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 ,
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 .
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,
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