Bene\v s condition for discontinuous exponential martingale

R. Liptser

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 144-154

Résumé

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.