Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

Atar, Rami; Budhiraja, Amarjit; Ramanan, Kavita

doi:10.1007/s00440-007-0104-z

Cited by 8 publications

(4 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, applied areas where Skorohod problems occur include heavy traffic analysis of queueing networks (see, e.g, [2,20,24,40,47,48,51,52]), control theory, game theory and mathematical economics (see, e.g., [3,37,49,50,58]), image processing (see, e.g., [8]) and molecular dynamics (see, e.g., [54][55][56]). For further results concerning Skorohod problems, as well as applications of Skorohod problems, we also refer to [1,4,5,14,23,25,29,31,35,38,44,46] and [60]. An important novelty of this article is that we conduct a thorough study of the Skorohod problem, and the subsequent applications to stochastic differential equations reflected at the boundary, in the setting of time-dependent domains.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Skorohod oblique reflection problem in time-dependent domains

Nyström¹,

Önskog²

2010

Ann. Probab.

View full text Add to dashboard Cite

The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka [61] and further investigated by P.-L. Lions and A.-S. Sznitman [42] in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of c\`{a}dl\`{a}g solutions $(x,\lambda)$ to the Skorohod problem, with oblique reflection, for $(D,\Gamma,w)$ assuming, in particular, that $D$ is a time-dependent domain (Theorem 1.2). In addition, we prove that if $w$ is continuous, then $x$ is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.Comment: Published in at http://dx.doi.org/10.1214/10-AOP538 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Skorohod oblique reflection problem in time-dependent domains

Nyström¹,

Önskog²

2010

Ann. Probab.

View full text Add to dashboard Cite

show abstract

“…In [42,Theorem 3.3], it was shown that Assumption 2.8 is a sufficient condition for Lipschitz continuity of the ESM as well. An analogue of Assumption 2.8 also serves as a sufficient condition for Lipschitz continuity of the map associated with the so-called constrained discontinuous media problem (see [3,Theorem 2.9]). A dual condition on the data {(d i , n i , c i ), i ∈ I} that implies the existence of a set B that satisfies Assumption 2.8 was introduced in [22,23].…”

Section: Lipschitz Continuitymentioning

confidence: 99%

On directional derivatives of Skorokhod maps in convex polyhedral domains

Lipshutz¹,

Ramanan²

2018

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging because the dynamics are discontinuous at the boundary of the domain and the boundary of the domain is not smooth. These difficulties can be addressed by studying directional derivatives of an associated extended Skorokhod map, which is a deterministic mapping that takes an unconstrained path to a suitably reflected version. In this work we develop an axiomatic framework for the analysis of directional derivatives of a large class of Lipschitz continuous extended Skorokhod maps in convex polyhedral domains with oblique directions of reflection. We establish existence of directional derivatives at a path whose reflected version satisfies a certain boundary jitter property, and also show that the right-continuous regularization of such a directional derivative can be characterized as the unique solution to a Skorokhod-type problem, where both the domain and directions of reflection vary (discontinuously) with time. A key ingredient in the proof is establishing certain contraction properties for a family of (oblique) derivative projection operators. As an application, we establish pathwise differentiability of reflected Brownian motion in the nonnegative quadrant with respect to the initial condition, drift vector, dispersion matrix and directions of reflection. The results of this paper are also used in subsequent work to establish pathwise differentiability of a much larger class of reflected diffusions in convex polyhedral domains.

show abstract

“…The results in [24] can be directly extended to càdlàg functions to support the above definition (see, for example, Theorem 2.1 in [15]). Generalizations of the ORM that are useful for more general networks are considered in [2,14,15,16,42] and references therein. Explicit formulas for the solution map associated with certain one-dimensional problems can be found in [28] and [46].…”

Section: Notation and Technical Paraphernaliamentioning

confidence: 99%

Asymptotically Optimal Controls for Time-Inhomogeneous Networks

Čudina¹,

Ramanan²

2011

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

A framework is introduced for the identification of controls for single-class timevarying queueing networks that are asymptotically optimal in the so-called uniform acceleration regime. A related, but simpler, first-order (or fluid) control problem is first formulated. For a class of performance measures that satisfy a certain continuity property, it is then shown that any sequence of policies whose performances approach the infimum in the fluid control problem is asymptotically optimal for the original network problem. Examples of performance measures with this property are described, and simulations implementing asymptotically optimal policies are presented. The use of directional derivatives of the reflection map for solving fluid control problems is also illustrated. This work serves to complement a large body of literature on asymptotically optimal controls for time-homogeneous networks.

show abstract

Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

Cited by 8 publications

References 24 publications

The Skorohod oblique reflection problem in time-dependent domains

The Skorohod oblique reflection problem in time-dependent domains

On directional derivatives of Skorokhod maps in convex polyhedral domains

Asymptotically Optimal Controls for Time-Inhomogeneous Networks

Contact Info

Product

Resources

About