## Electronic Journal of Probability

### On Wong-Zakai type approximations of reflected diffusions

Leszek Slominski

#### Abstract

We study weak and strong convergence of Wong-Zakai type approximations  of reflected stochastic differential equations on general domains satisfying the conditions (A) and (B)introduced by Lions and Sznitman. We assume that the diffusion coefficient is Lipschitz continuous but the drift coefficient need not be even continuous. In the case  where the  drift coefficient is also Lipschitz continuous we show that the rate of convergence is exactly the same as for usual Euler type approximation.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 118, 15 pp.

Dates
Accepted: 19 December 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065760

Digital Object Identifier
doi:10.1214/EJP.v19-3425

Mathematical Reviews number (MathSciNet)
MR3296534

Zentralblatt MATH identifier
1325.60097

Subjects
Primary: 60H20: Stochastic integral equations
Secondary: 60G17: Sample path properties

Rights

#### Citation

Slominski, Leszek. On Wong-Zakai type approximations of reflected diffusions. Electron. J. Probab. 19 (2014), paper no. 118, 15 pp. doi:10.1214/EJP.v19-3425. https://projecteuclid.org/euclid.ejp/1465065760

#### References

• Aida, Shigeki; Sasaki, Kosuke. Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces. Stochastic Process. Appl. 123 (2013), no. 10, 3800–3827.
• Aldous, David. Stopping times and tightness. Ann. Probability 6 (1978), no. 2, 335–340.
• Doss, Halim; Priouret, Pierre. Support d'un processus de réflexion. (French) [Support of a reflection process] Z. Wahrsch. Verw. Gebiete 61 (1982), no. 3, 327–345.
• Evans, Lawrence Christopher; Stroock, Daniel W. An approximation scheme for reflected stochastic differential equations. Stochastic Process. Appl. 121 (2011), no. 7, 1464–1491.
• Fischer, Markus; Nappo, Giovanna. On the moments of the modulus of continuity of Itô processes. Stoch. Anal. Appl. 28 (2010), no. 1, 103–122.
• Gyongy, Istvan; Krylov, Nicolai. Existence of strong solutions for Itô's stochastic equations via approximations. Probab. Theory Related Fields 105 (1996), no. 2, 143–158.
• Jakubowski, A.; Mémin, J.; Pagès, G. Convergence en loi des suites d'intégrales stochastiques sur l'espace ${\bf D}^ 1$ de Skorokhod. (French) [Convergence in law of sequences of stochastic integrals on the Skorokhod space ${\bf D}^ 1$] Probab. Theory Related Fields 81 (1989), no. 1, 111–137.
• Kohatsu-Higa, Arturo. Stratonovich type SDE's with normal reflection driven by semimartingales. SankhyÄ Ser. A 63 (2001), no. 2, 194–228.
• Kurtz, Thomas G.; Protter, Philip. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (1991), no. 3, 1035–1070.
• Lions, P.-L.; Sznitman, A.-S. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984), no. 4, 511–537.
• Mackevicius, Vigirdas. $S^ p$ stability of solutions of symmetric stochastic differential equations with discontinuous driving semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 4, 575–592.
• Melnikov, A. V. Stochastic equations and Krylov's estimates for semimartingales. Stochastics 10 (1983), no. 2, 81–102.
• Pettersson, Roger. Wong-Zakai approximations for reflecting stochastic differential equations. Stochastic Anal. Appl. 17 (1999), no. 4, 609–617.
• Pratelli, Maurizio. Majorations dans $L^ p$ du type Métivier-Pellaumail pour les semimartingales. (French) [Upper bounds in $L^ p$ of Metivier-Pellaumail type for semimartingales] Seminar on probability, XVII, 125–131, Lecture Notes in Math., 986, Springer, Berlin, 1983.
• Ren, Jiagang; Xu, Siyan. A transfer principle for multivalued stochastic differential equations. J. Funct. Anal. 256 (2009), no. 9, 2780–2814.
• Ren, Jiagang; Xu, Siyan. Support theorem for stochastic variational inequalities. Bull. Sci. Math. 134 (2010), no. 8, 826–856.
• Rozkosz, Andrzej; Słomiński, Leszek. On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients. Stochastic Process. Appl. 37 (1991), no. 2, 187–197.
• Rozkosz, Andrzej; Słomiński, Leszek. On stability and existence of solutions of SDEs with reflection at the boundary. Stochastic Process. Appl. 68 (1997), no. 2, 285–302.
• Saisho, Yasumasa. Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Related Fields 74 (1987), no. 3, 455–477.
• Seau, Alina. Discrete approximations of strong solutions of reflecting SDEs with discontinuous coefficients. Bull. Pol. Acad. Sci. Math. 57 (2009), no. 2, 169–180.
• Słomiński, Leszek. On existence, uniqueness and stability of solutions of multidimensional SDEs with reflecting boundary conditions. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), no. 2, 163–198.
• Słomiński, Leszek. On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stochastic Process. Appl. 50 (1994), no. 2, 197–219.
• Słomiński, Leszek. Euler's approximations of solutions of SDEs with reflecting boundary. Stochastic Process. Appl. 94 (2001), no. 2, 317–337.
• Słomiński, L.: On reflected Stratonovich stochastic differential equations. Stochastic Process. Appl., 125, (2015), 759–779. http://dx.doi.org/10.1016/j.spa.2014.10.003
• Stroock, Daniel W.; Varadhan, S. R. S. Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 1971 147–225.
• Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163–177.
• Wong, Eugene; Zakai, Moshe. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 1965 1560–1564.
• Wong, Eugene; Zakai, Moshe. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 1965 213–229.
• Zhang, Tu Sheng. On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary. Stochastic Process. Appl. 50 (1994), no. 1, 135–147.
• Zhang, Tusheng. Strong convergence of Wong-Zakai approximations of reflected SDEs in a multidimensional general domain. Potential Anal. 41 (2014), no. 3, 783–815.