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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:

  • The Stratonovich integral adds noise induced spurious drift.

  • The Stratonovich integral changes under coordinate transformations in the usual way we are familiar with in calculus, while the Itô integral obeys a more complicated rule, called Itô's formula.

  • The Stratonovich integral is defined for a smaller class of integrands than the Itô integral, and needs more differentiability for the chain rule.

  • On the other hand, the Stratonovich integral is more robust under perturbation of the integration process and “thus a useful tool in modelling” (see reference 1, p. 156).

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:

  • For modelling turbulence at small length scales, people use the Itô integral, because they want all the technical tools that it comes with.

  • For doing stochastic calculus on (smooth) manifolds, the simpler transformation rule of the Stratonovich integral makes people use that. For example, for constructing Brownian motion on manifolds, you can directly translate the differential equation of Cartan development into a Stratonovich integral equation.

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.