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is frequently useful in problem formulation (particularly in engineering disciplines). The
Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. The main benefit of the Stratonovich integral is that it obeys the usual
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does not hold for the
Stratonovich integral; consequently it is very difficult to prove results without re-expressing the integrals in Itô form.
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Besides the classical Itô and Fisk–Stratonovich integrals, many different notion of stochastic integrals exist such as the
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processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in
1960:
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Fima C Klebaner, 2012, Introduction to
Stochastic Calculus with Application (3rd Edition). World Scientific Publishing,
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1118:. For technical reasons the Itô integral is the most useful for general classes of processes, but the related
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Szabados, T. S.; Székely, B. Z. (2008). "Stochastic
Integration Based on Simple, Symmetric Random Walks".
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of stochastic processes with respect to stochastic processes. This field was created and started by the
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1394:{\displaystyle \int _{0}^{t}X_{s-}\circ dY_{s}:=\int _{0}^{t}X_{s-}dY_{s}+{\frac {1}{2}}\left_{t}^{c},}
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1576:{\displaystyle \left_{t}^{c}:=_{t}-\sum \limits _{s\leq t}\Delta X_{s}\Delta Y_{s}}
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The best-known stochastic process to which stochastic calculus is applied is the
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to model the evolution in time of stock prices and bond interest rates.
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The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
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1056:. It allows a consistent theory of integration to be defined for
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European
Community on Computational Methods in Applied Sciences
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The
Stratonovich integral or Fisk–Stratonovich integral of a
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is central to the study of stochastic calculus. The integral
2578:
International
Council for Industrial and Applied Mathematics
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An important application of stochastic calculus is in
2197:
Numerical methods for ordinary differential equations
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Société de Mathématiques
Appliquées et Industrielles
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Japan
Society for Industrial and Applied Mathematics
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Numerical methods for partial differential equations
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1661:, in which asset prices are often assumed to follow
2389:
1649:is also used to denote the Stratonovich integral.
1639:{\displaystyle \int _{0}^{t}X_{s}\,\partial Y_{s}}
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1110:The main flavours of stochastic calculus are the
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194:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
57:but its sources remain unclear because it lacks
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1246:can be defined in terms of the Itô integral as
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2379:Supersymmetric theory of stochastic dynamics
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88:Learn how and when to remove this message
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2647:Definitions of mathematical integration
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557:Differentiating under the integral sign
14:
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1669:prices options as if they follow a
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1828:Industrial and applied mathematics
1765:Journal of Theoretical Probability
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103:Part of a series of articles about
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2058:Stochastic differential equations
1663:stochastic differential equations
1114:and its variational relative the
2374:Supersymmetric quantum mechanics
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27:Calculus on stochastic processes
2256:Stochastic variational calculus
2048:Ordinary differential equations
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1926:Arbitrary-precision arithmetic
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1083:), which is used for modeling
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1414:of the continuous parts of
1095:in 1905 and other physical
906:Calculus on Euclidean space
329:Logarithmic differentiation
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1683:Hitsuda–Skorokhod integral
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1180:{\displaystyle \int H\,dX}
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1671:geometric Brownian motion
1587:The alternative notation
640:Summand limit (term test)
2192:Numerical linear algebra
324:Implicit differentiation
314:Differentiation notation
241:Inverse function theorem
43:This article includes a
1931:Finite element analysis
1881:Constraint satisfaction
782:Helmholtz decomposition
72:more precise citations.
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2160:Multivariable calculus
2043:Differential equations
1886:Constraint programming
1876:Computational geometry
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2439:Supersymmetry algebra
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2065:Differential geometry
1946:Optimization software
1918:Mathematical software
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1410:denotes the optional
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1212:Stratonovich integral
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1101:financial mathematics
1000:Mathematical analysis
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596:arithmetico-geometric
442:Leibniz integral rule
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2637:Mathematical finance
2491:Mathematical finance
2476:Social choice theory
2391:Algebraic structures
2340:in quantum mechanics
2276:Analytical mechanics
2242:Stochastic processes
2214:Variational calculus
2026:Approximation theory
1951:Statistical software
1677:Stochastic integrals
1659:mathematical finance
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1194:and locally bounded
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1054:stochastic processes
1005:Nonstandard analysis
478:Lebesgue integration
348:Rules and identities
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2632:Stochastic calculus
2466:Operations research
2335:Perturbation theory
2133:Multilinear algebra
2104:Functional analysis
1961:Numerical libraries
1893:Computational logic
1667:Black–Scholes model
1665:. For example, the
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1079:(named in honor of
1046:Stochastic calculus
676:Cauchy condensation
483:Contour integration
209:Fundamental theorem
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18:Stochastic analysis
2603:Mathematics portal
2500:Other applications
2224:Probability theory
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2080:Geometric analysis
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432:Lists of integrals
236:Mean value theorem
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45:list of references
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2031:Clifford analysis
2008:Discrete geometry
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2036:Clifford algebra
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1091:in 1900 and by
1089:Louis Bachelier
1085:Brownian motion
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2266:Mathematical
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2092:Chaos theory
2075:Gauge theory
2003:Graph theory
1898:Cryptography
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1720:Itô calculus
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1155:Itô integral
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1149:Itô calculus
1143:Itô integral
1132:
1112:Itô calculus
1109:
1074:
1070:World War II
1045:
1044:
949:
519:Substitution
281:Differential
254:Differential
84:
75:
64:Please help
56:
2611:topics list
2545:Mathematics
2461:Game theory
2362:Topological
2328:Topological
2323:Statistical
2286:Hamiltonian
1771:: 203–219.
1725:Itô's lemma
1196:predictable
1129:Itô's lemma
1050:mathematics
975:Precalculus
968:Miscellanea
933:Specialized
840:Definitions
607:Alternating
450:Definitions
263:Definitions
78:August 2011
70:introducing
2626:Categories
2517:Psychology
2481:Statistics
2281:Lagrangian
1908:Statistics
1844:Algorithms
1747:References
1693:and more.
1125:chain rule
1066:Kiyosi Itô
955:Variations
950:Stochastic
940:Fractional
809:Formalisms
772:Divergence
741:Identities
721:Divergence
271:Derivative
222:Continuity
2522:Sociology
2512:Chemistry
2308:Effective
2303:Conformal
2298:Classical
2170:Geometric
2143:Geometric
1778:0712.3908
1624:∂
1599:∫
1561:Δ
1548:Δ
1540:≤
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1529:−
1418:and
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1305:∫
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1280:−
1258:∫
1165:∫
1105:economics
1097:diffusion
1058:integrals
945:Malliavin
832:Geometric
731:Laplacian
681:Dirichlet
592:Geometric
177:−
124:∫
2597:Category
2246:analysis
2165:Exterior
2138:Exterior
2018:Analysis
1980:Discrete
1854:analysis
1792:Preprint
1697:See also
1198:process
1062:Japanese
985:Glossary
895:Advanced
873:Jacobian
827:Exterior
757:Gradient
749:Theorems
716:Gradient
655:Integral
617:Binomial
602:Harmonic
467:improper
463:Integral
420:Integral
402:Reynolds
377:Quotient
306:Concepts
142:′
109:Calculus
2607:outline
2538:Related
2507:Biology
2357:Bosonic
2318:Quantum
2268:physics
2234: (
1966:Solvers
1462:, i.e.
1068:during
980:History
878:Hessian
767:Stokes'
762:Green's
594: (
521: (
465: (
387:Inverse
362:Product
273: (
66:improve
2180:Vector
2175:Tensor
2153:Vector
2148:Tensor
1849:design
1756:
1689:, the
1685:, the
1404:where
1135:. The
822:Tensor
817:Matrix
704:Vector
622:Taylor
580:Series
217:Limits
2313:Gauge
1773:arXiv
645:Ratio
612:Power
531:Euler
509:Discs
504:Parts
372:Power
367:Chain
296:total
51:, or
1754:ISBN
1442:and
1153:The
1103:and
726:Curl
686:Abel
650:Root
1783:doi
1202:.
357:Sum
2628::
1781:.
1769:22
1767:.
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1350:1
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1175:X
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183:(
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168:(
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159:t
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149:t
146:(
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128:a
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85:(
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76:(
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20:)
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