If $W_s$ and $W_t$ are wiener processes, we have that the probability that $W_s$ and $W_t$ attain maximum is (I am concluding this from "running maximum", but I am not sure) $$P(W_s<a)=\frac{2}{\sqrt{2\pi}}\int_0^a e^{\frac{-x^2}{2}}dx$$ and $$P(W_t<a)=\frac{2}{\sqrt{2\pi}}\int_0^a e^{\frac{-x^2}{2}}dx$$ I am trying to compute the probability that $$P(\sqrt{(|W_s|^2+|W_t|^2)}<r)$$
More precisely, I would appreciate hints for computing the probability that a 2-dimensional wiener process lies inside the disc of radius $r$.
If $X$ and $Y$ are independent standard normal random variables and $R=\sqrt{X^2+Y^2}$ then, for every nonnegative $r$, $$ P(R\leqslant r)=\iint_{x^2+y^2\leqslant r^2}\frac1{2\pi}\mathrm e^{-(x^2+y^2)/2}\mathrm dx\mathrm dy. $$ The change of variable $(x,y)=(s\cos t,s\sin t)$ with $s\geqslant0$ and $t$ in $[0,2\pi)$ is rather ubiquitous in such a gaussian context and yields $$ P(R\leqslant r)=\int_0^{2\pi}\int_0^r\frac1{2\pi}\mathrm e^{-s^2/2}s\mathrm ds\mathrm dt=\int_0^rs\mathrm e^{-s^2/2}\mathrm ds=\mathrm e^{-r^2/2}, $$ which shows at the same time that the CDF $F_R$ and the PDF $f_R$ of $R$ are defined on $r\geqslant0$ respectively by $$ F_R(r)=\mathrm e^{-r^2/2},\qquad f_R(r)=r\mathrm e^{-r^2/2}. $$ The same reasoning shows that, for every $n\geqslant1$, the random variable $$R_n=\sqrt{X_1^2+X_2^2+\cdots+X_n^2},$$ with the analogue conventions, has a density $f_n$ proportional to $g_n$, where $$ g_n(r)=r^{n-1}\mathrm e^{-r^2/2}. $$