Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

Daniel Ševčovič,Cyril Izuchukwu Udeani

doi:10.3390/math9131463

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 Lévy 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 Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model 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.

Highlights

IntroductionOur goal is to prove existence and uniqueness of a solution to a nonlinear parabolic partial integro-differential equation (PIDE) with the form: Received: 19 May 2021
We investigated the existence and uniqueness of a solution to the nonlocal nonlinear partial integro-differential equation (PIDE) arising from financial modeling
We considered a call/put option pricing model on underlying asset that follows a Lévy process with jumps in the multidimensional space

Summary

Introduction

Our goal is to prove existence and uniqueness of a solution to a nonlinear parabolic partial integro-differential equation (PIDE) with the form: Received: 19 May 2021. Employed the theory of abstract semilinear parabolic equations to obtain a solution to the nonlinear non-local PIDE (1) in the one-dimensional case. They employed the framework of a scale of Bessel potential spaces for general Lévy measures by considering classical. We analyze the generalized nonlinear non-local PIDE (2), where the shift function ξ = ξ (τ, x, z) depends on the variables x, z ∈ Rn. Section 4 presents an application of the proposed results in the one-dimensional space for pricing of options on underlying asset that follows Lévy processes. Cruz and Ševčovič for an overview of various examples of admissible Lévy measures in the one-dimensional case

Existence and Uniqueness Results

Applications to Option Pricing

Conclusions