Suppose $W=(W_t)$ is a Brownian Motion with respect to a filtration $(\mathcal{F}_t)$. How can I compute the conditional distribution of $W_{t+h}$ given $\mathcal{F}_t$.
I started like this: $W_{t+h}-W_t$ is idependent of $\mathcal{F}_t$ and normald distributed with mean $0$ and variance $h$. Then I wrote $W_{t+h}=W_t+ (W_{t+h}-W_t)$, hence I have to compute:
$P[W_{t+h}=W_t+ (W_{t+h}-W_t)\in A|\mathcal{F}_t]$
For $A\in \mathcal{B}(\mathbb{R})$. I wrote the conditional probability as a expectation of an indicator function. The result should be a normal distribution with mean $W_t$ and variance $h$. Thanks for your help
math