Liquidity risk modeling and quantification methods applied to sequential stochastic control.

Summary This thesis is divided into two parts that may be read independently. The first part is about the mathematical modelling of liquidity risk. The aspect of illiquidity studied here is the constraint on the trading dates, meaning that in opposition to the classical models where investors may trade continuously, we assume that trading is only possible at discrete random times. We then use optimal control techniques (dynamic programming and Hamilton-Jacobi-Bellman equations) to identify the value functions and optimal investment strategies under these constraints. The first chapter focuses on a utility maximisation problem in finite horizon, in a framework inspired by energy markets. In the second chapter we study an illiquid market with regime-switching, and in the third chapter we consider a market in which the agent has the possibility to invest in a liquid asset and an illiquid asset which are correlated. In the second part we present probabilistic quantization methods to solve numerically an optimal switching problem. We first consider a discrete time approximation of our problem and prove a convergence rate. Then we propose two numerical quantization methods : a markovian approach where we quantize the gaussian in the Euler scheme, and, in the case where the underlying diffusion is not controlled, a marginal quantization approach inspired by numerical methods for the optimal stopping problem.