Density minimization for hopping diffusions: Malliavin calculation for pure hopping processes, applications to finance.

Summary This thesis gives two applications of Malliavin's calculus for jump processes. In the first part, we deal with the density minimization of hopping diffusions whose continuous part is directed by a Brownian motion. For this purpose, we use a conditional integration by parts formula based on the Brownian motion only. We then treat the computation of financial options whose underlying price is a pure jump process. In the second part, we develop an abstract Malliavin-type calculus based on non-independent random variables of discontinuous conditional density. We establish an integration by parts formula that we apply to the amplitudes and jump times of the considered jump processes. In the third part, we use this integration by parts to compute the Delta of European and Asian options and the price and Delta of American options.