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
- 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