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Concept
Stratonovich integral
Summary
In stochastic processes, the Stratonovich integral or Fisk–Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics.
In some circumstances, integrals in the Stratonovich definition are easier to manipulate. Unlike the Itô calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds.
Perhaps the most common situation in which these are encountered is as the solution to Stratonovich stochastic differential equations (SDEs). These are equivalent to Itô SDEs and it is possible to convert between the two whenever one definition is more convenient.
The Stratonovich integral can be defined in a manner similar to the Riemann integral, that is as a limit of Riemann sums. Suppose that is a Wiener process and is a semimartingale adapted to the natural filtration of the Wiener process. Then the Stratonovich integral
is a random variable defined as the limit in mean square of
as the mesh of the partition of tends to 0 (in the style of a Riemann–Stieltjes integral).
Many integration techniques of ordinary calculus can be used for the Stratonovich integral, e.g.: if is a smooth function, then
and more generally, if is a smooth function, then
This latter rule is akin to the chain rule of ordinary calculus.
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, and variations of these are used to solve Stratonovich SDEs .
Note however that the most widely used Euler scheme (the Euler–Maruyama method) for the numeric solution of Langevin equations requires the equation to be in Itô form.
If , and are stochastic processes such that
for all , we also write
This notation is often used to formulate stochastic differential equations (SDEs), which are really equations about stochastic integrals.
Official source
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