Properties of perpetual integral functionals of Brownian motion with drift (Q2485315)

Given a nonnegative, measurable function \(f\) and some parameter \(\mu> 0\), let \(B^{(\mu)}\) be a Brownian motion with drift \(\mu\) and set \(I_\infty(f)= \int^\infty_0 f(B^{(\mu)}_t)\,dt\). In the financial mathematical framework, \(I_\infty(f)\) is interpreted as a continuous perpetuity. The authors give here some conditions under which \(I_\infty\) (i) is finite a.s., (ii) has all the moments, (iii) has some exponential moments, (iv) has bounded potential.