Suppose we have a standard Brownian motion $B(t)$ (with $B(0)=0$) and we condition on the event (which I call $R$) that:
$$B(t)\ge0 \ \forall t\in[0,K].$$
Then, how do we calculate $E(B(K)|R)$, and is it the same as $E(B(K)|B(K)\ge0)$?
2026-03-25 04:41:36.1774413696
What is the expectation of a Brownian motion at time t given that its entire sample path exceeds H?
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