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Time-homogeneous diffusions with a given marginalat a random time

Published online by Cambridge University Press:  15 October 2010

Alexander M.G. Cox ,
David Hobson  and
Jan Obłój
Alexander M.G. Cox
Affiliation:
Dept. of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. [email protected]; web: www.maths.bath.ac.uk/~mapamgc/
David Hobson
Affiliation:
Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK; [email protected]; web: www.warwick.ac.uk/go/dhobson/
Jan Obłój
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK; [email protected]; web: www.maths.ox.ac.uk/~obloj/
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Abstract

We solve explicitly the following problem: for a given probability measure μ, we specify a generalised martingale diffusion (Xt) which, stopped at an independent exponential time T, is distributed according to μ. The process (Xt) is specified via its speed measure m. We present two heuristic arguments and three proofs. First we show how the result can be derived from the solution of [Bertoin and Le Jan, Ann. Probab. 20 (1992) 538–548.] to the Skorokhod embedding problem. Secondly, we give a proof exploiting applications of Krein's spectral theory of strings to the study of linear diffusions. Finally, we present a novel direct probabilistic proof based on a coupling argument.

Keywords

Time-homogeneous diffusiongeneralised diffusionexponential timeSkorokhod embedding problemBertoin-Le Jan stopping time
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. S11 - S24
Copyright
© EDP Sciences, SMAI, 2011

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References

Bertoin, J. and Le Jan, Y., Representation of measures by balayage from a regular recurrent point. Ann. Probab. 20 (1992) 538–548. CrossRef
P. Carr, Local Variance Gamma. Private communication (2008).
P. Carr and D. Madan, Determining volatility surfaces and option values from an implied volatility smile, in Quantitative Analysis of Financial Markets II, edited by M. Avellaneda. World Scientific (1998) 163–191.
Chacon, R.V., Potential processes. Trans. Amer. Math. Soc. 226 (1977) 39–58. CrossRef
A.M.G. Cox, Extending Chacon-Walsh: minimality and generalised starting distributions, in Séminaire de Probabilités XLI. Lecture Notes in Math. 1934, Springer, Berlin (2008) 233–264.
J.L. Doob, Measure theory. Graduate Texts Math. 143, Springer-Verlag, New York (1994).
Dupire, B., Pricing with a smile. Risk 7 (1994) 18–20.
H. Dym and H.P. McKean, Gaussian processes, function theory, and the inverse spectral problem. Probab. Math. Statist. 31. Academic Press (Harcourt Brace Jovanovich Publishers), New York (1976).
D. Hobson, The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices, in Paris-Princeton Lectures on Mathematical Finance 2010, edited by R.A. Carmona, E. Çinlar, I. Ekeland, E. Jouini, J.A. Scheinkman and N. Touzi. Lecture Notes in Math. 2003, Springer (2010) 267–318. www.warwick.ac.uk/go/dhobson/
K. Itô, Essentials of stochastic processes. Translations Math. Monographs 231, American Mathematical Society, Providence, RI, (2006), translated from the 1957 Japanese original by Yuji Ito.
Jiang, L. and Tao, Y., Identifying the volatility of underlying assets from option prices. Inverse Problems 17 (2001) 137–155.
Kac, I.S. and Kreĭn, M.G., Criteria for the discreteness of the spectrum of a singular string. Izv. Vysš. Učebn. Zaved. Matematika 2 (1958) 136–153.
F.B. Knight, Characterization of the Levy measures of inverse local times of gap diffusion, in Seminar on Stochastic Processes (Evanston, Ill., 1981). Progr. Probab. Statist. 1, Birkhäuser Boston, Mass. (1981) 53–78.
S. Kotani and S. Watanabe, Kreĭn's spectral theory of strings and generalized diffusion processes, in Functional analysis in Markov processes (Katata/Kyoto, 1981). Lecture Notes in Math. 923, Springer, Berlin (1982) 235–259.
Kreĭn, M.G., On a generalization of investigations of Stieltjes. Doklady Akad. Nauk SSSR (N.S.) 87 (1952) 881–884.
U. Küchler and P. Salminen, On spectral measures of strings and excursions of quasi diffusions, in Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372, Springer, Berlin (1989) 490–502.
Madan, D.B. and Yor, M., Making Markov martingales meet marginals: with explicit constructions. Bernoulli 8 (2002) 509–536.
Monroe, I., On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 (1972) 1293–1311. CrossRef
Obłój, J., The Skorokhod embedding problem and its offspring. Prob. Surveys 1 (2004) 321–392. CrossRef
L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales, volume 2, Itô Calculus. Cambridge University Press, Cambridge, reprint of the second edition of 1994 (2000).