# Time-homogeneous diffusions with a given marginal at a random time

Alexander M.G. Cox; David Hobson; Jan Obłój

ESAIM: Probability and Statistics (2011)

- Volume: 15, page S11-S24
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topCox, Alexander M.G., Hobson, David, and Obłój, Jan. "Time-homogeneous diffusions with a given marginal at a random time." ESAIM: Probability and Statistics 15 (2011): S11-S24. <http://eudml.org/doc/197757>.

@article{Cox2011,

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.
},

author = {Cox, Alexander M.G., Hobson, David, Obłój, Jan},

journal = {ESAIM: Probability and Statistics},

keywords = {Time-homogeneous diffusion; generalised diffusion; exponential time; Skorokhod embedding problem; Bertoin-Le Jan stopping time},

language = {eng},

month = {5},

pages = {S11-S24},

publisher = {EDP Sciences},

title = {Time-homogeneous diffusions with a given marginal at a random time},

url = {http://eudml.org/doc/197757},

volume = {15},

year = {2011},

}

TY - JOUR

AU - Cox, Alexander M.G.

AU - Hobson, David

AU - Obłój, Jan

TI - Time-homogeneous diffusions with a given marginal at a random time

JO - ESAIM: Probability and Statistics

DA - 2011/5//

PB - EDP Sciences

VL - 15

SP - S11

EP - S24

AB -
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.

LA - eng

KW - Time-homogeneous diffusion; generalised diffusion; exponential time; Skorokhod embedding problem; Bertoin-Le Jan stopping time

UR - http://eudml.org/doc/197757

ER -

## References

top- J. Bertoin and Y. Le Jan, Representation of measures by balayage from a regular recurrent point. Ann. Probab.20 (1992) 538–548. Zbl0749.60038
- 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. Zbl1012.91017
- R.V. Chacon, Potential processes. Trans. Amer. Math. Soc.226 (1977) 39–58. Zbl0366.60106
- 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. Zbl1165.60019
- J.L. Doob, Measure theory. Graduate Texts Math.143, Springer-Verlag, New York (1994). Zbl0791.28001
- B. Dupire, Pricing with a smile. Risk7 (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). Zbl0327.60029
- 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. Zbl1214.91113URIwww.warwick.ac.uk/go/dhobson/
- K. Itô, Essentials of stochastic processes. Translations Math. Monographs231, American Mathematical Society, Providence, RI, (2006), translated from the 1957 Japanese original by Yuji Ito.
- L. Jiang and Y. Tao, Identifying the volatility of underlying assets from option prices. Inverse Problems17 (2001) 137–155. Zbl0997.91024
- I.S. Kac and M.G. Kreĭn, Criteria for the discreteness of the spectrum of a singular string. Izv. Vysš. Učebn. Zaved. Matematika2 (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.
- M.G. Kreĭn, 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. Zbl0731.60071
- D.B. Madan and M. Yor, Making Markov martingales meet marginals: with explicit constructions. Bernoulli8 (2002) 509–536. Zbl1009.60037
- I. Monroe, On embedding right continuous martingales in Brownian motion. Ann. Math. Statist.43 (1972) 1293–1311. Zbl0267.60050
- J. Obłój, The Skorokhod embedding problem and its offspring. Prob. Surveys1 (2004) 321–392.
- 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). Zbl0977.60005

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.