As usual, let be a filtered probability space which satisfies the usual conditions. It is often useful to use the language of Stratonovitch ‘s integration to study stochastic differential equations because the Itō’s formula takes a much nicer form. If , , is an -adapted real valued local martingale and if is an -adapted continuous semimartingale satisfying , then by definition the Stratonovitch integral of with respect to is defined as

where:

- is the Itō integral of against ;
- is the quadratic covariation at time between and .

By using Stratonovitch integral instead of Itō’s, the Itō formula reduces to the classical change of variable formula.

**Theorem.*** Let be a – dimensional continuous semimartingale. Let now be a function. We have
*

Let be a non empty open set. A smooth vector field on is simply a smooth map

The vector field defines a differential operator acting on smooth functions as follows:

We note that is a derivation, that is a map on , linear over , satisfying for ,

An interesting result is that, conversely, any derivation on is a vector field.

Let now be a -dimensional Brownian motion and consider vector fields , , . By using the language of vector fields and Stratonovitch integrals, the fundamental theorem for the existence and the uniqueness of solutions for stochastic differential equations is the following:

**Theorem. ***Assume that are bounded vector fields with bounded derivatives up to order 2. Let . On , there exists a unique continuous and adapted process such that for ,
with the convention that .*

Thanks to Itō’s formula the corresponding Itō’s formulation is

where for , is the vector field given by

If is a function, from Itō’s formula, we have for ,

and the process

is a local martingale where is the second order differential operator

my sincere thanks for this blog.

I’d like to know why it prefers the integral of Stratonovich rather than Itô on solving stochastic differential equations.

Thanks for the interest. We sometimes prefer to use Stratonovitch integrals for SDEs because the Ito’s formula takes a very simple form. Also, it is the correct way to define SDEs on manifolds.

This is what my observations while flying over your book “An Introduction to the Geometry of Stochastic Flows.”

This is a very encouraging and promising Blog.

thank you again.