Strict Local Martingales and the Khasminskii test for Explosions

We exhibit sufficient conditions such that components of a multidimensional SDE giving rise to a local martingale $M$ are strict local martingales or martingales. We assume that the equations have diffusion coefficients of the form $\sigma (M_t,v_t),$ with $v_t$ being a stochastic volatility term.