Stability and convergence of Galerkin schemes for parabolic equations with application to Kolmogorov pricing equations in time-inhomogeneous L\'evy models

TL;DR

This paper establishes stability and convergence results for fully discrete Galerkin schemes applied to linear parabolic equations, including applications to complex option pricing models in finance.

Contribution

It extends stability and convergence analysis to time-dependent operators and Galerkin spaces, covering a broad class of integro-differential equations in finance.

Findings

Proves stability and convergence for fully discrete schemes

Applies results to time-inhomogeneous Lévy models

Supports a wide range of option types and models

Abstract

Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time dependent coercive operators. We consider abstract Galerkin approximations in space combined with theta-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular the results apply to partial integro differential equations for option pricing in time-inhomogeneous L\'evy models and allows for a large variety of option types and models. The derivation builds on the strong foundation laid out by von Petersdorff and Schwab (2003) who provide the respective results for the time-homogeneous case. We discuss the assumptions in the context of option pricing.