Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach

Abstract

A class of nonlinear integro-differential Cauchy problems is studied by means of the viscosity solutions approach. In view of financial applications, we are interested in continuous initial data with exponential growth at infinity. Existence and uniqueness of solution is obtained through Perron's method, via a comparison principle; besides, a first order regularity result is given. This extension of the standard theory of viscosity solutions allows to price derivatives in jump--diffusion markets with correlated assets, even in the presence of a large investor, by means of the PDE's approach. In particular, derivatives may be perfectly hedged in a completed market.