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

Let $mathfrak{z}$ be a stochastic exponential, i.e., $mathfrak{z}_t=1+int_0^tmathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $triangle M_t>-1$. Then $mathfrak{z}$ is a nonnegative local martingale with $Emathfrak{z}_tle 1$. If $Emathfrak{z}_T= 1$, 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 $Emathfrak{z}_{_T}=1$ 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}.