Geodesic
Beneš condition for discontinuous exponential martingale
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 144-154 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

It is known that the Girsanov exponent $\mathfrak z_t$, being solution of Doléans-Dade equation $\mathfrak z_t=1+\int_0^t\mathfrak z_s\alpha(s)\,dB_s$ generated by Brownian motion $B_t$ and a random process $\alpha(t)$ with $\int_0^t\alpha^2(s)\,ds<\infty$ a.s., is the martingale provided that the Beneš condition $$ |\alpha(t)|^2\le\mathrm{const.}\big[1+\sup_{s\in[0,t]}B^2_s\big],\quad\forall\ t>0, $$ holds true. In this paper, we show that $\int_0^t\alpha(s)\,dB_s$ can be replaced by a purely discontinuous square integrable martingale $M_t$ paths from the Skorokhod space $ \mathbb D_{[0,\infty)}$ having jumps $\alpha(s)\triangle M_t>-1$. The method of proof differs from the original Beneš proof. 
@article{ZNSL_2011_396_a8,
     author = {R. Liptser},
     title = {Bene\v{s} condition for discontinuous exponential martingale},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {144--154},
     year = {2011},
     volume = {396},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a8/}
}
TY  - JOUR
AU  - R. Liptser
TI  - Beneš condition for discontinuous exponential martingale
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 144
EP  - 154
VL  - 396
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a8/
LA  - en
ID  - ZNSL_2011_396_a8
ER  - 
%0 Journal Article
%A R. Liptser
%T Beneš condition for discontinuous exponential martingale
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 144-154
%V 396
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a8/
%G en
%F ZNSL_2011_396_a8
R. Liptser. Beneš condition for discontinuous exponential martingale. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 144-154. http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a8/

[1] L. B. Andersen, V. V. Piterbarg, “Moment explosions in stochastic volatility models”, Finance Stoch., 11 (2007), 29–50 | DOI | MR | Zbl

[2] V. E. Benes, “Existence of optimal stochastic control laws”, SIAM J. Control, 9 (1971), 446–475 | DOI | MR

[3] I. V. Girsanov, “On transforming a certan class of stochastic processes by absolutely continuous substitution of measures”, Theory Probab. Appl., 5 (1960), 285–301 | DOI | MR

[4] J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Springer-Verlag, Berlin, 2003 | MR

[5] M. Hitsuda, “Representation of Gaussian processes equivalent to Wiener process”, Osaka J. Math., 5 (1968), 299–312 | MR | Zbl

[6] I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York–Berlin–Heidelberg, 1991 | MR | Zbl

[7] N. Kazamaki, “On a problem of Girsanov”, Tôhoku Math. J., 29 (1977), 597–600 | DOI | MR | Zbl

[8] N. Kazamaki, T. Sekiguchi, “On the transformation of some classes of martingales by a change of law”, Tôhoku Math. J., 31 (1979), 261–279 | DOI | MR | Zbl

[9] N. V. Krylov, A simple proof of a result of A. Novikov, May 8, 2009, arXiv: math/0207013v2[math.PR]

[10] R. Sh. Liptser, A. N. Shiryayev, Theory of Martingales, Kluwer Acad. Publ., 1989 | MR | Zbl

[11] R. Sh. Liptser, A. N. Shiryaev, Statistics of Random Processes, v. I, 2nd ed., Springer-Verlag, Berlin–New York, 2000 | Zbl

[12] A. A. Novikov, “On the conditions of the uniform integrability of the continuous nonnegative martingales”, Theory Probab. Appl., 24:4 (1979), 821–825 | MR | Zbl

[13] A. Üstünel, M. Zakai, Transformation of measure on Wiener space, Springer-Verlag, Berlin–New York, 2000 | MR