Abstract
In this paper, we study a numerical approximation scheme for reflected stochastic differential equations (SDEs) with non-Lipschitzian coefficients in a bounded convex domain. It is shown, under some mild conditions, that the approximation scheme converges in uniform \({{L}}^2 \) to the solution of reflected SDEs. Moreover, we move from local to global monotonicity conditions and consider the rate of convergence for our approximation scheme to reflected SDEs with coefficients which have at most polynomial growth.
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Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. RGMIA Monographs. Victoria University, Footscray (2002)
Dupuis, P., Ishii, H.: SDEs with oblique reflection on nonsmooth domain. Ann. Probab. 21, 554–580 (1993)
Fang, S.Z., Zhang, T.S.: A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Relat. Fields 132, 356–390 (2005)
Gegout-Petit, A., Pardoux, E.: Equations différentielles stochastiques rétrogrades réfléchies dans un convexe. Stoch. Stoch. Rep. 57, 111–128 (1996)
Łaukaitys, W., Słomiński, L.: Penalization methods for the Skorokhod problem and reflecting SDEs with jumps. Bernoulli 19, 1750–1775 (2013)
Lan, G.Q., Wu, J.-L.: New sufficient conditions of existence, moment estimations and non confluence for SDEs with non-Lipschitzian coefficients. Stoch. Process. Appl. 124, 4030–4049 (2014)
Liu, Y.: Numerical approaches to stochastic differential equations with boundary conditions. Ph.D. thesis. Purdue University (1993)
Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary condition. Commun. Pure Appl. Math. 37, 511–537 (1987)
Menaldi, J.L.: Stochastic variational inequality for reflected diffusion. Indiana Univ. Math. J. 32, 733–744 (1983)
McKean, H.P.: Skorohod’s integral equation for a reflecting barrier diffusion. J. Math. Kyôto Univ. 3, 86–88 (1963)
McKean, H.P.: Stochastic Integrals. Academic Press, New York (1969)
Pettersson, R.: Penalization schemes for reflecting stochastic differential equations. Bernoulli 3, 403–414 (1997)
Prévöt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, 1905, Springer, Berlin (2007)
Sabanis, S.: Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26, 2083–2105 (2016)
Saisho, Y.: Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Relat. Fields 74, 455–477 (1987)
Skorohod, A.V.: Stochastic equations for diffusion process in a bound region. Theory Probab. Appl. 6, 264–274 (1961)
Słomiński, L.: Euler’s apprximations of solutions of SDEs with reflecting boundary. Stoch. Process. Appl. 94, 317–337 (2001)
Słomiński, L.: Weak and strong approximations of reflected diffusions via penalization methods. Stoch. Process. Appl. 123, 752–763 (2013)
Słomiński, L.: On existence, uniqueness and stability of solutions of multidimensional SDE’s with reflecting boundary conditions. Ann. Inst. H. Poincaré. 29, 163–198 (1993)
Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163–177 (1979)
Acknowledgements
The authors would like to thank the anonymous referee for careful reading and valuable comments that led to improvement of this work.
This work was partially supported by NSF of China (No.11871476) and NSF of Hunan Province (No. 2019JJ40356).
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Duan, J., Peng, J. An Approximation Scheme for Reflected Stochastic Differential Equations with Non-Lipschitzian Coefficients. J Theor Probab 35, 575–602 (2022). https://doi.org/10.1007/s10959-020-01052-7
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DOI: https://doi.org/10.1007/s10959-020-01052-7
Keywords
- Stochastic differential equations
- Reflecting boundary
- Non-Lipschitzian coefficients
- Penalization Euler scheme