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Zapiski Nauchnykh Seminarov POMI, 2011, Volume 396, Pages 144–154
(Mi znsl4656)
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Beneš condition for discontinuous exponential martingale
R. Liptser Department of Electrical Engineering Systems, Tel Aviv University, Tel Aviv, Israel
Abstract:
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.
Key words and phrases:
Girsanov's exponential martingale, uniform integrability.
Received: 29.08.2011
Citation:
R. Liptser, “Beneš condition for discontinuous exponential martingale”, Probability and statistics. Part 17, Zap. Nauchn. Sem. POMI, 396, POMI, St. Petersburg, 2011, 144–154; J. Math. Sci. (N. Y.), 188:6 (2013), 717–723
Linking options:
https://www.mathnet.ru/eng/znsl4656 https://www.mathnet.ru/eng/znsl/v396/p144
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Abstract page: | 238 | Full-text PDF : | 65 | References: | 45 |
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