The **Stratonovich Integral** is a stochastic integral, the commonest alternative to the Itō integral. ItÃ´ calculus, named after Kiyoshi ItÃ´, treats mathematical operations on stochastic processes. ...
In some circumstances integrals in the Stratonovich definition are easier to manipulate. Unlike the Itō calculus, it is defined such that the chain rule of ordinary calculus holds for these stochastic integrals. ItÃ´ calculus, named after Kiyoshi ItÃ´, treats mathematical operations on stochastic processes. ...
In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...
Perhaps the commonest situation in which these are encountered is as the solution to Stratonovich Stochastic Differential Equations (SDE). These are equivalent to Itō SDEs and it is possible to convert between the two whenever one definition is more convenient.
## Definition
The integral can be defined in a manner similar to the Riemann Integral, that is as a limit of Riemann Sums. Supposing that *W*_{t} is a Wiener process and *X*_{t} is a *W*_{t}-adapted stochastic process, the integral is the limit in probability of A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ...
In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ...
In probability theory, there exist several different notions of convergence of random variables. ...
c.f. the similar limit in the Ito definition. ItÃ´ calculus, named after Kiyoshi ItÃ´, treats mathematical operations on stochastic processes. ...
With probability 1, a general stochastic processes does *not* satisfy the criteria for convergence in the Riemann sense. If it did the Ito and Stratonovich definitions would converge to the same solution. As it is, for integrals with respect to Wiener processes, they are distinct. In the mathematics of probability, a stochastic process can be thought of as a random function. ...
## Advantages of the Stratonovich Integral ### Numerical methods Stochastic integrals can rarely be solved in analytic form, making stochastic numerical integration an important topic in all uses of stochastic integrals. Various numerical approximations converge to the Stratonovich integral, making this important in numerical solutions of SDEs (see Kloeden and Platen). In numerical analysis, the term numerical integration is used to describe a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe numerical algorithms for solving differential equations. ...
## Disadvantages of the Stratonovich Integral ## References - Bernt K. Øksendal
*Stochastic Differential Equations* Springer, (5th ed.) ISBN 3540637206 - Gardiner, Crispin W.
*Handbook of Stochastic Methods* Springer, (3rd ed.) ISBN 3-540-20882-8 - Kloeden, Peter E. and Platen, Eckhard
*Numerical Solution of Stochastic Differential Equations* Springer, Berlin, ISBN 3540540628 |