Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Skorokhod embeddings via stochastic flows on the space of Gaussian measures

Ronen Eldan

Full-text: Open access


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.


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Skorokhod embedding Brownian motion Log concave measure Markov chain


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.

Export citation


  • [1] S. Ankirchner, G. Heyne and P. Imkeller. A BSDE approach to the Skorokhod embedding problem for the Brownian motion with drift. Stoch. Dyn. 8 (2008) 35–46.
  • [2] S. Ankirchner and P. Strack. Skorokhod embeddings in bounded time. Stoch. Dyn. 11 (2–3) (2011) 215–226.
  • [3] J. Azéma and M. Yor. Une solution simple au probléme de Skorokhod. In Séminaire de Probabilités XIII 90–115. Lecture Notes in Math. 721. Springer, Berlin, 1979.
  • [4] R. F. Bass. Skorokhod imbedding via stochastic integrals. In Séminaire de Probabilités XVII 221–224. Lecture Notes in Math. 986. Springer, Berlin, 1983.
  • [5] H. J. Brascamp and E. H. Lieb. On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 (4) (1976) 366–389.
  • [6] E. A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
  • [7] R. Eldan. Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geometric Aspects of Functional Analysis 23 (2) (2013) 532–569.
  • [8] R. Eldan. A two-sided estimate for the Gaussian noise stability deficit. Invent. Math. 201 (2015) 561–624.
  • [9] W. Hall. On the Skorokhod embedding theorem. Technical Report 33, Dept. Statistics, Stanford Univ., 1968.
  • [10] L. Lovász and S. Vempala. The geometry of logconcave functions and sampling algorithms. Random Structures Algorithms 30 (3) (2007) 307–358.
  • [11] R. M. Loynes. Stopping times on Brownian motion: Some properties of Root’s construction. Z. Wahrsch. Verw. Gebiete 16 (1970) 211–218.
  • [12] J. K. Obłój. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004) 321–392.
  • [13] B. Oksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin, 2003.
  • [14] D. H. Root. The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40 (1969) 715–718.
  • [15] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999.
  • [16] A. V. Skorokhod. Studies in the Theory of Random Processes. Izdat. Kiev. Univ., Kiev, 1961. (In Russian.)
  • [17] P. Vallois. Le probléme de Skorokhod sur R: Une approche avec le temps local. In Séminaire de Probabilités XVII 227–239. Lecture Notes in Math. 986. Springer, Berlin, 1983.
  • [18] M. Veraar. The stochastic Fubini theorem revisited. Stochastics 84 (2012) 543–551.