Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Stochastic integral equations for Walsh semimartingales

Tomoyuki Ichiba, Ioannis Karatzas, Vilmos Prokaj, and Minghan Yan

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Abstract

We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison–Shepp-type equations and a change-of-variable formula in the spirit of Freidlin–Sheu for these so-called “Walsh semimartingales”. We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.

Résumé

Nous construisons des semimartingales planaires qui incluent le mouvement brownien de Walsh comme cas particulier, et nous établissons pour ces « semimartingales de Walsh » des équations de type Harrison–Shepp, et une formule de changement de variable dans l’esprit de Freidlin–Sheu. Dans des cadres markoviens appropriés, nous étudions deux types de relations aux problèmes de martingale, des questions d’unicité en loi pour de tels processus, et quelques exemples.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 726-756.

Dates
Received: 19 October 2015
Revised: 20 November 2016
Accepted: 16 December 2016
First available in Project Euclid: 25 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1524643227

Digital Object Identifier
doi:10.1214/16-AIHP819

Mathematical Reviews number (MathSciNet)
MR3795064

Zentralblatt MATH identifier
06897966

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Skew and Walsh Brownian motions Spider and Walsh semimartingales Skorokhod reflection Planar skew unfolding Harrison–Shepp equations Freidlin–Sheu formula Martingale problems Local time

Citation

Ichiba, Tomoyuki; Karatzas, Ioannis; Prokaj, Vilmos; Yan, Minghan. Stochastic integral equations for Walsh semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 726--756. doi:10.1214/16-AIHP819. https://projecteuclid.org/euclid.aihp/1524643227


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References

  • [1] M. Barlow, J. Pitman and M. Yor. Une extension multidimensionnelle de la loi de l’arc sinus. In Séminaire de Probabilités, XXIII 294–314. Lecture Notes in Math. 1372. Springer, Berlin, 1989.
  • [2] M. T. Barlow, M. Émery, F. B. Knight, S. Song and M. Yor. Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes. In Séminaire de Probabilités, XXXII 264–305. Lecture Notes in Math. 1686. Springer, Berlin, 1998.
  • [3] P. Biane and M. Yor. Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2) 111 (1987) 23–101.
  • [4] Z.-Q. Chen and M. Fukushima. One-point reflection. Stochastic Process. Appl. 125 (2015) 1368–1393.
  • [5] H. J. Engelbert and W. Schmidt. On one-dimensional stochastic differential equations with generalized drift. In Stochastic Differential Systems (Marseille-Luminy, 1984) 143–155. Lecture Notes in Control and Inform. Sci. 69. Springer, Berlin, 1985.
  • [6] S. N. Evans and R. B. Sowers. Pinching and twisting Markov processes. Ann. Probab. 31 (2003) 486–527.
  • [7] P. J. Fitzsimmons and K. E. Kuter. Harmonic functions on Walsh’s Brownian motion. Stochastic Process. Appl. 124 (2014) 2228–2248.
  • [8] M. Freidlin and S.-J. Sheu. Diffusion processes on graphs: Stochastic differential equations, large deviation principle. Probab. Theory Related Fields 116 (2000) 181–220.
  • [9] H. Hajri. Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16 (2011) no. 58, 1563–1599.
  • [10] H. Hajri and W. Touhami. Itô’s formula for Walsh’s Brownian motion and applications. Statist. Probab. Lett. 87 (2014) 48–53.
  • [11] J. M. Harrison and L. A. Shepp. On skew Brownian motion. Ann. Probab. 9 (1981) 309–313.
  • [12] T. Ichiba and I. Karatzas. Skew-unfolding the Skorokhod reflection of a continuous semimartingale. In Stochastic Analysis and Applications 2014 349–376. Springer Proc. Math. Stat. 100. Springer, Cham, 2014.
  • [13] T. Ichiba, V. Papathanakos, A. Banner, I. Karatzas and R. Fernholz. Hybrid atlas models. Ann. Appl. Probab. 21 (2011) 609–644.
  • [14] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
  • [15] K. Itô and H. P. McKean Jr.. Brownian motions on a half line. Illinois J. Math. 7 (1963) 181–231.
  • [16] J. Jacod. A general theorem of representation for martingales. In Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976) 37–53. Amer. Math. Soc., Providence, RI, 1977.
  • [17] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer-Verlag, New York, 1991.
  • [18] R. Mansuy and M. Yor. Random Times and Enlargements of Filtrations in a Brownian Setting. Lecture Notes in Mathematics 1873. Springer-Verlag, Berlin, 2006.
  • [19] J. Picard. Stochastic calculus and martingales on trees. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005) 631–683.
  • [20] V. Prokaj. Unfolding the Skorohod reflection of a semimartingale. Statist. Probab. Lett. 79 (2009) 534–536.
  • [21] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer-Verlag, Berlin, 1999.
  • [22] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition.
  • [23] P. Salminen, P. Vallois and M. Yor. On the excursion theory for linear diffusions. Jpn. J. Math. 2 (2007) 97–127.
  • [24] W. Schmidt. On stochastic differential equations with reflecting barriers. Math. Nachr. 142 (1989) 135–148.
  • [25] D. W. Stroock and S. R. S. Varadhan. Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 (1971) 147–225.
  • [26] B. Tsirelson. Triple points: From non-Brownian filtrations to harmonic measures. Geom. Funct. Anal. 7 (1997) 1096–1142.
  • [27] J. B. Walsh. A diffusion with a discontinuous local time. Astérisque 52–53 (1978) 37–45.
  • [28] S. Watanabe. The existence of a multiple spider martingale in the natural filtration of a certain diffusion in the plane. In Séminaire de Probabilités, XXXIII 277–290. Lecture Notes in Math. 1709. Springer, Berlin, 1999.
  • [29] M. Yor. Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997.