×

zbMATH — the first resource for mathematics

Two Brownian particles with rank-based characteristics and skew-elastic collisions. (English) Zbl 1296.60148
Summary: We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems.
The analysis depends crucially on properties of a skew Brownian motion with two-valued drift of the bang-bang type, which we also discuss in some detail. These properties allow us to compute the transition probabilities of the original planar diffusion, and to study its behavior under time reversal.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
60J55 Local time and additive functionals
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Anulova, S. V., Diffusion processes with singular characteristics, (Stochastic Differential Systems, Filtering and Control: Proceedings of an I.F.I.P.-W.G. Conference, Vilnius, 1978, Lithuania, Lecture Notes in Control and Information Systems, vol. 25, (1980), Springer-Verlag New York), 264-269 · Zbl 0539.60077
[2] Appuhamillage, Th.; Vrushali, B.; Thomann, E.; Waymire, E.; Wood, B., Occupation and local times for skew Brownian motion with application to dispersion across an interface, Ann. Appl. Probab., 21, 183-214, (2011), 2050-2051. Correction · Zbl 1226.60113
[3] Banner, A.; Fernholz, E. R.; Karatzas, I., Atlas models of equity markets, Ann. Appl. Probab., 15, 2296-2330, (2005) · Zbl 1099.91056
[4] Bass, R.; Chen, Z. Q., One-dimensional stochastic differential equations with singular and degenerate coëfficients, Sankhyā, 67, 19-45, (2005) · Zbl 1192.60081
[5] Burdzy, K.; Nualart, D., Brownian motion reflected on Brownian motion, Probab. Theory Related Fields, 122, 471-493, (2002) · Zbl 0995.60078
[6] Chitashvili, R. J.; Lazrieva, N. L., Strong solutions of stochastic differential equations with boundary conditions, Stochastics, 5, 255-309, (1981) · Zbl 0479.60062
[7] Engelbert, H. J.; Schmidt, W., On one-dimensional stochastic differential equations with generalized drift, (Lecture Notes in Control and Information Systems, vol. 69, (1985), Springer-Verlag NY), 143-155 · Zbl 0583.60052
[8] Fernholz, E. R., Stochastic portfolio theory, (2002), Springer-Verlag New York · Zbl 1049.91067
[9] E.R. Fernholz, T. Ichiba, I. Karatzas, A second-order stock market model, Ann. Finance (in press). · Zbl 1298.91136
[10] E.R. Fernholz, T. Ichiba, I. Karatzas, V. Prokaj, A planar diffusion with rank-based characteristics, and perturbed Tanaka equations, Probab. Theory Related Fields (in press). Available at the site http://arxiv.org/pdf/1108.3992 (posted: 19.08.11).
[11] Fernholz, E. R.; Karatzas, I., Stochastic portfolio theory: an overview, (Handbook of Numerical Analysis, vol. 15, (2009), Elsevier), 89-167 · Zbl 1180.91267
[12] Fukushima, M.; Stroock, D., Reversibility of solutions to martingale problems, (Probability, Statistical Mechanics, and Number Theory, Adv. Math. Suppl. Stud., vol. 9, (1986), Academic Press Orlando, FL), 107-123 · Zbl 0613.60066
[13] Harrison, J. M.; Reiman, M. I., Reflected Brownian motion on an orthant, Ann. Probab., 9, 302-308, (1981) · Zbl 0462.60073
[14] Harrison, J. M.; Shepp, L. A., On skew Brownian motion, Ann. Probab., 9, 309-313, (1981) · Zbl 0462.60076
[15] Itô, K.; McKean, H. P., Brownian motion on a half-line, Illinois J. Math., 7, 181-231, (1963) · Zbl 0114.33601
[16] Itô, K.; McKean, H. P., Diffusion processes and their sample paths, (1974), Springer-Verlag New York, Second printing (corrected) · Zbl 0285.60063
[17] I. Karatzas, S. Pal, M. Shkolnikov, Systems of Brownian particles with asymmetric collisions (submitted for publication). Available at the site http://arxiv.org/abs/1210.0259 (posted: 30.09.12). · Zbl 1333.60206
[18] Karatzas, I.; Shreve, S. E., Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control, Ann. Probab., 12, 819-828, (1984) · Zbl 0544.60069
[19] Karatzas, I.; Shreve, S. E., Brownian motion and stochastic calculus, (1991), Springer-Verlag New York · Zbl 0734.60060
[20] Lang, R., Effective conductivity and skew Brownian motion, J. Stat. Phys., 80, 125-146, (1995) · Zbl 1081.82603
[21] Le Gall, J. F., Applications des temps locaux aux équations différentielles stochastiques unidimensionelles, (Lecture Notes in Mathematics, vol. 986, (1983), Springer-Verlag New York), 15-31
[22] Le Gall, J. F., One-dimensional stochastic differential equations involving the local times of the unknown process, (Lecture Notes in Mathematics, vol. 1095, (1984), Springer-Verlag New York), 51-82 · Zbl 0551.60059
[23] Lejay, A., On the constructions of the skew Brownian motion, Probab. Surv., 3, 413-466, (2006) · Zbl 1189.60145
[24] Nakao, S., On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka J. Math., 9, 513-518, (1972) · Zbl 0255.60039
[25] Oshima, Y., Some singular diffusion processes and their associated stochastic differential equations, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 59, 249-276, (1982) · Zbl 0465.60069
[26] Ouknine, Y.; Rutkowski, M., Local times of functions of continuous semimartingales, Stoch. Anal. Appl., 13, 211-231, (1995) · Zbl 0826.60064
[27] Pardoux, E., Grossissement d’une filtration et retournement du temps, (Séminaire de Probabilités XX, 1984-1985, Lecture Notes in Mathematics, vol. 1204, (1986), Springer-Verlag New York), 48-55
[28] Petit, F., Time reversal and reflected diffusion, Stochastic Process. Appl., 69, 25-53, (1997) · Zbl 0911.60059
[29] Portenko, N. I., Generalized diffusion processes, (Lecture Notes in Mathematics, vol. 550, (1976), Springer-Verlag New York), 500-523 · Zbl 0387.60086
[30] Portenko, N. I., Diffusion processes with generalized drift coëfficients, Theory Probab. Appl., 24, 62-78, (1979) · Zbl 0432.60094
[31] Portenko, N. I., Stochastic differential equations with generalized drift vector, Theory Probab. Appl., 24, 338-353, (1979) · Zbl 0434.60062
[32] Portenko, N. I., (Generalized Diffusion Processes, Translations of Mathematical Monographs, (1990), American Mathematical Society Providence, RI)
[33] V. Prokaj, The solution of the perturbed Tanaka equation is pathwise unique, Ann. Probab. (2011) (in press). arXiv:1104.0740. · Zbl 1284.60134
[34] Soucaliuc, F.; Tóth, B.; Werner, W., Reflection and coalescence between independent one-dimensional Brownian motions, Ann. Inst. Henri Poincaré, Sec. B, 36, 509-545, (2000) · Zbl 0968.60072
[35] Soucaliuc, F.; Werner, W., A note on reflecting Brownian motions, Electron. Commun. Probab., 7, 117-122, (2002) · Zbl 1009.60068
[36] Sznitman, A. S.; Varadhan, S. R.S., A multidimensional process involving local time, Probab. Theory Related Fields, 71, 553-579, (1986) · Zbl 0613.60050
[37] Takanobu, S., On the existence of solutions of stochastic differential equations with singular drifts, Probab. Theory Related Fields, 74, 295-315, (1987) · Zbl 0586.60048
[38] Tomisaki, M., A construction of diffusion processes with singular product measures, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 53, 51-70, (1980) · Zbl 0433.60077
[39] Varadhan, S. R.S.; Williams, R. J., Brownian motion in a wedge with oblique reflection, Comm. Pure Appl. Math., 38, 405-443, (1985) · Zbl 0579.60082
[40] Walsh, J. B., A diffusion with discontinuous local time, (Temps Locaux, Astérisque, vol. 52-53, (1978)), 37-45
[41] Williams, R. J., Reflected Brownian motion with skew symmetric data in a polyhedral domain, Probab. Theory Related Fields, 75, 459-485, (1987) · Zbl 0608.60074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.