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What is $P(W_t>0,W_s>0)$?

Since increments are independent it may be helpful to split it into two processes as $W_t=W_{t-s}+W_s$.

Each of it has Gaussian probability density function and I think that pdf of sum of two independent processes may be evaluated as in case of independent random variables: $f_{W_s+W_{t-s}} (t) = \int\limits_{-\infty}^{+\infty} f_{W_s}(s) f_{W_{t-s}}(t-s) ds = \frac{1}{\sqrt{4\pi \sigma^2}}e^{\frac{-(-2\mu+t)^2}{4\sigma^2}} \sim N(2\mu,2\sigma)$.

What should I do next? If I integrate it over the area $0<t<+\infty$ than I will find probability that $W_s+W_{t-s}>0$, but I can't proceed forward.

Hasek
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    You made several mistakes: 1. Assume wlog $t\geq s$. You then have $W_t = (W_t-W_s) + W_s$, where the increment $(W_t-W_s)$ is independent of $W_s$ and not $W_t= W_{t-s}+W_s$. 2. Moreover $W_{t-s}$ and $W_s$ would be correlated since $E(W_{t-s}W_s)= cov(W_{t-s},W_s) = \min(t-s,s) \neq 0$ however, the sum would still be gaussian, since each two time points will have a multivariate gaussian distribution (gaussian process). i think by using the correct decomposition of $W_t$ your task will be much easier. – EliKa Mar 21 '17 at 11:30
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    The proof is here: http://math.stackexchange.com/questions/1687795/correlated-joint-normal-distribution-calculating-a-probability/1688568#1688568 –  Mar 21 '17 at 11:36

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