On the integral of the sum of squared Ornstein-Uhlenbeck processes (Q2736835)

Given \((y(t))_{t\geq 0}\), \((z(t))_{t\geq 0}\) two independent Ornstein-Uhlenbeck processes starting from \(y,z\in {\mathbb R}\), let NEWLINE\[NEWLINEx(t) = \int_0^t (y^2(s)+z^2(s)) ds, \quad t\geq 0, NEWLINE\]NEWLINE and let \(T\) denote the first time \((y(t),z(t))_{t\geq 0}\) reaches either of two circles of radii \(r\) and \(R\), with \(0\leq r < R \leq +\infty\) and \(r^2 <y^2+z^2 < R^2\). The main results of this paper are the explicit computations of the moment generating function of \(x(T)\) and of the mean of \(x(T)\).