An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians

TL;DR

This paper introduces an approximation scheme for semilinear parabolic PDEs with convex, coercive Hamiltonians, combining splitting methods and established formulas to ensure convergence and determine the rate of approximation.

Contribution

It presents a novel splitting-based approximation scheme for a class of PDEs linked to stochastic processes, with proven convergence and explicit rate analysis.

Findings

The scheme converges to the true solution.

Convergence rate is explicitly determined.

Applicable to PDEs in finance and stochastic control.

Abstract

We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac and the Hopf-Lax formulae. We establish the convergence of the scheme and determine the convergence rate, combining Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation.