Pathwise differentiability of reflected diffusions in convex polyhedral domains

Abstract

Reflected diffusions in convex polyhedral domains arise in a variety of applications, including interacting particle systems, queueing networks, biochemical reaction networks and mathematical finance. Under suitable conditions on the data, we establish pathwise differentiability of such a reflected diffusion with respect to its defining parameters -namely, its initial condition, drift and diffusion coefficients, and (oblique) directions of reflection along the boundary of the domain. We characterize the right-continuous regularization of a pathwise derivative of the reflected diffusion as the pathwise unique solution to a constrained linear stochastic differential equation with jumps whose drift and diffusion coefficients, domain and directions of reflection depend on the state of the reflected diffusion. The proof of this result relies on properties of directional derivatives of the associated (extended) Skorokhod reflection map and their characterization in terms of a so-called derivative problem, and also involves establishing certain path properties of the reflected diffusion at nonsmooth parts of the boundary of the polyhedral domain, which may be of independent interest. As a corollary, we obtain a probabilistic representation for derivatives of expectations of functionals of reflected diffusions, which is useful for sensitivity analysis of reflected diffusions. Date

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

“…The remainder of this paper is organized as follows. In Section 2 we give precise definitions of a reflected diffusion and its associated derivative process, and we recall the probabilistic representation of sensitivities of reflected diffusions from [15]. In Section 3 we define an Euler approximation for a reflected diffusion and its derivative process, state our main convergence result and describe a numerical algorithm for estimating sensitivities.…”

“…Remark 2.2. In [15, Definition 2.1] the authors define a family of reflected diffusions in which the drift and dispersion coefficients and directions of reflection are parameterized by α ∈ U , but the initial condition is parameterized by x ∈ G. In [15], this allowed for a characterization of pathwise derivatives of flows of reflected diffusions and was a convenient representation in the proofs. In contrast, here we will find it more convenient to assume that the initial condition is a continuously differentiable function x 0 (·) on U taking values in G.…”

“…The equation 2.9 for the derivative process can be viewed as a linearized version of the equation (2.2) for the reflected diffusion Z α , with the key feature, established in [15], that R (α)L α + K α serves as the appropriate linearization of the constraining process R(α)L α . [e m , x 0 (α)e m ] for all t ≥ 0 (where the right-hand side of the equality is written in the notation of [15]).…”

“…The main contribution of this work is to develop a broadly applicable asymptotically unbiased estimator of a large class of sensitivities of reflected diffusions, which can be used to approximate sensitivities of reflected diffusions via a Monte Carlo method. Our estimator is based on a probabilistic representation for sensitivities of reflected diffusions that was obtained in [15,Corollary 3.15]. This representation expresses the sensitivity as the expectation of a functional of a reflected diffusion and its associated derivative process.…”

“…This representation expresses the sensitivity as the expectation of a functional of a reflected diffusion and its associated derivative process. The derivative process satisfies a constrained linear stochastic differential equations with jumps whose coefficients, domain and directions of reflection are modulated by the reflected diffusion, and it was shown in [15,Theorem 3.13] that the pathwise derivatives of a reflected diffusion can be described via the derivative process. While this representation provides an unbiased estimator for sensitivities of a reflected diffusion, computation of sensitivities via this representation would entail simulation of the reflected diffusion and its associated derivative process, which typically involves discrete-time approximations.…”