Preservation of martingale property under absolutely continuous monotone time change

Question

Suppose I have a martingale $M(t)$ adapted to some filtration $\mathbb{F}=\{\mathcal{F}_t\}_{t\geq 0}$ and a positive monotonically increasing time change $T(t)=\int_0^tv(s)\mathrm{d}s$ with an $\mathbb{F}$-adapted process $v(s)$ which is not necessarily independent of $M(t)$. Is the time changed process $M(T(t))$ still a martingale? This is true when we have independence of $M(t)$ and $v(t)$ because then we can integrate things out by conditioning. But does it still hold when the processes are not independent? And if so, to which filtration is the time changed martingale adapted? (I allow the filtration to be enlarged).

Answer 1

No, even under continuous time changes, the martingale property fails to be preserved. Sin 1998, Theorem 3.2, finds a stochastic volatility model (so a time changed model of the form $T_t=\int_0^tv_sds$) which under some parametrict assumption is a strict local martingale. In general, after time changing with a continuous time change only the local martingale property is preserved. By looking into Jacod 1979, chapter 10, one gets the full picture.