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I know that in general it is very difficult to construct Green function. Recently, I read a paper called "On the first passage time density of a continuous martingale over a moving boundary". It suggest a way to construct Green function of a PDE:

$$-\frac{\partial u}{\partial t}=-\beta^{'}(t)a u(t,a)+\frac{h^{2}(t)}{2}\frac{\partial^{2} u}{\partial a^{2}}+h^{2}(t)\left(\frac{1}{a}-\frac{a}{\int_{t}^{s}h^{2}(u)du}\right)\frac{\partial u}{\partial a}$$

The idea is to solve simultaneously both the backward and forward Kolmogorov equations and the obtain the Green's function. The author first assume that the solution takes the form $u=u^{1}\frac{B(t)}{a}e^{A(t)a^{2}}$ and soln for forward Kolmogorov equation to be $v=v^{1}\tilde{B}(\tau)b e^{\tilde{A}(\tau)a^{2}}$ and reduce the PDE $u^{1}$ to

$$-\frac{\partial u^{1}}{\partial t}+\beta^{'}(t)a u^{1}(t,a)=\frac{h^{2}(t)}{2}\frac{\partial^{2} u}{\partial a^{2}}$$

I understand the procedure and it gives that the Green's function for the above PDE, which is

$$\exp\{b\tilde{\pi}(\tau)+a\pi(\tau)+\frac{1}{2}\int_{t}^{\tau}h^{2}(u)\pi^{2}(u)du\}\tilde{G}$$. I do not state too much here due to copyright. Here is the link https://arxiv.org/pdf/0905.1975.pdf. I want to ask if the way he gets the Green's function is correct? Also, when I take $\tau$ to be very large, the Green's function is infinity, which is not reasonable. What's wrong here?

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