[Stochastic control and applications to option hedging with illiquidity : theoritical and numerical aspects].

Summary We study some applications of stochastic control to option hedging in the presence of illiquidity. In the first part, we consider an option overhedging problem in a stochastic volatility model. The originality of this problem is that the asset used to hedge the volatility is illiquid and that the agent will have to make a finite amount of transactions. The second part concerns option hedging in the presence of uncertain volatility whose dynamics are not specified. We introduce a criterion for non-trivial option prices, allowing the agent to lose money for volatility realizations that he considers unlikely. Finally, in a third part, we study an impulse control problem for which the controls take effect with delay. This study applies in particular to the hedging of options on hedge funds, for which buy and sell orders are executed with delay. In each part, we characterize the value function of the problem as the unique viscosity solution of a partial differential equation. In the first and third parts, we introduce in a second chapter algorithms for numerically solving these PDEs by finite differences. The convergence of these algorithms is proved theoretically.