The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Costantini, Cristina

doi:10.1007/bf01194489

Cited by 39 publications

(64 citation statements)

References 19 publications

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“…B. Constrained ODE on G R n Consider now the case where F (x) = {f (x)} for every x ∈ G. When f is the constant function, this coincides with the SP, which has been studied extensively in past works [2,5,7,10,12,18,21,24]. Thus, below we only summarise results when f has non-trivial state-dependence.…”

Section: Summary Of Some Prior Resultsmentioning

confidence: 99%

“…It was shown in Theorem 1.3 of [22] that the SM has a closed graph if there is no x ∈ ∂G for which there exists a unit vector d with {d, −d} ⊂ D(x). Under this condition, the paper [7] shows that the SM preserves relative compactness (and thus Assumption 2 holds) when certain oscillation estimates (see equations (2.17) and (2.18) therein) are satisfied. Such estimates are available for broad families of SPs.…”

Section: Assumption 2 Consider the Sp Associated With The Data (G Dmentioning

confidence: 99%

“…, n}, these oscillation estimates were shown in [5] to hold if the constraint matrix (the n × n matrix with d i as its ith column) is completely-S (and in this case Assumption 1 holds as well). Other settings where Assumptions 1 and 2 hold can be found in [7,8].…”

Section: Assumption 2 Consider the Sp Associated With The Data (G Dmentioning

confidence: 99%

“…Assumption 1 reduces to Assumption 3.1 of [10] in the setting of that paper (also, in a more general setting, see Definition 4.1 of [7]). As shown in those papers, this assumption is necessary for existence of solutions to the corresponding SP on D[0, T ].…”

Section: Assumption 2 Consider the Sp Associated With The Data (G Dmentioning

confidence: 99%

“…We refer to this formulation as the discontinuous media problem (DMP). For the case where G is a proper subset of R n , a natural framework for constraining the dynamics to G is through the notion of the Skorokhod problem (SP) [7,10,21,23,24]. Roughly speaking, given a domain and a constraint vector field D on its boundary, the SP associates to any given trajectory ψ a version that is restricted to the domain.…”

Section: General Frameworkmentioning

confidence: 99%

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Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

Atar

Budhiraja²,

Ramanan

2008

Probab. Theory Relat. Fields

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We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in R n according to a constraint vector field D(·) specified on the boundary ∂G of the domain. Specifically, we consider equations of the formfor u in an appropriate class of functions, where η is the "constraining term" in the Skorokhod problem specified by (G, D) and F is the setvalued upper semicontinuous envelope of f . The case G = R n (when there is no constraining mechanism) and u is absolutely continuous corresponds to the well known setting of differential inclusions (DI). We provide a general sufficient condition for uniqueness of solutions and Lipschitz continuity of the solution map, in the form of existence of a Lyapunov set. Here we assume (i) G is convex and admits the representation G = ∪ i C i , where {C i , i ∈ I} is a finite collection of disjoint, open, convex, polyhedral cones in R n , each having its vertex at the origin; (ii) f = b + f c is a vector field defined on G such that b assumes a constant value on each of the given cones and f c is Lipschitz continuous on G; and (iii) D is an upper semicontinuous, cone-valued vector field that is constant on each face of ∂G. We also provide existence results under much weaker conditions (where no Lyapunov set condition is imposed). For stochastic differential equations (SDE) (possibly, reflected) that have drift coefficient f and a Lipschitz continuous (possibly degenerate) diffusion coefficient, we establish strong existence and pathwise uniqueness under appropriate conditions. Our approach yields new existence and uniqueness results for both DI and SDE even in the case G = R n . The work has applications in the study of scaling limits of stochastic networks.

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