Finite element scheme for integro-partial differential equations

TL;DR

This paper develops a finite element-like scheme for solving complex non-linear integro-partial differential equations in finance, ensuring convergence and providing error bounds even under challenging conditions.

Contribution

It introduces a monotone, robust finite element scheme for fully non-linear integro-PDEs with convergence proofs and error bounds applicable to degenerate, multi-dimensional, and low-regularity cases.

Findings

Scheme converges in general situations including degenerate equations

Provides error bounds for unstructured grids and singular integral terms

Applicable to various jump-process models in finance

Abstract

We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in Finance. The schemes are monotone and robust. We prove that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types of jump-models used in Finance. In all cases we provide (probably optimal) error bounds. These bounds apply when grids are unstructured and integral terms are very singular, two features that are new or highly unusual in this setting.