What are the SDE's satisfied by the following processes?
- $X_t = B_t^q$
- $X_t = (\sin B_t)^q$
- $X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the equations need only be valid up to a (nontrivial) stopping time. For each of these, what is the appropriate $C^1$ process $C_t$ such that $M_t = C_tX_t$ is a local martingale and what is the function $R_t$ such that$$dM_t = R_t\,M_t\,dB_t?$$
