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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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.


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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