This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (August 2011) (Learn how and when to remove this template message)

Part of a series of articles about |

Calculus |
---|

**Stochastic calculus** is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field is created and started by Kiyoshi Itô during World War 2.

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. 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**^{n}.
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.

Main article: Itô calculus |

The Itô integral is central to the study of stochastic calculus. The integral is defined for a semimartingale *X* and locally bounded **predictable** process *H*.^{[citation needed]}

Main article: Stratonovich integral |

The Stratonovich integral of a semimartingale against another semimartingale *Y* can be defined in terms of the Itô integral as

where [*X*, *Y*]_{t}^{c} denotes the quadratic covariation of the continuous parts of *X*
and *Y*. The alternative notation

is also used to denote the Stratonovich integral.

An important application of stochastic calculus is in mathematical finance, in which asset prices are often assumed to follow stochastic differential equations. In the Black–Scholes model, prices are assumed to follow geometric Brownian motion.