Applications of probabilistic and stochastic control methods to mathematical finance.

Summary The present work consists of six chapters divided into three parts and has as a common feature probabilistic and stochastic control applications to the evaluation of incompletely running contingent assets. The first chapter deals with the optimal stopping time problem of a jump-controlled diffusion process and shows, generalizing the results of Lions (1983), that the value function is characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation from dynamic programming, with appropriate boundary conditions. The second chapter establishes existence and uniqueness results in the class of regular functions c#1#,#2 when the preceding integrodifferential operator is linear, i.e., when there is no control on the jump diffusion process. The second part focuses on the contingent asset coverage and valuation problem in an incomplete walk setting. Using quadratic optimization criteria, we determine in Chapter 3 the pricing and hedging strategy that best duplicates a given contingent asset, when the pricing processes are semi martingales. From an equilibrium pricing approach, Chapter 4, in a stochastic volatility model framework, gives necessary and sufficient conditions on an Arrow-Debreu price system given to be consistent with an additive multi-agent intertemporal equilibrium model. Finally, the third part deals with option pricing in a diffusion model with jumps. Using the results of part 1, we study the regularity of the price of a European option (chapter 5) and that of an American option (chapter 6) and characterize them as solutions of parabolic integrodifferential equations of the second order, with appropriate boundary conditions. We establish some properties of an option, related to the incompleteness of the walk caused by the presence of jumps.