As usual, we consider a filtered probability space which satisfies the usual conditions and on which is defined a -dimensional Brownian motion . Our purpose here, is to prove that solutions of stochastic differential equations are differentiable in the sense of Malliavin.
The following lemma is easy to prove by using the Wiener chaos expansion.
Lemma. Let be a progressively measurable process such that for every , and
Proof. We make the proof when and use the notations introduced in the Wiener chaos expansion Lecture. For , we have
But we can write,
with . Since
we get the result when can be written as . By continuity of the Malliavin derivative on the space of chaos of order , we conclude that the formula is true if is a chaos of order . The result finally holds in all generality by using the Wiener chaos expansion
We consider two functions and and we assume that and are with derivatives at any order (more than 1) bounded.
As we know, there exists a bicontinuous process such that for ,
Moreover, for every , and
Theorem. For every , , and for ,
where is the -th component of . If , then .
Proof. We first prove that for every . We consider the Picard approximations given by and
By induction, it is easy to see that and that for every , we have
Then, we observe that converges to in and that the sequence is bounded. As a consequence for every . The equation for the Malliavin derivative is obtained by differentiating the equation satisfied by . Higher order derivatives may be treated in a similar way with a few additional work
Combining this theorem with the uniqueness property for solutions of linear stochastic differential equations, we obtain the following representation for the Malliavin derivative of a solution of a stochastic differential equation:
where is the first variation process defined by
We now fix as the initial condition for our equation and denote by the Malliavin matrix of . From the previous corollary, we deduce that
We are now finally in position to state the main theorem of the section:
Theorem. Assume that there exists such that for every ,
then for every and , the random variable has a smooth density with respect to the Lebesgue measure.
We want to prove that is invertible with inverse in for . Since is invertible and that its inverse solves a linear equation, we deduce that for every ,
We conclude that it is enough to prove that is invertible with inverse in where
By the uniform ellipticity assumption, we have
where the inequality is understood in the sense that the difference of the two symmetric matrices is non negative. This implies that is invertible. Moreover, it is an easy exercise to prove that if is a continuous map taking its values in the set of positive definite matrices, then we have
As a consequence, we obtain
Since has moments in for all , we conclude that is invertible with inverse in