Contributions to the theory of viscosity solutions of Hamilton-Jacobi-Bellman equations and applications to finance.

Summary The thesis gathers three papers which all establish the existence and uniqueness of viscosity solutions for some partial differential equations coming from optimal control problems or mathematical economics models. Most of these equations break out of the classical framework by the presence of either a constraint on the gradient or a nonlinear integral term. We also focus on the determination of an asymptotic criterion compatible with uniqueness results. Thus the first work shows that there exists a unique minor solution for first order Hamilton-Jacobi equations, when the Hamiltonian is convex and nonlinear. In a second work, we establish that the value function of a singular control problem (investment-consumption model with local substitution) is the unique solution of the associated dynamic programming equation, in an unbounded open, with state constraint at the edge. In the last work, we are interested in a class of integro-differential equations whose nonlinearity comes essentially from the integral term. Again, the origin of this problem is economic since the unique solution is the recursive utility function when the uncertainty is generated by a Brownian motion and a jump process.