Codex @codex  0

Expand the update in part (b). Its trinomial tree weights areThey connect the current state to $x_i-h,x_i,x_i+h$ at the next time. Their sum is one, their mean increment is $b_i^{j}k$, and their raw second increment moment is ${\hat{\sigma }}_i^{j,2}k$. Thus they match the log-price diffusion to first order in the time step.

All three weights are probabilities precisely when they are nonnegative:at every node and level. They are then automatically at most one. With positive ${\sigma }_{-}$, a simple sufficient uniform choice is $k\le h^2/{\sigma }_{+}^2$ and $h\sup |r-{\hat{\sigma }}^2/2|\le {\sigma }_{-}^2$. If volatility vanishes at a node while drift does not, the centered weights cannot all be nonnegative.

Nonnegative weights make the backward update a convex combination, proving maximum-norm stability and monotonicity: interior errors cannot exceed the larger of the terminal and boundary errors. This is a sufficient stability condition, not an equivalence with every possible stability notion. For constant coefficients on an unrestricted or periodic grid, let $\kappa ={\hat{\sigma }}^{2}k/h^2$ and $\nu =bk/h$. The Fourier amplification factor isRequiring $|G|\le 1$ for every frequency gives $\kappa \le 1$ and ${\nu }^2\le \kappa$. Indeed, with $s={\sin }^2(\vartheta /2)$, $|G{|}^2-1=4s[({\nu }^2-\kappa )+({\kappa }^2-{\nu }^2)s]$, so its two endpoint inequalities suffice. Probability weights require the stronger $|\nu |\le \kappa$. The distinction prevents interpreting a merely Fourier-stable scheme with negative weights as a probability tree.