- Stochastic calculus
**Stochastic calculus**is a branch ofmathematics that operates onstochastic process es. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.The best-known stochastic process to which stochastic calculus is applied is the

Wiener process (named in honor ofNorbert Wiener ), which is used for modelingBrownian motion as described byAlbert Einstein and other physicaldiffusion processes in space of particles subject to random forces. Since the 1970's, the Wiener process has been widely applied infinancial mathematics to model the evolution in time of stock and bond prices.The main flavours of stochastic calculus are the

Itō calculus and its variational relative theMalliavin calculus . For technical reasons the Itō integral is the most useful for general classes of processes but the relatedStratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines.) The Stratonovich integral can readily be expressed in terms of the Itō integral. Another benefit of the Stratonovich integral is that it enables some problems to be expressed in a co-ordinate system invariant form and is therefore invaluable when developing stochastic calculus on manifolds other than**R**^{"n"}.TheDominated convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itō form.**Itō integral**The

Itō integral is central to the study of stochastic calculus. The integral $int\; H,dX$ is defined for asemimartingale "X" and locally bounded**predictable**process "H".**tratonovich integral**The Stratonovich integral can be defined in terms of the Itō integral as

:$int\_0^t\; X\_s\; circ\; d\; Y\_s\; :\; =\; int\_0^t\; X\_s\; d\; Y\_s\; +\; frac\{1\}\{2\}\; left\; [\; X,\; Y\; ight]\; \_t.$

The alternative notation

:$int\_0^t\; X\_s\; partial\; Y\_s$

is also used to denote the Stratonovich integral.

**Applications**A very important application of stochastic calculus is in

quantitative finance .**External links*** [

*http://www.chiark.greenend.org.uk/~alanb/stoc-calc.pdf Notes on Stochastic Calculus*] — A short elementary description of the basic Itō integral.* [

*http://arXiv.org/abs/0712.3908/ T. Szabados and B. Szekely, Stochastic integration based on simple, symmetric random walks*] - A new approach which the authors hope is more transparent and technically less demanding.

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