EDSPR Fortement couplées et contrôle optimal stochastique

Abstract

This thesis contains two themes. The first topic considers the problem of the well-posedness for a kind of fully coupled forward backward stochastic differential equations driven by Teugels martingales associated with some Lévy processes. The second one is devoted to the stochastic optimal control for systems driven by stochastic differential equations (SDE for short). In the first part which contains two papers, we provide and prove some existence and uniqueness results in two different cases: (i) The final time is assumed to be fixed and large; (ii) the final time is allowed to be random. The Second part of this thesis is concerned with the stochastic control problems to optimize an insurance firm problem in the case where its cash-balance process is assumed to be governed by a stochastic differential equation driven by Teugels martingales. We deal with several cases according to the interest rate process; we first suppose that the insurance firm only invests in a money account with compounded interest rate. Then we discuss this optimal premium problem, in the case where the interest rate is allowed to be stochastic. More precisely, we consider the case in which the payment function and the stochastic interest rate are given by the same Brownian motion, in addition to the case where we assume that they are given by different and independent Brownian motions.