Annals of Probability

Embedding laws in diffusions by functions of time

A. M. G. Cox and G. Peskir

Full-text: Open access


We present a constructive probabilistic proof of the fact that if $B=(B_{t})_{t\ge0}$ is standard Brownian motion started at $0$, and $\mu$ is a given probability measure on $\mathbb{R}$ such that $\mu(\{0\})=0$, then there exists a unique left-continuous increasing function $b:(0,\infty)\rightarrow\mathbb{R}\cup\{+\infty\}$ and a unique left-continuous decreasing function $c:(0,\infty)\rightarrow\mathbb{R}\cup\{-\infty\}$ such that $B$ stopped at $\tau_{b,c}=\inf\{t>0\vert B_{t}\ge b(t)\mbox{ or }B_{t}\le c(t)\}$ has the law $\mu$. The method of proof relies upon weak convergence arguments arising from Helly’s selection theorem and makes use of the Lévy metric which appears to be novel in the context of embedding theorems. We show that $\tau_{b,c}$ is minimal in the sense of Monroe so that the stopped process $B^{\tau_{b,c}}=(B_{t\wedge\tau_{b,c}})_{t\ge0}$ satisfies natural uniform integrability conditions expressed in terms of $\mu$. We also show that $\tau_{b,c}$ has the smallest truncated expectation among all stopping times that embed $\mu$ into $B$. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.

Article information

Ann. Probab., Volume 43, Number 5 (2015), 2481-2510.

Received: January 2013
Revised: May 2014
First available in Project Euclid: 9 September 2015

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] 60J65: Brownian motion [See also 58J65]
Secondary: 60F05: Central limit and other weak theorems 60J60: Diffusion processes [See also 58J65]

Skorokhod embedding Brownian motion diffusion process Markov process Helly’s selection theorem weak convergence Lévy metric reversed barrier minimal stopping time


Cox, A. M. G.; Peskir, G. Embedding laws in diffusions by functions of time. Ann. Probab. 43 (2015), no. 5, 2481--2510. doi:10.1214/14-AOP941.

Export citation


  • [1] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [2] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel.
  • [3] Chacon, R. M. (1985). Barrier stopping times and the filling scheme. Ph.D. dissertation, Univ. Washington, Seattle.
  • [4] Chacon, R. V. and Ornstein, D. S. (1960). A general ergodic theorem. Illinois J. Math. 4 153–160.
  • [5] Cox, A. M. G. and Hobson, D. G. (2006). Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Related Fields 135 395–414.
  • [6] Cox, A. M. G. and Wang, J. (2013). Root’s barrier: Construction, optimality and applications to variance options. Ann. Appl. Probab. 23 859–894.
  • [7] Dinges, H. (1974). Stopping sequences. In Séminaire de Probabilitiés, VIII (Univ. Strasbourg, Année Universitaire 19721973). Lecture Notes in Math. 381 27–36. Springer, Berlin.
  • [8] Dubins, L. E. (1968). On a theorem of Skorohod. Ann. Math. Statist. 39 2094–2097.
  • [9] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris-Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 267–318. Springer, Berlin.
  • [10] Itô, K. and McKean, H. P. Jr. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.
  • [11] Kleptsyn, V. and Kurtzmann, A. (2012). A counter-example to the Cantelli conjecture. Submitted. Available at arXiv:1202.2250v1.
  • [12] Loynes, R. M. (1970). Stopping times on Brownian motion: Some properties of Root’s construction. Z. Wahrsch. Verw. Gebiete 16 211–218.
  • [13] McConnell, T. R. (1991). The two-sided Stefan problem with a spatially dependent latent heat. Trans. Amer. Math. Soc. 326 669–699.
  • [14] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293–1311.
  • [15] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
  • [16] Pedersen, J. L. and Peskir, G. (2001). The Azéma–Yor embedding in non-singular diffusions. Stochastic Process. Appl. 96 305–312.
  • [17] Peskir, G. (1999). Designing options given the risk: The optimal Skorokhod-embedding problem. Stochastic Process. Appl. 81 25–38.
  • [18] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • [19] Root, D. H. (1969). The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40 715–718.
  • [20] Rost, H. (1971). The stopping distributions of a Markov process. Invent. Math. 14 1–16.
  • [21] Rost, H. (1976). Skorokhod stopping times of minimal variance. In Séminaire de Probabilités, X (Première Partie, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975). Lecture Notes in Math. 511 194–208. Springer, Berlin.
  • [22] Skorokhod, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.