y_{t}}
13021:
13019:{\displaystyle X_{t}}
12994:
12945:
12925:
12839:
12705:
12650:
12622:
12589:
12587:{\displaystyle |c|=1}
12553:
12522:
12500:
12467:
12465:{\displaystyle Y_{t}}
12440:
12438:{\displaystyle X_{t}}
12413:
12350:Using this fact, the
12308:
12249:
12192:other cases (such as
12187:
12117:
12044:
11981:
11905:is a Wiener process.
11880:
11807:
11733:
11669:but distributed like
11621:
11538:
11503:
11436:
11326:
11201:
11119:
11083:
11041:
11005:
10833:
10720:
10694:
10536:
10473:
10328:
10166:
10133:
10107:
10077:
10009:
9868:
9802:
9791:) and without drift (
9786:
9775:) and without drift (
9770:
9754:
9719:
9647:
9570:
9443:
9290:
9267:
9222:
9220:{\displaystyle T_{s}}
9184:
9158:
9096:
9053:
9033:
9007:
8984:
8982:{\displaystyle TR(D)}
8948:
8890:
8695:
8525:
8346:
8227:Modulus of continuity
8218:
7960:
7835:
7790:
7754:is almost surely not
7749:
7720:
7634:
7572:
7534:
7434:
7381:
7303:
7173:
7003:
6956:
6901:
6765:
6625:
6529:
6467:
6443:with the Markov time
6438:
6418:
6385:
6356:
6320:
6281:
6247:
6173:
6146:
6118:
6075:
6039:
5976:
5816:
5753:
5500:
5465:
5408:
5380:
5352:Projective invariance
5343:
5259:
5192:
5108:
5048:
5033:
4771:
4769:{\displaystyle W_{t}}
4732:
4700:
4698:{\displaystyle W_{t}}
4673:
4651:
4466:. The expectation is
4455:
4119:
4083:
3887:
3813:
3781:
3740:. The scaled process
3735:
3703:
3582:
3550:
3485:
3437:
3407:Wiener representation
3394:
3286:
3163:
2977:
2923:
2847:
2530:
2428:
2171:
2129:
1897:
1856:
1773:
1715:
1664:
1539:
1523:
1521:{\displaystyle W_{n}}
1496:
1466:
1425:
1363:
1336:
1334:{\displaystyle W_{n}}
1309:
1181:
1091:
955:LĂ©vy characterisation
854:
834:
832:{\displaystyle W_{t}}
807:
786:
712:
692:
672:
626:
605:
576:
574:{\displaystyle W_{s}}
549:
521:
472:
439:
415:
378:
376:{\displaystyle W_{t}}
179:
171:
158:
120:
14866:Time series analysis
14821:Mathematical finance
14706:Reflection principle
14033:Regenerative process
13833:FlemingâViot process
13648:Stochastic processes
13359:Vervaat, W. (1979).
13071:
13030:
13003:
12957:
12934:
12848:
12719:
12663:
12639:
12598:
12562:
12542:
12509:
12480:
12449:
12422:
12366:
12320:â â) almost surely.
12258:
12196:
12125:
12055:
11989:
11926:
11827:
11742:
11682:
11555:
11512:
11445:
11334:
11214:
11128:
11092:
11050:
11014:
10841:
10729:
10703:
10548:
10498:
10400:
10372:). Then the process
10349:Brownian martingales
10339:Dirac delta function
10251:
10142:
10116:
10090:
10023:
9944:
9825:
9728:
9656:
9579:
9451:
9299:
9279:
9234:
9204:
9167:
9109:
9066:
9042:
9016:
8996:
8961:
8899:
8814:
8578:
8360:
8236:
8122:
7863:
7803:
7799:, and almost surely
7758:
7747:{\displaystyle w(t)}
7729:
7703:
7694:Weierstrass function
7581:
7543:
7443:
7389:
7316:
7186:
7072:
6974:
6910:
6802:
6633:
6537:
6476:
6465:{\displaystyle S(t)}
6447:
6427:
6416:{\displaystyle Y(t)}
6398:
6383:{\displaystyle f(D)}
6365:
6329:
6294:
6288:holomorphic function
6256:
6182:
6155:
6127:
6084:
6073:{\displaystyle W(t)}
6055:
5989:
5985:, in the sense that
5825:
5766:
5513:
5474:
5417:
5389:
5378:{\displaystyle W(t)}
5360:
5301:
5272:is distributed like
5209:
5137:
5053:
4780:
4753:
4709:
4682:
4662:
4470:
4139:
4094:
3905:
3834:
3790:
3744:
3712:
3559:
3446:
3419:
3313:
3174:
2986:
2932:
2858:
2539:
2439:
2180:
2140:
1906:
1880:
1788:
1734:
1679:
1569:
1505:
1479:
1434:
1376:
1345:
1318:
1198:
1144:
1110:continuous functions
1050:
843:
816:
796:
721:
701:
681:
635:
615:
585:
558:
529:
481:
452:
428:
391:
360:
343:, in particular the
329:Schrödinger equation
327:, a solution to the
292:and disturbances in
240:evolutionary biology
236:quantitative finance
137:
109:
14861:Stochastic analysis
14701:Quadratic variation
14696:Prokhorov's theorem
14631:FeynmanâKac formula
14101:Markov random field
13749:Birthâdeath process
12883:
12777:
12759:
12736:
12341:diffusion processes
12293:
12275:
12231:
12213:
12169:
12151:
12099:
12081:
12006:
11943:
11895:quadratic variation
11792:
11777:
11699:
10920:
10780:
10641:
10587:
10448:
10343:RayâKnight theorems
10290:
9511:
9383:
9368:
9058:large enough and a
8782:corresponding to a
8649:
8595:
8561:pushforward measure
8541:Hausdorff dimension
8101:Hausdorff dimension
8072:quadratic variation
8065:unbounded variation
7840:-Hölder continuous.
7616:
7601:
7560:
7519:
7504:
7466:
7130:
7010:quadratic variation
6991:
6706:
6569:
4749:by the known value
4566:
4512:
4269:
4196:
3260:
2830:
1782:normal distribution
1555:normal distribution
973:quadratic variation
355:The Wiener process
325:FeynmanâKac formula
271:applied mathematics
259:diffusion processes
255:stochastic calculus
228:applied mathematics
85:
14831:Probability theory
14711:Skorokhod integral
14681:Malliavin calculus
14264:Korn-Kreer-Lenssen
14148:Time series models
14111:PitmanâYor process
13077:
13043:
13016:
12989:
12940:
12920:
12869:
12834:
12763:
12745:
12722:
12700:
12645:
12617:
12584:
12548:
12517:
12495:
12462:
12435:
12408:
12303:
12279:
12261:
12244:
12217:
12199:
12182:
12155:
12137:
12112:
12085:
12067:
12039:
12025:
11992:
11976:
11962:
11929:
11875:
11802:
11778:
11763:
11728:
11685:
11616:
11533:
11498:
11431:
11321:
11196:
11114:
11078:
11036:
11000:
10906:
10828:
10766:
10715:
10689:
10627:
10573:
10531:
10468:
10434:
10323:
10276:
10194:modification of a
10161:
10128:
10102:
10072:
10004:
9891:Brownian excursion
9863:
9819:
9797:
9781:
9749:
9714:
9642:
9565:
9497:
9438:
9369:
9354:
9285:
9262:
9217:
9179:
9153:
9103:mean squared error
9091:
9048:
9028:
9002:
8979:
8943:
8885:
8690:
8629:
8581:
8555:on under the map
8520:
8427:
8381:
8341:
8257:
8213:
8143:
8067:on every interval.
8040:is increasing on (
7955:
7881:
7830:
7819:
7785:
7774:
7744:
7715:
7629:
7602:
7587:
7567:
7546:
7529:
7505:
7490:
7452:
7429:
7376:
7298:
7182:is the polynomial
7168:
7116:
7030:time of first exit
6998:
6977:
6951:
6896:
6760:
6683:
6620:
6555:
6524:
6462:
6433:
6413:
6380:
6351:
6315:
6276:
6242:
6168:
6141:
6113:
6070:
6034:
5971:
5811:
5805:
5748:
5746:
5495:
5460:
5438:
5403:
5375:
5338:
5254:
5187:
5119:
5103:
5028:
4869:
4766:
4727:
4695:
4668:
4646:
4552:
4498:
4450:
4448:
4252:
4179:
4114:
4078:
3882:
3871:
3808:
3776:
3730:
3698:
3545:
3432:
3389:
3281:
3239:
3158:
2972:
2918:
2842:
2840:
2809:
2525:
2423:
2166:
2124:
2122:
1892:
1851:
1768:
1710:
1659:
1561:, at a fixed time
1549:The unconditional
1542:
1518:
1491:
1461:
1420:
1358:
1331:
1304:
1265:
1176:
1086:
981:(which means that
849:
829:
802:
781:
707:
687:
667:
621:
600:
571:
544:
516:
467:
434:
410:
373:
337:physical cosmology
313:Langevin equations
197:stochastic process
182:
174:
153:
115:
83:
14915:Martingale theory
14897:
14896:
14851:Signal processing
14570:Doob's upcrossing
14565:Doob's martingale
14529:EngelbertâSchmidt
14472:Donsker's theorem
14406:Feller-continuous
14274:RendlemanâBartter
14064:Dirichlet process
13981:Branching process
13950:Telegraph process
13843:Geometric process
13823:Empirical process
13813:Diffusion process
13669:Branching process
13664:Bernoulli process
13407:www.quantopia.net
13327:978-0-521-76018-8
13302:978-0-387-40101-0
13269:Wolfram Mathworld
13161:
13160:
13080:{\displaystyle t}
12943:{\displaystyle U}
12659:then the process
12648:{\displaystyle f}
12551:{\displaystyle c}
12345:change of measure
12331:Change of measure
12325:local martingales
12010:
11947:
11604:
11577:
11319:
10184:LĂ©vy distribution
9981:
9763:Related processes
9724:. In particular,
9543:
9482:
9423:
9417:
9352:
9288:{\displaystyle D}
9274:mean square error
9051:{\displaystyle T}
9005:{\displaystyle D}
8861:
8780:singular function
8738:of the interval (
8551:The image of the
8515:
8500:
8499:
8382:
8363:
8336:
8321:
8320:
8239:
8208:
8193:
8192:
8125:
7944:
7866:
7818:
7797:Hölder continuous
7773:
7658:local martingales
7270:
7241:
7228:
6865:
6836:
6823:
6436:{\displaystyle D}
5962:
5935:
5905:
5420:
5169:
5085:
4996:
4993:
4985:
4979:
4935:
4926:
4848:
4671:{\displaystyle t}
4644:
4643:
4615:
4585:
4584:
4426:
4396:
4395:
4360:
4313:
4310:
4124:, integrate over
4046:
3999:
3996:
3850:
3770:
3752:
3696:
3685:
3646:
3580:
3543:
3483:
3381:
2982:are independent,
2115:
2114:
2100:
2099:
2082:
1617:
1616:
1473:Donsker's theorem
1235:
1233:
1232:
1035:Donsker's theorem
1012:definite integral
852:{\displaystyle t}
805:{\displaystyle W}
710:{\displaystyle u}
690:{\displaystyle 0}
624:{\displaystyle W}
437:{\displaystyle W}
333:eternal inflation
321:quantum mechanics
192:is a real-valued
166:
165:
118:{\displaystyle 0}
81:
80:
73:
14927:
14871:Machine learning
14758:Usual hypotheses
14641:Girsanov theorem
14626:Dynkin's formula
14391:Continuous paths
14299:Actuarial models
14239:GarmanâKohlhagen
14209:BlackâKarasinski
14204:BlackâDermanâToy
14191:Financial models
14057:Fields and other
13986:Gaussian process
13935:Sigma-martingale
13739:Additive process
13641:
13634:
13627:
13618:
13617:
13613:
13576:
13563:
13554:
13534:
13518:
13506:
13483:
13482:
13466:(6): 2859â2867,
13454:
13448:
13445:
13439:
13436:
13430:
13423:
13417:
13416:
13414:
13413:
13399:
13393:
13392:
13382:
13356:
13350:
13347:
13341:
13338:
13332:
13331:
13313:
13307:
13306:
13288:
13282:
13279:
13273:
13272:
13261:
13255:
13254:
13236:
13212:
13206:
13205:
13184:
13178:
13175:
13096:
13086:
13084:
13083:
13078:
13052:
13050:
13049:
13044:
13042:
13041:
13025:
13023:
13022:
13017:
13015:
13014:
12998:
12996:
12995:
12990:
12988:
12987:
12972:
12971:
12949:
12947:
12946:
12941:
12929:
12927:
12926:
12921:
12916:
12910:
12909:
12904:
12898:
12897:
12888:
12882:
12877:
12843:
12841:
12840:
12835:
12833:
12832:
12808:
12807:
12798:
12797:
12782:
12778:
12776:
12771:
12758:
12753:
12735:
12730:
12709:
12707:
12706:
12701:
12681:
12680:
12654:
12652:
12651:
12646:
12626:
12624:
12623:
12618:
12616:
12615:
12593:
12591:
12590:
12585:
12577:
12569:
12557:
12555:
12554:
12549:
12526:
12524:
12523:
12518:
12516:
12504:
12502:
12501:
12496:
12494:
12493:
12488:
12471:
12469:
12468:
12463:
12461:
12460:
12444:
12442:
12441:
12436:
12434:
12433:
12417:
12415:
12414:
12409:
12407:
12406:
12391:
12390:
12378:
12377:
12339:(especially, of
12335:A wide class of
12312:
12310:
12309:
12304:
12292:
12287:
12274:
12269:
12253:
12251:
12250:
12245:
12230:
12225:
12212:
12207:
12191:
12189:
12188:
12183:
12168:
12163:
12150:
12145:
12121:
12119:
12118:
12113:
12098:
12093:
12080:
12075:
12048:
12046:
12045:
12040:
12035:
12034:
12024:
12005:
12000:
11985:
11983:
11982:
11977:
11972:
11971:
11961:
11942:
11937:
11884:
11882:
11881:
11876:
11874:
11873:
11852:
11851:
11839:
11838:
11811:
11809:
11808:
11803:
11798:
11791:
11786:
11776:
11771:
11737:
11735:
11734:
11729:
11727:
11726:
11698:
11693:
11625:
11623:
11622:
11617:
11615:
11614:
11609:
11605:
11603:
11595:
11594:
11585:
11578:
11576:
11559:
11542:
11540:
11539:
11534:
11507:
11505:
11504:
11499:
11488:
11487:
11457:
11456:
11440:
11438:
11437:
11432:
11418:
11417:
11399:
11398:
11368:
11367:
11346:
11345:
11330:
11328:
11327:
11322:
11320:
11318:
11305:
11304:
11285:
11272:
11271:
11244:
11243:
11224:
11205:
11203:
11202:
11197:
11174:
11173:
11140:
11139:
11123:
11121:
11120:
11115:
11104:
11103:
11087:
11085:
11084:
11079:
11062:
11061:
11045:
11043:
11042:
11037:
11026:
11025:
11009:
11007:
11006:
11001:
10919:
10914:
10890:
10889:
10862:
10861:
10837:
10835:
10834:
10829:
10820:
10819:
10779:
10774:
10750:
10749:
10724:
10722:
10721:
10716:
10698:
10696:
10695:
10690:
10688:
10687:
10640:
10635:
10595:
10586:
10581:
10560:
10559:
10540:
10538:
10537:
10532:
10477:
10475:
10474:
10469:
10447:
10442:
10421:
10420:
10332:
10330:
10329:
10324:
10312:
10311:
10289:
10284:
10263:
10262:
10242:
10200:right-continuous
10170:
10168:
10167:
10162:
10154:
10153:
10137:
10135:
10134:
10129:
10111:
10109:
10108:
10103:
10086:with parameters
10081:
10079:
10078:
10073:
10071:
10070:
10069:
10068:
10051:
10050:
10035:
10034:
10013:
10011:
10010:
10005:
10000:
9999:
9998:
9997:
9982:
9977:
9973:
9972:
9962:
9872:
9870:
9869:
9864:
9862:
9861:
9837:
9836:
9812:
9811:
9807:
9794:
9790:
9778:
9774:
9758:
9756:
9755:
9750:
9745:
9740:
9739:
9723:
9721:
9720:
9715:
9668:
9667:
9651:
9649:
9648:
9643:
9641:
9640:
9622:
9574:
9572:
9571:
9566:
9555:
9551:
9544:
9536:
9510:
9505:
9496:
9495:
9483:
9478:
9477:
9468:
9463:
9462:
9447:
9445:
9444:
9439:
9428:
9424:
9419:
9418:
9410:
9392:
9382:
9377:
9367:
9362:
9353:
9348:
9347:
9338:
9330:
9329:
9317:
9316:
9294:
9292:
9291:
9286:
9271:
9269:
9268:
9263:
9252:
9251:
9226:
9224:
9223:
9218:
9216:
9215:
9188:
9186:
9185:
9180:
9162:
9160:
9159:
9154:
9152:
9151:
9124:
9123:
9100:
9098:
9097:
9092:
9090:
9089:
9062:of no more than
9057:
9055:
9054:
9049:
9037:
9035:
9034:
9029:
9011:
9009:
9008:
9003:
8988:
8986:
8985:
8980:
8952:
8950:
8949:
8944:
8942:
8941:
8914:
8913:
8894:
8892:
8891:
8886:
8881:
8880:
8862:
8860:
8847:
8846:
8833:
8804:information rate
8798:Information rate
8722:) is called the
8699:
8697:
8696:
8691:
8686:
8671:
8670:
8648:
8640:
8622:
8594:
8589:
8573:
8563:) has a density
8553:Lebesgue measure
8529:
8527:
8526:
8521:
8516:
8513:
8501:
8492:
8469:
8468:
8467:
8435:
8429:
8426:
8380:
8350:
8348:
8347:
8342:
8337:
8334:
8322:
8313:
8284:
8283:
8282:
8265:
8259:
8256:
8222:
8220:
8219:
8214:
8209:
8206:
8194:
8170:
8169:
8168:
8151:
8145:
8142:
8088:of the function
7964:
7962:
7961:
7956:
7945:
7943:
7942:
7928:
7922:
7921:
7889:
7883:
7880:
7858:
7847:of the function
7839:
7837:
7836:
7831:
7820:
7811:
7794:
7792:
7791:
7786:
7775:
7766:
7753:
7751:
7750:
7745:
7724:
7722:
7721:
7716:
7655:
7641:About functions
7638:
7636:
7635:
7630:
7622:
7615:
7610:
7600:
7595:
7577:on is equal to
7576:
7574:
7573:
7568:
7559:
7554:
7538:
7536:
7535:
7530:
7525:
7518:
7513:
7503:
7498:
7483:
7482:
7477:
7473:
7465:
7460:
7438:
7436:
7435:
7430:
7425:
7424:
7385:
7383:
7382:
7377:
7372:
7371:
7366:
7362:
7355:
7354:
7307:
7305:
7304:
7299:
7276:
7272:
7271:
7269:
7268:
7267:
7254:
7253:
7244:
7242:
7234:
7229:
7227:
7216:
7177:
7175:
7174:
7169:
7161:
7146:
7145:
7129:
7124:
7103:
7102:
7084:
7083:
7067:
7049:
7027:
7023:
7007:
7005:
7004:
6999:
6990:
6985:
6960:
6958:
6957:
6952:
6941:
6940:
6922:
6921:
6905:
6903:
6902:
6897:
6871:
6867:
6866:
6864:
6863:
6862:
6849:
6848:
6839:
6837:
6829:
6824:
6822:
6811:
6794:
6769:
6767:
6766:
6761:
6749:
6748:
6743:
6719:
6711:
6705:
6691:
6663:
6662:
6629:
6627:
6626:
6621:
6612:
6611:
6606:
6582:
6574:
6568:
6563:
6533:
6531:
6530:
6525:
6471:
6469:
6468:
6463:
6442:
6440:
6439:
6434:
6422:
6420:
6419:
6414:
6389:
6387:
6386:
6381:
6360:
6358:
6357:
6352:
6347:
6346:
6324:
6322:
6321:
6316:
6285:
6283:
6282:
6277:
6275:
6251:
6249:
6248:
6243:
6217:
6194:
6193:
6177:
6175:
6174:
6169:
6167:
6166:
6150:
6148:
6147:
6142:
6140:
6122:
6120:
6119:
6114:
6112:
6079:
6077:
6076:
6071:
6043:
6041:
6040:
6035:
6030:
6029:
6014:
6013:
6004:
6003:
5981:which defines a
5980:
5978:
5977:
5972:
5967:
5963:
5955:
5940:
5936:
5928:
5910:
5906:
5904:
5890:
5876:
5837:
5836:
5820:
5818:
5817:
5812:
5810:
5809:
5757:
5755:
5754:
5749:
5747:
5734:
5698:
5697:
5656:
5655:
5651:
5619:
5618:
5599:
5535:
5534:
5504:
5502:
5501:
5496:
5469:
5467:
5466:
5461:
5437:
5412:
5410:
5409:
5404:
5402:
5384:
5382:
5381:
5376:
5347:
5345:
5344:
5339:
5337:
5336:
5332:
5313:
5312:
5288:
5280:
5271:
5263:
5261:
5260:
5255:
5253:
5252:
5240:
5239:
5221:
5220:
5196:
5194:
5193:
5188:
5186:
5185:
5170:
5165:
5163:
5149:
5148:
5132:
5122:Brownian scaling
5112:
5110:
5109:
5104:
5102:
5101:
5086:
5081:
5079:
5065:
5064:
5037:
5035:
5034:
5029:
5024:
5023:
4994:
4991:
4983:
4982:
4981:
4980:
4975:
4971:
4970:
4948:
4933:
4924:
4920:
4916:
4915:
4914:
4868:
4844:
4843:
4842:
4841:
4807:
4806:
4805:
4804:
4803:
4802:
4775:
4773:
4772:
4767:
4765:
4764:
4736:
4734:
4733:
4730:{\displaystyle }
4728:
4704:
4702:
4701:
4696:
4694:
4693:
4677:
4675:
4674:
4669:
4655:
4653:
4652:
4647:
4645:
4639:
4631:
4630:
4618:
4617:
4616:
4614:
4606:
4605:
4596:
4586:
4583:
4572:
4571:
4565:
4560:
4532:
4531:
4530:
4529:
4511:
4506:
4491:
4490:
4459:
4457:
4456:
4451:
4449:
4429:
4428:
4427:
4425:
4417:
4416:
4407:
4397:
4394:
4383:
4382:
4374:
4363:
4362:
4361:
4359:
4351:
4350:
4349:
4324:
4314:
4312:
4311:
4300:
4294:
4271:
4268:
4263:
4226:
4225:
4224:
4223:
4211:
4210:
4195:
4190:
4162:
4161:
4160:
4159:
4134:
4123:
4121:
4120:
4115:
4113:
4112:
4111:
4110:
4087:
4085:
4084:
4079:
4049:
4048:
4047:
4045:
4037:
4036:
4035:
4010:
4000:
3998:
3997:
3986:
3980:
3957:
3937:
3936:
3935:
3934:
3922:
3921:
3900:
3891:
3889:
3888:
3883:
3881:
3880:
3870:
3846:
3845:
3817:
3815:
3814:
3811:{\displaystyle }
3809:
3785:
3783:
3782:
3777:
3775:
3771:
3763:
3753:
3748:
3739:
3737:
3736:
3733:{\displaystyle }
3731:
3707:
3705:
3704:
3699:
3697:
3695:
3691:
3687:
3686:
3678:
3664:
3663:
3659:
3652:
3648:
3647:
3639:
3614:
3612:
3611:
3601:
3596:
3581:
3576:
3571:
3570:
3554:
3552:
3551:
3546:
3544:
3542:
3534:
3517:
3515:
3514:
3504:
3499:
3484:
3479:
3471:
3470:
3458:
3457:
3441:
3439:
3438:
3433:
3431:
3430:
3402:
3398:
3396:
3395:
3390:
3382:
3380:
3379:
3367:
3366:
3357:
3352:
3351:
3350:
3349:
3332:
3331:
3330:
3329:
3308:
3290:
3288:
3287:
3282:
3277:
3276:
3264:
3259:
3254:
3253:
3252:
3222:
3221:
3220:
3219:
3202:
3201:
3200:
3199:
3167:
3165:
3164:
3159:
3148:
3147:
3146:
3145:
3128:
3127:
3126:
3125:
3096:
3095:
3094:
3093:
3067:
3063:
3059:
3058:
3057:
3056:
3039:
3038:
3037:
3036:
3016:
3015:
3014:
3013:
2981:
2979:
2978:
2973:
2971:
2970:
2969:
2968:
2951:
2950:
2949:
2948:
2927:
2925:
2924:
2919:
2917:
2916:
2915:
2914:
2897:
2896:
2895:
2894:
2877:
2876:
2875:
2874:
2851:
2849:
2848:
2843:
2841:
2834:
2829:
2824:
2823:
2822:
2795:
2791:
2787:
2786:
2785:
2784:
2767:
2766:
2765:
2764:
2744:
2743:
2742:
2741:
2710:
2706:
2702:
2698:
2697:
2696:
2695:
2675:
2674:
2673:
2672:
2655:
2654:
2653:
2652:
2629:
2628:
2627:
2626:
2591:
2590:
2589:
2588:
2571:
2570:
2569:
2568:
2534:
2532:
2531:
2526:
2524:
2523:
2522:
2521:
2501:
2500:
2499:
2498:
2481:
2480:
2479:
2478:
2458:
2457:
2456:
2455:
2432:
2430:
2429:
2424:
2419:
2415:
2414:
2413:
2412:
2411:
2394:
2393:
2392:
2391:
2363:
2359:
2352:
2351:
2350:
2349:
2323:
2322:
2321:
2320:
2294:
2293:
2292:
2291:
2265:
2264:
2263:
2262:
2228:
2227:
2226:
2225:
2208:
2207:
2206:
2205:
2175:
2173:
2172:
2167:
2165:
2164:
2152:
2151:
2133:
2131:
2130:
2125:
2123:
2116:
2107:
2106:
2101:
2092:
2088:
2083:
2081:
2080:
2079:
2078:
2077:
2063:
2062:
2061:
2060:
2045:
2041:
2040:
2028:
2027:
2008:
1996:
1995:
1983:
1982:
1944:
1943:
1931:
1930:
1901:
1899:
1898:
1893:
1860:
1858:
1857:
1852:
1826:
1825:
1813:
1812:
1800:
1799:
1777:
1775:
1774:
1769:
1755:
1754:
1729:
1719:
1717:
1716:
1711:
1700:
1699:
1668:
1666:
1665:
1660:
1655:
1654:
1641:
1636:
1635:
1618:
1606:
1602:
1588:
1587:
1586:
1585:
1564:
1545:Basic properties
1527:
1525:
1524:
1519:
1517:
1516:
1500:
1498:
1497:
1492:
1475:asserts that as
1470:
1468:
1467:
1462:
1429:
1427:
1426:
1421:
1410:
1409:
1388:
1387:
1367:
1365:
1364:
1359:
1357:
1356:
1340:
1338:
1337:
1332:
1330:
1329:
1313:
1311:
1310:
1305:
1275:
1274:
1264:
1234:
1228:
1224:
1210:
1209:
1185:
1183:
1182:
1177:
1169:
1168:
1156:
1155:
1123:
1116:
1108:on the space of
1099:
1095:
1093:
1092:
1087:
1085:
1084:
1080:
1079:
1065:
1064:
1020:Gaussian process
995:
980:
970:
945:
919:
893:
858:
856:
855:
850:
838:
836:
835:
830:
828:
827:
811:
809:
808:
803:
790:
788:
787:
782:
762:
761:
752:
751:
739:
738:
716:
714:
713:
708:
696:
694:
693:
688:
676:
674:
673:
668:
666:
665:
653:
652:
630:
628:
627:
622:
609:
607:
606:
601:
580:
578:
577:
572:
570:
569:
553:
551:
550:
545:
525:
523:
522:
517:
512:
511:
499:
498:
476:
474:
473:
468:
443:
441:
440:
435:
419:
417:
416:
411:
403:
402:
382:
380:
379:
374:
372:
371:
290:filtering theory
278:Gaussian process
263:potential theory
162:
160:
159:
154:
149:
148:
124:
122:
121:
116:
96:
86:
82:
76:
69:
65:
62:
56:
51:this article by
42:inline citations
29:
28:
21:
14935:
14934:
14930:
14929:
14928:
14926:
14925:
14924:
14900:
14899:
14898:
14893:
14875:
14836:Queueing theory
14779:
14721:Skorokhod space
14584:
14575:KunitaâWatanabe
14546:
14512:Sanov's theorem
14482:Ergodic theorem
14455:
14451:Time-reversible
14369:
14332:Queueing models
14326:
14322:SparreâAnderson
14312:CramĂ©râLundberg
14293:
14279:SABR volatility
14185:
14142:
14094:Boolean network
14052:
14038:Renewal process
13969:
13918:Non-homogeneous
13908:Poisson process
13798:Contact process
13761:Brownian motion
13731:Continuous time
13725:
13719:Maximal entropy
13650:
13645:
13608:
13584:
13569:Proc Japan Acad
13551:
13515:
13497:Kleinert, Hagen
13492:
13487:
13486:
13455:
13451:
13446:
13442:
13437:
13433:
13424:
13420:
13411:
13409:
13401:
13400:
13396:
13357:
13353:
13348:
13344:
13339:
13335:
13328:
13318:Brownian motion
13314:
13310:
13303:
13289:
13285:
13280:
13276:
13263:
13262:
13258:
13213:
13209:
13202:
13185:
13181:
13176:
13172:
13167:
13162:
13093:
13072:
13069:
13068:
13065:
13059:
13037:
13033:
13031:
13028:
13027:
13010:
13006:
13004:
13001:
13000:
12983:
12979:
12967:
12963:
12958:
12955:
12954:
12935:
12932:
12931:
12912:
12905:
12900:
12899:
12893:
12889:
12884:
12878:
12873:
12849:
12846:
12845:
12819:
12815:
12803:
12799:
12793:
12789:
12772:
12767:
12754:
12749:
12744:
12740:
12731:
12726:
12720:
12717:
12716:
12676:
12672:
12664:
12661:
12660:
12657:entire function
12640:
12637:
12636:
12633:
12611:
12607:
12599:
12596:
12595:
12573:
12565:
12563:
12560:
12559:
12543:
12540:
12539:
12533:
12531:Self-similarity
12512:
12510:
12507:
12506:
12489:
12484:
12483:
12481:
12478:
12477:
12456:
12452:
12450:
12447:
12446:
12429:
12425:
12423:
12420:
12419:
12402:
12398:
12386:
12382:
12373:
12369:
12367:
12364:
12363:
12360:
12333:
12288:
12283:
12270:
12265:
12259:
12256:
12255:
12226:
12221:
12208:
12203:
12197:
12194:
12193:
12164:
12159:
12146:
12141:
12126:
12123:
12122:
12094:
12089:
12076:
12071:
12056:
12053:
12052:
12030:
12026:
12014:
12001:
11996:
11990:
11987:
11986:
11967:
11963:
11951:
11938:
11933:
11927:
11924:
11923:
11920:
11860:
11856:
11847:
11843:
11834:
11830:
11828:
11825:
11824:
11819:In general, if
11794:
11787:
11782:
11772:
11767:
11743:
11740:
11739:
11713:
11709:
11694:
11689:
11683:
11680:
11679:
11652:
11636:
11610:
11596:
11590:
11586:
11584:
11580:
11579:
11563:
11558:
11556:
11553:
11552:
11513:
11510:
11509:
11508:corresponds to
11474:
11470:
11452:
11448:
11446:
11443:
11442:
11413:
11409:
11394:
11390:
11363:
11359:
11341:
11337:
11335:
11332:
11331:
11300:
11296:
11286:
11267:
11263:
11239:
11235:
11225:
11223:
11215:
11212:
11211:
11169:
11165:
11135:
11131:
11129:
11126:
11125:
11099:
11095:
11093:
11090:
11089:
11057:
11053:
11051:
11048:
11047:
11021:
11017:
11015:
11012:
11011:
10915:
10910:
10885:
10881:
10857:
10853:
10842:
10839:
10838:
10815:
10811:
10775:
10770:
10745:
10741:
10730:
10727:
10726:
10704:
10701:
10700:
10683:
10679:
10636:
10631:
10588:
10582:
10577:
10555:
10551:
10549:
10546:
10545:
10499:
10496:
10495:
10443:
10438:
10407:
10403:
10401:
10398:
10397:
10394:
10377:
10362:
10351:
10307:
10303:
10285:
10280:
10258:
10254:
10252:
10249:
10248:
10241:
10225:
10216:
10192:left-continuous
10178:a single point
10176:time of hitting
10149:
10145:
10143:
10140:
10139:
10117:
10114:
10113:
10091:
10088:
10087:
10061:
10057:
10056:
10052:
10043:
10039:
10030:
10026:
10024:
10021:
10020:
9993:
9989:
9968:
9964:
9963:
9961:
9951:
9947:
9945:
9942:
9941:
9940:can be written
9887:Brownian bridge
9857:
9853:
9832:
9828:
9826:
9823:
9822:
9809:
9805:
9804:
9792:
9788:
9776:
9772:
9765:
9741:
9735:
9731:
9729:
9726:
9725:
9663:
9659:
9657:
9654:
9653:
9633:
9629:
9618:
9580:
9577:
9576:
9535:
9519:
9515:
9506:
9501:
9491:
9487:
9473:
9469:
9467:
9458:
9454:
9452:
9449:
9448:
9409:
9393:
9391:
9387:
9378:
9373:
9363:
9358:
9343:
9339:
9337:
9325:
9321:
9312:
9308:
9300:
9297:
9296:
9280:
9277:
9276:
9247:
9243:
9235:
9232:
9231:
9211:
9207:
9205:
9202:
9201:
9168:
9165:
9164:
9129:
9125:
9119:
9115:
9110:
9107:
9106:
9073:
9069:
9067:
9064:
9063:
9043:
9040:
9039:
9038:, there exists
9017:
9014:
9013:
8997:
8994:
8993:
8962:
8959:
8958:
8919:
8915:
8909:
8905:
8900:
8897:
8896:
8873:
8869:
8842:
8838:
8837:
8832:
8815:
8812:
8811:
8810:, is given by
8800:
8716:
8709:
8682:
8666:
8662:
8641:
8633:
8618:
8590:
8585:
8579:
8576:
8575:
8572:
8564:
8549:
8537:
8512:
8488:
8463:
8431:
8430:
8428:
8386:
8367:
8361:
8358:
8357:
8333:
8309:
8278:
8261:
8260:
8258:
8243:
8237:
8234:
8233:
8230:
8205:
8164:
8147:
8146:
8144:
8129:
8123:
8120:
8119:
8117:
8110:
8008:), and second,
7938:
7924:
7923:
7917:
7885:
7884:
7882:
7870:
7864:
7861:
7860:
7856:
7809:
7804:
7801:
7800:
7764:
7759:
7756:
7755:
7730:
7727:
7726:
7704:
7701:
7700:
7678:
7666:
7642:
7618:
7611:
7606:
7596:
7591:
7582:
7579:
7578:
7555:
7550:
7544:
7541:
7540:
7521:
7514:
7509:
7499:
7494:
7478:
7461:
7456:
7451:
7447:
7446:
7444:
7441:
7440:
7420:
7416:
7390:
7387:
7386:
7367:
7350:
7346:
7345:
7341:
7340:
7317:
7314:
7313:
7263:
7259:
7255:
7249:
7245:
7243:
7233:
7220:
7215:
7214:
7210:
7187:
7184:
7183:
7157:
7141:
7137:
7125:
7120:
7098:
7094:
7079:
7075:
7073:
7070:
7069:
7054:
7045:
7025:
7017:
6986:
6981:
6975:
6972:
6971:
6936:
6932:
6917:
6913:
6911:
6908:
6907:
6858:
6854:
6850:
6844:
6840:
6838:
6828:
6815:
6810:
6809:
6805:
6803:
6800:
6799:
6781:
6775:
6744:
6739:
6738:
6712:
6707:
6692:
6687:
6655:
6651:
6634:
6631:
6630:
6607:
6602:
6601:
6575:
6570:
6564:
6559:
6538:
6535:
6534:
6477:
6474:
6473:
6448:
6445:
6444:
6428:
6425:
6424:
6399:
6396:
6395:
6366:
6363:
6362:
6342:
6338:
6330:
6327:
6326:
6295:
6292:
6291:
6271:
6257:
6254:
6253:
6213:
6189:
6185:
6183:
6180:
6179:
6162:
6158:
6156:
6153:
6152:
6136:
6128:
6125:
6124:
6108:
6085:
6082:
6081:
6056:
6053:
6052:
6049:
6022:
6018:
6009:
6005:
5999:
5995:
5990:
5987:
5986:
5954:
5950:
5927:
5923:
5891:
5877:
5875:
5871:
5832:
5828:
5826:
5823:
5822:
5804:
5803:
5798:
5792:
5791:
5786:
5776:
5775:
5767:
5764:
5763:
5745:
5744:
5730:
5713:
5708:
5693:
5689:
5686:
5685:
5647:
5640:
5636:
5634:
5629:
5608:
5604:
5601:
5600:
5595:
5550:
5545:
5524:
5520:
5516:
5514:
5511:
5510:
5475:
5472:
5471:
5424:
5418:
5415:
5414:
5398:
5390:
5387:
5386:
5361:
5358:
5357:
5354:
5328:
5324:
5320:
5308:
5304:
5302:
5299:
5298:
5295:
5282:
5278:
5273:
5265:
5248:
5244:
5229:
5225:
5216:
5212:
5210:
5207:
5206:
5203:
5178:
5174:
5164:
5159:
5144:
5140:
5138:
5135:
5134:
5127:
5124:
5113:for decreasing
5094:
5090:
5080:
5075:
5060:
5056:
5054:
5051:
5050:
5043:
5041:Self-similarity
5019:
5015:
4966:
4962:
4949:
4947:
4940:
4936:
4910:
4906:
4852:
4837:
4833:
4832:
4828:
4827:
4823:
4798:
4794:
4793:
4789:
4788:
4784:
4781:
4778:
4777:
4760:
4756:
4754:
4751:
4750:
4710:
4707:
4706:
4689:
4685:
4683:
4680:
4679:
4663:
4660:
4659:
4632:
4629:
4607:
4601:
4597:
4595:
4591:
4587:
4576:
4570:
4561:
4556:
4525:
4521:
4520:
4516:
4507:
4502:
4486:
4482:
4471:
4468:
4467:
4447:
4446:
4418:
4412:
4408:
4406:
4402:
4398:
4387:
4381:
4372:
4371:
4352:
4345:
4341:
4325:
4323:
4319:
4315:
4299:
4295:
4272:
4270:
4264:
4256:
4219:
4215:
4206:
4202:
4201:
4197:
4191:
4183:
4172:
4155:
4151:
4150:
4146:
4142:
4140:
4137:
4136:
4125:
4106:
4102:
4101:
4097:
4095:
4092:
4091:
4038:
4031:
4027:
4011:
4009:
4005:
4001:
3985:
3981:
3958:
3956:
3930:
3926:
3917:
3913:
3912:
3908:
3906:
3903:
3902:
3898:
3893:
3876:
3872:
3854:
3841:
3837:
3835:
3832:
3831:
3828:
3826:Running maximum
3791:
3788:
3787:
3762:
3758:
3747:
3745:
3742:
3741:
3713:
3710:
3709:
3677:
3670:
3666:
3665:
3638:
3631:
3627:
3626:
3622:
3615:
3613:
3607:
3603:
3597:
3586:
3575:
3566:
3562:
3560:
3557:
3556:
3535:
3518:
3516:
3510:
3506:
3500:
3489:
3478:
3466:
3462:
3453:
3449:
3447:
3444:
3443:
3426:
3422:
3420:
3417:
3416:
3409:
3400:
3375:
3371:
3362:
3358:
3356:
3345:
3341:
3340:
3336:
3325:
3321:
3320:
3316:
3314:
3311:
3310:
3307:
3300:
3294:
3272:
3268:
3255:
3248:
3244:
3243:
3235:
3215:
3211:
3210:
3206:
3195:
3191:
3190:
3186:
3175:
3172:
3171:
3141:
3137:
3136:
3132:
3121:
3117:
3116:
3112:
3089:
3085:
3084:
3080:
3052:
3048:
3047:
3043:
3032:
3028:
3027:
3023:
3009:
3005:
3004:
3000:
2999:
2995:
2987:
2984:
2983:
2964:
2960:
2959:
2955:
2944:
2940:
2939:
2935:
2933:
2930:
2929:
2910:
2906:
2905:
2901:
2890:
2886:
2885:
2881:
2870:
2866:
2865:
2861:
2859:
2856:
2855:
2839:
2838:
2825:
2818:
2814:
2813:
2805:
2780:
2776:
2775:
2771:
2760:
2756:
2755:
2751:
2737:
2733:
2732:
2728:
2727:
2723:
2708:
2707:
2691:
2687:
2686:
2682:
2668:
2664:
2663:
2659:
2648:
2644:
2643:
2639:
2622:
2618:
2617:
2613:
2612:
2608:
2595:
2584:
2580:
2579:
2575:
2564:
2560:
2559:
2555:
2542:
2540:
2537:
2536:
2517:
2513:
2512:
2508:
2494:
2490:
2489:
2485:
2474:
2470:
2469:
2465:
2451:
2447:
2446:
2442:
2440:
2437:
2436:
2407:
2403:
2402:
2398:
2387:
2383:
2382:
2378:
2377:
2373:
2345:
2341:
2340:
2336:
2316:
2312:
2311:
2307:
2287:
2283:
2282:
2278:
2258:
2254:
2253:
2249:
2245:
2241:
2221:
2217:
2216:
2212:
2201:
2197:
2196:
2192:
2181:
2178:
2177:
2160:
2156:
2147:
2143:
2141:
2138:
2137:
2121:
2120:
2105:
2087:
2073:
2069:
2068:
2064:
2056:
2052:
2051:
2047:
2046:
2036:
2032:
2023:
2019:
2009:
2007:
2000:
1991:
1987:
1978:
1974:
1962:
1961:
1948:
1939:
1935:
1926:
1922:
1909:
1907:
1904:
1903:
1881:
1878:
1877:
1866:
1821:
1817:
1808:
1804:
1795:
1791:
1789:
1786:
1785:
1750:
1746:
1735:
1732:
1731:
1727:
1695:
1691:
1680:
1677:
1676:
1637:
1631:
1627:
1623:
1619:
1601:
1581:
1577:
1576:
1572:
1570:
1567:
1566:
1562:
1547:
1534:
1512:
1508:
1506:
1503:
1502:
1480:
1477:
1476:
1435:
1432:
1431:
1405:
1401:
1383:
1379:
1377:
1374:
1373:
1352:
1348:
1346:
1343:
1342:
1325:
1321:
1319:
1316:
1315:
1270:
1266:
1239:
1223:
1205:
1201:
1199:
1196:
1195:
1164:
1160:
1151:
1147:
1145:
1142:
1141:
1138:
1130:Wiener integral
1118:
1112:
1106:probability law
1097:
1075:
1071:
1070:
1066:
1057:
1053:
1051:
1048:
1047:
1046:, meaning that
1044:scale invariant
990:
982:
975:
968:
962:
944:
943:
932:
931:
921:
918:
917:
906:
905:
895:
892:
885:
878:
871:
864:
844:
841:
840:
823:
819:
817:
814:
813:
797:
794:
793:
757:
756:
747:
743:
728:
724:
722:
719:
718:
702:
699:
698:
682:
679:
678:
661:
657:
642:
638:
636:
633:
632:
616:
613:
612:
586:
583:
582:
565:
561:
559:
556:
555:
530:
527:
526:
507:
503:
488:
484:
482:
479:
478:
453:
450:
449:
429:
426:
425:
398:
394:
392:
389:
388:
367:
363:
361:
358:
357:
353:
301:Brownian motion
205:Brownian motion
194:continuous-time
144:
140:
138:
135:
134:
110:
107:
106:
91:
77:
66:
60:
57:
47:Please help to
46:
30:
26:
19:
12:
11:
5:
14933:
14923:
14922:
14920:LĂ©vy processes
14917:
14912:
14910:Wiener process
14895:
14894:
14892:
14891:
14886:
14884:List of topics
14880:
14877:
14876:
14874:
14873:
14868:
14863:
14858:
14853:
14848:
14843:
14841:Renewal theory
14838:
14833:
14828:
14823:
14818:
14813:
14808:
14806:Ergodic theory
14803:
14798:
14796:Control theory
14793:
14787:
14785:
14781:
14780:
14778:
14777:
14776:
14775:
14770:
14760:
14755:
14750:
14745:
14740:
14739:
14738:
14728:
14726:Snell envelope
14723:
14718:
14713:
14708:
14703:
14698:
14693:
14688:
14683:
14678:
14673:
14668:
14663:
14658:
14653:
14648:
14643:
14638:
14633:
14628:
14623:
14618:
14613:
14608:
14603:
14598:
14592:
14590:
14586:
14585:
14583:
14582:
14577:
14572:
14567:
14562:
14556:
14554:
14548:
14547:
14545:
14544:
14525:BorelâCantelli
14514:
14509:
14504:
14499:
14494:
14489:
14484:
14479:
14474:
14469:
14463:
14461:
14460:Limit theorems
14457:
14456:
14454:
14453:
14448:
14443:
14438:
14433:
14428:
14423:
14418:
14413:
14408:
14403:
14398:
14393:
14388:
14383:
14377:
14375:
14371:
14370:
14368:
14367:
14362:
14357:
14352:
14347:
14342:
14336:
14334:
14328:
14327:
14325:
14324:
14319:
14314:
14309:
14303:
14301:
14295:
14294:
14292:
14291:
14286:
14281:
14276:
14271:
14266:
14261:
14256:
14251:
14246:
14241:
14236:
14231:
14226:
14221:
14216:
14211:
14206:
14201:
14195:
14193:
14187:
14186:
14184:
14183:
14178:
14173:
14168:
14163:
14158:
14152:
14150:
14144:
14143:
14141:
14140:
14135:
14130:
14129:
14128:
14123:
14113:
14108:
14103:
14098:
14097:
14096:
14091:
14081:
14079:Hopfield model
14076:
14071:
14066:
14060:
14058:
14054:
14053:
14051:
14050:
14045:
14040:
14035:
14030:
14025:
14024:
14023:
14018:
14013:
14008:
13998:
13996:Markov process
13993:
13988:
13983:
13977:
13975:
13971:
13970:
13968:
13967:
13965:Wiener sausage
13962:
13960:Wiener process
13957:
13952:
13947:
13942:
13940:Stable process
13937:
13932:
13930:Semimartingale
13927:
13922:
13921:
13920:
13915:
13905:
13900:
13895:
13890:
13885:
13880:
13875:
13873:Jump diffusion
13870:
13865:
13860:
13855:
13850:
13848:Hawkes process
13845:
13840:
13835:
13830:
13828:Feller process
13825:
13820:
13815:
13810:
13805:
13800:
13795:
13793:Cauchy process
13790:
13789:
13788:
13783:
13778:
13773:
13768:
13758:
13757:
13756:
13746:
13744:Bessel process
13741:
13735:
13733:
13727:
13726:
13724:
13723:
13722:
13721:
13716:
13711:
13706:
13696:
13691:
13686:
13681:
13676:
13671:
13666:
13660:
13658:
13652:
13651:
13644:
13643:
13636:
13629:
13621:
13615:
13614:
13606:
13600:
13595:
13590:
13583:
13582:External links
13580:
13579:
13578:
13564:
13555:
13549:
13536:
13526:
13513:
13491:
13488:
13485:
13484:
13449:
13440:
13431:
13418:
13394:
13373:(1): 143â149.
13351:
13342:
13333:
13326:
13308:
13301:
13283:
13274:
13256:
13227:(4): 585â614.
13207:
13200:
13179:
13169:
13168:
13166:
13163:
13159:
13158:
13154:
13153:
13148:
13136:
13135:
13134:
13129:
13124:
13119:
13114:
13109:
13094:
13092:
13089:
13076:
13063:Brownian sheet
13061:Main article:
13058:
13057:Brownian sheet
13055:
13040:
13036:
13013:
13009:
12986:
12982:
12978:
12975:
12970:
12966:
12962:
12939:
12919:
12915:
12908:
12903:
12896:
12892:
12887:
12881:
12876:
12872:
12868:
12865:
12862:
12859:
12856:
12853:
12831:
12828:
12825:
12822:
12818:
12814:
12811:
12806:
12802:
12796:
12792:
12788:
12785:
12781:
12775:
12770:
12766:
12762:
12757:
12752:
12748:
12743:
12739:
12734:
12729:
12725:
12699:
12696:
12693:
12690:
12687:
12684:
12679:
12675:
12671:
12668:
12644:
12632:
12629:
12614:
12610:
12606:
12603:
12583:
12580:
12576:
12572:
12568:
12547:
12532:
12529:
12515:
12492:
12487:
12459:
12455:
12432:
12428:
12405:
12401:
12397:
12394:
12389:
12385:
12381:
12376:
12372:
12359:
12356:
12332:
12329:
12302:
12299:
12296:
12291:
12286:
12282:
12278:
12273:
12268:
12264:
12243:
12240:
12237:
12234:
12229:
12224:
12220:
12216:
12211:
12206:
12202:
12181:
12178:
12175:
12172:
12167:
12162:
12158:
12154:
12149:
12144:
12140:
12136:
12133:
12130:
12111:
12108:
12105:
12102:
12097:
12092:
12088:
12084:
12079:
12074:
12070:
12066:
12063:
12060:
12038:
12033:
12029:
12023:
12020:
12017:
12013:
12012:lim sup
12009:
12004:
11999:
11995:
11975:
11970:
11966:
11960:
11957:
11954:
11950:
11949:lim inf
11946:
11941:
11936:
11932:
11918:
11872:
11869:
11866:
11863:
11859:
11855:
11850:
11846:
11842:
11837:
11833:
11801:
11797:
11790:
11785:
11781:
11775:
11770:
11766:
11762:
11759:
11756:
11753:
11750:
11747:
11725:
11722:
11719:
11716:
11712:
11708:
11705:
11702:
11697:
11692:
11688:
11648:
11635:
11632:
11613:
11608:
11602:
11599:
11593:
11589:
11583:
11575:
11572:
11569:
11566:
11562:
11532:
11529:
11526:
11523:
11520:
11517:
11497:
11494:
11491:
11486:
11483:
11480:
11477:
11473:
11469:
11466:
11463:
11460:
11455:
11451:
11430:
11427:
11424:
11421:
11416:
11412:
11408:
11405:
11402:
11397:
11393:
11389:
11386:
11383:
11380:
11377:
11374:
11371:
11366:
11362:
11358:
11355:
11352:
11349:
11344:
11340:
11317:
11314:
11311:
11308:
11303:
11299:
11295:
11292:
11289:
11284:
11281:
11278:
11275:
11270:
11266:
11262:
11259:
11256:
11253:
11250:
11247:
11242:
11238:
11234:
11231:
11228:
11222:
11219:
11195:
11192:
11189:
11186:
11183:
11180:
11177:
11172:
11168:
11164:
11161:
11158:
11155:
11152:
11149:
11146:
11143:
11138:
11134:
11113:
11110:
11107:
11102:
11098:
11077:
11074:
11071:
11068:
11065:
11060:
11056:
11035:
11032:
11029:
11024:
11020:
10999:
10996:
10992:
10989:
10986:
10983:
10980:
10977:
10974:
10971:
10968:
10965:
10962:
10959:
10956:
10953:
10950:
10947:
10944:
10941:
10938:
10935:
10932:
10929:
10926:
10923:
10918:
10913:
10909:
10905:
10902:
10899:
10896:
10893:
10888:
10884:
10880:
10877:
10874:
10871:
10868:
10865:
10860:
10856:
10852:
10849:
10846:
10827:
10824:
10818:
10814:
10810:
10807:
10804:
10801:
10798:
10795:
10792:
10789:
10786:
10783:
10778:
10773:
10769:
10765:
10762:
10759:
10756:
10753:
10748:
10744:
10740:
10737:
10734:
10714:
10711:
10708:
10686:
10682:
10678:
10674:
10671:
10668:
10665:
10662:
10659:
10656:
10653:
10650:
10647:
10644:
10639:
10634:
10630:
10626:
10623:
10620:
10616:
10613:
10610:
10607:
10604:
10601:
10598:
10594:
10591:
10585:
10580:
10576:
10572:
10569:
10566:
10563:
10558:
10554:
10530:
10527:
10524:
10521:
10518:
10515:
10512:
10509:
10506:
10503:
10467:
10464:
10460:
10457:
10454:
10451:
10446:
10441:
10437:
10433:
10430:
10427:
10424:
10419:
10416:
10413:
10410:
10406:
10393:
10390:
10375:
10360:
10350:
10347:
10322:
10319:
10315:
10310:
10306:
10302:
10299:
10296:
10293:
10288:
10283:
10279:
10275:
10272:
10269:
10266:
10261:
10257:
10228:
10223:
10160:
10157:
10152:
10148:
10127:
10124:
10121:
10101:
10098:
10095:
10067:
10064:
10060:
10055:
10049:
10046:
10042:
10038:
10033:
10029:
10003:
9996:
9992:
9988:
9985:
9980:
9976:
9971:
9967:
9960:
9957:
9954:
9950:
9879:LĂ©vy processes
9860:
9856:
9852:
9849:
9846:
9843:
9840:
9835:
9831:
9764:
9761:
9748:
9744:
9738:
9734:
9713:
9710:
9707:
9704:
9701:
9698:
9695:
9692:
9689:
9686:
9683:
9680:
9677:
9674:
9671:
9666:
9662:
9639:
9636:
9632:
9628:
9625:
9621:
9617:
9614:
9611:
9608:
9605:
9602:
9599:
9596:
9593:
9590:
9587:
9584:
9564:
9561:
9558:
9554:
9550:
9547:
9542:
9539:
9534:
9531:
9528:
9525:
9522:
9518:
9514:
9509:
9504:
9500:
9494:
9490:
9486:
9481:
9476:
9472:
9466:
9461:
9457:
9437:
9434:
9431:
9427:
9422:
9416:
9413:
9408:
9405:
9402:
9399:
9396:
9390:
9386:
9381:
9376:
9372:
9366:
9361:
9357:
9351:
9346:
9342:
9336:
9333:
9328:
9324:
9320:
9315:
9311:
9307:
9304:
9284:
9261:
9258:
9255:
9250:
9246:
9242:
9239:
9214:
9210:
9178:
9175:
9172:
9150:
9147:
9144:
9141:
9138:
9135:
9132:
9128:
9122:
9118:
9114:
9105:in recovering
9088:
9085:
9082:
9079:
9076:
9072:
9047:
9027:
9024:
9021:
9001:
8978:
8975:
8972:
8969:
8966:
8940:
8937:
8934:
8931:
8928:
8925:
8922:
8918:
8912:
8908:
8904:
8884:
8879:
8876:
8872:
8868:
8865:
8859:
8856:
8853:
8850:
8845:
8841:
8836:
8831:
8828:
8825:
8822:
8819:
8799:
8796:
8714:
8707:
8689:
8685:
8680:
8677:
8674:
8669:
8665:
8661:
8658:
8655:
8652:
8647:
8644:
8639:
8636:
8632:
8628:
8625:
8621:
8616:
8613:
8610:
8607:
8604:
8601:
8598:
8593:
8588:
8584:
8568:
8548:
8545:
8536:
8531:
8519:
8510:
8507:
8504:
8498:
8495:
8491:
8487:
8484:
8481:
8478:
8475:
8472:
8466:
8462:
8459:
8456:
8453:
8450:
8447:
8444:
8441:
8438:
8434:
8425:
8422:
8419:
8416:
8413:
8410:
8407:
8404:
8401:
8398:
8395:
8392:
8389:
8385:
8379:
8376:
8373:
8370:
8366:
8365:lim sup
8340:
8331:
8328:
8325:
8319:
8316:
8312:
8308:
8305:
8302:
8299:
8296:
8293:
8290:
8287:
8281:
8277:
8274:
8271:
8268:
8264:
8255:
8252:
8249:
8246:
8242:
8241:lim sup
8229:
8224:
8212:
8203:
8200:
8197:
8191:
8188:
8185:
8182:
8179:
8176:
8173:
8167:
8163:
8160:
8157:
8154:
8150:
8141:
8138:
8135:
8132:
8128:
8127:lim sup
8116:
8111:
8109:
8106:
8105:
8104:
8083:
8068:
8057:
7966:
7954:
7951:
7948:
7941:
7937:
7934:
7931:
7927:
7920:
7916:
7913:
7910:
7907:
7904:
7901:
7898:
7895:
7892:
7888:
7879:
7876:
7873:
7869:
7841:
7829:
7826:
7823:
7817:
7814:
7808:
7784:
7781:
7778:
7772:
7769:
7763:
7743:
7740:
7737:
7734:
7714:
7711:
7708:
7697:
7686:
7677:
7674:
7665:
7662:
7628:
7625:
7621:
7614:
7609:
7605:
7599:
7594:
7590:
7586:
7566:
7563:
7558:
7553:
7549:
7528:
7524:
7517:
7512:
7508:
7502:
7497:
7493:
7489:
7486:
7481:
7476:
7472:
7469:
7464:
7459:
7455:
7450:
7428:
7423:
7419:
7415:
7412:
7409:
7406:
7403:
7400:
7397:
7394:
7375:
7370:
7365:
7361:
7358:
7353:
7349:
7344:
7339:
7336:
7333:
7330:
7327:
7324:
7321:
7297:
7294:
7291:
7288:
7285:
7282:
7279:
7275:
7266:
7262:
7258:
7252:
7248:
7240:
7237:
7232:
7226:
7223:
7219:
7213:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7167:
7164:
7160:
7155:
7152:
7149:
7144:
7140:
7136:
7133:
7128:
7123:
7119:
7115:
7112:
7109:
7106:
7101:
7097:
7093:
7090:
7087:
7082:
7078:
7044:) is equal to
6997:
6994:
6989:
6984:
6980:
6950:
6947:
6944:
6939:
6935:
6931:
6928:
6925:
6920:
6916:
6895:
6892:
6889:
6886:
6883:
6880:
6877:
6874:
6870:
6861:
6857:
6853:
6847:
6843:
6835:
6832:
6827:
6821:
6818:
6814:
6808:
6795:satisfies the
6774:
6771:
6759:
6756:
6753:
6747:
6742:
6737:
6734:
6731:
6728:
6725:
6722:
6718:
6715:
6710:
6704:
6701:
6698:
6695:
6690:
6686:
6682:
6679:
6675:
6672:
6669:
6666:
6661:
6658:
6654:
6650:
6647:
6644:
6641:
6638:
6619:
6616:
6610:
6605:
6600:
6597:
6594:
6591:
6588:
6585:
6581:
6578:
6573:
6567:
6562:
6558:
6554:
6551:
6548:
6545:
6542:
6523:
6520:
6517:
6514:
6511:
6508:
6505:
6502:
6499:
6496:
6493:
6490:
6487:
6484:
6481:
6461:
6458:
6455:
6452:
6432:
6412:
6409:
6406:
6403:
6379:
6376:
6373:
6370:
6350:
6345:
6341:
6337:
6334:
6314:
6311:
6308:
6305:
6302:
6299:
6274:
6270:
6267:
6264:
6261:
6241:
6238:
6235:
6232:
6229:
6226:
6223:
6220:
6216:
6212:
6209:
6206:
6203:
6200:
6197:
6192:
6188:
6165:
6161:
6139:
6135:
6132:
6111:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6069:
6066:
6063:
6060:
6048:
6045:
6033:
6028:
6025:
6021:
6017:
6012:
6008:
6002:
5998:
5994:
5970:
5966:
5961:
5958:
5953:
5949:
5946:
5943:
5939:
5934:
5931:
5926:
5922:
5919:
5916:
5913:
5909:
5903:
5900:
5897:
5894:
5889:
5886:
5883:
5880:
5874:
5870:
5867:
5864:
5861:
5858:
5855:
5852:
5849:
5846:
5843:
5840:
5835:
5831:
5808:
5802:
5799:
5797:
5794:
5793:
5790:
5787:
5785:
5782:
5781:
5779:
5774:
5771:
5743:
5740:
5737:
5733:
5729:
5726:
5723:
5720:
5717:
5714:
5712:
5709:
5707:
5704:
5701:
5696:
5692:
5688:
5687:
5684:
5681:
5678:
5674:
5671:
5668:
5665:
5662:
5659:
5654:
5650:
5646:
5643:
5639:
5635:
5633:
5630:
5628:
5625:
5622:
5617:
5614:
5611:
5607:
5603:
5602:
5598:
5594:
5591:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5549:
5546:
5544:
5541:
5538:
5533:
5530:
5527:
5523:
5519:
5518:
5494:
5491:
5488:
5485:
5482:
5479:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5436:
5433:
5430:
5427:
5423:
5401:
5397:
5394:
5374:
5371:
5368:
5365:
5353:
5350:
5335:
5331:
5327:
5323:
5319:
5316:
5311:
5307:
5294:
5293:Time inversion
5291:
5276:
5251:
5247:
5243:
5238:
5235:
5232:
5228:
5224:
5219:
5215:
5202:
5199:
5184:
5181:
5177:
5173:
5168:
5162:
5158:
5155:
5152:
5147:
5143:
5123:
5120:
5100:
5097:
5093:
5089:
5084:
5078:
5074:
5071:
5068:
5063:
5059:
5042:
5039:
5027:
5022:
5018:
5014:
5011:
5008:
5005:
5002:
4999:
4989:
4978:
4974:
4969:
4965:
4961:
4958:
4955:
4952:
4946:
4943:
4939:
4932:
4929:
4923:
4919:
4913:
4909:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4867:
4864:
4861:
4858:
4855:
4851:
4847:
4840:
4836:
4831:
4826:
4822:
4819:
4816:
4813:
4810:
4801:
4797:
4792:
4787:
4763:
4759:
4726:
4723:
4720:
4717:
4714:
4692:
4688:
4667:
4642:
4638:
4635:
4628:
4625:
4622:
4613:
4610:
4604:
4600:
4594:
4590:
4582:
4579:
4575:
4569:
4564:
4559:
4555:
4551:
4548:
4545:
4541:
4538:
4535:
4528:
4524:
4519:
4515:
4510:
4505:
4501:
4497:
4494:
4489:
4485:
4481:
4478:
4475:
4445:
4442:
4439:
4436:
4432:
4424:
4421:
4415:
4411:
4405:
4401:
4393:
4390:
4386:
4380:
4377:
4375:
4373:
4370:
4367:
4358:
4355:
4348:
4344:
4340:
4337:
4334:
4331:
4328:
4322:
4318:
4309:
4306:
4303:
4298:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4267:
4262:
4259:
4255:
4251:
4248:
4245:
4241:
4238:
4235:
4232:
4229:
4222:
4218:
4214:
4209:
4205:
4200:
4194:
4189:
4186:
4182:
4178:
4175:
4173:
4171:
4168:
4165:
4158:
4154:
4149:
4145:
4144:
4109:
4105:
4100:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4052:
4044:
4041:
4034:
4030:
4026:
4023:
4020:
4017:
4014:
4008:
4004:
3995:
3992:
3989:
3984:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3955:
3952:
3949:
3946:
3943:
3940:
3933:
3929:
3925:
3920:
3916:
3911:
3896:
3879:
3875:
3869:
3866:
3863:
3860:
3857:
3853:
3849:
3844:
3840:
3827:
3824:
3807:
3804:
3801:
3798:
3795:
3774:
3769:
3766:
3761:
3757:
3751:
3729:
3726:
3723:
3720:
3717:
3694:
3690:
3684:
3681:
3676:
3673:
3669:
3662:
3658:
3655:
3651:
3645:
3642:
3637:
3634:
3630:
3625:
3621:
3618:
3610:
3606:
3600:
3595:
3592:
3589:
3585:
3579:
3574:
3569:
3565:
3541:
3538:
3533:
3530:
3527:
3524:
3521:
3513:
3509:
3503:
3498:
3495:
3492:
3488:
3482:
3477:
3474:
3469:
3465:
3461:
3456:
3452:
3429:
3425:
3413:Fourier series
3408:
3405:
3388:
3385:
3378:
3374:
3370:
3365:
3361:
3355:
3348:
3344:
3339:
3335:
3328:
3324:
3319:
3305:
3298:
3280:
3275:
3271:
3267:
3263:
3258:
3251:
3247:
3242:
3238:
3234:
3231:
3228:
3225:
3218:
3214:
3209:
3205:
3198:
3194:
3189:
3185:
3182:
3179:
3157:
3154:
3151:
3144:
3140:
3135:
3131:
3124:
3120:
3115:
3111:
3108:
3105:
3102:
3099:
3092:
3088:
3083:
3079:
3076:
3073:
3070:
3066:
3062:
3055:
3051:
3046:
3042:
3035:
3031:
3026:
3022:
3019:
3012:
3008:
3003:
2998:
2994:
2991:
2967:
2963:
2958:
2954:
2947:
2943:
2938:
2913:
2909:
2904:
2900:
2893:
2889:
2884:
2880:
2873:
2869:
2864:
2837:
2833:
2828:
2821:
2817:
2812:
2808:
2804:
2801:
2798:
2794:
2790:
2783:
2779:
2774:
2770:
2763:
2759:
2754:
2750:
2747:
2740:
2736:
2731:
2726:
2722:
2719:
2716:
2713:
2711:
2709:
2705:
2701:
2694:
2690:
2685:
2681:
2678:
2671:
2667:
2662:
2658:
2651:
2647:
2642:
2638:
2635:
2632:
2625:
2621:
2616:
2611:
2607:
2604:
2601:
2598:
2596:
2594:
2587:
2583:
2578:
2574:
2567:
2563:
2558:
2554:
2551:
2548:
2545:
2544:
2535:we arrive at:
2520:
2516:
2511:
2507:
2504:
2497:
2493:
2488:
2484:
2477:
2473:
2468:
2464:
2461:
2454:
2450:
2445:
2422:
2418:
2410:
2406:
2401:
2397:
2390:
2386:
2381:
2376:
2372:
2369:
2366:
2362:
2358:
2355:
2348:
2344:
2339:
2335:
2332:
2329:
2326:
2319:
2315:
2310:
2306:
2303:
2300:
2297:
2290:
2286:
2281:
2277:
2274:
2271:
2268:
2261:
2257:
2252:
2248:
2244:
2240:
2237:
2234:
2231:
2224:
2220:
2215:
2211:
2204:
2200:
2195:
2191:
2188:
2185:
2163:
2159:
2155:
2150:
2146:
2119:
2113:
2110:
2104:
2098:
2095:
2091:
2086:
2076:
2072:
2067:
2059:
2055:
2050:
2044:
2039:
2035:
2031:
2026:
2022:
2018:
2015:
2012:
2006:
2003:
2001:
1999:
1994:
1990:
1986:
1981:
1977:
1973:
1970:
1967:
1964:
1963:
1960:
1957:
1954:
1951:
1949:
1947:
1942:
1938:
1934:
1929:
1925:
1921:
1918:
1915:
1912:
1911:
1891:
1888:
1885:
1865:
1862:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1824:
1820:
1816:
1811:
1807:
1803:
1798:
1794:
1767:
1764:
1761:
1758:
1753:
1749:
1745:
1742:
1739:
1709:
1706:
1703:
1698:
1694:
1690:
1687:
1684:
1658:
1653:
1650:
1647:
1644:
1640:
1634:
1630:
1626:
1622:
1615:
1612:
1609:
1605:
1600:
1597:
1594:
1591:
1584:
1580:
1575:
1546:
1543:
1533:
1530:
1515:
1511:
1490:
1487:
1484:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1419:
1416:
1413:
1408:
1404:
1400:
1397:
1394:
1391:
1386:
1382:
1355:
1351:
1328:
1324:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1278:
1273:
1269:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1238:
1231:
1227:
1222:
1219:
1216:
1213:
1208:
1204:
1175:
1172:
1167:
1163:
1159:
1154:
1150:
1137:
1134:
1102:Wiener measure
1083:
1078:
1074:
1069:
1063:
1060:
1056:
986:
966:
941:
937:
929:
925:
915:
911:
903:
899:
890:
883:
876:
869:
861:
860:
848:
826:
822:
801:
791:
780:
777:
774:
771:
768:
765:
760:
755:
750:
746:
742:
737:
734:
731:
727:
706:
686:
664:
660:
656:
651:
648:
645:
641:
620:
610:
599:
596:
593:
590:
568:
564:
543:
540:
537:
534:
515:
510:
506:
502:
497:
494:
491:
487:
466:
463:
460:
457:
433:
423:
409:
406:
401:
397:
370:
366:
352:
349:
294:control theory
286:Brownian noise
213:LĂ©vy processes
201:Norbert Wiener
190:Wiener process
164:
163:
152:
147:
143:
132:
126:
125:
114:
104:
98:
97:
89:
84:Wiener Process
79:
78:
33:
31:
24:
17:
9:
6:
4:
3:
2:
14932:
14921:
14918:
14916:
14913:
14911:
14908:
14907:
14905:
14890:
14887:
14885:
14882:
14881:
14878:
14872:
14869:
14867:
14864:
14862:
14859:
14857:
14854:
14852:
14849:
14847:
14844:
14842:
14839:
14837:
14834:
14832:
14829:
14827:
14824:
14822:
14819:
14817:
14814:
14812:
14809:
14807:
14804:
14802:
14799:
14797:
14794:
14792:
14789:
14788:
14786:
14782:
14774:
14771:
14769:
14766:
14765:
14764:
14761:
14759:
14756:
14754:
14751:
14749:
14746:
14744:
14743:Stopping time
14741:
14737:
14734:
14733:
14732:
14729:
14727:
14724:
14722:
14719:
14717:
14714:
14712:
14709:
14707:
14704:
14702:
14699:
14697:
14694:
14692:
14689:
14687:
14684:
14682:
14679:
14677:
14674:
14672:
14669:
14667:
14664:
14662:
14659:
14657:
14654:
14652:
14649:
14647:
14644:
14642:
14639:
14637:
14634:
14632:
14629:
14627:
14624:
14622:
14619:
14617:
14614:
14612:
14609:
14607:
14604:
14602:
14599:
14597:
14594:
14593:
14591:
14587:
14581:
14578:
14576:
14573:
14571:
14568:
14566:
14563:
14561:
14558:
14557:
14555:
14553:
14549:
14542:
14538:
14534:
14533:HewittâSavage
14530:
14526:
14522:
14518:
14517:Zeroâone laws
14515:
14513:
14510:
14508:
14505:
14503:
14500:
14498:
14495:
14493:
14490:
14488:
14485:
14483:
14480:
14478:
14475:
14473:
14470:
14468:
14465:
14464:
14462:
14458:
14452:
14449:
14447:
14444:
14442:
14439:
14437:
14434:
14432:
14429:
14427:
14424:
14422:
14419:
14417:
14414:
14412:
14409:
14407:
14404:
14402:
14399:
14397:
14394:
14392:
14389:
14387:
14384:
14382:
14379:
14378:
14376:
14372:
14366:
14363:
14361:
14358:
14356:
14353:
14351:
14348:
14346:
14343:
14341:
14338:
14337:
14335:
14333:
14329:
14323:
14320:
14318:
14315:
14313:
14310:
14308:
14305:
14304:
14302:
14300:
14296:
14290:
14287:
14285:
14282:
14280:
14277:
14275:
14272:
14270:
14267:
14265:
14262:
14260:
14257:
14255:
14252:
14250:
14247:
14245:
14242:
14240:
14237:
14235:
14232:
14230:
14227:
14225:
14222:
14220:
14217:
14215:
14214:BlackâScholes
14212:
14210:
14207:
14205:
14202:
14200:
14197:
14196:
14194:
14192:
14188:
14182:
14179:
14177:
14174:
14172:
14169:
14167:
14164:
14162:
14159:
14157:
14154:
14153:
14151:
14149:
14145:
14139:
14136:
14134:
14131:
14127:
14124:
14122:
14119:
14118:
14117:
14116:Point process
14114:
14112:
14109:
14107:
14104:
14102:
14099:
14095:
14092:
14090:
14087:
14086:
14085:
14082:
14080:
14077:
14075:
14074:Gibbs measure
14072:
14070:
14067:
14065:
14062:
14061:
14059:
14055:
14049:
14046:
14044:
14041:
14039:
14036:
14034:
14031:
14029:
14026:
14022:
14019:
14017:
14014:
14012:
14009:
14007:
14004:
14003:
14002:
13999:
13997:
13994:
13992:
13989:
13987:
13984:
13982:
13979:
13978:
13976:
13972:
13966:
13963:
13961:
13958:
13956:
13953:
13951:
13948:
13946:
13943:
13941:
13938:
13936:
13933:
13931:
13928:
13926:
13923:
13919:
13916:
13914:
13911:
13910:
13909:
13906:
13904:
13901:
13899:
13896:
13894:
13891:
13889:
13886:
13884:
13881:
13879:
13876:
13874:
13871:
13869:
13866:
13864:
13863:ItĂŽ diffusion
13861:
13859:
13856:
13854:
13851:
13849:
13846:
13844:
13841:
13839:
13838:Gamma process
13836:
13834:
13831:
13829:
13826:
13824:
13821:
13819:
13816:
13814:
13811:
13809:
13806:
13804:
13801:
13799:
13796:
13794:
13791:
13787:
13784:
13782:
13779:
13777:
13774:
13772:
13769:
13767:
13764:
13763:
13762:
13759:
13755:
13752:
13751:
13750:
13747:
13745:
13742:
13740:
13737:
13736:
13734:
13732:
13728:
13720:
13717:
13715:
13712:
13710:
13709:Self-avoiding
13707:
13705:
13702:
13701:
13700:
13697:
13695:
13694:Moran process
13692:
13690:
13687:
13685:
13682:
13680:
13677:
13675:
13672:
13670:
13667:
13665:
13662:
13661:
13659:
13657:
13656:Discrete time
13653:
13649:
13642:
13637:
13635:
13630:
13628:
13623:
13622:
13619:
13611:
13607:
13604:
13601:
13599:
13596:
13594:
13591:
13589:
13586:
13585:
13574:
13570:
13565:
13561:
13556:
13552:
13550:0-13-020071-9
13546:
13542:
13537:
13532:
13527:
13525:
13522:
13516:
13514:981-238-107-4
13510:
13505:
13504:
13498:
13494:
13493:
13481:
13477:
13473:
13469:
13465:
13461:
13453:
13444:
13435:
13428:
13422:
13408:
13404:
13398:
13390:
13386:
13381:
13376:
13372:
13368:
13367:
13362:
13355:
13346:
13337:
13329:
13323:
13319:
13312:
13304:
13298:
13294:
13287:
13278:
13270:
13266:
13260:
13252:
13248:
13244:
13240:
13235:
13230:
13226:
13222:
13218:
13211:
13203:
13201:9781108591034
13197:
13193:
13189:
13188:Durrett, Rick
13183:
13174:
13170:
13157:
13152:
13149:
13147:
13144:
13143:
13142:
13141:
13137:
13133:
13130:
13128:
13125:
13123:
13120:
13118:
13115:
13113:
13110:
13108:
13105:
13104:
13103:
13102:
13101:Generalities:
13098:
13097:
13088:
13074:
13064:
13054:
13038:
13034:
13011:
13007:
12999:is not (here
12984:
12980:
12976:
12973:
12968:
12964:
12960:
12951:
12937:
12917:
12906:
12894:
12890:
12879:
12874:
12870:
12866:
12863:
12857:
12851:
12826:
12820:
12816:
12812:
12809:
12804:
12800:
12794:
12790:
12786:
12783:
12779:
12773:
12768:
12764:
12760:
12755:
12750:
12746:
12741:
12737:
12732:
12727:
12723:
12715:
12711:
12694:
12688:
12685:
12677:
12673:
12666:
12658:
12642:
12628:
12612:
12608:
12604:
12601:
12581:
12578:
12570:
12545:
12536:
12528:
12490:
12475:
12457:
12453:
12430:
12426:
12403:
12399:
12395:
12392:
12387:
12383:
12379:
12374:
12370:
12355:
12353:
12348:
12346:
12342:
12338:
12328:
12326:
12321:
12319:
12314:
12297:
12294:
12289:
12280:
12276:
12271:
12262:
12241:
12235:
12232:
12227:
12218:
12214:
12209:
12200:
12179:
12173:
12170:
12165:
12156:
12152:
12147:
12138:
12134:
12128:
12109:
12103:
12100:
12095:
12086:
12082:
12077:
12068:
12064:
12058:
12049:
12036:
12031:
12027:
12015:
12007:
12002:
11993:
11973:
11968:
11964:
11952:
11944:
11939:
11930:
11921:
11914:
11910:
11906:
11904:
11900:
11896:
11892:
11888:
11867:
11861:
11857:
11853:
11848:
11844:
11840:
11835:
11831:
11822:
11817:
11815:
11799:
11788:
11783:
11779:
11773:
11768:
11764:
11760:
11757:
11751:
11745:
11720:
11714:
11710:
11706:
11703:
11700:
11695:
11690:
11686:
11678:
11674:
11672:
11668:
11664:
11660:
11656:
11651:
11647:
11643:
11639:
11631:
11629:
11611:
11606:
11600:
11597:
11591:
11587:
11581:
11573:
11570:
11567:
11564:
11560:
11550:
11546:
11530:
11527:
11521:
11515:
11492:
11481:
11478:
11471:
11467:
11461:
11453:
11449:
11422:
11414:
11410:
11403:
11400:
11395:
11391:
11387:
11378:
11375:
11372:
11364:
11360:
11353:
11350:
11347:
11342:
11338:
11309:
11301:
11297:
11290:
11287:
11276:
11268:
11264:
11260:
11254:
11251:
11248:
11240:
11236:
11229:
11226:
11220:
11217:
11209:
11193:
11190:
11187:
11184:
11178:
11170:
11166:
11162:
11159:
11156:
11150:
11147:
11144:
11136:
11132:
11108:
11100:
11096:
11072:
11069:
11066:
11058:
11054:
11030:
11022:
11018:
10997:
10994:
10984:
10978:
10975:
10969:
10963:
10951:
10945:
10942:
10936:
10933:
10930:
10924:
10916:
10911:
10907:
10903:
10894:
10886:
10882:
10878:
10872:
10869:
10866:
10858:
10854:
10847:
10844:
10825:
10822:
10816:
10805:
10799:
10796:
10790:
10784:
10776:
10771:
10767:
10763:
10754:
10746:
10742:
10735:
10732:
10712:
10709:
10706:
10684:
10680:
10676:
10666:
10660:
10657:
10651:
10645:
10637:
10632:
10628:
10624:
10621:
10618:
10611:
10605:
10599:
10592:
10589:
10583:
10578:
10574:
10570:
10564:
10556:
10552:
10542:
10525:
10522:
10519:
10510:
10507:
10504:
10501:
10493:
10489:
10485:
10481:
10465:
10462:
10455:
10449:
10444:
10439:
10435:
10431:
10425:
10414:
10411:
10404:
10389:
10387:
10383:
10378:
10371:
10367:
10363:
10356:
10346:
10344:
10340:
10336:
10320:
10317:
10308:
10304:
10300:
10297:
10291:
10286:
10281:
10277:
10273:
10267:
10259:
10255:
10246:
10239:
10235:
10231:
10226:
10219:
10215:
10210:
10208:
10204:
10201:
10197:
10193:
10189:
10185:
10181:
10177:
10172:
10158:
10155:
10150:
10146:
10125:
10122:
10119:
10099:
10096:
10093:
10085:
10065:
10062:
10058:
10053:
10047:
10044:
10040:
10036:
10031:
10027:
10017:
10014:
10001:
9994:
9990:
9986:
9983:
9978:
9974:
9969:
9965:
9958:
9955:
9952:
9948:
9939:
9934:
9932:
9928:
9924:
9920:
9916:
9912:
9908:
9904:
9900:
9896:
9892:
9888:
9882:
9880:
9876:
9858:
9854:
9850:
9847:
9844:
9841:
9838:
9833:
9829:
9816:
9801:
9785:
9769:
9760:
9746:
9742:
9736:
9732:
9705:
9699:
9696:
9693:
9690:
9681:
9675:
9669:
9664:
9660:
9637:
9634:
9623:
9619:
9615:
9612:
9606:
9603:
9600:
9594:
9588:
9582:
9562:
9559:
9556:
9552:
9548:
9545:
9540:
9537:
9532:
9526:
9520:
9516:
9507:
9502:
9498:
9492:
9488:
9484:
9479:
9474:
9470:
9464:
9459:
9455:
9435:
9432:
9429:
9425:
9420:
9414:
9411:
9406:
9400:
9394:
9388:
9384:
9379:
9374:
9370:
9364:
9359:
9355:
9349:
9344:
9340:
9334:
9326:
9322:
9318:
9313:
9309:
9302:
9282:
9275:
9272:and expected
9256:
9253:
9248:
9244:
9237:
9230:
9212:
9208:
9199:
9195:
9190:
9176:
9173:
9170:
9145:
9142:
9139:
9133:
9130:
9120:
9116:
9104:
9083:
9077:
9074:
9070:
9061:
9045:
9025:
9022:
9019:
8999:
8991:
8973:
8967:
8964:
8957:of less than
8956:
8935:
8932:
8929:
8923:
8920:
8910:
8906:
8882:
8877:
8874:
8870:
8866:
8863:
8857:
8854:
8851:
8848:
8843:
8839:
8834:
8829:
8823:
8817:
8809:
8805:
8795:
8791:
8789:
8785:
8781:
8777:
8773:
8769:
8765:
8761:
8757:
8753:
8749:
8745:
8741:
8737:
8733:
8729:
8725:
8721:
8717:
8710:
8703:
8687:
8675:
8667:
8663:
8656:
8650:
8642:
8634:
8630:
8626:
8623:
8608:
8602:
8596:
8591:
8586:
8582:
8571:
8567:
8562:
8558:
8554:
8544:
8542:
8535:
8530:
8517:
8514:almost surely
8508:
8505:
8502:
8493:
8489:
8485:
8479:
8476:
8473:
8470:
8457:
8451:
8448:
8442:
8436:
8423:
8420:
8417:
8414:
8411:
8408:
8405:
8402:
8399:
8396:
8393:
8390:
8387:
8377:
8374:
8368:
8355:
8351:
8338:
8335:almost surely
8329:
8326:
8323:
8314:
8310:
8306:
8300:
8297:
8294:
8291:
8288:
8285:
8272:
8266:
8253:
8250:
8244:
8228:
8223:
8210:
8207:almost surely
8201:
8198:
8195:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8158:
8152:
8136:
8130:
8115:
8102:
8098:
8095:
8094:nowhere dense
8091:
8087:
8084:
8081:
8077:
8073:
8069:
8066:
8062:
8059:The function
8058:
8055:
8051:
8047:
8043:
8039:
8035:
8031:
8027:
8023:
8019:
8015:
8011:
8007:
8003:
7999:
7995:
7991:
7987:
7983:
7979:
7975:
7971:
7968:The function
7967:
7952:
7935:
7932:
7929:
7911:
7905:
7902:
7896:
7890:
7877:
7871:
7854:
7850:
7846:
7845:local maximum
7842:
7824:
7821:
7815:
7812:
7798:
7779:
7776:
7770:
7767:
7738:
7732:
7712:
7709:
7706:
7698:
7695:
7691:
7688:The function
7687:
7684:
7680:
7679:
7673:
7671:
7661:
7659:
7653:
7649:
7645:
7639:
7626:
7623:
7612:
7607:
7603:
7597:
7592:
7588:
7584:
7564:
7561:
7556:
7551:
7547:
7526:
7515:
7510:
7506:
7500:
7495:
7491:
7487:
7484:
7479:
7474:
7470:
7467:
7462:
7457:
7453:
7448:
7426:
7421:
7417:
7413:
7410:
7404:
7401:
7398:
7392:
7373:
7368:
7363:
7359:
7356:
7351:
7347:
7342:
7337:
7331:
7328:
7325:
7319:
7312:
7308:
7295:
7289:
7286:
7283:
7277:
7273:
7264:
7260:
7250:
7238:
7235:
7230:
7224:
7211:
7207:
7201:
7198:
7195:
7189:
7181:
7165:
7162:
7150:
7147:
7142:
7138:
7131:
7126:
7121:
7117:
7113:
7107:
7104:
7099:
7095:
7088:
7085:
7080:
7076:
7065:
7061:
7057:
7051:
7048:
7043:
7039:
7035:
7031:
7021:
7015:
7011:
6995:
6992:
6987:
6982:
6978:
6970:
6966:
6964:
6945:
6942:
6937:
6933:
6926:
6923:
6918:
6914:
6893:
6890:
6884:
6881:
6878:
6872:
6868:
6859:
6855:
6845:
6833:
6830:
6825:
6819:
6806:
6798:
6792:
6788:
6784:
6780:
6770:
6757:
6754:
6751:
6745:
6729:
6723:
6716:
6713:
6699:
6693:
6688:
6684:
6680:
6677:
6673:
6667:
6659:
6656:
6652:
6648:
6642:
6636:
6617:
6614:
6608:
6592:
6586:
6579:
6576:
6565:
6560:
6556:
6552:
6546:
6540:
6512:
6506:
6500:
6494:
6491:
6485:
6479:
6456:
6450:
6430:
6423:is Wiener in
6407:
6401:
6393:
6374:
6368:
6343:
6339:
6332:
6312:
6309:
6303:
6297:
6289:
6265:
6262:
6259:
6239:
6233:
6230:
6224:
6218:
6210:
6207:
6204:
6195:
6190:
6186:
6163:
6159:
6133:
6130:
6105:
6102:
6099:
6093:
6087:
6064:
6058:
6044:
6031:
6026:
6023:
6019:
6015:
6010:
6000:
5996:
5984:
5968:
5964:
5959:
5956:
5951:
5947:
5944:
5941:
5937:
5932:
5929:
5924:
5920:
5917:
5914:
5911:
5907:
5901:
5898:
5895:
5892:
5887:
5884:
5881:
5878:
5872:
5868:
5862:
5859:
5856:
5853:
5847:
5841:
5833:
5829:
5806:
5800:
5795:
5788:
5783:
5777:
5772:
5769:
5761:
5741:
5735:
5731:
5727:
5724:
5718:
5715:
5710:
5702:
5694:
5690:
5682:
5679:
5676:
5672:
5666:
5663:
5657:
5652:
5648:
5644:
5641:
5637:
5631:
5623:
5615:
5612:
5609:
5605:
5592:
5589:
5585:
5579:
5573:
5570:
5564:
5561:
5558:
5552:
5547:
5539:
5531:
5528:
5525:
5521:
5508:
5507:Takenaka 1988
5492:
5489:
5483:
5477:
5457:
5454:
5448:
5442:
5439:
5431:
5425:
5395:
5392:
5369:
5363:
5349:
5333:
5329:
5325:
5321:
5317:
5314:
5309:
5305:
5290:
5286:
5279:
5269:
5249:
5245:
5241:
5236:
5233:
5230:
5226:
5222:
5217:
5213:
5201:Time reversal
5198:
5182:
5179:
5175:
5166:
5160:
5156:
5150:
5145:
5141:
5130:
5116:
5098:
5095:
5091:
5082:
5076:
5072:
5066:
5061:
5057:
5047:
5038:
5020:
5016:
5012:
5009:
5000:
4997:
4987:
4976:
4967:
4963:
4959:
4956:
4950:
4944:
4941:
4937:
4930:
4927:
4921:
4917:
4911:
4907:
4903:
4897:
4891:
4888:
4885:
4882:
4876:
4870:
4865:
4862:
4859:
4856:
4853:
4845:
4838:
4834:
4829:
4824:
4817:
4811:
4799:
4795:
4790:
4785:
4761:
4757:
4748:
4744:
4740:
4721:
4718:
4715:
4690:
4686:
4665:
4656:
4640:
4636:
4633:
4626:
4623:
4620:
4611:
4608:
4602:
4598:
4592:
4588:
4580:
4577:
4573:
4567:
4557:
4553:
4549:
4546:
4543:
4536:
4526:
4522:
4517:
4513:
4503:
4499:
4495:
4487:
4483:
4476:
4465:
4460:
4443:
4440:
4437:
4434:
4430:
4422:
4419:
4413:
4409:
4403:
4399:
4391:
4388:
4384:
4378:
4376:
4368:
4365:
4356:
4353:
4346:
4338:
4335:
4332:
4329:
4320:
4316:
4307:
4304:
4301:
4296:
4288:
4285:
4282:
4279:
4273:
4265:
4257:
4253:
4249:
4246:
4243:
4236:
4233:
4230:
4220:
4216:
4212:
4207:
4203:
4198:
4192:
4184:
4180:
4176:
4174:
4166:
4156:
4152:
4147:
4133:
4129:
4107:
4103:
4098:
4088:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4050:
4042:
4039:
4032:
4024:
4021:
4018:
4015:
4006:
4002:
3993:
3990:
3987:
3982:
3974:
3971:
3968:
3965:
3959:
3953:
3947:
3944:
3941:
3931:
3927:
3923:
3918:
3914:
3909:
3899:
3877:
3873:
3867:
3864:
3861:
3858:
3855:
3847:
3842:
3838:
3823:
3821:
3802:
3799:
3796:
3772:
3767:
3764:
3759:
3755:
3749:
3724:
3721:
3718:
3692:
3688:
3682:
3679:
3674:
3671:
3667:
3660:
3656:
3653:
3649:
3643:
3640:
3635:
3632:
3628:
3623:
3619:
3616:
3608:
3604:
3593:
3590:
3587:
3583:
3577:
3572:
3567:
3563:
3539:
3536:
3531:
3528:
3525:
3522:
3519:
3511:
3507:
3496:
3493:
3490:
3486:
3480:
3475:
3472:
3467:
3463:
3459:
3454:
3450:
3427:
3423:
3414:
3404:
3386:
3383:
3376:
3372:
3368:
3363:
3359:
3353:
3346:
3342:
3337:
3333:
3326:
3322:
3317:
3304:
3297:
3291:
3278:
3273:
3269:
3265:
3261:
3256:
3249:
3245:
3240:
3236:
3232:
3226:
3216:
3212:
3207:
3203:
3196:
3192:
3187:
3180:
3177:
3168:
3155:
3152:
3142:
3138:
3133:
3129:
3122:
3118:
3113:
3106:
3100:
3090:
3086:
3081:
3074:
3068:
3064:
3053:
3049:
3044:
3040:
3033:
3029:
3024:
3017:
3010:
3006:
3001:
2996:
2992:
2965:
2961:
2956:
2952:
2945:
2941:
2936:
2911:
2907:
2902:
2898:
2891:
2887:
2882:
2878:
2871:
2867:
2862:
2852:
2835:
2831:
2826:
2819:
2815:
2810:
2806:
2802:
2796:
2792:
2781:
2777:
2772:
2768:
2761:
2757:
2752:
2745:
2738:
2734:
2729:
2724:
2720:
2714:
2712:
2703:
2692:
2688:
2683:
2679:
2669:
2665:
2660:
2656:
2649:
2645:
2640:
2630:
2623:
2619:
2614:
2609:
2605:
2599:
2597:
2585:
2581:
2576:
2572:
2565:
2561:
2556:
2549:
2518:
2514:
2509:
2505:
2495:
2491:
2486:
2482:
2475:
2471:
2466:
2459:
2452:
2448:
2443:
2435:Substituting
2433:
2420:
2416:
2408:
2404:
2399:
2395:
2388:
2384:
2379:
2374:
2370:
2364:
2360:
2346:
2342:
2337:
2330:
2324:
2317:
2313:
2308:
2301:
2288:
2284:
2279:
2272:
2266:
2259:
2255:
2250:
2242:
2238:
2232:
2222:
2218:
2213:
2209:
2202:
2198:
2193:
2186:
2183:
2161:
2157:
2153:
2148:
2144:
2134:
2117:
2111:
2108:
2102:
2096:
2093:
2089:
2084:
2074:
2070:
2065:
2057:
2053:
2048:
2037:
2033:
2029:
2024:
2020:
2013:
2010:
2004:
2002:
1992:
1988:
1984:
1979:
1975:
1968:
1965:
1958:
1955:
1952:
1950:
1940:
1936:
1932:
1927:
1923:
1916:
1913:
1889:
1886:
1883:
1875:
1871:
1861:
1848:
1842:
1839:
1836:
1830:
1827:
1822:
1818:
1814:
1809:
1805:
1801:
1796:
1792:
1783:
1778:
1765:
1762:
1759:
1751:
1747:
1740:
1737:
1725:
1720:
1707:
1704:
1696:
1692:
1685:
1674:
1669:
1656:
1648:
1645:
1638:
1632:
1628:
1624:
1620:
1613:
1610:
1607:
1603:
1598:
1592:
1582:
1578:
1573:
1560:
1556:
1552:
1538:
1529:
1513:
1509:
1482:
1474:
1455:
1452:
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1446:
1443:
1437:
1414:
1406:
1402:
1398:
1392:
1384:
1380:
1371:
1353:
1349:
1326:
1322:
1301:
1295:
1292:
1289:
1283:
1280:
1276:
1271:
1267:
1258:
1255:
1249:
1246:
1243:
1240:
1229:
1225:
1220:
1214:
1206:
1202:
1193:
1189:
1173:
1170:
1165:
1161:
1157:
1152:
1148:
1133:
1131:
1127:
1121:
1115:
1111:
1107:
1103:
1081:
1076:
1072:
1067:
1061:
1058:
1054:
1045:
1040:
1036:
1032:
1028:
1027:scaling limit
1023:
1021:
1017:
1013:
1008:
1006:
1002:
997:
994:
989:
985:
979:
974:
965:
960:
956:
951:
949:
940:
936:
928:
924:
914:
910:
902:
898:
889:
882:
875:
868:
846:
824:
820:
799:
792:
778:
772:
769:
766:
753:
748:
744:
740:
735:
732:
729:
725:
704:
697:and variance
684:
662:
658:
654:
649:
646:
643:
639:
618:
611:
597:
594:
591:
588:
566:
562:
541:
538:
535:
532:
513:
508:
504:
500:
495:
492:
489:
485:
464:
461:
458:
455:
447:
431:
424:
422:
421:almost surely
407:
404:
399:
395:
387:
386:
385:
383:
368:
364:
348:
346:
345:BlackâScholes
342:
338:
334:
330:
326:
322:
318:
314:
310:
309:FokkerâPlanck
306:
302:
297:
295:
291:
287:
283:
279:
276:
272:
268:
264:
260:
256:
252:
247:
245:
241:
237:
233:
229:
225:
222:
218:
214:
210:
206:
202:
198:
195:
191:
187:
178:
170:
150:
145:
141:
131:
127:
112:
103:
99:
95:
87:
75:
72:
64:
61:February 2010
54:
50:
44:
43:
37:
32:
23:
22:
16:
14801:Econometrics
14763:Wiener space
14651:ItĂŽ integral
14552:Inequalities
14441:Self-similar
14411:GaussâMarkov
14401:Exchangeable
14381:CĂ dlĂ g paths
14317:Risk process
14269:LIBOR market
14138:Random graph
14133:Random field
13959:
13945:Superprocess
13883:LĂ©vy process
13878:Jump process
13853:Hunt process
13760:
13689:Markov chain
13572:
13568:
13559:
13540:
13530:
13524:
13502:
13463:
13459:
13452:
13443:
13434:
13421:
13410:. Retrieved
13406:
13397:
13370:
13364:
13354:
13345:
13336:
13317:
13311:
13292:
13286:
13277:
13268:
13259:
13224:
13220:
13210:
13191:
13182:
13173:
13155:
13139:
13138:
13127:Brownian web
13100:
13099:
13066:
12952:
12713:
12712:
12634:
12594:the process
12537:
12534:
12361:
12349:
12334:
12322:
12317:
12315:
12050:
11916:
11908:
11907:
11902:
11898:
11890:
11886:
11820:
11818:
11813:
11676:
11675:
11670:
11666:
11662:
11658:
11654:
11649:
11645:
11641:
11640:
11637:
11548:
11545:ItĂŽ isometry
11441:The case of
11207:
10543:
10491:
10487:
10483:
10479:
10395:
10373:
10369:
10365:
10358:
10354:
10352:
10334:
10244:
10237:
10233:
10229:
10221:
10217:
10211:
10203:modification
10196:LĂ©vy process
10187:
10179:
10173:
10018:
10015:
9935:
9930:
9926:
9922:
9918:
9914:
9910:
9906:
9902:
9898:
9894:
9883:
9874:
9873:is called a
9820:
9191:
8801:
8792:
8787:
8775:
8771:
8767:
8763:
8759:
8755:
8751:
8747:
8743:
8739:
8735:
8731:
8727:
8719:
8712:
8705:
8701:
8569:
8565:
8556:
8550:
8538:
8352:
8231:
8118:
8089:
8079:
8075:
8060:
8053:
8049:
8045:
8041:
8037:
8033:
8029:
8025:
8021:
8017:
8013:
8009:
8005:
8001:
7997:
7993:
7989:
7985:
7981:
7977:
7973:
7969:
7852:
7848:
7689:
7682:
7669:
7667:
7651:
7647:
7643:
7640:
7439:the process
7310:
7309:
7179:
7063:
7059:
7055:
7052:
7046:
7041:
7037:
7033:
7024:is equal to
7019:
7013:
6968:
6967:
6790:
6786:
6782:
6776:
6050:
5983:group action
5355:
5297:The process
5296:
5284:
5274:
5267:
5205:The process
5204:
5133:the process
5128:
5125:
5114:
4657:
4461:
4131:
4127:
4089:
3894:
3829:
3410:
3302:
3295:
3292:
3169:
2853:
2434:
2135:
1867:
1779:
1721:
1670:
1558:
1548:
1430:is close to
1369:
1191:
1139:
1129:
1119:
1113:
1101:
1039:neighborhood
1024:
1015:
1009:
1000:
998:
992:
987:
983:
977:
963:
954:
952:
950:increments.
947:
938:
934:
926:
922:
912:
908:
900:
896:
887:
880:
873:
866:
862:
448:: for every
356:
354:
298:
248:
209:Robert Brown
189:
183:
67:
58:
39:
15:
14846:Ruin theory
14784:Disciplines
14656:ItĂŽ's lemma
14431:Predictable
14106:Percolation
14089:Potts model
14084:Ising model
14048:White noise
14006:Differences
13868:ItĂŽ process
13808:Cox process
13704:Loop-erased
13699:Random walk
12631:Time change
12474:independent
11634:Time change
10247:. Formally
9060:binary code
8955:binary code
8097:perfect set
6392:Lawler 2005
4747:conditioned
4658:If at time
1874:correlation
1673:expectation
1031:random walk
275:white noise
251:martingales
186:mathematics
53:introducing
14904:Categories
14856:Statistics
14636:Filtration
14537:Kolmogorov
14521:Blumenthal
14446:Stationary
14386:Continuous
14374:Properties
14259:HullâWhite
14001:Martingale
13888:Local time
13776:Fractional
13754:pure birth
13490:References
13412:2017-05-14
12558:such that
11911:(See also
11909:Corollary.
11124:by taking
10699:Then, for
10478:is called
10386:filtration
10382:martingale
10214:local time
10207:first exit
9813:times the
8724:local time
8547:Local time
8024:) for all
7996:) for all
7980:): first,
7843:Points of
6963:martingale
6779:polynomial
5126:For every
1870:covariance
1553:follows a
959:martingale
221:stationary
36:references
14768:Classical
13781:Geometric
13771:Excursion
13521:PDF-files
13243:0091-1798
12871:∫
12761:−
12686:−
12605:⋅
12301:∞
12285:∞
12272:−
12267:∞
12239:∞
12223:∞
12210:−
12205:∞
12177:∞
12161:∞
12148:−
12143:∞
12132:∞
12129:−
12107:∞
12091:∞
12078:−
12073:∞
12062:∞
12059:−
12022:∞
12019:→
11998:∞
11959:∞
11956:→
11940:−
11935:∞
11901:on , and
11893:) is the
11841:−
11765:∫
11701:−
11479:−
11404:
11388:−
11354:
11291:
11230:
11191:⋅
11163:⋅
11010:In fact,
10976:−
10943:−
10908:∫
10848:
10797:−
10768:∫
10736:
10658:−
10629:∫
10575:∫
10505:∧
10436:∫
10412:−
10301:−
10292:δ
10278:∫
10147:σ
10120:μ
10094:θ
10045:−
9987:σ
9966:σ
9959:−
9953:μ
9851:σ
9842:μ
9700:
9670:
9635:−
9616:φ
9613:π
9607:
9589:φ
9560:φ
9549:θ
9533:−
9527:φ
9499:∫
9460:θ
9433:φ
9421:θ
9407:−
9401:φ
9385:
9356:∫
9327:θ
9229:code rate
9177:ε
9174:−
9134:∈
9020:ε
8924:∈
8875:−
8864:≈
8852:
8840:π
8784:nonatomic
8646:∞
8638:∞
8635:−
8631:∫
8583:∫
8574:. Thus,
8494:ε
8480:
8474:ε
8449:−
8424:ε
8421:≤
8415:−
8403:≤
8391:≤
8372:→
8369:ε
8315:ε
8301:
8295:
8289:ε
8273:ε
8248:→
8245:ε
8187:
8181:
8140:∞
8134:→
8078:over is
7950:∞
7947:→
7933:−
7903:−
7875:→
7825:ϵ
7822:−
7780:ϵ
7707:ϵ
7589:∫
7562:−
7492:∫
7485:−
7468:−
7357:−
7257:∂
7247:∂
7222:∂
7218:∂
7118:∫
7114:−
6993:−
6852:∂
6842:∂
6826:−
6817:∂
6813:∂
6694:σ
6685:∫
6657:−
6637:σ
6557:∫
6507:σ
6269:→
6208:≥
6187:τ
6160:τ
6134:⊂
6106:∈
5942:−
5912:−
5725:−
5677:σ
5664:σ
5642:−
5638:σ
5616:σ
5593:∈
5571:−
5435:∞
5432:±
5429:→
5396:∈
5242:−
5234:−
4960:−
4942:−
4931:−
4889:∣
4883:≤
4863:≤
4857:≤
4641:π
4593:−
4578:π
4563:∞
4554:∫
4509:∞
4500:∫
4477:
4438:≥
4404:−
4389:π
4336:−
4321:−
4305:π
4286:−
4261:∞
4258:−
4254:∫
4188:∞
4185:−
4181:∫
4070:≤
4058:≥
4022:−
4007:−
3991:π
3972:−
3865:≤
3859:≤
3693:π
3675:−
3654:π
3636:−
3620:
3605:ξ
3599:∞
3584:∑
3537:π
3526:π
3523:
3508:ξ
3502:∞
3487:∑
3464:ξ
3424:ξ
3384:⋅
3369:−
3233:
3181:
3130:−
3107:
3101:⋅
3075:
3041:−
3018:⋅
2993:
2953:−
2899:−
2803:
2769:−
2746:⋅
2721:
2657:−
2631:⋅
2606:
2573:⋅
2550:
2483:−
2396:⋅
2371:
2331:
2325:−
2302:⋅
2273:
2267:−
2239:
2187:
2154:≤
2066:σ
2049:σ
2014:
1969:
1917:
1887:≤
1828:∼
1815:−
1741:
1686:
1675:is zero:
1625:−
1611:π
1489:∞
1486:→
1453:−
1399:−
1350:ξ
1284:∈
1268:ξ
1262:⌋
1253:⌊
1250:≤
1244:≤
1237:∑
1174:…
1162:ξ
1149:ξ
1073:α
1059:−
1055:α
754:∼
741:−
655:−
536:≥
501:−
305:diffusion
261:and even
232:economics
142:σ
14889:Category
14773:Abstract
14307:BĂŒhlmann
13913:Compound
13499:(2004).
13091:See also
12714:Example:
11677:Example.
11661:) where
11642:Example:
10593:′
9198:sampling
8953:using a
8746:) where
8356:(LĂ©vy):
7699:For any
7311:Example:
7018:[0,
6969:Example:
6717:′
6580:′
6231:∉
5760:PSL(2,R)
4741:). The
4126:ââ <
1724:variance
1126:integral
323:(by the
307:via the
130:Variance
14396:Ergodic
14284:VaĆĄĂÄek
14126:Poisson
13786:Meander
13575:: 41â44
13480:5911584
13429:, 2009.
13425:Forum,
13389:2242845
13251:2243125
13122:Fractal
12254:
10337:is the
10190:) is a
9933:) = 0.
9808:⁄
8774:(while
7036:from (â
6325:, then
6123:. Let
1876:(where
1122:(0) = 0
1117:, with
1104:is the
244:physics
49:improve
14736:Tanaka
14421:Mixing
14416:Markov
14289:Wilkie
14254:HoâLee
14249:Heston
14021:Super-
13766:Bridge
13714:Biased
13547:
13511:
13478:
13387:
13324:
13299:
13249:
13241:
13198:
13156:
12844:where
12655:is an
12418:where
11915:) Let
11885:where
11738:where
11547:. The
11206:where
11088:given
10333:where
10198:. The
10138:, and
9575:where
9194:encode
8092:are a
8063:is of
7178:where
6472:where
5131:> 0
4995:
4992:
4984:
4934:
4925:
4776:, is:
3399:where
2854:Since
1188:i.i.d.
1100:. The
242:, and
217:cĂ dlĂ g
188:, the
38:, but
14589:Tools
14365:M/M/c
14360:M/M/1
14355:M/G/1
14345:Fluid
14011:Local
13533:, AMS
13476:S2CID
13385:JSTOR
13247:JSTOR
13165:Notes
12505:with
8559:(the
8086:Zeros
8004:â Δ,
7859:then
7022:]
6961:is a
6777:If a
6286:is a
4737:(cf.
3818:(cf.
3415:. If
3301:<
3170:Thus
1029:of a
961:with
894:then
886:<
872:<
284:(see
269:. In
14541:LĂ©vy
14340:Bulk
14224:Chen
14016:Sub-
13974:Both
13545:ISBN
13509:ISBN
13322:ISBN
13297:ISBN
13239:ISSN
13196:ISBN
13026:and
12930:and
12472:are
12445:and
12295:<
12277:<
12153:<
12101:<
12065:<
11812:and
10710:>
10490:(0,
10353:Let
10212:The
10174:The
9905:) =
9789:blue
9773:blue
9652:and
9023:>
8990:bits
8867:0.29
8802:The
8766:and
8750:and
8397:<
8070:The
8028:in (
8016:) â„
8000:in (
7988:) â€
7710:>
6051:Let
5680:>
5283:0 â€
5281:for
5266:0 â€
5264:for
5001:>
3892:and
3555:and
2928:and
1966:corr
1872:and
1868:The
1722:The
1671:The
1140:Let
971:and
920:and
865:0 â€
592:<
459:>
444:has
311:and
102:Mean
14121:Cox
13468:doi
13375:doi
13229:doi
12635:If
11897:of
11673:).
11401:Var
11351:Var
11288:Var
11227:cov
10845:cov
10733:Var
10514:min
10482:or
10240:â„ 0
10220:= (
9793:red
9777:red
9697:log
9685:max
9661:log
9604:sin
9513:min
9371:log
8730:of
8726:at
8477:log
8384:sup
8298:log
8292:log
8184:log
8178:log
8074:of
7868:lim
7032:of
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6252:If
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5821:is
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969:= 0
335:in
319:of
184:In
14906::
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14535:,
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13573:64
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13464:55
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13267:.
13245:.
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13219:.
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12327:.
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11653:=
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10541:.
10432::=
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10232:â
10171:.
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9189:.
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8742:,
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257:,
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238:,
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13470::
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13371:7
13330:.
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13271:.
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13225:6
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13075:t
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12918:s
12914:d
12907:2
12902:|
12895:s
12891:Z
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12880:t
12875:0
12867:4
12864:=
12861:)
12858:t
12855:(
12852:A
12830:)
12827:t
12824:(
12821:A
12817:U
12813:=
12810:i
12805:t
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12795:t
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12787:2
12784:+
12780:)
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12738:=
12733:2
12728:t
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12695:0
12692:(
12689:f
12683:)
12678:t
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12670:(
12667:f
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12602:c
12582:1
12579:=
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9712:}
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9465:=
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8627:=
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8503:=
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7399:x
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7364:)
7360:t
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7332:t
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7320:p
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7212:(
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7042:c
7038:c
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6938:t
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6891:=
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6873:p
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6733:)
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6714:f
6709:|
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6700:t
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6689:0
6681:=
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6668:t
6665:(
6660:1
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6649:=
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6643:t
6640:(
6618:s
6615:d
6609:2
6604:|
6599:)
6596:)
6593:s
6590:(
6587:W
6584:(
6577:f
6572:|
6566:t
6561:0
6553:=
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6547:t
6544:(
6541:S
6522:)
6519:)
6516:)
6513:t
6510:(
6504:(
6501:W
6498:(
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6492:=
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6486:t
6483:(
6480:Y
6460:)
6457:t
6454:(
6451:S
6431:D
6411:)
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6405:(
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6390:(
6378:)
6375:D
6372:(
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6349:)
6344:t
6340:W
6336:(
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6313:0
6310:=
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6304:0
6301:(
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6273:C
6266:D
6263::
6260:f
6240:.
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6234:D
6228:)
6225:t
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6219:W
6215:|
6211:0
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6110:C
6103:0
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6094:0
6091:(
6088:W
6068:)
6065:t
6062:(
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6032:.
6027:h
6024:g
6020:W
6016:=
6011:h
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6001:g
5997:W
5993:(
5969:,
5965:)
5960:d
5957:b
5952:(
5948:W
5945:d
5938:)
5933:c
5930:a
5925:(
5921:W
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5902:d
5899:+
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5882:t
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5873:(
5869:W
5866:)
5863:d
5860:+
5857:t
5854:c
5851:(
5848:=
5845:)
5842:t
5839:(
5834:g
5830:W
5807:]
5801:d
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5789:b
5784:a
5778:[
5773:=
5770:g
5742:.
5739:)
5736:t
5732:/
5728:1
5722:(
5719:W
5716:t
5711:=
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5703:t
5700:(
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5683:0
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5670:)
5667:t
5661:(
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5653:2
5649:/
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5632:=
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5624:t
5621:(
5613:,
5610:2
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5597:R
5590:s
5586:,
5583:)
5580:s
5577:(
5574:W
5568:)
5565:s
5562:+
5559:t
5556:(
5553:W
5548:=
5543:)
5540:t
5537:(
5532:s
5529:,
5526:1
5522:W
5493:0
5490:=
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5484:0
5481:(
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5458:0
5455:=
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5449:t
5446:(
5443:W
5440:t
5426:t
5400:R
5393:t
5373:)
5370:t
5367:(
5364:W
5334:t
5330:/
5326:1
5322:W
5318:t
5315:=
5310:t
5306:V
5285:t
5277:t
5275:W
5268:t
5250:1
5246:W
5237:t
5231:1
5227:W
5223:=
5218:t
5214:V
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5180:c
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5172:)
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5161:/
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5151:=
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5115:c
5099:t
5096:c
5092:W
5088:)
5083:c
5077:/
5073:1
5070:(
5067:=
5062:t
5058:V
5026:)
5021:t
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5007:(
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4988:,
4977:t
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4968:t
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4957:m
4954:(
4951:m
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4938:e
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4922:=
4918:)
4912:t
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4904:=
4901:)
4898:t
4895:(
4892:W
4886:m
4880:)
4877:s
4874:(
4871:W
4866:t
4860:s
4854:0
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4839:t
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4830:M
4825:(
4818:=
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4812:m
4809:(
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4786:F
4762:t
4758:W
4725:]
4722:t
4719:,
4716:0
4713:[
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4637:t
4634:2
4627:=
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4621:d
4612:t
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4599:m
4589:e
4581:t
4574:2
4568:m
4558:0
4550:=
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4537:m
4534:(
4527:t
4523:M
4518:f
4514:m
4504:0
4496:=
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4488:t
4484:M
4480:[
4474:E
4444:,
4441:0
4435:m
4431:,
4423:t
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4410:m
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4330:2
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4317:e
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4289:w
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4280:2
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4250:=
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4228:(
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4213:,
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4061:0
4055:m
4051:,
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4040:2
4033:2
4029:)
4025:w
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3988:2
3983:t
3978:)
3975:w
3969:m
3966:2
3963:(
3960:2
3954:=
3951:)
3948:w
3945:,
3942:m
3939:(
3932:t
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3924:,
3919:t
3915:M
3910:f
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3878:s
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3868:t
3862:s
3856:0
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3839:M
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3803:c
3800:,
3797:0
3794:[
3773:)
3768:c
3765:t
3760:(
3756:W
3750:c
3728:]
3725:1
3722:,
3719:0
3716:[
3689:)
3683:2
3680:1
3672:n
3668:(
3661:)
3657:t
3650:)
3644:2
3641:1
3633:n
3629:(
3624:(
3609:n
3594:1
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3540:n
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3529:n
3512:n
3497:1
3494:=
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3481:2
3476:+
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3460:=
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3451:W
3428:n
3401:Z
3387:Z
3377:1
3373:t
3364:2
3360:t
3354:+
3347:1
3343:t
3338:W
3334:=
3327:2
3323:t
3318:W
3306:2
3303:t
3299:1
3296:t
3279:.
3274:1
3270:t
3266:=
3262:]
3257:2
3250:1
3246:t
3241:W
3237:[
3230:E
3227:=
3224:)
3217:2
3213:t
3208:W
3204:,
3197:1
3193:t
3188:W
3184:(
3153:=
3150:]
3143:1
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3123:2
3119:t
3114:W
3110:[
3104:E
3098:]
3091:1
3087:t
3082:W
3078:[
3072:E
3069:=
3065:]
3061:)
3054:1
3050:t
3045:W
3034:2
3030:t
3025:W
3021:(
3011:1
3007:t
3002:W
2997:[
2990:E
2966:1
2962:t
2957:W
2946:2
2942:t
2937:W
2912:0
2908:t
2903:W
2892:1
2888:t
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2879:=
2872:1
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2863:W
2836:.
2832:]
2827:2
2820:1
2816:t
2811:W
2807:[
2800:E
2797:+
2793:]
2789:)
2782:1
2778:t
2773:W
2762:2
2758:t
2753:W
2749:(
2739:1
2735:t
2730:W
2725:[
2718:E
2715:=
2704:]
2700:)
2693:1
2689:t
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2680:+
2677:)
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2666:t
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2650:2
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2641:W
2637:(
2634:(
2624:1
2620:t
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2610:[
2603:E
2600:=
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2586:2
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2562:t
2557:W
2553:[
2547:E
2519:1
2515:t
2510:W
2506:+
2503:)
2496:1
2492:t
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2476:2
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2463:(
2460:=
2453:2
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2444:W
2421:.
2417:]
2409:2
2405:t
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2389:1
2385:t
2380:W
2375:[
2368:E
2365:=
2361:]
2357:)
2354:]
2347:2
2343:t
2338:W
2334:[
2328:E
2318:2
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2305:(
2299:)
2296:]
2289:1
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2280:W
2276:[
2270:E
2260:1
2256:t
2251:W
2247:(
2243:[
2236:E
2233:=
2230:)
2223:2
2219:t
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2210:,
2203:1
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2190:(
2162:2
2158:t
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2145:t
2118:.
2112:t
2109:s
2103:=
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2094:s
2090:s
2085:=
2075:t
2071:W
2058:s
2054:W
2043:)
2038:t
2034:W
2030:,
2025:s
2021:W
2017:(
2005:=
1998:)
1993:t
1989:W
1985:,
1980:s
1976:W
1972:(
1959:,
1956:s
1953:=
1946:)
1941:t
1937:W
1933:,
1928:s
1924:W
1920:(
1890:t
1884:s
1849:.
1846:)
1843:t
1840:,
1837:0
1834:(
1831:N
1823:0
1819:W
1810:t
1806:W
1802:=
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1793:W
1766:.
1763:t
1760:=
1757:)
1752:t
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1744:(
1728:t
1705:=
1702:]
1697:t
1693:W
1689:[
1683:E
1657:.
1652:)
1649:t
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1643:(
1639:/
1633:2
1629:x
1621:e
1614:t
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1604:1
1599:=
1596:)
1593:x
1590:(
1583:t
1579:W
1574:f
1563:t
1559:t
1514:n
1510:W
1483:n
1459:)
1456:s
1450:t
1447:,
1444:0
1441:(
1438:N
1418:)
1415:s
1412:(
1407:n
1403:W
1396:)
1393:t
1390:(
1385:n
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1302:.
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1287:[
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1230:n
1226:1
1221:=
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1212:(
1207:n
1203:W
1192:n
1171:,
1166:2
1158:,
1153:1
1120:g
1114:g
1098:α
1082:t
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1016:t
1001:N
993:t
988:t
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978:t
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847:t
825:t
821:W
800:W
779:.
776:)
773:u
770:,
767:0
764:(
759:N
749:t
745:W
736:u
733:+
730:t
726:W
705:u
685:0
663:t
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650:u
647:+
644:t
640:W
619:W
598:.
595:t
589:s
567:s
563:W
542:,
539:0
533:u
514:,
509:t
505:W
496:u
493:+
490:t
486:W
465:,
462:0
456:t
432:W
408:0
405:=
400:0
396:W
369:t
365:W
215:(
151:t
146:2
113:0
74:)
68:(
63:)
59:(
45:.
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