General problem:
Given a positive function $g(x,s)\geq 0$ with $\int_{0}^{\infty}g(x,s)ds=\infty$ can we show the existence or lack thereof of some positive process $\{\lambda(t)\}_{t=0}^{\infty}$ such that $$ E\left[\lambda(t+s)|X(t)=x\right]=g(x,s)\\ \lambda(t) \geq 0\\ $$
(I.e the expected trajectory of some random hazard given data is given by some function of data)
Particular problem:
In particular, I'm trying to generate some interesting data from unknown $\lambda,X$ where I want $g$ to be given by $$ g(x,s)=b(x)a(x) s^{b(x)-1} $$ Or equivalently $$ g(x,s)=\frac{\beta(x)}{\alpha(x)} \left(\frac{s}{\alpha(x)}\right)^{\beta(x)-1} $$ Where $a,b$ or $\alpha,\beta$ are known strictly positive functions.
(I.e the expected trajectory of some random hazard given data is Weibull)
Does such a process exist? If so any (possibly trivial) suggestions? Any reading tips/hints on where this type of problem might have been studied?
Edit: Possibly helpful summary of properties of $g$. For all $s\geq0$ and real valued $x$: $$ g(x,s)\geq 0 \\ \int_{0}^{s}g(x,\tau)d\tau=G(x,s)\\ G(x,\infty)=\infty\\ G(x,0)=0\\ $$ Also note that for each positive distribution parametrized by $x$ there's a $G$ s.t $Pr(T>t)=e^{-G(x,t)}$ and conversely $G(x,t)$ induces a probability distribution