Some applications of viscosity solution theory in image processing and finance.

Summary We apply the theory of viscosity solutions due to Michael Grain Crandall and Pierre-Louis Lions, to the numerical solution of two nonlinear partial differential equations. The first part is devoted to the study of the Horn equation, which models the shape-from-shading problem. It concerns the reconstruction of a surface illuminated by light sources, from a single gray level coded photograph. The theory of viscosity solutions provides us with a framework in which the problem is mathematically well posed: we prove the uniqueness of the viscosity solution of the Horn equation verifying appropriate edge conditions. Then, in order to compute a first order approximation of the viscosity solution, we construct a monotone finite difference scheme, obtained by discretization of the dynamic programming principle, due to Bellman. In the second part, we propose a finite difference scheme that approximates the unique viscosity solution of the variational inequality that models the stochastic control problem: portfolio management with transaction costs. The algorithm allows not only the computation of an approximation of the value function but also the search for free boundaries that delimit the region in which no transactions are made.