I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\frac{\partial f}{\partial \theta}+\frac{1}{2}\sigma^2(\theta)\frac{\partial^2 f}{\partial \theta^2} = \frac{\partial f}{\partial t} + f(0,\theta)f(t,\theta)$$
$P$ represents the probability distribution that an event will happen $t$ seconds from now. $\theta_s$ is a stochastic process representing a changing parameter. The constraint expresses the reflexion principle.
There are of course infinitely many solutions, but I'm looking for a non-trivial one. If $f$ is an exponential distribution, for instance, the only solution is $\mu = \sigma = 0$, not a particularly interesting case.
I tried a multivariate generalization on the gamma distribution and the Weibull distribution and their two parameters. It didn't work either (I think).
