Contrôle stochastique appliqué à la finance

This PhD thesis considers the optimal trading problem from the stochastic control approach and consists of four parts. In the first part, we begin with the study of the impacts generated by volumes on the price process. We introduce a structural model in which price movements are due to not only the last trade’s volume but also to those of earlier trades, weakened by a decay factor. Considering a similar continuous version, we provide a condition ensuring the optimality of a strategy for the minimization of the execution cost in a mean-variance framework, and solve it numerically. In the second part, we propose a general model to optimize the way trading algorithms are used. Using an impulse control approach, we model the execution of a large order by a sequence (τi,δi,Ei)i, which is defined so that the i-th slice is executed in [τi,τi+δi] with parameter Ei. We characterize the value function as a viscosity solution of a system of PDE. We provide a numerical scheme and prove its convergence. Numerical illustrations are given for a real case. We deal with the problem of pricing an option on the book liquidation in presence of impact where the classical pricing by neutral risk measure fails. We begin with an abstract model generalized from the work of Bouchard- Eile-Touzi (2008), and then apply to compute the price of a VWAP guaranteed contract. We establish in the last part an equivalence result between stochastic target problems and standard optimal control. We derive the classical HJB equation from the PDE obtained in the stochastic target framework.