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

Lipshutz, David; Ramanan, Kavita

doi:10.1287/stsy.2019.0031

Cited by 3 publications

(11 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The representation (3.17) suggests pathwise methods for estimating Θ ′ (α, x), which we develop in subsequent work [21]. Pathwise estimators (also referred to as infinitesimal perturbation analysis estimators) are usually preferable when available (see, e.g., the discussion at the end of [14,Chapter 7]).…”

Section: Resultsmentioning

confidence: 99%

Pathwise differentiability of reflected diffusions in convex polyhedral domains

Lipshutz¹,

Ramanan²

2019

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Resultsmentioning

confidence: 99%

Pathwise differentiability of reflected diffusions in convex polyhedral domains

Lipshutz¹,

Ramanan²

2019

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

show abstract

“…See the appendix of [24] for an explicit expression for π α in the case Assumptions 2.2, 2.4 and 2.5 hold.…”

Section: 1mentioning

confidence: 99%

“…The boundary jitter property plays a crucial role in characterizing directional derivatives of the SP (see [26,Theorem 3.11]). The stronger version 2' of condition 2 is used in [24] to prove a continuity property of the derivative map stated in Proposition 4.17 below. The latter is used in the next section to prove the joint process is Feller continuous.…”

Section: 1mentioning

confidence: 99%

“…This is potentially useful for developing efficient numerical algorithms for computing sensitivities of such functionals. For example, in related work [24], the joint process was shown to be useful for estimating sensitivities of functionals of RBMs on finite time intervals.…”

mentioning

confidence: 99%

“…The proofs of these results entail several novel arguments. The proof of the Feller Markov property relies on continuity properties of the so-called derivative map, a deterministic map on path space that is useful in the analysis of the derivative process, which were established in [24,26]. The proof of ergodicity of the joint process (under the stability condition on the RBM) entails a careful analysis of the derivative process, whose dynamics are unusual due to the fact that the process jumps at the singular set of times when the RBM hits the boundary of the cone.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Sensitivity analysis for the stationary distribution of reflected Brownian motion in a convex polyhedral cone

Lipshutz¹,

Ramanan²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

Reflected Brownian motion (RBM) in a convex polyhedral cone arises in a variety of applications ranging from the theory of stochastic networks to mathematical finance, and under general stability conditions, it has a unique stationary distribution. In such applications, to implement a stochastic optimization algorithm or quantify robustness of a model, it is useful to characterize the dependence of stationary performance measures on model parameters. In this work we characterize parametric sensitivities of the stationary distribution of an RBM in a simple convex polyhedral cone; that is, sensitivities to perturbations of the parameters that define the RBM -namely, the covariance matrix, drift vector and directions of reflection along the boundary of the polyhedral cone. In order to characterize these sensitivities we study the long time behavior of the joint process consisting of an RBM along with its so-called derivative process, which characterizes pathwise derivatives of RBMs on finite time intervals. We show that the joint process is positive recurrent, has a unique stationary distribution, and parametric sensitivities of the stationary distribution of an RBM can be expressed in terms of the stationary distribution of the joint process. This can be thought of as establishing an interchange of the differential operator and the limit in time. The analysis of ergodicity of the joint process is significantly more complicated than that of the RBM due to its degeneracy and the fact that the derivative process exhibits jumps that are modulated by the RBM. The proofs of our results rely on path properties of coupled RBMs and contraction properties related to the geometry of the polyhedral cone and directions of reflection along the boundary. Our results are potentially useful for developing efficient numerical algorithms for computing sensitivities of functionals of stationary RBMs.

show abstract

Sensitivity Analysis for the Stationary Distribution of Reflected Brownian Motion in a Convex Polyhedral Cone

Lipshutz

Ramanan

2021

Mathematics of OR

View full text Add to dashboard Cite

Reflected Brownian motion (RBM) in a convex polyhedral cone arises in a variety of applications ranging from the theory of stochastic networks to mathematical finance, and under general stability conditions, it has a unique stationary distribution. In such applications, to implement a stochastic optimization algorithm or quantify robustness of a model, it is useful to characterize the dependence of stationary performance measures on model parameters. In this paper, we characterize parametric sensitivities of the stationary distribution of an RBM in a simple convex polyhedral cone, that is, sensitivities to perturbations of the parameters that define the RBM—namely the covariance matrix, drift vector, and directions of reflection along the boundary of the polyhedral cone. In order to characterize these sensitivities, we study the long-time behavior of the joint process consisting of an RBM along with its so-called derivative process, which characterizes pathwise derivatives of RBMs on finite time intervals. We show that the joint process is positive recurrent and has a unique stationary distribution and that parametric sensitivities of the stationary distribution of an RBM can be expressed in terms of the stationary distribution of the joint process. This can be thought of as establishing an interchange of the differential operator and the limit in time. The analysis of ergodicity of the joint process is significantly more complicated than that of the RBM because of its degeneracy and the fact that the derivative process exhibits jumps that are modulated by the RBM. The proofs of our results rely on path properties of coupled RBMs and contraction properties related to the geometry of the polyhedral cone and directions of reflection along the boundary. Our results are potentially useful for developing efficient numerical algorithms for computing sensitivities of functionals of stationary RBMs.

show abstract

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

Cited by 3 publications

References 38 publications

Pathwise differentiability of reflected diffusions in convex polyhedral domains

Pathwise differentiability of reflected diffusions in convex polyhedral domains

Sensitivity analysis for the stationary distribution of reflected Brownian motion in a convex polyhedral cone

Sensitivity Analysis for the Stationary Distribution of Reflected Brownian Motion in a Convex Polyhedral Cone

Contact Info

Product

Resources

About