Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 3 (2006) open journal systems 

On the constructions of the skew Brownian motion

Antoine Lejay, Projet OMEGA, INRIA

This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.

AMS 2000 subject classifications: Primary 60J60; secondary 60H10, 60J55.

Keywords: skew Brownian motion, PDE with singular drift, PDE with transmission condition, SDE with local time, excursions of Brownian motion, scale function and speed measure, mathematical modelling, Monte Carlo methods.

Creative Common LOGO

Full Text: PDF

Lejay, Antoine, On the constructions of the skew Brownian motion, Probability Surveys, 3, (2006), 413-466 (electronic).


[1]   Bardou, O. (2005). Contrôle dynamique des erreurs de simulation et d’estimation de processus de diffusion. Ph.D. thesis, Université de Nice and INRIA Sophia-Antipolis.

[2]   Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2000). Variably skewed Brownian motion. Electron. Comm. Probab. 5, 57–66. MR1752008

[3]   Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2001). Coalescence of skew Brownian motions. In Séminaire de Probabilités, XXXV. Lecture Notes in Math., Vol. 1755. Springer, 202–205. MR1837288

[4]   Barlow, M., Pitman, J. and Yor, M. (1989a). On Walsh’s Brownian motions. In Séminaire de Probabilités, XXIII. Lecture Notes in Math., Vol. 1372. Springer, 275–293. MR1022917

[5]   Barlow, M., Pitman, J. and Yor, M. (1989b). Une extension multidimensionnelle de la loi de l’arc sinus. In Séminaire de Probabilités, XXIII. Lecture Notes in Math., Vol. 1372. Springer, 294–314. MR1022918

[6]   Barlow, M. T. (1988). Skew Brownian motion and a one-dimensional stochastic differential equation. Stochastics 25, 1, 1–2. MR1008231

[7]   Barlow, M. T., Émery, M., Knight, F. B., Song, S. and Yor, M. (1998). Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes. In Séminaire de Probabilités, XXXII. Lecture Notes in Math., Vol. 1686. Springer, 264–305.

[8]   Bass, R. and Chen, Z.-Q. (2002). Stochastic differential equations for dirichlet processes. Probab. Theory Related Fields 121, 3, 422–446. <doi:10.1007/s004400100151>. MR1867429

[9]   Bass, R. and Chen, Z.-Q. (2005). One-dimensional stochastic differential equations with singular and degenerate coefficients. Sankhyˉa
      67, 19–45. MR2203887

[10]   Bass, R. F. and Chen, Z.-Q. (2003). Brownian motion with singular drift. Ann. Probab. 31, 2, 791–817. MR1964949

[11]   Blumenthal, R. (1992). Excursions of Markov Processes. Probability and Its Applications. Birkhäuser. MR1138461

[12]   Breiman, L. (1981). Probability. Addison-Wesley.

[13]   Burdzy, K. and Chen, Z.-Q. (2001). Local time flow related to skew Brownian motion. Ann. Probab. 29, 4, 1693–1715. MR1880238

[14]   Burdzy, K. and Kaspi, H. (2004). Lenses in skew Brownian flow. Ann. Probab. 32, 4, 3085–3115. MR2094439

[15]   Cantrell, R. and Cosner, C. (1999). Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design. Theoritical Population Biology 55, 2, 189–207.

[16]   Chan, K. S. and Stramer, O. (1998). Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients. Stochastic Process. Appl. 76, 1, 33–44. MR1638015

[17]   Cherny, A., Shiryaev, A. and Yor, M. (2004). Limit behavior of the “horizontal-vertial” random walk and some extension of the Donsker-Prokhorov invariance principle. Theory Probab. Appl. 47, 3, 377–394.

[18]   Cox, A.M.G. and Hobson, D.G. (2005). A unifying class of Skorokhod embeddings: connecting the Azema-Yor and Vallois embeddings. Preprint, <arxiv:math.PR/0506040>. MR2213778

[19]   Decamps, M., de Schepper, A. and Goovaerts, M. (2004). Applications of δ-perturbation to the pricing of derivative securities. Phys. A 342, 3–4, 677–692. MR2105796

[20]   Decamps, M., de Schepper, A., Goovaerts, M. and Schoutens, W. (2005). A note on some new perpetuities. Scand. Actuar. J. 2005, 4, 261–270. <doi:10.1080/03461230510009772>.

[21]   Decamps, M., Goovaerts, M. and Schoutens, W. (2006). Asymetric skew Bessel processes and related processes with applications in finance. J. Comput. Appl. Math. 186, 1, 130–147. MR2190302

[22]   Decamps, M., Goovaerts, M. and Schoutens, W. (2006). Self Exciting Threshold Interest Rates Model. Int. J. Theor. Appl. Finance 9, 7, 1093-1122, <doi:10.1142/S0219024906003937>.

[23]   Desbois, J. (2002). Occupation times distribution for Brownian motion on graphs. J. Phys. A: Math. Gen. 35, L673–L678. MR1947234

[24]   Engelbert, H. J. and Schmidt, W. (1985). On one-dimensional stochastic differential equations with generalized drift. In Stochastic differential systems (Marseille-Luminy, 1984). Lecture Notes in Control and Inform. Sci., Vol. 69. Springer, 143–155. MR0798317

[25]   Engelbert, H.-J. and Wolf, J. (1999). Strong Markov local Dirichlet processes and stochastic differential equations. Theory Probab. Appl. 43, 2, 189–202. MR1679006

[26]   Enriquez, N. and Kifer, Y. (2001). Markov chains on graphs and Brownian motion. J. Theoret. Probab. 14, 2, 495–510. MR1838739

[27]   Étoré, P. (2006). On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients. Electron. J. Probab. 11, 9, 249–275. MR2217816

[28]   Étoré, P. and Lejay, A. (2006). A Donsker theorem to simulate one-dimensional diffusion processes with measurable coefficients. Preprint, Institut Élie Cartan, Nancy, France.

[29]   Étoré, P. (2006). Approximation de processus de diffusion à coefficients discontinus en dimension un et applications à la simulation. Ph.D. thesis, Université Nancy 1.

[30]   Flandoli, F., Russo, F. and Wolf, J. (2003). Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40, 2, 493–542. MR1988703

[31]   Folland, G. B. (1995). Introduction to partial differential equations, 2 ed. Princeton University Press, Princeton, NJ. MR1357411

[32]   Freidlin, M., Mayergoyz, I.D. and Pfeifer, R. (2000). Noise in hysteretic systems and stochastic processes on graphs. Phys. Rev. E 61, 2, 1850–1855.

[33]   Freidlin, M. and Sheu, S.-J. (2000). Diffusion processes on graphs: stochastic differential equations, large deviation principle. Probab. Theory Related Fields 116, 2, 181–220. MR1743769

[34]   Freidlin, M. and Weber, M. (1998). Random perturbations of nonlinear oscillators. Ann. Probab. 26, 3, 925–967. MR1634409

[35]   Freidlin, M. and Wentzell, A. (1993). Diffusion processes on graphs and the averaging principle. Ann. Probab. 21, 4, 2215–2245. MR1245308

[36]   Freidlin, M. and Wentzell, A. (1994a). Necessary and sufficient conditions for weak convergence of one-dimensional Markov processes. In Festschrift dedicated to 70th Birthday of Professor E.B. Dynkin, M. Freidlin, Ed. Birkhäuser, 95–109. MR1311713

[37]   Freidlin, M. I. and Wentzell, A. D. (1994b). Random perturbations of Hamiltonian systems. Mem. Amer. Math. Soc. 109, 523. MR1201269

[38]   Friedman, A. (1983). Partial Differential Equations of Parabolic Type. Robert E. Krieger Publishing Company. MR0454266

[39]   Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Process. De Gruyter. MR1303354

[40]   Gaveau, B., Okada, M. and Okada, T. (1987). Second order differential operators and Dirichlet integrals with singular coefficients. I: Functional calculus of one-dimensional operators. Tôhoku Math. J.(2) 39, 4, 465–504. MR0917463

[41]   Harrison, J. and Shepp, L. (1981). On skew Brownian motion. Ann. Probab. 9, 2, 309–313. MR0606993

[42]   Hausenblas, E. (1999). A numerical scheme using Itô excursions for simulating local time resp. stochastic differential equations with reflection. Osaka J. Math. 36, 1, 105–137. MR99m:60087. MR1670754

[43]   Ikeda, N. and Watanabe, S. (1989). Stochastic differential equations and diffusion processes 2 ed. North-Holland Mathematical Library, 24. MR1011252

[44]   Itô, K. and McKean, H. (1974). Diffusion and Their Sample Paths, 2 ed. Springer-Verlag.

[45]   Janssen, R. (1984). Difference methods for stochastic differential equations with discontinuous coefficients. Stochastics 13, 3, 199–212. MR0762816

[46]   Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus, 2 ed. Springer-Verlag. MR1121940

[47]   Kurtz, T. and Protter, P. (1996). Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, D. Talay and L. Tubaro, Eds. Lecture Notes in Math., Vol. 1627. Springer-Verlag, 1–41. MR1431298

[48]   Ladyženskaja, O., Solonnikov, V. and Ural’ceva, N. (1968). Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society.

[49]   Ladyženskaja, O. A., Rivkind, V. J. and Ural’ceva, N. N. (1966). Solvability of diffraction problems in the classical sense. Trudy Mat. Inst. Steklov. 92, 116–146. MR0211050

[50]   Lang, R. (1995). Effective conductivity and Skew Brownian motion. J. Statist. Phys. 80, 125–145. MR1340556

[51]   Le Gall, J.-F. (1982). Applications du temps local aux équations différentielles stochastiques unidimensionnelles. In Séminaire de Probabilités XVII. Lecture Notes in Math., Vol. 986. Springer-Verlag, 15–31.

[52]   Le Gall, J.-F. (1985). One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic Analysis and Applications. Lecture Notes in Math., Vol. 1095. Springer-Verlag, 51–82. MR0777514

[53]   Lee, S. (1994). Optimal drift on [0,1]. Trans. Amer. Math. Soc. 346, 1, 159–175. MR1254190

[54]   Lejay, A. (2000). Méthodes probabilistes pour l’homogénéisation des opérateurs sous forme-divergence : cas linéaires et semi-linéaires. Ph.D. thesis, Université de Provence, Marseille, France.

[55]   Lejay, A. (2001). On the decomposition of excursions measures of processes whose generators have diffusion coefficients discontinuous at one point. Markov Process. Related Fields 8, 1, 117–126. MR1897608

[56]   Lejay, A. (2003). Simulating a diffusion on a graph. application to reservoir engineering. Monte Carlo Methods Appl. 9, 3, 241–256. MR2009371

[57]   Lejay, A. (2004). Monte Carlo methods for fissured porous media: a gridless approach. Monte Carlo Methods Appl. 10, 3–4, 385–392. Conference proceeding of IV IMACS Seminar on Monte Carlo Methods, . MR2105066

[58]   Lejay, A. and Martinez, M. (2006). A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 16, 1, 107–139. <doi:10.1214/105051605000000656>. MR2209338

[59]   Ma, Z. and Röckner, M. (1991). Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer-Verlag. MR1106853

[60]   Mandelbaum, A., Shepp, L. A. and Vanderbei, R. J. (1990). Optimal switching between a pair of Brownian motions. Ann. Probab. 18, 3, 1010–1033. MR1062057

[61]   Martinez, M. (2004). Interprétations probabilistes d’opérateurs sous forme divergence et analyse de méthodes numériques associées. Ph.D. thesis, Université de Provence / INRIA Sophia-Antipolis.

[62]   Martinez, M. and Talay, D. (2006). Discrétisation d’équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu. C. R. Math. Acad. Sci. Paris 342, 1, 51–56. MR2193396

[63]   Nakao, S. (1972). On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9, 513–518. MR0326840

[64]   Nicaise, S. (1985). Some results on spectral theory over networks, applied to nerve impulse transmission. In Orthogonal polynomials and applications (Bar-le-Duc, 1984). Lecture Notes in Math., Vol. 1171. Springer, 532–541. MR0839024

[65]   Okada, T. (1993). Asymptotic behavior of skew conditional heat kernels on graph networks. Canad. J. Math. 45, 4, 863–878. MR1227664

[66]   ˉO    shima, Y. (1982). Some singular diffusion processes and their associated stochastic differential equations. Z. Wahrsch. Verw. Gebiete 59, 2, 249–276. MR0650616

[67]   Ouknine, Y. (1990). “Skew-Brownian motion” and derived processes. Theory Probab. Appl. 35, 1, 163–169. MR1050069

[68]   Peskir, G. (2005). A change-of-variable formula with local time on curves. J. Theoret. Probab. 18, 3, 499–535. <doi:10.1007/s10959-005-3517-6>. MR2167640

[69]   Portenko, N. (1979a). Diffusion processes with a generalized drift coefficient. Theory Probab. Appl. 24, 1, 62–78. MR0522237

[70]   Portenko, N. (1979b). Stochastic differential equations with generalized drift vector. Theory Probab. Appl. 24, 2, 338–353. MR0532446

[71]   Portenko, N. I. (1990). Generalized diffusion processes. Translations of Mathematical Monographs, Vol. 83. American Mathematical Society, Providence, RI. MR1104660

[72]   Portenko, N. I. (2001). A probabilistic representation for the solution to one problem of mathematical physics. Ukrainian Math. J. 52 (2000), 9, 1457–1469. MR1816940

[73]   Ramirez, J.M., Thomann, E.A., Waymire, E.C., Haggerty R. and Wood B. (2005). A generalization Taylor-Aris formula and Skew Diffusion Multiscale Model. Simul. 5, 3, 786–801, <doi:10.1137/050642770>.

[74]   Rogers, L. and Williams, D. (2000). Itô Calculus, 2 ed. Diffusions, Markov Processes, and Martingales, Vol. 2. Cambridge University Press.

[75]   Rosenkrantz, W. A. (1974/75). Limit theorems for solutions to a class of stochastic differential equations. Indiana Univ. Math. J. 24, 613–625. MR0368143

[76]   Rozkosz, A. (1996). Weak convergence of diffusions corresponding to divergence form operator. Stochastics Stochastics Rep. 57, 129–157. MR1407951

[77]   Sznitman, A.-S. and Varadhan, S. R. S. (1986). A multidimensional process involving local time. Probab. Theory Relat. Fields 71, 4, 553–579. MR0833269

[78]   Stroock, D. (1988). Diffusion semigroups corresponding to uniformly elliptic divergence form operator. In Séminaire de Probabilités XXII. Lecture Notes in Math., Vol. 1321. Springer-Verlag, 316–347. MR0960535

[79]   Stroock, D. and Varadhan, S. (1979). Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften, Vol. 233. Springer-Verlag, Berlin. MR0532498

[80]   Stroock, D. and Zheng, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincar Probab. Statist. 33, 5, 619–649. MR1473568

[81]   Takanobu, S. (1986). On the uniqueness of solutions of stochastic differential equations with singular drifts. Publ. Res. Inst. Math. Sci.22, 5, 813–848. MR0866659

[82]   Takanobu, S. (1987). On the existence of solutions of stochastic differential equations with singular drifts. Probab. Theory Related Fields 74, 2, 295–315. MR0871257

[83]   Tomisaki, M. (1980). Superposition of diffusion processes. J. Math. Soc. Japan 32, 4, 671–696. MR0589106

[84]   Trutnau, G. (2005). Multidimensional skew reflected diffusions. Stochastic analysis: classical and quantum, 228–244, World Sci. Publ., Hackensack, NJ. MR2233163

[85]   Tsirelson, B. (1997). Triple points: from non-Brownian filtrations to harmonic measures. Geom. Funct. Anal. 7, 6, 1096–1142. MR1487755

[86]   Vuolle-Apiala, J. (1996). Skew Brownian motion-type of extensions. J. Theor. Probab. 9, 4, 853–861. MR1419866

[87]   Walsh, J. (1978). A diffusion with a discontinuous local time. In Temps locaux. Astérisque. Société Mathématique de France, 37–45. MR0509476

[88]   Walsh, J. (1981). Optional increasing paths. In Two-index random processes (Paris, 1980). Lecture Notes in Math., Vol. 863. Springer, 172–201. MR0630313

[89]   Watanabe, S. (1984). Excursion point processes and diffusions. In Proceedings of the International Congress of Mathematicians (Warsaw, 1983), Vol. 1, 2, 1117–1124, PWN, Warsaw. MR0804763

[90]   Watanabe, S. (1995). Generalized arc-sine laws for one-dimensional diffusion processes and random walks. In Stochastic analysis (Ithaca, NY, 1993). Proc. Sympos. Pure Math., Vol. 57. Amer. Math. Soc., 157–172. MR1335470

[91]   Weinryb, S. (1982). Étude d’une équation différentielle stochastique avec temps local. In Séminaire de Probabilités XVII. Lecture Notes in Math., Vol. 986. Springer-Verlag, 72–77. MR0770397

[92]   Weinryb, S. (1984). Homogénéisation pour des processus associés à des frontières perméables. Ann. Inst. H. Poincaré Probab. Statist. 20, 4, 373–407. MR0771896

[93]   Yan, L. (2002). The Euler scheme with irregular coefficients. Ann. Probab. 30, 3, 1172–1194. MR1920104

[94]   Yor, M. (1997). Some aspects of Brownian motion. Part II: Some recent martingale problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag. MR1442263

[95]   Zaitseva, L. (2000). On a probabilistic approach to the construction of the generalized diffusion processes. Theory Stoch. Process. 6, 1–2, 141–146. MR2007732

[96]   Zaitseva, L. (2005). On stochastic continuity of generalized diffusion processes constructed as the strong solution to an SDE. Theory Stoch. Process. 11(27), 1–2, 125–135. <arxiv:math.PR/0609305>. MR2007732

[97]   Zaitseva, L. (2005). On the Markov property of strong solutions to SDE with generalized coefficients. Theory Stoch. Process. 11(27), 3–4, 140–146. <arxiv:math.PR/0609307>.

[98]   Zhang, M. (2000). Calculation of diffusive shock acceleration of charged particles by skew Brownian motion. Astrophys. J. 541, 428–435.

[99]   Zhikov, V., Kozlov, S. and Oleinik, O. (1981). G-convergence of parabolic operators. Russian Math. Survey 36, 1, 9–60. MR0608940

[100]   Zhikov, V., Kozlov, S., Oleinik, O. and T’en Ngoan, K. (1979). Averaging and G-convergence of differential operators. Russian Math. Survey 34, 5, 69–147. MR0562800

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787