ε-Monotone Fourier Methods for Optimal Stochastic Control in Finance

Abstract

Stochastic control problems in finance often involve complex controls at discrete times. As a result, numerically solving such problems using, for example, methods based on partial differential or integrodifferential equations inevitably gives rise to low-order (usually at most second-order) accuracy. In many cases, Fourier methods can be used to efficiently advance solutions between control monitoring dates, and numerical optimization methods can then be applied across decision times. However, Fourier methods are not monotone, and as a result they 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 that 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), ℓ ∞-stable and satisfies an ε-discrete comparison principle. In addition, the algorithm has the same complexity per step as a standard Fourier method and second-order accuracy for smooth problems.

Keywords: monotonicity, Fourier methods, discrete comparison, optimal stochastic control, finance