Consider the stochastic integral
$$ Z_t = 1+\int_0^tZ_{s^{-}}\,dX_s $$
where $X$ is a Martingale. In the textbook by Shreve (see here pages 493-493) it is said that since $Z_{s^{-}}$ is left-continuous and the integrator $X$ is a martingale then $Z_t$ is a martingale. I cannot figure out why this should be the case, any suggestions?
In fact, one can show that for any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of p-integrable martingales. If your process is a bounded left-continuous process, then it holds. You could have more info on the following link : https://almostsure.wordpress.com/2010/04/06/the-burkholder-davis-gundy-inequality/