Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pathwise differentiability of reflected diffusions in convex polyhedral domains

David Lipshutz and Kavita Ramanan

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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. Previous work in the multidimensional context has been largely restricted to the study of differentiability of stochastic flows for (normally) reflected Brownian motions. A key difficulty is to identify a suitable linearization of the dynamics of the local time process, especially in the presence of a non-smooth boundary. We take a new approach that uses properties of directional derivatives of the associated extended Skorokhod map, and their characterization in terms of the so-called derivative problem. The proof involves establishing certain path properties of the reflected diffusion at nonsmooth parts of the boundary of the polyhedral domain, which may be independent interest, and proving that pathwise derivatives of reflected diffusions can be characterized in terms of directional derivatives of the extended Skorokhod map. 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.

Résumé

Les diffusions réfléchies dans les domaines polyhédriques convexes apparaissent dans diverses applications, notamment les systèmes de particules, les réseaux de files d’attente, les réseaux de réaction biochimique et la finance mathématique. Dans des conditions appropriées sur les données, nous établissons la différentiabilité selon la trajectoire d’une telle diffusion réfléchie par rapport à ses paramètres de définition, tel que sa condition initiale, ses coefficients de dérive et de diffusion et ses directions (obliques) de réflexion le long des limites du domaine. Nous caractérisons la régularisation continue droite d’une dérivée selon la trajectorie de la diffusion réfléchie comme solution unique à une équation différentielle stochastique linéaire contrainte avec sauts dont les coefficients de dérive et de diffusion, le domaine et les directions de réflexion dépendent de l’état de la diffusion réfléchie. Les travaux antérieurs dans le contexte multidimensionnel ont été en grande partie limités à l’étude de la différentiabilité des flux stochastiques des mouvements browniens (normalement) réfléchis. Une difficulté essentielle consiste à identifier une linéarisation appropriée de la dynamique du processus de temps local, en particulier en présence d’une frontière non lisse. Nous adoptons une nouvelle approche qui utilise les propriétés des dérivées directionnelles de l’application de Skorokhod étendue associée et leur caractérisation en termes d’une problème dérivé. La preuve implique l’établissement de certaines propriétés des trajectoires de la diffusion réfléchie aux frontieres non lisses du domaine polyhédral, qui peut être un resultat d’intérêt indépendant, et de démontrer que les dérivées selon la trajectorie des diffusions réfléchies peuvent être caractérisées en termes de dérivées directionnelles de l’application de Skorokhod étendu. En corollaire, nous obtenons une représentation probabiliste pour les dérivées des valeurs d’attendues des fonctionnelles de diffusions réfléchies, ce qui est utile pour l’analyse de sensibilité des diffusions réfléchies.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1439-1476.

Dates
Received: 4 October 2017
Revised: 22 July 2018
Accepted: 20 August 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398875

Digital Object Identifier
doi:10.1214/18-AIHP924

Mathematical Reviews number (MathSciNet)
MR4010941

Subjects
Primary: 60G17: Sample path properties 90C31: Sensitivity, stability, parametric optimization 93B35: Sensitivity (robustness)
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

Keywords
Reflected diffusion Reflected Brownian motion Boundary jitter property Derivative process Pathwise differentiability Stochastic flow Sensitivity analysis Directional derivative of the extended Skorokhod map Derivative problem

Citation

Lipshutz, David; Ramanan, Kavita. Pathwise differentiability of reflected diffusions in convex polyhedral domains. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1439--1476. doi:10.1214/18-AIHP924. https://projecteuclid.org/euclid.aihp/1569398875


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