Stratonovich integral

In , the Stratonovich integral (developed simultaneously by Ruslan L. Stratonovich and D. L. Fisk) is a , 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 (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 ${\displaystyle W:[0,T]\times \Omega \to \mathbb {R} }$ is a Wiener process and ${\displaystyle X:[0,T]\times \Omega \to \mathbb {R} }$ is a semimartingale adapted to the natural filtration of the Wiener process. Then the Stratonovich integral

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