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Suppose we have the following parabolic PDE in $X(s, t)$:

$\frac{\partial X}{\partial t} + sM_1 \frac{\partial X}{\partial s} + \frac{1}{2} s^2 M_2 \frac{\partial^2 X}{\partial s^2} + (M_3 - M_1)X = F(s, t)$.

This can be concisely written as $\frac{\partial X}{\partial t} + \mathbb{A}X = F(s, t)$. ($\mathbb{A}$ is a differential operator)

Some conditions are: $X(s, t), M_1, M_2, M_3, F(s, t)$ are $n \times n$ matrices, $F$ is a "known" function but we don't know how it looks like exactly. $M_1, M_2$ are diagonal matrices, and $M_3$ is a rate matrix, i.e., all its rows add up to zero, and all of its non-diagonal entries are non-negative.

My question is: what conditions on $F(s, t)$ must hold so that a solution to the above PDE exists? So far I know that $F(s, t)$ has at most polynomial growth w.r.t. $s$.

I'd also be grateful if someone could direct me to a reference where the above problem is dealt with

Thanks

ul15524
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  • Isn't this a bit like a Kolmogorov backward equation for a stochastic process with one diffusion and a continuous-time Markov chain? (With $F=0$.) – Kirill Jul 03 '14 at 10:45
  • @Kirill : I'm not sure about that (as in I don't know). It would, of course, help if it's like some well-known PDE. But as far as I know, it's a parabolic PDE and the "source" term $F(s, t)$ is important. – ul15524 Jul 03 '14 at 15:26

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