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I'm reading some articles about PDE and I found the following PDE, with $q_1,A >0$:
$g_t(t,y)+ \beta^2yg_y(t,y)+\frac{1}{2}\beta^2y^2g_{yy}(t,y)-q_1 g(t,y)=0 \quad (t,y) \in [0,T), \times (0,+\infty)$.
with boundary conditions $\begin{cases} g(t,0)=F, t \in [0,T] \\ g(T,y)= \max\{F-Ay,S\} \end{cases}$
The solution proposed is the following:
$g(t,y)=Ae^{(\beta^2-q_1)(T-t)}p_{put}(t,y)+Se^{-q_1(T-t)}$
where $p_{put}$ is the price of a European put option with strike price $\frac{1}{A} (F-S)$ in a BS market with volatility of risky asset $\beta$ and rate of riskless asset $\beta^2$.
My question is the following. I cannot understand actually the solution becasue if I substitute in the PDE it solves but it doesn't satisfy the condition $g(t,0)=F$ becasue I obtain $g(t,0)=Fe^{-q_1(T-t)}$. Can anyone help me?

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