Skip to main content
SpringerLink
Log in

Penalty method for obliquely reflected diffusions

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider a multidimensional normally or obliquely reflected diffusion in a smooth domain. We approximate it by solutions of stochastic differential equations without reflection using the penalty method, that is, we approximate the reflection term with an additional drift term. In the existing literature, usually a specific approximating sequence is provided to prove the existence of a reflected diffusion. In this paper, we provide general sufficient conditions on the approximating coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S Aida and K. Sasaki, Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in euclidean spaces, Stochastic Processes Appl., 123(10):3800–3827, 2013.

  2. A.D. Banner, E.R. Fernholz, and I. Karatzas, Atlas models of equity markets, Ann. Appl. Probab., 15(4):2996–2330, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  3. R.F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab., 19(2):486–508, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, 1999.

    Book  MATH  Google Scholar 

  5. C. Bruggeman and A. Sarantsev, Penalty method for reflected diffusions on the half-line, Stochastics, 89(2):485–509, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Cepa, Problème de Skorohod multivoque, Ann. Probab., 26(2):500–532, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Constantini and T.G. Kurtz, Diffusion approximation for transport processes with general reflection boundary conditions, Math. Models Methods Appl. Sci., 16(5):717–762, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Costantini, The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations, Probab. Theory Relat. Fields, 91(1):43–70, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  9. J.G. Dai and R.J. Williams, Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra, Theory Probab. Appl., 40(1):3–53, 1995.

    MathSciNet  MATH  Google Scholar 

  10. P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics, 35(1):31–62, 1991.

    MathSciNet  MATH  Google Scholar 

  11. P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains, Ann. Probab., 21(1):554–580, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  12. L.C. Evans and D.W. Stroock, An approximation scheme for reflected stochastic differential equations, Stochastic Processes Appl., 121(7):1464–1491, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  13. G.B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, New York, 1999.

    MATH  Google Scholar 

  14. R.L. Foote, Regularity of the distance function, Proc. Am. Math. Soc., 92(1):153–155, 1984.

    MathSciNet  MATH  Google Scholar 

  15. M. Fradon, Stochastic differential equations on domains defined by multiple constraints, Electron. Commun. Probab., 18(26):1–13, 2013.

    MathSciNet  MATH  Google Scholar 

  16. M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps, Probab. Theory Relat. Fields, 106(4):521–557, 1996.

    Article  MATH  Google Scholar 

  17. J.M. Harrison and I.M. Reiman, Reflected Brownian motion on an orthant, Ann. Probab., 9(2):302–308, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.M. Harrison and R.J. Williams, Brownian models of open queueing networks with homogeneous customer populations, Stochastics, 22(2):77–115, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  19. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1989.

    MATH  Google Scholar 

  20. W. Kang and R.J. Williams, An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries, Ann. Appl. Probab., 17(2):741–779, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  21. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1991.

    MATH  Google Scholar 

  22. N. El Karoui, C. Kapoudjian, E. Pardoux, S. Ge Peng, and M.-C. Quenez, Reflected solutions of backward SDEs and related obstacle problems for PDEs, Ann. Probab., 25(2):702–737, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. Klimsiak, A. Rozkosz, and L. Slominski, Reflected BSDEs in time-dependent convex regions, Stochastic Processes Appl., 125(2):571–596, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  24. S.G. Krantz and H.R. Parks, Distance to Ck hypersurfaces, J. Differ. Equations, 40(1):116–120, 1981.

    Article  MATH  Google Scholar 

  25. P. Lakner, J. Reed, and F. Simatos, Scaling limit of a limit order book via the regenerative characterization of Lévy trees, Stoch. Syst., 7(2):342–373, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  26. W. Łaukajtys and L. Słomiński, Penalization methods for reflecting stochastic differential equations with jumps, Stochastics Stochastics Rep., 75(5):275–293, 2003.

  27. W. Łaukajtys and L. Słomiński, Penalization methods for the Skorokhod problem and reflecting SDEs with jumps, Bernoulli, 19(5A):1750–1775, 2013.

  28. P.-L. Lions, J.L. Menaldi, and A.-S. Sznitman, Construction de processus de diffusion réfléchis par pénalisation du domaine, C. R. Acad. Sci., Paris, Sér. I, Math, 292:559–562, 1981.

  29. P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Commun. Pure Appl. Math., 37(4):511–537, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Maticiuc, A. Rascanu, L. Slominski, and M. Topolevski, Càdlàg Skorokhod problem driven by a maximal monotone operator, J. Math. Anal. Appl., 429(2):1305–1346, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  31. H.P. McKean Jr., Skorohod’s stochastic integral equation for a reflecting barrier diffusion, J. Math. Kyoto Univ., 3:85–88, 1963.

    MathSciNet  MATH  Google Scholar 

  32. J.-L. Menaldi, Stochastic variational inequality for reflected diffusions, Indiana Univ. Math. J., 32(5):733–744, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  33. J.-L. Menaldi, Stochastic differential equations with jumps, 2014, http://www.clas.wayne.edu/Multimedia/menaldi/Files/m-book-06-2013.pdf.

    Google Scholar 

  34. J.-L. Menaldi and M. Robin, Construction and control of reflected diffusion with jumps, in Stochastic Differential Systems Filtering and Control, Springer, Berlin, Heidelberg, 1985, pp. 309–322.

    Google Scholar 

  35. J.-L. Menaldi and M. Robin, Reflected diffusion processes with jumps, Ann. Probab., 13(2):319–341, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  36. N. O’Connell and J. Ortmann, Product-form invariant measures for Brownian motion with drift satisfying a skewsymmetry condition, ALEA, Lat. Am. J. Probab. Math. Stat., 11(1):307–329, 2014.

    MathSciNet  MATH  Google Scholar 

  37. E. Pardoux and R.J.Williams, Symmetric reflected diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., 30(1):13–62, 1994.

  38. R. Petterson, Approximations for stochastic differential equation with reflecting convex boundaries, Stochastic Processes Appl., 59(2):295–308, 1995.

    Article  MathSciNet  Google Scholar 

  39. R. Petterson, Penalization schemes for reflecting stochastic differential equations, Bernoulli, 3(4):403–414, 1997.

    Article  MathSciNet  Google Scholar 

  40. R. Petterson, Wong–Zakai approximations for reflecting stochastic differential equations, Stochastic Anal. Appl., 17(4):609–617, 2007.

    Article  MathSciNet  Google Scholar 

  41. A. Pilipenko, Properties of the flows generated by a stochastic equation with reflection, Ukr. Math. J., 57(8):1262–1274, 2005.

    Article  MathSciNet  Google Scholar 

  42. A. Pilipenko, An Introduction to Stochastic Differential Equations with Reflection, Potsdam Univ. Press, Potsdam, 2014.

    MATH  Google Scholar 

  43. N. Portenko, Diffusion processes with generalized drift coefficients, Theory Probab. Appl., 24(1):62–78, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  44. N. Portenko, Stochastic differential equations with generalized drift vector, Theory Probab. Appl., 24(2):338–353, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Rozkosz and L. Słomiński, Lp solutions of reflected BSDEs under monotonicity condition, Stochastic Processes Appl., 122(12):3875–3900, 2012.

  46. Y. Saisho, Stochastic differential equations for multidimensional domain with reflecting boundary, Probab. Theory Relat. Fields, 74(3):455–477, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Sarantsev, On a class of diverse market models, Ann. Finance, 10(2):291–314, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Sarantsev, Triple and simultaneous collisions of competing Brownian particles, Electron. J. Probab., 20(29):1–28, 2015.

    MathSciNet  MATH  Google Scholar 

  49. A. Sarantsev, Weak convergence of obliquely reflected diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., 54(3): 1408–1431, 2016.

  50. A. Sarantsev, Infinite systems of competing Brownian particles, Ann. Inst. Henri Poincaré, Probab. Stat., 53(4):2279–2315, 2017.

  51. T. Sasamoto and H. Spohn, Point-interacting Brownian motions in the KPZ universality class, Electron. J. Probab., 20(87):1–28, 2015.

    MathSciNet  MATH  Google Scholar 

  52. V.A. Shalaumov, On the behavior of a diffusion process with a large drift coefficient in a halfspace, Theory Probab. Appl., 24(3):592–598, 1980.

    Article  MATH  Google Scholar 

  53. A.V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl., 6(3):264–274, 1961.

    Article  MathSciNet  Google Scholar 

  54. A.V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl., 7(1):3–23, 1962.

    Article  Google Scholar 

  55. L. Słomiński, Weak and strong approximations of reflected diffusions via penalization methods, Stochastic Processes Appl., 123(3):752–763, 2013.

  56. L. Słomiński, On Wong–Zakai type approximations for reflected diffusions, Electron. J. Probab., 19(118):1–15, 2014.

  57. A. Storm, Stochastic differential equations with a convex constraint, Stochastics Stochastics Rep., 53(3–4):241–274, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  58. D.W. Stroock and S.R.S. Varadhan, Diffusion processes with boundary conditions, Commun. Pure Appl. Math., 24(2):147–225, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  59. H. Tanaka, Stochastic differential equations with reflecting boundary conditions in convex regions, Hiroshima Math. J., 9(1):163–177, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  60. S.Watanabe, On stochastic differential equations for multidimensional diffusion processes with boundary conditions, J. Math. Kyoto Univ., 11(1):169–180, 1971.

    MathSciNet  MATH  Google Scholar 

  61. S.Watanabe, On stochastic differential equations for multidimensional diffusion processes with boundary conditions ii, J. Math. Kyoto Univ., 11(3):545–551, 1971.

    MathSciNet  MATH  Google Scholar 

  62. R.J.Williams, Reflected Brownian motion with skew-symmetric data in a polyhedral domain, Probab. Theory Relat. Fields, 75(4):459–485, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  63. R.J. Williams and W.A. Zheng, On reflecting Brownian motion—a weak convergence approach, Ann. Inst. Henri Poincaré, Probab. Stat., 26(3):461–488, 1990.

Download references

Author information

Authors and Affiliations

  1. University of Nevada, 1664 North Virginia Street, Reno, NV, 89557, USA

    Andrey Sarantsev

Authors
  1. Andrey Sarantsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrey Sarantsev.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was partially supported by NSF grants DMS 1007563, DMS 1308340, DMS 1405210, and DMS 1409434.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sarantsev, A. Penalty method for obliquely reflected diffusions. Lith Math J 61, 518–549 (2021). https://doi.org/10.1007/s10986-021-09542-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-021-09542-9

MSC

Keywords

Search

Navigation