Bene$\check{\bf S}$ condition for discontinuous exponential martingale

Abstract

It is known the Girsanov exponent $\mathfrak{z}_t$, being solution of Doleans-Dade equation $ \mathfrak{z_t}=1+\int_0^t\alpha(\omega,s)dB_s $ generated by Brownian motion $B_t$ and a random process $\alpha(\omega,t)$ with $\int_0^t\alpha^2(\omega,s)ds<\infty$ a.s., is the martingale provided that the Bene${\rm \check{s}}$ condition $$ |\alpha(\omega,t)|^2\le \text{\rm const.}\big[1+\sup_{s\in[0,t]}B^2_s\big], \forall t>0, $$ holds true. In this paper, we show $B_t$ can be replaced by by a homogeneous purely discontinuous square integrable martingale $M_t$ with independent increments and paths from the Skorokhod space $ \mathbb{D}_{[0,\infty)} $ having positive jumps $\triangle M_t$ with $\E\sum_{s\in[0,t]}(\triangle M_s)^3<\infty$. A function $\alpha(\omega,t)$ is assumed to be nonnegative and predictable. Under this setting $\mathfrak{z}_t$ is the martingale provided that $$ \alpha^2(\omega,t)\le \text{\rm const.}\big[1+\sup_{s\in[0,t]}M^2_{s-}\big], \ \forall t>0. $$ The method of proof differs from the original Bene${\rm \check{s}}$ one and is compatible for both setting with $B_t$ and $M_t$.