## Abstract

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 \({\mathbb{R}}^{n}\) according to a constraint vector field *D*(·) specified on the boundary \(\partial G\) of the domain. Specifically, we consider equations of the form \(\phi = \theta + \eta + u , \quad \dot{\theta}(t) \in F(\phi(t)), \quad \mbox{a.e. } t\) for *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 set-valued upper semicontinuous envelope of *f*. The case \(G ={\mathbb{R}}^{n}\) (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 \(G=\cup_i\overline{C_i}\) , where \(\{C_i,i\in {\mathbb{I}}\}\) is a finite collection of disjoint, open, convex, polyhedral cones in \({\mathbb{R}}^{n}\) , 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 \(G = {\mathbb{R}}^{n}.\) The work has applications in the study of scaling limits of stochastic networks.

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## References

- 1.
Alanyali M. and Hajek B. (1998). On large deviations of Markov processes with discontinuous statistics.

*Ann. Appl. Probab.*8(1): 45–66 - 2.
Anderson R. and Orey S. (1976). Small random perturbations of dynamical systems with reflecting boundary.

*Nagoya Math. J.*60: 189–216 - 3.
Aubin J.P. and Cellina A. (1984). Differential Inclusions. Springer, Berlin

- 4.
Bernard A. and El Kharroubi A. (1991). Regulation de processus dans le premier orthant de

*R*^{n}.*Stochastics*34: 149–167 - 5.
Cepa E. (1998). Problème de Skorokhod multivoque.

*Ann. Probab.*26: 500–532 - 6.
Costantini C. (1992). The Skorokhod oblique reflection problem in domains with corners and application to stochastic differential equations.

*Probab. Theor. Rel. Fields*91: 43–70 - 7.
Dai J.G. and Williams R.J. (1995). Existence and uniqueness of semimartingale reflecting Brownian motion in convex polyhedrons.

*Theor. Probab. Appl.*40: 1–40 - 8.
Dupuis P. and Ellis R.S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York

- 9.
Dupuis P. and Ishii H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications.

*Stochastics*35: 31–62 - 10.
Dupuis P. and Ishii H. (1993). SDEs with oblique reflection on nonsmooth domains.

*Ann. Probab.*21(1): 554–580 - 11.
Dupuis P. and Ramanan K. (1999). Convex duality and the Skorokhod problem I.

*Probab. Theor. Rel. Fields*115: 153–195 - 12.
Dupuis P. and Ramanan K. (1999). Convex duality and the Skorokhod problem II.

*Probab. Theor. Rel. Fields*115: 197–236 - 13.
Dupuis P. and Ramanan K. (2000). A multiclass feeback queueing network with a regular Skorokhod problem.

*Queueing Syst.*36: 327–349 - 14.
Ethier S.N. and Kurtz T.G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York

- 15.
Filippov A.F. (1988). Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht

- 16.
Harrison J.M. and Reiman M. (1981). Reflected Brownian motion on an orthant.

*Ann. Probab.*9: 302–308 - 17.
Karatzas I. and Shreve S. (2000). Brownian Motion and Stochastic Calculus. Springer, Heidelberg

- 18.
Korostelev A.P. and Leonov S.L. (1993). Action functional for diffusions in discontinuous media.

*Probab. Theor. Rel. Fields*94(3): 317–333 - 19.
Lions P.-L. and Sznitman A.-S. (1984). Stochastic differential equations with reflecting boundary conditions.

*Commun. Pure Appl. Math.*37(4): 511–537 - 20.
Ramanan K. (2006). Reflected diffusions defined via the extended Skorokhod map.

*Elec. J. Probab.*11: 934–992 - 21.
Skorokhod A.V. (1961). Stochastic equations for diffusions in a bounded region.

*Theor. Probab. Appl.*6: 264–274 - 22.
Tanaka H. (1979). Stochastic differential equations with reflecting boundary conditions.

*Hiroshima Math. J.*9: 163–177 - 23.
Veretennikov A.Ju. (1980). Strong solutions and explicit formulas for solutions of stochastic integral equations.

*Math. Sb. (N.S.)*111(153)(3): 434–452 (480) - 24.
Veretennikov A.Ju. (1983). Stochastic equations with diffusion that degenerates with respect to part of the variables.

*Izv. Akad. Nauk SSSR Ser. Mat.*47(1): 189–196

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## Additional information

R. Atar was partially supported by the Israel Science Foundation (grant 126/02), the NSF (grant DMS-0600206), and the fund for promotion of research at the Technion.

A. Budhiraja was partially supported by the ARO (grants W911NF-04-1-0230,W911NF-0-1-0080).

K. Ramanan was partially Supported by the NSF (grants DMS-0406191, DMI-0323668-0000000965, DMS-0405343).

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Atar, R., Budhiraja, A. & Ramanan, K. Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity.
*Probab. Theory Relat. Fields* **142, **249–283 (2008). https://doi.org/10.1007/s00440-007-0104-z

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### Keywords

- Discontinous drift
- Ordinary differential equations
- Differential inclusions
- Stochastic differential equations
- Stochastic differential inclusions
- Reflected diffusions
- Skorokhod map
- Skorokhod problem

### Mathematics Subject Classifications (2000)

- Primary: 34A60
- 60H10
- Secondary: 60J60
- 34F05
- 34A36