I'm trying to gain an intuitive understanding of conditional expectation of a function of brownian motion. Let $B_t$ be a brownian motion starting at $x$, and $f$ a bounded function. Is it right to state $$ E[f(B_t) \mid B_0 = x] = E[f(B_t - x) \mid B_0 = 0] = E[f(B_t - x)] $$ ? Since you shift the brownian motion path down $x$ unit, you have a new brownian motion starting at $0$. But since the path stays the same, so the conditional expectation of $f(B_t)$ stay the same?
2026-04-06 11:34:37.1775475277
Interpreting conditional expectation of brownian motion?
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