Malliavin calculus and applications in stochastic differential equations

Overview & Synopsis

This thesis is dedicated to the study of stochastic differential equations using Malliavin calculus. In the first chapter, the continuity at the terminal time of the minimal solution to a backward stochastic differential equation with a singular terminal condition is examined. Malliavin calculus is used to control the various terms that appear in the computations, specifically by using the relationship between the Malliavin derivative of the process of interest and the process adapted to the filtration of the motion. In the second chapter, it is shown that the Malliavin derivative of the solution exhibits a singularity at the terminal time. In the third chapter, we add a jump term to the same equation as in the first chapter and show that these jumps can prevent the continuity of the solution. The fourth chapter concludes with the development of a local Malliavin calculus with respect to a Hawkes process. This is applied to obtain a density criterion for the solution of a stochastic differential equation and for the computation of Greeks.