Electronic Communications in Probability

From minimal embeddings to minimal diffusions

Alexander Cox and Martin Klimmek

Full-text: Open access


We show that there is a one-to-one correspondence between diffusions and the solutions of the Skorokhod Embedding Problem due to Bertoin and Le-Jan. In particular, the minimal embedding corresponds to a "minimal local martingale diffusion", which is a notion we introduce in this article. Minimality is closely related to the martingale property. A diffusion is minimal if it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local time characteristics of time-homogeneous diffusions.

Article information

Electron. Commun. Probab. Volume 19 (2014), paper no. 34, 13 pp.

Accepted: 11 June 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J55: Local time and additive functionals

diffusion minimality local-martingales Skorokhod embedding problem

This work is licensed under a Creative Commons Attribution 3.0 License.


Cox, Alexander; Klimmek, Martin. From minimal embeddings to minimal diffusions. Electron. Commun. Probab. 19 (2014), paper no. 34, 13 pp. doi:10.1214/ECP.v19-2889. https://projecteuclid.org/euclid.ecp/1465316736

Export citation


  • Albanese, C.; Kuznetsov, A. Transformations of Markov processes and classification scheme for solvable driftless diffusions. Markov Process. Related Fields 15 (2009), no. 4, 563–574.
  • Azema, Jacques; Yor, Marc. Une solution simple au probleme de Skorokhod. (French) Seminaire de Probabilites, XIII (Univ. Strasbourg, Strasbourg, 1977/78), pp. 90–115, Lecture Notes in Math., 721, Springer, Berlin, 1979.
  • Azema, Jacques; Yor, Marc. Le problème de Skorokhod: compléments à ”Une solution simple au problème de Skorokhod”, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 625–633.
  • Bertoin, J.; Le Jan, Y. Representation of measures by balayage from a regular recurrent point. Ann. Probab. 20 (1992), no. 1, 538–548.
  • Chacon, R. V.; Walsh, J. B. One-dimensional potential embedding. Seminaire de Probabilites, X (Premiere partie, Univ. Strasbourg, Strasbourg, annee universitaire 1974/1975), pp. 19–23. Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976.
  • Cox, Alexander M. G.; Wang, Jiajie. Root's barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23 (2013), no. 3, 859–894.
  • Cox, A. M. G. Extending Chacon-Walsh: minimality and generalised starting distributions. Seminaire de probabilites XLI, 233–264, Lecture Notes in Math., 1934, Springer, Berlin, 2008.
  • Cox, A. M. G.; Hobson, D. G. Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Related Fields 135 (2006), no. 3, 395–414.
  • Cox, Alexander M. G.; Hobson, David; ObłÃ³j, Jan. Time-homogeneous diffusions with a given marginal at a random time. ESAIM Probab. Stat. 15 (2011), In honor of Marc Yor, suppl., S11–S24.
  • F. Delbaen and H. Shirakawa, No arbitrage condition for positive diffusion price processes, Asia-Pacific Financial Markets 9 (2002), no. 3, 159–168.
  • Feller, William. The birth and death processes as diffusion processes. J. Math. Pures Appl. (9) 38 1959 301–345.
  • T. Huillet, On the Karlin–Kimura approaches to the Wright–Fisher diffusion with fluctuating selection, Journal of Statistical Mechanics: Theory and Experiment 2011 (2011), no. 02, P02016.
  • Ito, Kiyosi; McKean, Henry P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp.
  • Klimmek, Martin. The Wronskian parametrises the class of diffusions with a given distribution at a random time. Electron. Commun. Probab. 17 (2012), no. 50, 8 pp.
  • Kotani, Shinichi. On a condition that one-dimensional diffusion processes are martingales. In memoriam Paul-Andre Meyer: Seminaire de Probabilites XXXIX, 149–156, Lecture Notes in Math., 1874, Springer, Berlin, 2006.
  • Kotani, S.; Watanabe, S. KreÄ­n's spectral theory of strings and generalized diffusion processes. Functional analysis in Markov processes (Katata/Kyoto, 1981), pp. 235–259, Lecture Notes in Math., 923, Springer, Berlin-New York, 1982.
  • Monroe, Itrel. On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 (1972), 1293–1311.
  • ObłÃ³j, Jan. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004), 321–390.
  • Perkins, Edwin. The Cereteli-Davis solution to the $H^ 1$-embedding problem and an optimal embedding in Brownian motion. Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985), 172–223, Progr. Probab. Statist., 12, Birkhauser Boston, Boston, MA, 1986.
  • Rogers, L. C. G. A guided tour through excursions. Bull. London Math. Soc. 21 (1989), no. 4, 305–341.
  • L.C.G. Rogers and D. Williams, Diffusions, Markov processes and Martingales, Volume 2, Cambridge University Press, 2000.
  • Salminen, P. Optimal stopping of one-dimensional diffusions. Math. Nachr. 124 (1985), 85–101.