Geodesic
Time-homogeneous diffusions with a given marginal at a random time
ESAIM: Probability and Statistics, Tome 15 (2011), pp. S11-S24.

Voir la notice de l'article provenant de la source Numdam

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.

DOI : 10.1051/ps/2010021
Classification : 60G40, 60J60
Keywords: time-homogeneous diffusion, generalised diffusion, exponential time, Skorokhod embedding problem, Bertoin-Le Jan stopping time 
@article{PS_2011__15__S11_0,
     author = {Cox, Alexander M. G. and Hobson, David and Ob{\l}\'oj, Jan},
     title = {Time-homogeneous diffusions with a given marginal at a random time},
     journal = {ESAIM: Probability and Statistics},
     pages = {S11--S24},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     doi = {10.1051/ps/2010021},
     mrnumber = {2817342},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2010021/}
}
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
PY  - 2011
SP  - S11
EP  - S24
VL  - 15
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/ps/2010021/
DO  - 10.1051/ps/2010021
LA  - en
ID  - PS_2011__15__S11_0
ER  - 
%0 Journal Article
%A Cox, Alexander M. G.
%A Hobson, David
%A Obłój, Jan
%T Time-homogeneous diffusions with a given marginal at a random time
%J ESAIM: Probability and Statistics
%D 2011
%P S11-S24
%V 15
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/ps/2010021/
%R 10.1051/ps/2010021
%G en
%F PS_2011__15__S11_0
Cox, Alexander M. G.; Hobson, David; Obłój, Jan. Time-homogeneous diffusions with a given marginal at a random time. ESAIM: Probability and Statistics, Tome 15 (2011), pp. S11-S24. doi : 10.1051/ps/2010021. http://geodesic.mathdoc.fr/articles/10.1051/ps/2010021/

[1] J. Bertoin and Y. Le Jan, Representation of measures by balayage from a regular recurrent point. Ann. Probab. 20 (1992) 538-548. | Zbl | MR

[2] P. Carr, Local Variance Gamma. Private communication (2008).

[3] 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. | Zbl | MR

[4] R.V. Chacon, Potential processes. Trans. Amer. Math. Soc. 226 (1977) 39-58. | Zbl | MR

[5] 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. | Zbl | MR

[6] J.L. Doob, Measure theory. Graduate Texts Math. 143, Springer-Verlag, New York (1994). | Zbl | MR

[7] B. Dupire, Pricing with a smile. Risk 7 (1994) 18-20.

[8] 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). | Zbl | MR

[9] 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/ | Zbl | MR

[10] 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. | Zbl | MR

[11] L. Jiang and Y. Tao, Identifying the volatility of underlying assets from option prices. Inverse Problems 17 (2001) 137-155. | Zbl | MR

[12] I.S. Kac and M.G. Kreĭn, Criteria for the discreteness of the spectrum of a singular string. Izv. Vysš. Učebn. Zaved. Matematika 2 (1958) 136-153. | MR

[13] 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. | Zbl | MR

[14] 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. | Zbl | MR

[15] M.G. Kreĭn, On a generalization of investigations of Stieltjes. Doklady Akad. Nauk SSSR (N.S.) 87 (1952) 881-884. | Zbl | MR

[16] 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. | Zbl | MR | mathdoc-id

[17] D.B. Madan and M. Yor, Making Markov martingales meet marginals: with explicit constructions. Bernoulli 8 (2002) 509-536. | Zbl | MR

[18] I. Monroe, On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 (1972) 1293-1311. | Zbl | MR

[19] J. Obłój, The Skorokhod embedding problem and its offspring. Prob. Surveys 1 (2004) 321-392. | Zbl | MR

[20] 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). | Zbl | MR

Cité par Sources :