Atypical exponential martingale

Question

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} \Delta M_s - \int_0^t a_s \mathrm{d}s $$ It is well known that if all the jumps of $M_t$ are greater than $-1$ then the process: $$ e^{M_t}\prod_{0<s\leq t} (1+\Delta M_s)e^{-\Delta M_s} $$ is a positive martingale. I would like to generalise this result a bit. Assume we have a paramter $\theta$ (feel free to introduce bounds on the values), and the process: $$ N_t = \prod_{0<s\leq t} (1+\Delta M_s)^\theta G_t $$ What should be the form of the process $\{G\}$ to make $\{N\}$ a positive martingale?