Probabilistic methods for artificial edge conditions of nonlinear partial differential equations in finance: optimal stopping problem for regular diffusion.

Summary In this thesis, we give a localization error control on the system of parabolic partial differential inequalities with Dirichlet edge conditions. This error control is done via the probabilistic interpretation of the variational inequalities in the form of stochastic backward differential equations (SDFEs). Thus, the viscosity solutions of localized variational inequations with Dirichlet conditions at the edge are interpreted as solutions of the reflected EDSRs with bounded random final time. We establish an existence and uniqueness theorem for this type and give a definition to the notion of viscosity solution for our problem. In the last part of this chapter, we apply this control to the American option pricing problem. Next, we establish the almost everywhere derivability of the reflected diffusion with respect to its initial value and give the derivative in the one-dimensional case. We give the representation of the almost everywhere derivatives of the solutions of variational inequalities with Neumann edge condition. From these representations, we give the localization error over an entire portfolio of American options. In the second part, we explicitly solve the optimal stopping problem with random discounting and an additive functional as the cost of observations for a regular linear diffusion. This result generalizes the work of Beibel and Lerche who solved (1997 and 1998) this type of problem without any additional additive functional. We use in our approach the h-transformed method, the martingale technique, the time change.