---
author:
- Jean Daniel Mukam
- |
    Jean Daniel Mukam (jean.d.mukam@aims-senegal.org)\
    African Institute for Mathematical Sciences (AIMS)\
    Senegal\
    \
    [ Supervised by : Dr. Antoine Tambue]{}\
    [ AIMS-South Africa and University of Cape Town ]{}\
    [ antonio@aims.ac.za]{}
date: |
    [ 13 June 2015]{}\
    [*Submitted in Partial Fulfillment of a Masters II at AIMS Senegal*]{}\
title: |
     Stochastic Calculus with Jumps Processes : Theory and Numerical Techniques\
    (Master Thesis)
---

Abstract {#abstract .unnumbered}
========

In this work we consider a stochastic differential equation (SDEs) with jump. We prove the existence and the uniqueness of solution of this equation in the strong sense under global Lipschitz condition. Generally, exact solutions of SDEs are unknowns. The challenge is to approach them numerically. There exist several numerical techniques. In this thesis, we present the compensated stochastic theta method (CSTM) which is already developed in the literature. We prove that under global Lipschitz condition, the CSTM converges strongly with standard order 0.5. We also investigated the stability behaviour of both CSTM and stochastic theta method (STM). Inspired by the tamed Euler scheme developed in [@Martin1], we propose a new scheme for SDEs with jumps called compensated tamed Euler scheme. We prove that under non-global 