Well, not quite. If X(t) is the mathematical VGSSD process, then [$]\phi_{X(t+1) - X(t)}(u)[$] is the ratio of the expression on the r.h.s of (15) of O'Sullivan at t+1 and t. Then, for the risk-neutral stock price process, which has a compensator drift, you need something like (**) [$]f_{t,t+1}(u) ...