pr.probability – Conditional probability distribution of a Brownian particle surviving forever Answer

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pr.probability – Conditional probability distribution of a Brownian particle surviving forever

Consider the drift Brownian motion $X_t:=1+bt+W_t$where $(W_t)_{t\ge 0}$ is a Brownian motion starting at zero. set $\tau:=\inf\{t\ge 0: X_t=0\}$. Assume $b>0$then $\mathbb P[\tau=\infty]>0$. What is the conditional law of $X_{\infty}$ knowing $\tau=\infty$?

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