Electronic Journal of Probability

Classes of measures which can be embedded in the Simple Symmetric Random Walk

Alexander Cox and Jan Obloj

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Abstract

We characterize the possible distributions of a stopped simple symmetric random walk $X_\tau$, where $\tau$ is a stopping time relative to the natural filtration of $(X_n)$. We prove that any probability measure on $\mathbb{Z}$ can be achieved as the law of $X_\tau$ where $\tau$ is a minimal stopping time, but the set of measures obtained under the further assumption that $(X_{n\land \tau}:n\geq 0)$ is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on $\mathbb{Z}$. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the Chacon-Walsh (1976) and Azema-Yor (1979) embeddings and show that they lead to yet smaller sets of achievable measures. Finally, we solve explicitly the Skorokhod embedding problem constructing, for a given measure $\mu$, a minimal stopping time $\tau$ which embeds $\mu$ and which further is uniformly integrable whenever a uniformly integrable embedding of $\mu$ exists.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 42, 1203-1228.

Dates
Accepted: 31 July 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819115

Digital Object Identifier
doi:10.1214/EJP.v13-516

Mathematical Reviews number (MathSciNet)
MR2430705

Zentralblatt MATH identifier
1196.60080

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 28A80: Fractals [See also 37Fxx]

Keywords
Skorokhod embedding problem random walk minimal stopping time Azema-Yor stopping time Chacon-Walsh stopping time iterated function system self-similar set fractal uniform integrability

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Cox, Alexander; Obloj, Jan. Classes of measures which can be embedded in the Simple Symmetric Random Walk. Electron. J. Probab. 13 (2008), paper no. 42, 1203--1228. doi:10.1214/EJP.v13-516. https://projecteuclid.org/euclid.ejp/1464819115


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References

  • Azéma, Jacques; Yor, Marc. Une solution simple au problème de Skorokhod. (French) Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), pp. 90–115, Lecture Notes in Math., 721, Springer, Berlin, 1979. (82c:60073a)
  • Barnsley, M. F.; Demko, Stephen. Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 243–275.
  • Chacon, R. V. Potential processes. Trans. Amer. Math. Soc. 226 (1977), 39–58.
  • Chacon, R. V.; Walsh, J. B. One-dimensional potential embedding. Séminaire de Probabilités, X (Prèmiere partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), pp. 19–23. Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976.
  • Cox, A.M.G.. Extending Chacon-Walsh: Minimality and generalised starting distributions. Séminaire de Probabilités, XLI, Lecture Notes in Math., 1934, Springer, Berlin, 2008. ISBN:978-3-540-77912-4
  • 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.
  • Dinges, Hermann. Stopping sequences. Journées de la Sociéte Mathématique de France de Probabilités, Strasbourg, 25 Mai 1973. Séminaire de Probabilitiés, VIII (Univ. Strasbourg, année universitaire 1972–1973), pp. 27–36. Lecture Notes in Math., Vol. 381, Springer, Berlin, 1974.
  • Falconer, Kenneth. Fractal geometry. Mathematical foundations and applications. Second edition. John Wiley & Sons, Inc., Hoboken, NJ, 2003. xxviii+337 pp. ISBN: 0-470-84861-8
  • Fujita, Tkahiko. Certain martingales of simple symmetric random walk and their applications. Private Communication (2004)
  • Hall, W.J. On the Skorokhod embedding theorem. Tech. Report 33 (1968), Stanford Univ., Dept. of Stat.
  • Jacka, S. D. Doob's inequalities revisited: a maximal $H\sp 1$-embedding. Stochastic Process. Appl. 29 (1988), no. 2, 281–290.
  • Meilijson, Isaac. On the Azéma-Yor stopping time. Seminar on probability, XVII, 225–226, Lecture Notes in Math., 986, Springer, Berlin, 1983.
  • Monroe, Itrel. On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 (1972), 1293–1311.
  • Obój, Jan. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004), 321–390 (electronic).
  • Obój, Jan; Pistorius, Martijn. An explicit Skorokhod embedding for spectrally negative Lévy processes. J. Theoret. Probab. (2008), to appear, DOI:10.1007/s10959-008-0157-7.
  • Obój, Jan; Yor, Marc. An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale. Stochastic Process. Appl. 110 (2004), no. 1, 83–110.
  • Perkins, Edwin. The Cereteli-Davis solution to the $H\sp 1$-embedding problem and an optimal embedding in Brownian motion. Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985), 172–223, Progr. Probab. Statist., 12, Birkhäuser Boston, Boston, MA, 1986.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
  • Root, D. H. The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40 1969 715–718.
  • Rost, Hermann. Darstellung einer Ordnung von Maßen durch Stoppzeiten. (German) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 15 1970 19–28.
  • Rost, Hermann. Markoff-Ketten bei sich füllenden Löchern im Zustandsraum. (German) Ann. Inst. Fourier (Grenoble) 21 (1971), no. 1, 253–270.
  • Rost, Hermann. The stopping distributions of a Markov Process. Invent. Math. 14 (1971), 1–16.
  • Skorokhod, A. V. Studies in the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965 viii+199 pp. (32 #3082b)