Consider a parametrised martingale as follows :
$$X^x_t := x+ \int_0^t\sqrt{2p_s} \, dW_s,$$
where $W$ is a standard Brownian motion and $(p_t)_{t\ge 0}$ is a locally square integrable process satisfying $p_t\ge 1$ for all $t\ge 0$. Set $\tau^x:=\{t\ge 0: X^x_t\notin [0,1]\}$ and define the function $F: [0,1]\to \mathbb R$ by
$$F(x):=\mathbb E \left[\int_0^{\min(1,\tau_x)} \big(1+\log(p_s)\big) \, ds \right].$$
Can we show $F$ is continuous, Hölder-continuous or Lipschitz?
