Electronic Journal of Probability

Existence and uniqueness of reflecting diffusions in cusps

Cristina Costantini and Thomas G. Kurtz

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We consider stochastic differential equations with (oblique) reflection in a 2-dimensional domain that has a cusp at the origin, i.e. in a neighborhood of the origin has the form $\{(x_1,x_2):0<x_1\leq \delta _0,\psi _1(x_1)<x_2<\psi _ 2(x_1)\}$, with $\psi _1(0)=\psi _2(0)=0$, $\psi _1'(0)=\psi _2'(0)=0$.

Given a vector field $g$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $g^i(0):=\lim _{x_1\rightarrow 0^{+}}g (x_1,\psi _i(x_1))$, $ i=1,2$, and assuming there exists a vector $e^{*}$ such that $\langle e^{*},g^i(0)\rangle >0$, $i=1,2$, and $e^{*}_1>0$, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin.

Our proof uses a new scaling result and a coupling argument.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 84, 21 pp.

Received: 17 November 2017
Accepted: 24 July 2018
First available in Project Euclid: 12 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J55: Local time and additive functionals 60G17: Sample path properties

oblique reflection stochastic differential equation diffusion process cusp boundary singularity

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Costantini, Cristina; Kurtz, Thomas G. Existence and uniqueness of reflecting diffusions in cusps. Electron. J. Probab. 23 (2018), paper no. 84, 21 pp. doi:10.1214/18-EJP204. https://projecteuclid.org/euclid.ejp/1536717743

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  • [1] Burdzy, K., Kang, W. and Ramanan, K.: The Skorokhod problem in a time-dependent interval. Stochastic Process. Appl. 119 (2009), no. 2, 428–452.
  • [2] Burdzy, K. and Toby, E.H.: A Skorohod-type lemma and a decomposition of reflected Brownian motion. Ann. Probab. 23 (1995), no. 2, 586–604.
  • [3] Costantini, C.: The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations. Probab. Theory Related Fields 91 (1992), no. 1, 43–70.
  • [4] Costantini, C. and Kurtz, T.G.: Viscosity methods giving uniqueness for martingale problems. Electron. J. Probab. 20 (2015), no. 67, 27.
  • [5] Crandall, M.G., Ishii, H. and Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67.
  • [6] Dai, J.G. and Williams, R.J.: Existence and uniqueness of semimartingale reflecting brownian motions in convex polyhedrons. Theory of Probability & Its Applications 40 (1996), no. 1, 1–40.
  • [7] DeBlassie, R.D. and Toby, E.H.: On the semimartingale representation of reflecting Brownian motion in a cusp. Probab. Theory Related Fields 94 (1993), no. 4, 505–524.
  • [8] DeBlassie, R.D. and Toby, E.H.: Reflecting Brownian motion in a cusp. Trans. Amer. Math. Soc. 339 (1993), no. 1, 297–321.
  • [9] Dupuis, P. and Ishii, H.: SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993), no. 1, 554–580.
  • [10] Fukushima, M. and Tomisaki, M.: Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps. Probab. Theory Related Fields 106 (1996), no. 4, 521–557.
  • [11] Kang, W.N., Kelly, F.P., Lee, N.H. and Williams, R.J.: State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 19 (2009), no. 5, 1719–1780.
  • [12] Kang, W.N. and Williams, R.J.: Diffusion approximation for an input-queued switch operating under a maximum weight matching policy. Stoch. Syst. 2 (2012), no. 2, 277–321.
  • [13] Kurtz, T.G.: Martingale problems for constrained Markov problems. Recent advances in stochastic calculus (College Park, MD, 1987), Progr. Automat. Info. Systems, Springer, New York, 1990, pp. 151–168.
  • [14] Kurtz, T.G.: Weak and strong solutions of general stochastic models. Electron. Commun. Probab. 19 (2014), no. 58, 16.
  • [15] Kurtz, T.G. and Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (1991), no. 3, 1035–1070.
  • [16] Lindvall, T. and Rogers, L.C.G.: Coupling of multidimensional diffusions by reflection. Ann. Probab. 14 (1986), no. 3, 860–872.
  • [17] Taylor, L.M. and Williams, R.J.: Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96 (1993), no. 3, 283–317.
  • [18] Varadhan, S.R.S. and Williams, R.J.: Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985), no. 4, 405–443.
  • [19] Yamada, T. and Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 (1971), 155–167.