Abstract : This thesis is concerned with
applications of Malliavin-like calculus for jump processes. In the
first part, we compute lower bounds for the density of jump
diffusions with a continuous part driven by a Brownian motion. We
use a Malliavin conditional integration by parts formula based on
Brownian increments only. We then deal with the computation of
financial options, when the asset price follows a pure jump process.
In the second part, we develop an abstract calculus of the Malliavin type based on random variables which are not independent and have discontinuous conditional densities. We settle an integration by parts formula that we apply then to the jump times and amplitudes of pure jump processes. In the third part, we use this integration by parts formula for the computation of the Delta of European and Asian options, and we derive representation formulas for conditional expectations and their gradients in order to compute the price and the Delta of American options.