Stochastic 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. 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 of Norbert Wiener), which is used for modeling Brownian motion as described by Albert Einstein and other physical diffusion processes in space of particles subject to random forces. Since the 1970's, the Wiener process has been widely applied in financial 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 the Malliavin calculus. For technical reasons the Itō 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 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 Rn. 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 Itō form.

Itō integral
The Itō integral is central to the study of stochastic calculus. The integral $$\int H\,dX$$ is defined for a semimartingale X and locally bounded predictable process H.

Stratonovich integral
The Stratonovich integral can be defined in terms of the Itō integral as


 * $$ \int_0^t X_s \circ d Y_s : = \int_0^t X_s d Y_s + \frac{1}{2} \left [ X, Y\right]_t.$$

The alternative notation


 * $$ \int_0^t X_s \partial Y_s $$

is also used to denote the Stratonovich integral.