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

TL;DR

This paper extends the Bene${ m reve{s}}$ condition for the martingale property of Girsanov exponent from Brownian motion to purely discontinuous martingales with jumps, broadening its applicability.

Contribution

It introduces a new Bene${ m reve{s}}$ condition for discontinuous martingales, replacing Brownian motion with a jump process, and provides a compatible proof method.

Findings

The Girsanov exponent remains a martingale under the new condition.

The condition applies to jump martingales with independent increments.

The proof method is adaptable to both continuous and discontinuous martingales.

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 $M_t$.