Pricing of American contingent claims with jump stock price and constrained portfolios

In this paper, we study the problem of pricing American contingent claims in an incomplete market where the stock price process is supposed to be driven by both a Wiener process and a Poisson random measure and the portfolios are constrained. We formulate this problem as to find the minimal solution of a backward stochastic differential equation (BSDE) with constraints. We use the penalization method to construct a sequence of BSDEs with respect to Wiener process and Poisson random measure, and we show that the solutions of these equations converge to the minimal solution we are interested in. Finally, in the Markovian case, we characterize the minimal hedging price as the minimal viscosity supersolution of an integral-partial differential inequality with constraints.