$\epsilon$-Monotone Fourier Methods for Optimal Stochastic Control in Finance

TL;DR

This paper introduces an $oldsymbol{ ext{ extit{ extepsilon}}}$-monotone Fourier method for stochastic control in finance, ensuring stability and accuracy while preventing arbitrage violations in numerical solutions.

Contribution

It proposes a preprocessing step that projects the Green's function onto linear basis functions, making Fourier methods monotone and stable without sacrificing second order accuracy.

Findings

Guarantees monotonicity within a specified tolerance.

Maintains second order accuracy for smooth problems.

Has the same computational complexity as standard Fourier methods.

Abstract

Stochastic control problems in finance often involve complex controls at discrete times. As a result numerically solving such problems, for example using methods based on partial differential or integro-differential equations, inevitably give rise to low order accuracy, usually at most second order. In many cases one can make use of Fourier methods to efficiently advance solutions between control monitoring dates and then apply numerical optimization methods across decision times. However Fourier methods are not monotone and as a result give rise to possible violations of arbitrage inequalities. This is problematic in the context of control problems, where the control is determined by comparing value functions. In this paper we give a preprocessing step for Fourier methods which involves projecting the Green's function onto the set of linear basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance), $\ell_\infty$-stable and satisfies an $\epsilon$-discrete comparison principle. In addition the algorithm has the same complexity per step as a standard Fourier method while at the same time having second order accuracy for smooth problems.