Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

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Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...