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

Abstract

It is known the Girsanov exponent ${\mathfrak{z}}_t$, being solution of Doleans-Dade equation ${\mathfrak{z}}_{\mathfrak{t}}=1+{\int }_0^t\alpha (\omega ,s)d{B}_s$ 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$\check{\mathrm{s}}$ condition $$|\alpha (\omega ,t){|}^2\le \text{const.}[1+\underset{s\in [0,t]}{\sup }{B}_s^2],\forall t>0,$$ 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 $△M_t$ 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 $${\alpha }^2(\omega ,t)\le \text{const.}[1+\underset{s\in [0,t]}{\sup }{M}_{s-}^2],\forall t>0.$$ The method of proof differs from the original Bene$\check{\mathrm{s}}$ one and is compatible for both setting with $B_t$ and $M_t$.

8 pages