On integral equations arising in the first-passage problem for Brownian motion
Let \(B_t, t \geq 0\), be a standard Brownian motion started at zero and \(g(t)\) be a continuous non-negative function defined on \((0, \infty)\). The first passage problem is considered. Namely, let \(\tau = \inf \{t>0: B_t \geq g(t)\}\) and \(F\) be the distribution function of \(\tau\). Then the following system of integral equations holds: \[ t^{n/2}H_n\left(\frac{g(t)}{\sqrt{t}}\right)= \int_x^{\infty} (t-s)^{n/2}H_n\left(\frac{g(t)-g(s)}{\sqrt{t-s}}\right)F(ds), \] where \(H_n(x)= \int_x^{\infty} H_{n-1}(z)dz, n \geq 0,\) and \(H_{-1}(x)= (1/\sqrt{2 \pi})e^{{-x^2}/2}\).
scientific article; zbMATH DE number 2059977
scientific article