A Monte Carlo Method for Estimating Sensitivities of Reflected Diffusions in Convex Polyhedral Domains

Abstract

We develop an effective Monte Carlo method for estimating sensitivities or gradients of expectations of sufficiently smooth functionals of a reflected diffusion in a convex polyhedral domain with respect to its defining parameters, namely its initial condition, drift, and diffusion coefficients and directions of reflection. Our method, which falls into the class of infinitesimal perturbation analysis (IPA) methods, uses a probabilistic representation for such sensitivities as the expectation of a functional of the reflected diffusion and its associated derivative process. The latter process is the unique solution to a constrained linear stochastic differential equation with jumps whose coefficients, domain, and directions of reflection are modulated by the reflected diffusion. We propose a consistent estimator for such sensitivities using an Euler approximation of the reflected diffusion and its associated derivative process. Proving that the Euler approximation converges is challenging because the derivative process jumps whenever the reflected diffusion hits the boundary of the domain. A key step in the proof is establishing a continuity property of the related derivative map, which is of independent interest. We compare the performance of our IPA estimator with a standard likelihood ratio estimator (whenever the latter is applicable) and provide numerical evidence that the variance of the former is substantially smaller than that of the latter. We illustrate our method with an example of a rank-based interacting diffusion model of equity markets. Interestingly, we show that estimating certain sensitivities of the rank-based interacting diffusion model using our method for a reflected Brownian motion description of the model outperforms a finite difference method for a stochastic differential equation description of the model.