Multidimensional linear and nonlinear partial integro-differential equation in Bessel potential spaces with applications in option pricing

TL;DR

This paper extends the analysis of multidimensional nonlinear partial integro-differential equations in Bessel potential spaces, with applications to complex option pricing models involving Lévy processes and large trader strategies.

Contribution

It generalizes known one-dimensional results to multidimensional cases and incorporates nonlinear shift functions for advanced option pricing models.

Findings

Proved existence and uniqueness of solutions in multidimensional settings.

Extended the theory to a broad class of Lévy measures with growth conditions.

Applied results to nonlinear option pricing models with large trader effects.

Abstract

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of L\'evy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black-Scholes models for option pricing on underlying assets following a L\'evy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from nonlinear option pricing models taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.