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

Daniel Ševčovič and Cyril Izuchukwu Udeani
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Daniel Ševčovič: Department of Applied Mathematics and Statistics, Comenius University in Bratislava, Mlynská Dolina, 84248 Bratislava, Slovakia
Cyril Izuchukwu Udeani: Department of Applied Mathematics and Statistics, Comenius University in Bratislava, Mlynská Dolina, 84248 Bratislava, Slovakia

Mathematics, 2021, vol. 9, issue 13, 1-12

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.

Keywords: lévy measure; option pricing; strong kernel; hölder continuity; partial integro-differential equation; bessel potential spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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