Category Archives: Stochastic Calculus lectures

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 … Continue reading →

As in the previous Lectures, we consider a filtered probability space on which is defined a Brownian motion , and we assume that is the usual completion of the natural filtration of . Our goal is here to write an … Continue reading →

More generally, by using the same methods as in the previous Lecture, we can introduce iterated derivatives. If , we set . We may then consider as a square integrable random process indexed by and valued in . By using … Continue reading →

The next Lectures will be devoted to the study of the problem of the existence of a density for solutions of stochastic differential equations. The basic tool to study such questions is the so-called Malliavin calculus. Let us consider a … Continue reading →

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 … Continue reading →

In the previous section, we have seen that if is the solution of a stochastic differential equation then is a Markov process, that is for every , where . It is remarkable that this property still holds when is now … Continue reading →

It is now time to give some applications of the theory of stochastic differential equations to parabolic second order partial differential equations. In particular we are going to prove that solutions of such equations can represented by using solutions of … Continue reading →

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 … Continue reading →