When a Stochastic Exponential is a True Martingale. Extension of a Method of Bene^s

Let $\mathfrak{z}$ be a stochastic exponential, i.e., ${\mathfrak{z}}_t=1+{\int }_0^t{\mathfrak{z}}_{s-}d{M}_s$, of a local martingale $M$ with jumps $\mathrm{△}{M}_t>-1$. Then $\mathfrak{z}$ is a nonnegative local martingale with $\text{E}{\mathfrak{z}}_t\le 1$. If $\text{E}{\mathfrak{z}}_T=1$, then $\mathfrak{z}$ is a martingale on the time interval $[0,T]$. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property $\text{E}{\mathfrak{z}}_{{}_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov \cite{Girs}. These conditions generalize the Bene^s idea, \cite{Benes}, and avoid the technology of piece-wise approximation. These conditions are applicable even if Novikov, \cite{Novikov}, and Kazamaki, \cite{Kaz}, conditions fail. They are effective for Markov processes that explode, Markov processes with jumps and also non Markov processes. Our approach is different to recently published papers \cite{CFY} and \cite{MiUr}.