Stochastic Impulse Control with Uncertainty in Finance and Insurance.

Summary This thesis is composed of three chapters that deal with impulse control problems. In the first chapter, we introduce a general framework for impulse control with uncertainty. Knowing an a priori law on unknown parameters, we explain how it should evolve and integrate it to the optimal control problem. We characterize the solution through a quasivariational parabolic equation that can be solved numerically and give examples of applications to finance. In the second chapter, we introduce an impulse control problem with uncertainty in an actuarial setting. An (re)insurer faces natural catastrophes and can issue CAT bonds to reduce the risk taken. We again characterize the optimal control problem through a numerically solvable quasi-variational parabolic equation and give some application examples. In the last chapter, we propose a model of the price through a completely endogenous order book. We solve impulse optimal control problems (order placement) of rational economic agents that we gather on a same market.