What is the explicit solution to this PDE?
Before I write the equation let $P(t,n)=P(X(t)=n) $ where P is a probabilty function such that for every $t$ it satisfis $\sum_{n=0}^{\infty} P(t,n) =1$.
X(t) is a stochastic process. For the purpose of this question it doesn't matter what $X$ is. I just wanted to tell where $P$ comes from!
The main question is as follows. Maybe you can skip previous statements and start from here!
Let $P: [0, \infty) \times \mathbb{N}\rightarrow [0,1]$ be a nice function {Continuous derivatives,...} satisfying this partial differential equation with initial conditions:
$$\partial_{t} P(t,n)=aP(t,n-1)+b(n)P(t,n) \quad \quad \quad ;n\geq1$$ $$P(t,0)=r_{t}$$ $$ P(0,0)=1$$ and $$ P(0,n)=0 \quad ;n\geq 1$$
Where $a$ is a constant and $b(n)$ is a functions which depends on $n$. I can give you $ b(n)$ or $r_{t}$ but they might be complicated functions so I believe we'd better write them in this form!
Can anyone find an explict (general ) solution of $P(t,n)$ in terms of $a$, $b(n)$, $r_{t}$ ?
Given $P(t,n-1)$, you have a first-order linear initial value problem for $P(t,n)$, whose solution is $$ P(t,n) = e^{b(n) t} \int_0^t a P(s,n-1) e^{-b(n) s} \; ds $$ In the absence of an explicit form for $P(t,0)$, there's not much hope of anything more explicit than this.