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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 $t$?

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