$$where generator $\mathcal{L}^a$ replaces directional derivative $\nabla V^\top f$, incorporating curvature term $\tfrac{1}{2}\mathrm{Tr}(\Sigma\Sigma^\top\nabla^2 V)$ from $W$'s quadratic variation. Remainder unchanged: substitute into dynamic programming, cancel $V(x)$, divide by $h$, and take $h\downarrow 0$ to obtain (1). $\quad\blacksquare$
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