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Encyclopedia > Stratonovich integral

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.

## Contents

The integral can be defined in a manner similar to the Riemann Integral, that is as a limit of Riemann Sums. Supposing that Wt is a Wiener process and Xt is a Wt-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. ...

## 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

Results from FactBites:

 Itō calculus - Wikipedia, the free encyclopedia (184 words) Its most important concept is the Itô stochastic integral. A crucial fact about this integral is Ito's lemma. The Stratonovich integral is another way to define stochastic integrals.
 Stochastic calculus - Wikipedia, the free encyclopedia (765 words) 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) and the integrals can readily be expressed in terms of the Itô integral. The key to the construction of a stochastic integral is the definition of a quadratic-variation process; the quadratic variation of a general L where the first integral is defined by the natural extension of the Itô integral from martingale integrators to local martingale integrators, and the second integral is understood in the usual Lebesgue-Stieltjes sense.
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