# Two Brownian Motions and stopping time: expected value and variance

Suppose I have the following two processes $$S_t= S_0 + \int^t_0\mu^{*}_s\,ds +\int^t_0\sigma^{*}_s \, dW_s$$ and $$P_t=P_0 + \int^{t^{-}}_0\mu_s \, ds +\int^{t^{-}}_0\sigma_s \, dW_s.$$ With $t-t^- = \tau$$I now define X_t = S_t-P_{t}, so that$$X_t = S_0 - P_0 +\int^t_{t^{-}}\mu^{*}_s\,ds+\int^t_{t^{-}}\sigma^{*}_s\,dW_s + \int^{t^{-}}_0(\mu^{*}_s - \mu_s) \, ds + \int^{t^{-}}_0 (\sigma_s^*-\sigma_s) dW_s.$$Furthermore if at any point$X_t$crosses/touches a barrier at$0$,$P_t=0$and$S_t=0$for all subsequent times after$t$. So I'd like to calculate$E[P_T]$and$V[P_T]$analytically. Obviously (ignoring the barrier) all variables are Gaussian with analytically known distributions. Also it is clear that a lower bound on the barrier crossing probability is given by the probability of process$X_t$being below$0$at time$t$. Also I am aware of the standard barrier probability crossing result for a Brownian motion with a drift. Question 1: Is the following true:$E[P_t] = \int^\infty_0 \phi(x)\int^{x-\tau}_0\mu_s \, ds \, dx$where$x$is the stopping time of$X_t$crossing the barrier at$0$and$\phi(x)$is the pdf of the stopping time? Question 2: Is the following true: Is$V[P_t] = \int^\infty_0 \varphi(x) \int^{x-\tau}_0 \sigma^2_s \, ds \, dx$? Question 3: Can stopping time distribution be determined analytically - eg by using the standard result for the stopping time of a Brownian motion with a drift, and for example scaling$\sigma_s$and$\mu_s$to get a new BM with adjusted drift and volatility to time$t\$?