Editing Stochastic calculus (section)

{{Short description|Calculus on stochastic processes}}
{{No footnotes|date=August 2011}}
{{Calculus |Specialized}}
'''Stochastic calculus''' is a branch of [[mathematics]] that operates on [[stochastic process]]es. It allows a consistent theory of integration to be defined for [[integrals]] of stochastic processes with respect to stochastic processes. This field was created and started by the [[Japanese people|Japanese]] mathematician [[Kiyosi Itô]] during [[World War II]].

The best-known stochastic process to which stochastic calculus is applied is the [[Wiener process]] (named in honor of [[Norbert Wiener]]), which is used for modeling [[Brownian motion]] as described by [[Louis Bachelier]] in 1900 and by [[Albert Einstein]] in 1905 and other physical [[diffusion]] processes in space of particles subject to random forces.  Since the 1970s, the Wiener process has been widely applied in [[financial mathematics]] and [[economics]] to model the evolution in time of stock prices and bond interest rates.

The main flavours of stochastic calculus are the [[Itô calculus]] and its variational relative the [[Malliavin calculus]].  For technical reasons the Itô integral is the most useful for general classes of processes, but the related [[Stratonovich integral]] 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 [[chain rule]] and therefore does not require [[Itô's lemma]]. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than '''R'''<sup>''n''</sup>.
The [[dominated convergence theorem]] does not hold for the Stratonovich integral; consequently it is very difficult to prove results without re-expressing the integrals in Itô form.