NONLINEAR DEGENERATE INTEGRO-PARTIAL DIFFERENTIAL EVOLUTION EQUATIONS RELATED TO GEOMETRIC LÉVY PROCESSES AND APPLICATIONS TO BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of nonlinear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the ``geometric form'' of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations and (ii) pricing of European and American derivatives via backward stochastic differential equations. Regarding (i), we extend previous results on backward stochastic differential equations in a Lévy setting and the connection to semilinear integro--partial differential equations.