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Stratonovich integral

Idea

The Fisk-Stratonovich integral or simply Stratonovich integral is a way to integrate stochastic functions which is an alternative to the Itô integral? in stochastic calculus.

Under some technical conditions that guarantee both Itô and Stratonovich integrals are defined, each can be computed from each other. Therefore one can always switch from one to the other, depending on the circumstances. Many references only discuss the Itô integral, because it has some good technical properties (it is non-anticipating, it is a martingale, it has a Fokker-Planck equation etc.).

However, they have different advantages and disadvantages:

There is a very simple practical criterion for deciding which to use. The Stratonovich integral obeys a simpler transformation rule, but the Itô integral comes with a lot of helpful technical tools like the Fokker-Planck equation. So, one must simply decide which matters more in the application at hand. For example:

Stratonovich versus Itô

In one dimension one has

X T=X 0+ 0 Ta(t,X t)dt+ 0 Tb(t,X t)dW t X_T = X_0 + \int_0^T a(t, X_t) d t + \int_0^T b(t, X_t) \circ d W_t

where the last integral is a Stratonovich integral and W tW_t is the Wiener process, iff one has

X T=X 0+ 0 Ta(t,X t)dt+ 0 Tb(t,X t)dW t+12 0 Tdbdxb(t,X t)dt X_T = X_0 + \int_0^T a(t, X_t) d t + \int_0^T b(t, X_t) d W_t + \frac{1}{2} \int_0^T \frac{d b}{d x} b(t, X_t) d t

as an Itô integral. (For a continuous function aa and a continuously differentiable bb on R +×R\R_+\times\R)

References

  1. Ioannis Karatzas and Steven E Shreve: Brownian Motion and Stochastic Calculus, 2nd ed., Springer, 1998. (ZMATH)

  2. Stratonovich integral, Wikipedia.