  In this work 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 an
asymptotically unbiased 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 to 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.
