Maximum norm stability for implicit Black-Scholes equation

Question

I am trying to prove maximum norm stability for the following implicit approximation to the Black-Scholes equation

$$\frac1{\Delta t}\left(U_j^{(n+1)}-U_j^{(n)}\right)+\frac{rS_j}{\Delta S}\left(U_{j+1}^{(n)}-U_j^{(n)}\right)+\frac{\sigma^2S_j^2}{2\Delta S^2}\left(U_{j+1}^{(n)}-2U_j^{(n)}+U_{j-1}\right)=rU_j^{(n)}$$

with the terminal conditions $U^{(N)}_j=u(S_j,T)$, $U^{(N)}_0=u(0,T)\mathrm{e}^{-r(T-t_n)}$, and $U_j^{(n)}\to0$ as $j\to\infty$. Defining $S_j=j\Delta S$ and $t_n=n\Delta t$ and rearranging, I obtain

\begin{align*} U_j^{(n+1)}&=-\left(\frac{\sigma^2j^2\Delta t}2\right)U_{j-1}^{(n)}+\left[1+r(1+j)\Delta t+\sigma^2j^2\Delta t\right]U_j^{(n)}-\left(rj\Delta t+\frac{\sigma^2j^2\Delta t}2\right)U_{j+1}^{(n)}\\ &=a_jU_{j-1}^{(n)}+b_jU_j^{(n)}+c_jU_{j+1}^{(n)}. \end{align*}

I am asked to prove that $(1+r\Delta t)\max_j|U_j^{(n)}|\leq\max|U_j^{(n+1)}|$, which in turn implies maximum norm stability.

All I know is that for implicit equations like these I must satisfy $a_j,\,c_j\leq0$ and $a_j+b_j+c_j\geq1$, which is definitely satisfied here, but I don't know how to justify the requested inequality. I know that the coefficient of the LHS is $a_j+b_j+c_j=1+r\Delta t$, but given the negativity of two of the coefficients, I am unclear of how to connect the two arguments (or anything with monotonicity/discrete maximum principle). Any advice is appreciated!

(NB: I have shifted this post over from MSE since it was probably not the most appropriate for it to be there.)

Answer 1

Note that \begin{align*} U_j^{(n)} &= \frac{U_j^{(n+1)} - a_jU_{j-1}^{(n)} - c_jU_{j+1}^{(n)}}{b_j}\\ &\le \frac{\max_j|U_j^{(n+1)}| - a_j\max_j|U_j^{(n)}| - c_j\max_j|U_j^{(n)}|}{b_j}. \end{align*} Moreover, there exists $j_0$ such that \begin{align*} &\ \frac{\max_j|U_j^{(n+1)}| - a_{j_0}\max_j|U_j^{(n)}| - c_{j_0}\max_j|U_j^{(n)}|}{b_{j_0}} \\ =&\ \max_j \frac{\max|U_j^{(n+1)}| - a_j\max_j|U_j^{(n)}| - c_j\max_j|U_j^{(n)}|}{b_j}. \end{align*} That is, \begin{align*} \max_j|U_j^{(n)}| &\le \max_j \frac{\max|U_j^{(n+1)}| - a_j\max_j|U_j^{(n)}| - c_j\max_j|U_j^{(n)}|}{b_j}\\ &= \frac{\max_j|U_j^{(n+1)}| - a_{j_0}\max_j|U_j^{(n)}| - c_{j_0}\max_j|U_j^{(n)}|}{b_{j_0}}. \end{align*} Then, \begin{align*} (1+r\Delta t)\max_j|U_j^{(n)}|\leq\max|U_j^{(n+1)}|. \end{align*}