I would like to solve the PDE$$P_{t}\left(t, r{(t)}\right)+ \Lambda r{(t)} P_{r}\left(t,r{(t)}\right)+\frac{1}{2}r{(t)}^2{\sigma}^2P_{rr}\left(t, r{(t)}\right)=r(t)P\left(t, r{(t)}\right), $$ where $$\Lambda :=\alpha_{1}+\alpha_{2}\ln r{(t)},$$ and $r(t)$ evolves according to the black-karasinki model. I want to solve this using the finite difference method. However the problem is I only know one boundary condition: $$P\left(T ,r; T\right)=1.$$ I would like to know the other possible boundary condition in order for the equation to be solved numerically.
2026-02-23 08:37:54.1771835874
Boundary conditions for zero coupond bond PDE
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