- Stochastic calculus
Stochastic calculus is a branch of
mathematicsthat 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 motionas described by Albert Einsteinand other physical diffusionprocesses in space of particles subject to random forces. Since the 1970's, the Wiener process has been widely applied in financial mathematicsto model the evolution in time of stock and bond prices.
The main flavours of stochastic calculus are the
Itō calculusand 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 integralis 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".The Dominated convergence theoremdoes not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itō form.
Itō integralis central to the study of stochastic calculus. The integral is defined for a semimartingale"X" and locally bounded predictable process "H".
The Stratonovich integral can be defined in terms of the Itō integral as
The alternative notation
is also used to denote the Stratonovich integral.
A very important application of stochastic calculus is in
* [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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