Geodesic
On the problem of stochastic integral representations of functionals of the Brownian motion. I
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 375-385 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For functionals $S=S(\omega)$ of the Brownian motion $B$, we propose a method for finding stochastic integral representations based on the Itô formula for the stochastic integral associated with $B$. As an illustration of the method, we consider functionals of the “maximal” type: $S_T$, $S_{T_{-a}}$, $S_{g_T}$, and $S_{\theta_T}$, where $S_T=\max_{t\le T}B_t$ , $S_{T_{-a}}=\max_{t\le T_{-a}}B_t$ with $T_{-a}=\inf\{t>0: B_t=-a\}$, $a>0$, and $S_{g_T}=\max_{t\le g_T} B_t$, $S_{\theta_T}=\max_{t\le \theta_T}B_t$, $g_T$ and $\theta_T$ are non-Markov times: $g_T$ is the time of the last zero of Brownian motion on $[0,T]$ and $\theta_T$ is a time when the Brownian motion achieves its maximal value on $[0,T]$.
Keywords: Brownian motion, Markov time, stochastic integral, stochastic integral representation
Mots-clés : non-Markov time, Itô formula. 
@article{TVP_2003_48_2_a8,
     author = {A. N. Shiryaev and M. Yor},
     title = {On the problem of stochastic integral representations of functionals of the {Brownian} {motion.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {375--385},
     year = {2003},
     volume = {48},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a8/}
}
TY  - JOUR
AU  - A. N. Shiryaev
AU  - M. Yor
TI  - On the problem of stochastic integral representations of functionals of the Brownian motion. I
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2003
SP  - 375
EP  - 385
VL  - 48
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a8/
LA  - ru
ID  - TVP_2003_48_2_a8
ER  - 
%0 Journal Article
%A A. N. Shiryaev
%A M. Yor
%T On the problem of stochastic integral representations of functionals of the Brownian motion. I
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2003
%P 375-385
%V 48
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a8/
%G ru
%F TVP_2003_48_2_a8
A. N. Shiryaev; M. Yor. On the problem of stochastic integral representations of functionals of the Brownian motion. I. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 375-385. http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a8/

[1] Clark I. M. C., “The representation of functionals of Brownian motion by stochastic integrals”, Ann. Math. Statist., 41:4 (1970), 1282–1295 | DOI | MR | Zbl

[2] Itô K., “Multiple Wiener integral”, J. Math. Soc. Japan, 3:1 (1951), 157–169 | DOI | MR | Zbl

[3] Revuz D., Yor M., Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, Heidelberg, 1994, 560 pp. | MR | Zbl

[4] Rogers L. C. G., Williams D., Diffusions, Markov Processes, and Martingales, v. 2, Itô Calculus, Wiley, Chichester, 1987, 475 pp. | MR

[5] Liptser R. Sh., Shiryaev A. N., Statistika sluchainykh protsessov, Nauka, M., 1974 | MR | Zbl

[6] Graversen S. E., Peskir G., Shiryaev A. N., “Stopping Brownian motion without anticipation as close as possible to its ultimate maximum”, Teoriya veroyatn. i ee primen., 45:1 (2000), 125–136 | MR | Zbl

[7] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 1, Fakty, modeli, FAZIS, M., 1998, 512 pp.; т. 2, Теория, 544 с.

[8] Kunita H., “Some extensions of Itô's formula”, Lecture Notes in Math., 850, 1981, 118–141 | MR | Zbl

[9] Venttsel A. D., “Ob uravneniyakh teorii uslovnykh markovskikh protsessov”, Teoriya veroyatn. i ee primen., 10:2 (1965), 390–393