Stochastic control problem under liquidity risk constraints.

Summary This thesis studies some stochastic control problems in a context of liquidity risk and impact on asset prices. The thesis is composed of four chapters.In the second chapter, we propose a modeling of a market making problem in a liquidity risk context in the presence of inventory constraints and regime shifts. This formulation can be considered as an extension of previous studies on this subject. The main result of this part is the characterization of the value function as a unique solution, in the sense of viscosity, of a system of Hamilton-Jacobi-Bellman equations . In the third chapter, we propose a numerical approximation scheme to solve a portfolio optimization problem in a context of liquidity risk and impact on asset prices. We show that the value function can be obtained as a limit of an iterative procedure where each iteration represents an optimal stopping problem and we use a numerical algorithm, based on optimal quantization, to compute the value function and the control policy. The convergence of the numerical scheme is obtained via monotonicity, stability and consistency criteria.In the fourth chapter, we focus on a coupled problem of singular control and impulse control in an illiquidity context. We propose a mathematical formulation to model the dividend distribution and the investment policy of a firm subject to liquidity constraints. We show that, under transaction costs and an impact on the price of illiquid assets, the firm's value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. An iterative numerical method is also proposed to compute the optimal buy, sell and dividend strategy.