Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Skorokhod embeddings via stochastic flows on the space of Gaussian measures

Ronen Eldan

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Abstract

We present a new construction of a Skorokhod embedding, namely, given a probability measure $\mu$ with zero expectation and finite variance, we construct an integrable stopping time $T$ adapted to a filtration $\mathcal{F}_{t}$, such that $W_{T}$ has the law $\mu$, where $W_{t}$ is a standard Wiener process adapted to the same filtration. We find several sufficient conditions for the stopping time $T$ to be bounded or to have a sub-exponential tail. In particular, our embedding seems rather natural for the case that $\mu$ is a log-concave measure and $T$ satisfies several tight bounds in that case. Our embedding admits the property that the stochastic measure-valued process $\{\mu_{t}\}_{t\geq0}$, where $\mu_{t}$ is as the law of $W_{T}$ conditioned on $\mathcal{F}_{t}$, is a Markov process. In view of this property, we will consider a more general family of Skorokhod embeddings which can be constructed via a kernel generating a stochastic flow on the space of measures. This family includes existing constructions such as the ones by Azéma–Yor (In Séminaire de Probabilités XIII (1979) 90–115 Springer) and by Bass (In Séminaire de Probabilités XVII (1983) 221–224 Springer), and thus suggests a new point of view on these constructions.

Résumé

Nous proposons une nouvelle construction d’un plongement de Skorokhod: étant donnée une mesure de probabilité $\mu$ avec espérance nulle et variance finie, nous construisons un temps d’arrêt intégrable $T$ adapté à la filtration $\mathcal{F}_{t}$, tel que $W_{T}$ possède la loi $\mu$ et $W$ est un processus de Wiener standard adapté à la même filtration. Nous trouvons plusieurs conditions suffisantes pour que le temps d’arrêt $T$ soit borné ou ait des queues sous-exponentielles. En particulier, notre plongement semble assez naturel dans le cas où $\mu$ est log-concave et $T$ satisfait plusieurs estimations fortes. Notre plongement a la propriété suivante : le processus stochastique à valeur dans les mesures $\{\mu_{t}\}_{t\geq0}$, où $\mu_{t}$ est la loi de $W_{T}$ conditionnée par $\mathcal{F}_{t}$, est un processus de Markov. Compte tenu de cette propriété, nous allons considérer une famille plus générale de plongements de Skorokhod qui peuvent être construits à l’aide d’un noyau générant un flot stochastique sur l’espace des mesures. Cette famille inclut des constructions déjà existantes comme celle d’Azéma–Yor (In Séminaire de Probabilités XIII (1979) 90–115 Springer) et celle de Bass (In Séminaire de Probabilités XVII (1983) 221–224 Springer), suggérant ainsi un point de vue nouveau sur ces constructions.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1259-1280.

Dates
Received: 18 May 2013
Revised: 8 April 2015
Accepted: 9 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723520

Digital Object Identifier
doi:10.1214/15-AIHP682

Mathematical Reviews number (MathSciNet)
MR3531709

Zentralblatt MATH identifier
1350.60039

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G44: Martingales with continuous parameter
Secondary: 60H20: Stochastic integral equations

Keywords
Skorokhod embedding Brownian motion Log concave measure Markov chain

Citation

Eldan, Ronen. Skorokhod embeddings via stochastic flows on the space of Gaussian measures. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1259--1280. doi:10.1214/15-AIHP682. https://projecteuclid.org/euclid.aihp/1469723520


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