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.
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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.
A.D. Banner, E.R. Fernholz, and I. Karatzas, Atlas models of equity markets, Ann. Appl. Probab., 15(4):2996–2330, 2005.
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.
P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, 1999.
C. Bruggeman and A. Sarantsev, Penalty method for reflected diffusions on the half-line, Stochastics, 89(2):485–509, 2017.
E. Cepa, Problème de Skorohod multivoque, Ann. Probab., 26(2):500–532, 1998.
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.
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.
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.
P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics, 35(1):31–62, 1991.
P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains, Ann. Probab., 21(1):554–580, 1993.
L.C. Evans and D.W. Stroock, An approximation scheme for reflected stochastic differential equations, Stochastic Processes Appl., 121(7):1464–1491, 2011.
G.B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, New York, 1999.
R.L. Foote, Regularity of the distance function, Proc. Am. Math. Soc., 92(1):153–155, 1984.
M. Fradon, Stochastic differential equations on domains defined by multiple constraints, Electron. Commun. Probab., 18(26):1–13, 2013.
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.
J.M. Harrison and I.M. Reiman, Reflected Brownian motion on an orthant, Ann. Probab., 9(2):302–308, 1981.
J.M. Harrison and R.J. Williams, Brownian models of open queueing networks with homogeneous customer populations, Stochastics, 22(2):77–115, 1987.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1989.
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.
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1991.
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.
T. Klimsiak, A. Rozkosz, and L. Slominski, Reflected BSDEs in time-dependent convex regions, Stochastic Processes Appl., 125(2):571–596, 2015.
S.G. Krantz and H.R. Parks, Distance to Ck hypersurfaces, J. Differ. Equations, 40(1):116–120, 1981.
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.
W. Łaukajtys and L. Słomiński, Penalization methods for reflecting stochastic differential equations with jumps, Stochastics Stochastics Rep., 75(5):275–293, 2003.
W. Łaukajtys and L. Słomiński, Penalization methods for the Skorokhod problem and reflecting SDEs with jumps, Bernoulli, 19(5A):1750–1775, 2013.
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.
P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Commun. Pure Appl. Math., 37(4):511–537, 1984.
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.
H.P. McKean Jr., Skorohod’s stochastic integral equation for a reflecting barrier diffusion, J. Math. Kyoto Univ., 3:85–88, 1963.
J.-L. Menaldi, Stochastic variational inequality for reflected diffusions, Indiana Univ. Math. J., 32(5):733–744, 1983.
J.-L. Menaldi, Stochastic differential equations with jumps, 2014, http://www.clas.wayne.edu/Multimedia/menaldi/Files/m-book-06-2013.pdf.
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.
J.-L. Menaldi and M. Robin, Reflected diffusion processes with jumps, Ann. Probab., 13(2):319–341, 1985.
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.
E. Pardoux and R.J.Williams, Symmetric reflected diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., 30(1):13–62, 1994.
R. Petterson, Approximations for stochastic differential equation with reflecting convex boundaries, Stochastic Processes Appl., 59(2):295–308, 1995.
R. Petterson, Penalization schemes for reflecting stochastic differential equations, Bernoulli, 3(4):403–414, 1997.
R. Petterson, Wong–Zakai approximations for reflecting stochastic differential equations, Stochastic Anal. Appl., 17(4):609–617, 2007.
A. Pilipenko, Properties of the flows generated by a stochastic equation with reflection, Ukr. Math. J., 57(8):1262–1274, 2005.
A. Pilipenko, An Introduction to Stochastic Differential Equations with Reflection, Potsdam Univ. Press, Potsdam, 2014.
N. Portenko, Diffusion processes with generalized drift coefficients, Theory Probab. Appl., 24(1):62–78, 1979.
N. Portenko, Stochastic differential equations with generalized drift vector, Theory Probab. Appl., 24(2):338–353, 1980.
A. Rozkosz and L. Słomiński, Lp solutions of reflected BSDEs under monotonicity condition, Stochastic Processes Appl., 122(12):3875–3900, 2012.
Y. Saisho, Stochastic differential equations for multidimensional domain with reflecting boundary, Probab. Theory Relat. Fields, 74(3):455–477, 1987.
A. Sarantsev, On a class of diverse market models, Ann. Finance, 10(2):291–314, 2014.
A. Sarantsev, Triple and simultaneous collisions of competing Brownian particles, Electron. J. Probab., 20(29):1–28, 2015.
A. Sarantsev, Weak convergence of obliquely reflected diffusions, Ann. Inst. Henri Poincaré, Probab. Stat., 54(3): 1408–1431, 2016.
A. Sarantsev, Infinite systems of competing Brownian particles, Ann. Inst. Henri Poincaré, Probab. Stat., 53(4):2279–2315, 2017.
T. Sasamoto and H. Spohn, Point-interacting Brownian motions in the KPZ universality class, Electron. J. Probab., 20(87):1–28, 2015.
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.
A.V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl., 6(3):264–274, 1961.
A.V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl., 7(1):3–23, 1962.
L. Słomiński, Weak and strong approximations of reflected diffusions via penalization methods, Stochastic Processes Appl., 123(3):752–763, 2013.
L. Słomiński, On Wong–Zakai type approximations for reflected diffusions, Electron. J. Probab., 19(118):1–15, 2014.
A. Storm, Stochastic differential equations with a convex constraint, Stochastics Stochastics Rep., 53(3–4):241–274, 1995.
D.W. Stroock and S.R.S. Varadhan, Diffusion processes with boundary conditions, Commun. Pure Appl. Math., 24(2):147–225, 1971.
H. Tanaka, Stochastic differential equations with reflecting boundary conditions in convex regions, Hiroshima Math. J., 9(1):163–177, 1979.
S.Watanabe, On stochastic differential equations for multidimensional diffusion processes with boundary conditions, J. Math. Kyoto Univ., 11(1):169–180, 1971.
S.Watanabe, On stochastic differential equations for multidimensional diffusion processes with boundary conditions ii, J. Math. Kyoto Univ., 11(3):545–551, 1971.
R.J.Williams, Reflected Brownian motion with skew-symmetric data in a polyhedral domain, Probab. Theory Relat. Fields, 75(4):459–485, 1987.
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.
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This research was partially supported by NSF grants DMS 1007563, DMS 1308340, DMS 1405210, and DMS 1409434.
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Sarantsev, A. Penalty method for obliquely reflected diffusions. Lith Math J 61, 518–549 (2021). https://doi.org/10.1007/s10986-021-09542-9
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DOI: https://doi.org/10.1007/s10986-021-09542-9