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May 2018 Stochastic integral equations for Walsh semimartingales
Tomoyuki Ichiba, Ioannis Karatzas, Vilmos Prokaj, Minghan Yan
Ann. Inst. H. Poincaré Probab. Statist. 54(2): 726-756 (May 2018). DOI: 10.1214/16-AIHP819


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.


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Tomoyuki Ichiba. Ioannis Karatzas. Vilmos Prokaj. Minghan Yan. "Stochastic integral equations for Walsh semimartingales." Ann. Inst. H. Poincaré Probab. Statist. 54 (2) 726 - 756, May 2018.


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

zbMATH: 06897966
MathSciNet: MR3795064
Digital Object Identifier: 10.1214/16-AIHP819

Primary: 60G42
Secondary: 60H10

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

Rights: Copyright © 2018 Institut Henri Poincaré


Vol.54 • No. 2 • May 2018
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