Denote $$M(t) = f(t, \alpha(t))\exp \bigg\{-\int_0^t g(u, \alpha (u)) \, du - \int_0^t h(u, \alpha(u)) \, d\xi(u)\bigg\}$$
Here $\xi: [0,\infty) \times \Omega \to \Bbb{R}$. If for each $\omega$ the function $\xi_\omega(t) := \xi(t, \omega) \in C^1$, then $M(t)$ is a martingale.
Question: if $\xi_\omega$ is only continuous and increasing is $M^n(t)$ still a martingale?
Here $\alpha: [0,\infty) \times \Omega $ is continuous and progressively measurable $f, g, h$ are continuous functions
Maybe one could consider $\xi^n \in C^1$ with the property that $d \xi^n \to d\xi$. But I can't see why that works.