From minimal embeddings to minimal diffusions

Alexander Matthew Gordon Cox (University of Bath)
Martin Klimmek (Mathematical Institute, University of Oxford)


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-13

Publication Date: June 11, 2014

DOI: 10.1214/ECP.v19-2889


  • 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. MR2598129
  • 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. MR0544782
  • 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. MR1143434
  • 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. MR0445598
  • 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. MR3076672
  • 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. MR2483735
  • 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. MR2240692
  • Cox, Alexander M. G.; Hobson, David; Obłój, 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. MR2817342
  • 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. MR0112192
  • 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. MR0345224
  • 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. MR2988396
  • 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. MR2276894
  • Kotani, S.; Watanabe, S. Kreĭn'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. MR0661628
  • Monroe, Itrel. On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 (1972), 1293--1311. MR0343354
  • Obłój, Jan. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004), 321--390. MR2068476
  • 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. MR0896743
  • Rogers, L. C. G. A guided tour through excursions. Bull. London Math. Soc. 21 (1989), no. 4, 305--341. MR0998631
  • L.C.G. Rogers and D. Williams, Diffusions, Markov processes and Martingales, Volume 2, Cambridge University Press, 2000. MR1780932
  • Salminen, P. Optimal stopping of one-dimensional diffusions. Math. Nachr. 124 (1985), 85--101. MR0827892

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