Special Issue in honor of John W. Neuberger. Electron. J. Diff. Eqns., Special Issue 02 (2023), pp. 239-253.

Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model

Peter Takac

Abstract:
We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the quasilinear diffusion term with the p-Laplace operator Δp for 1 < p < ∞. The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics.

Published March 27, 2023.
Math Subject Classifications: 35K92, 91B02, 35J92, 91G20.
Key Words: Nonlinear parabolic equation; p-Laplacian; nonlinear semigroup; Black-Scholes equation; option pricing.
DOI: https://doi.org/10.58997/ejde.sp.02.t1

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Peter Takáç
Universität Rostock
Institut für Mathematik
Ulmenstraße 69, Haus 3
D-18057 Rostock, Germany
email: peter.takac@uni-rostock.de

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