Solving an integral equation

Hello everyone,

I am trying to solve a seemingly simple integral equation:

$\displaystyle g(z)=\int_{z-1}^{\infty}\frac{g(t)}{t}dt$

where $\displaystyle g(z)$ is known to be zero for $\displaystyle z\le 1$.

$\displaystyle g(z)$ is a probability distribution function of a random process

$\displaystyle z(t+1)=a(t)z(t)+1$ where $\displaystyle a(t)=U(0,1)$ are iid random variables from a uniform distribution. I ran simulations of this process and got that $\displaystyle g(z)$ is constant for $\displaystyle 1>z\ge 2$ and decays for $\displaystyle z>2$. The decay does not seem to be exponential or 1/z. I did not manage to find an analytical solution or some transform (Laplace/Mellin) that can help.

Any suggestions would be welcome!