I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "Stochastic Calculus for finance II, continuous time model" from Shreve. Just a different underlying process.
I do not understand the last boundary condition.
Here the underlying process is given by
\begin{align*}
dX_t=(-a_1X_t+\gamma_t)dt+dW_t\\
\end{align*}
\begin{align*}
Y_t=\int_{0}^{t}X_udu\
\end{align*}
The boundary condtion is given by
\begin{align*}
v(t,0,y)=e^{-r(T-t)}\max(\frac{y}{T}-K,0).
\end{align*}
Derivation from Shreve ($dX_t=rX_tdt+X_tdW_t$):
If $X_t=0$ and $Y_t=y$ for some variable $t$, then $X_u=0, \ \forall u \in[t,T]$ and so $Y_u$ is constant on $[t,T]$ and therefore $Y_T=y$ and the value of the Asian call option at time t is $e^{-r(T-t)}\max(\frac{y}{T}-K,0)$.
I understand his derivation (see https://quant.stackexchange.com/questions/78206/pricing-pde-of-asian-option-by-shreve)
Problem:
The author of my paper gives no explanation to this boundary condition. But I don't think I can apply Shreve's explanation to my process, since it has no $X_t$ before the $dW_t$ term and can therefore take non-zero values again once it was zero?!
Thank you very much for your help!