Pathwise Differentiability for SDEs in a Smooth Domain with Reflection

Sebastian Andres (University of British Columbia)

Abstract


In this paper we study a Skorohod SDE in a smooth domain with normal reflection at the boundary, in particular we prove that the solution is pathwise differentiable with respect to the deterministic starting point. The resulting derivatives evolve according to an ordinary differential equation, when the process is in the interior of the domain, and they are projected to the tangent space, when the process hits the boundary.

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Pages: 845-879

Publication Date: April 22, 2011

DOI: 10.1214/EJP.v16-872

References

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  1. Airault, HÈlËne. Perturbations singuliËres et solutions stochastiques de problËmes de D. Neumann-Spencer. (French) J. Math. Pures Appl. (9) 55 (1976), no. 3, 233--267. MR0501184 (58 #18606)

  1. Anderson, Robert F.; Orey, Steven. Small random perturbation of dynamical systems with reflecting boundary. Nagoya Math. J. 60 (1976), 189--216. MR0397893 (53 #1749)

  1. Andres, Sebastian. Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection. Ann. Inst. Henri PoincarÈ Probab. Stat. 45 (2009), no. 1, 104--116. MR2500230 (2010e:60118)

  1. Burdzy, Krzysztof. Differentiability of stochastic flow of reflected Brownian motions. Electron. J. Probab. 14 (2009), no. 75, 2182--2240. MR2550297 (2011b:60321)

  1. Burdzy, Krzysztof; Chen, Zhen-Qing. Coalescence of synchronous couplings. Probab. Theory Related Fields 123 (2002), no. 4, 553--578. MR1921013 (2003e:60184)

  1. Burdzy, Krzysztof; Chen, Zhen-Qing; Jones, Peter. Synchronous couplings of reflected Brownian motions in smooth domains. Illinois J. Math. 50 (2006), no. 1-4, 189--268 (electronic). MR2247829 (2007i:60102)

  2. Burdzy, Krzysztof; Lee, John M. Multiplicative functional for reflected Brownian motion via deterministic ODE. Preprint, Math. Review number not available.

  1. Cranston, M.; Le Jan, Y. On the noncoalescence of a two point Brownian motion reflecting on a circle. Ann. Inst. H. PoincarÈ Probab. Statist. 25 (1989), no. 2, 99--107. MR1001020 (90k:60154)

  1. Cranston, M.; Le Jan, Y. Noncoalescence for the Skorohod equation in a convex domain of R2. Probab. Theory Related Fields 87 (1990), no. 2, 241--252. MR1080491 (92e:60116)

  1. Denisov, I.-V.  A random walk and a Wiener process near a maximum. Theor. Prob. Appl. 28 (1984), 821--824.  Math. Review number not available.

  1. Deuschel, Jean-Dominique; Zambotti, Lorenzo. Bismut-Elworthy's formula and random walk representation for SDEs with reflection. Stochastic Process. Appl. 115 (2005), no. 6, 907--925. MR2134484 (2006e:60080)

  1. Hsu, Elton P. Multiplicative functional for the heat equation on manifolds with boundary. Michigan Math. J. 50 (2002), no. 2, 351--367. MR1914069 (2003f:58067)

  1. Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. xiv+464 pp. ISBN: 0-444-86172-6 MR0637061 (84b:60080)

  1. Imhof, J.-P. Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21 (1984), no. 3, 500--510. MR0752015 (85j:60152)

  1. ItÙ, Kiyosi; McKean, Henry P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp. MR0345224 (49 #9963)

  1. Jacod, Jean; MÈmin, Jean. Weak and strong solutions of stochastic differential equations: existence and stability. Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), pp. 169--212, Lecture Notes in Math., 851, Springer, Berlin-New York, 1981. MR0620991 (83h:60062)

  1. Kunita, Hiroshi. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990. xiv+346 pp. ISBN: 0-521-35050-6 MR1070361 (91m:60107)

  1. Lions, P.-L.; Sznitman, A.-S. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984), no. 4, 511--537. MR0745330 (85m:60105)

  1. Pilipenko, A. Yu. Stochastic flows with reflection. Preprint, available on arXiv:0810.4644. Math. Review number not available.

  1. Pilipenko, A. Yu. Properties of flows generated by stochastic equations with reflection. (Russian) UkraÔn. Mat. Zh. 57 (2005), no. 8, 1069--1078; translation in Ukrainian Math. J. 57 (2005), no. 8, 1262--1274 MR2218469 (2007g:60066)

  1. Pilipenko, A. Yu. Stochastic flows with reflection. (Russian) Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2005, no. 10, 23--28. MR2218498 (2007j:60089)

  1. Pilipenko, A. Yu. On the generalized differentiability with initial data of a flow generated by a stochastic equation with reflection. (Ukrainian) Teor. ?movīr. Mat. Stat. No. 75 (2006), 127--139; translation in Theory Probab. Math. Statist. No. 75 (2007), 147--160 MR2321188 (2008e:60179)

  1. Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357 (2000h:60050)

  1. Sheu, Shey Shiung. Noncoalescence of Brownian motion reflecting on a sphere. Stochastic Anal. Appl. 19 (2001), no. 4, 545--554. MR1841943 (2002f:60161)

  1. Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163--177. MR0529332 (80k:60075)



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