Wiener Process. Where can I find proof.

Question

I looking for proof, that $P({W_s >0,W_t >0})=\frac{1}{4}+\frac{1}{2\pi}\arcsin{\sqrt\frac{s}{t}}$, where $W_k$ is a wiener process and $t>s>0$. Do you know where I can find it?

Answer 1

By scaling, $P({W_s >0,W_t >0})=P({W_{s/t} >0,W_1 >0})$ so it suffices to consider the case $t=1$. Let $\theta=\arcsin(\sqrt{s})$. Since $W_s$ is independent of $W_1-W_s$, we can write $W_s= X \sin{\theta}$ and $W_1=X \sin{\theta}+Y \cos{\theta}$, where $X,Y$ are independent standard normal variables and $\theta \in (0, \pi/2)$. Therefore, $$ P({W_s >0,\, W_1 >0})= P(X>0,\, Y+X\tan{\theta}>0)=P(X>0,\, Y>0)+P(X>0, \,\, 0>Y>-X\tan{\theta})=\frac{1}{4}+\frac{\theta}{2\pi} \,,$$ since the joint distribution of $(X,Y)$ is rotation invariant.

This formula goes back to [1], see also [2].

[1] Sheppard, W. F. (1899). III. On the application of the theory of error to cases of normal distribution and normal correlation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, (192), 101-167.

[2] Kendall, M.G., Stuart, A. and Ord, J.K., 1979. The advanced theory of statistics (Vol. 2, 4th edn). Griffin, London.