We now turn to the theory of stochastic differential equations. Stochastic differential equations are the differential equations corresponding to the theory of the stochastic integration.

As usual, we consider a filtered probability space which satisfies the usual conditions and on which is defined a -dimensional Brownian motion . Let , and be functions.

**Theorem.** * Let us assume that there exists such that
Then, for every , there exists a unique continuous and adapted process such that for
Moreover, for every ,
*

**Proof.**

Let us first observe that from our assumptions, there exists such that

- , ;
- ), .

The idea is to apply a fixed point theorem in a convenient Banach space.

For , let us consider the space of continuous and adapted processes such that

.

We endow that space with the norm

It is easily seen that is a Banach space.

**Step one:** We first prove that if a continuous and adapted process is a solution of the equation then, for every , .

Let us fix and consider for the stopping times For ,

Therefore, by using the inequality , we get

Thus, we have

By using our assumptions, we first estimate

By using our assumptions and Doob’s inequality, we now estimate

Therefore, from the inequality , we get

We may now apply Gronwall’s lemma to the function and deduce

where is a constant that does not depend on . Fatou’s lemma implies by passing to the limit when that

We conclude, as expected, that

More generally, by using the same arguments we can observe that if a continuous and adapted process satisfies

with , then .

**Step 2:** We now show existence and uniqueness of solutions for the equation on a time interval where is small enough.

Let us consider the application that sends a continuous and adapted process to the process By using successively the inequalities , Cauchy-Schwarz inequality and Doob’s inequality, we get . Moreover, arguing the same way as above, we can prove

Therefore, if is small enough is a Lipschitz map whose Lipshitz constant is strictly less than 1. Consequently, it has a unique fixed point. This fixed point is, of course the unique solution of the equation on the time interval . Here again, we can observe that the same reasoning applies if is replaced by a random variable that satisfies .

**Step 3.**

In order to get a solution of the equation on , we may apply the previous step to get a solution on intervals , where is small enough and . This will provide a solution of the equation on . This solution is unique, from the uniqueness on each interval

**Definition:** *An equation like in the previous theorem is called a stochastic differential equation.*

**Exercise:*** (Ornstein-Uhlenbeck process) Let . We consider the following stochastic differential equation,
*

- Show that it admits a unique solution that is given by

- Show that is Gaussian process. Compute its mean and covariance function.
- Show that if then, when , converges in distribution toward a Gaussian distribution.

**Exercise.***(Brownian bridge) We consider for the following stochastic differential equation
*

- Show that

is the unique solution. - Deduce that is Gaussian process. Compute its mean and covariance function.
- Show that in , when , .

**Exercise.** *Let and . We consider the following stochastic differential equation,
Show that
is the unique solution.*

The next proposition shows that solutions of stochastic differential equations are intrinsically related to a second order differential operator. This connection will later be investigated in more details.

**Proposition.*** Let be the solution of a stochastic differential equation
where and are Borel functions. Let now be a function. The process
is a local martingale, where is the second order differential operator
and .*