I'm considering the following PDE in $\Phi$:
$\frac{\partial \Phi(t,s)}{\partial t}$ + $sR\frac{\partial \Phi(t,s)}{\partial s}$ + $\frac{1}{2} s^2 M \frac{\partial^2 \Phi(t,s)}{\partial s^2}$ + $\Lambda \Phi(t,s) = R \Phi(t,s)$
with boundary conditions $\Phi_i(T,s) = max(0, s-K)$, where $K$ is a constant, and $\Phi_i(t, 0) = 0$ for all $t, i$.
$R$ and $M$ are diagonal $k$ x $k$ matrices, $\Lambda$ is a $k$ x $k$ matrix and $\Phi$ is a $k$-vector. It is known that this Cauchy problem has a unique solution in the class of $C([0, T] \times \mathbb{R}) \cap C^{1, 2}([0, T] \times \mathbb{R})$ functions. (not sure whether this fact is important but I'm stating it anyway)
I am trying to prove the continuity of the map $\Lambda \mapsto X_{\Lambda}$, where $X_{\Lambda}$ is the classical solution to the above PDE.
I was thinking of using the epsilon-delta criterion, that is, give $\epsilon > 0$, there exists $\delta$ such that $||\Lambda_1 - \Lambda_2|| < \delta \implies ||X_{\Lambda_1} - X_{\Lambda_2}|| < \epsilon$. I tried subtracting 2 PDE's : one with $\Lambda_1$ and $\Phi_1$, and the other with $\Lambda_2$ and $\Phi_2$, but try as I might I'm still not getting it.
Could someone please explain how I should proceed? I'm at my wit's end!