• Bernoulli
  • Volume 21, Number 3 (2015), 1341-1360.

Mimicking self-similar processes

Jie Yen Fan, Kais Hamza, and Fima Klebaner

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We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable Lévy processes as well as an example of an artificial process having the marginals of $t^{\kappa}V$ for some symmetric random variable $V$. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.

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Bernoulli, Volume 21, Number 3 (2015), 1341-1360.

Received: September 2012
Revised: August 2013
First available in Project Euclid: 27 May 2015

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Lévy processes martingales with given marginals self-similar


Fan, Jie Yen; Hamza, Kais; Klebaner, Fima. Mimicking self-similar processes. Bernoulli 21 (2015), no. 3, 1341--1360. doi:10.3150/13-BEJ588.

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  • [1] Albin, J.M.P. (2008). A continuous non-Brownian motion martingale with Brownian motion marginal distributions. Statist. Probab. Lett. 78 682–686.
  • [2] Baker, D., Donati-Martin, C. and Yor, M. (2011). A sequence of Albin type continuous martingales with Brownian marginals and scaling. In Séminaire de Probabilités XLIII. Lecture Notes in Math. 2006 441–449. Berlin: Springer.
  • [3] Bertoin, J. (1999). Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 1–91. Berlin: Springer.
  • [4] Dupire, B. (1994). Pricing with a smile. Risk 7 18–20.
  • [5] Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton Series in Applied Mathematics. Princeton, NJ: Princeton Univ. Press.
  • [6] Fan, J.Y., Hamza, K. and Klebaner, F.C. (2012). On the Markov property of some Brownian martingales. Stochastic Process. Appl. 122 3506–3512.
  • [7] Fitzsimmons, P.J. (2001). Hermite martingales. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 153–157. Berlin: Springer.
  • [8] Hamza, K. and Klebaner, F.C. (2006). On nonexistence of non-constant volatility in the Black–Scholes formula. Discrete Contin. Dyn. Syst. Ser. B 6 829–834 (electronic).
  • [9] Hamza, K. and Klebaner, F.C. (2007). A family of non-Gaussian martingales with Gaussian marginals. J. Appl. Math. Stoch. Anal. 2007 Art. ID 92723, 19.
  • [10] Hirsch, F., Profeta, C., Roynette, B. and Yor, M. (2011). Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series 3. Milan: Springer.
  • [11] Hirsch, F. and Roynette, B. (2012). A new proof of Kellerer’s theorem. ESAIM Probab. Stat. 16 48–60.
  • [12] Kellerer, H.G. (1972). Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198 99–122.
  • [13] Klebaner, F.C. (2012). Introduction to Stochastic Calculus with Applications, 3rd ed. London: Imperial College Press.
  • [14] Madan, D.B. and Yor, M. (2002). Making Markov martingales meet marginals: With explicit constructions. Bernoulli 8 509–536.
  • [15] Oleszkiewicz, K. (2008). On fake Brownian motions. Statist. Probab. Lett. 78 1251–1254.
  • [16] Plucińska, A. (1998). A stochastic characterization of Hermite polynomials. J. Math. Sci. (New York) 89 1541–1544. Stability problems for stochastic models, Part 1 (Moscow, 1996).
  • [17] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original, Revised by the author.
  • [18] Shreve, S.E. (2004). Stochastic Calculus for Finance. II. Continuous-Time Models. Springer Finance. New York: Springer.