In this lecture, we study the regularity of the solution of a stochastic differential equation with respect to its initial condition. The key tool is a multimensional parameter extension of the Kolmogorov continuity theorem whose proof is almost identical to the one-dimensional case.

**Theorem.** * Let be a -dimensional stochastic process such that there exist positive constants such that for every
There exists a modification of the process such that for every there exists a finite random variable such that for every
*

As above, we consider two functions and and we assume that there exists such that

As we already know, for every , there exists a continuous and adapted process such that for ,

**Proposition.*** Let . For every , there exists a constant such that for every and ,
.
As a consequence, there exists a modification of the process such that for , ,
and such that is continuous for almost every .*

**Proof.** As before, we can find such that

, ;

and ), .

We fix and . Let

By using the inequality , we obtain

We now have

and from Burkholder-Davis-Gundy inequality

As a conclusion we obtain

Gronwall’s inequality yields then

where is a continuous function.

We have for ,

,

and

The conclusion then easily follows by combining the two previous estimates

In the sequel, of course, we shall always work with this bicontinuous version of the solution.

**Definition.***The continuous process of continuous maps is called the stochastic flow associated to the equation.*

If the maps and are moreover , then the stochastic flow is itself differentiable and the equation for the derivative can be obtained by formally differentiating the equation with respect to its initial condition. We willl admit this result without proof:

**Theorem.** *Let us assume that and are Lipschitz functions, then for every , the flow associated to the equation is a flow of differentiable maps. Moreover, the first variation process which is defined as the Jacobian matrix is the unique solution of the matrix stochastic differential equation:
*