A Positivity-Preserving Expectation Scheme for Hamilton--Jacobi--Bellman Equations with Oblique Robin Boundary Conditions

Abstract

For anisotropic diffusion with mixed derivatives, standard compact coordinate-aligned stencils and some local finite-volume or finite-element constructions can lose nonnegative coefficients unless suitable coefficient or mesh conditions are imposed. Here nonnegative coefficients are generated directly from conditional expectation rather than from an algebraic stencil decomposition. We construct a positivity-preserving scheme for possibly degenerate Hamilton--Jacobi--Bellman equations with controlled oblique Robin boundary conditions. At interior nodes the construction stems from a one-step reflected Feynman--Kac identity: an $m$-dimensional Rademacher vector generates $P=2^m$ equally probable weak-Euler branches, an exterior branch is mirrored about its oblique projection, and twice its overshoot serves as the discrete boundary local time $D$. The Robin coefficient enters only through the attenuation factors $e^{-κD}$ and $De^{-κD/2}g$, leaving the equal branch probabilities unchanged. Boundary nodes use a separate same-level closure over a spatial offset $\ell_h\asymp h$, which may be implicit; uniform obliqueness and $\mathbb P_1$ interpolation give a mesh-independent positive weight on interior nodes, so fixed linear Robin data yield a sparse nonsingular $M$-matrix system and the general control set yields a monotone contraction. Positivity is preserved for nonnegative data without a diagonal-dominance condition and without any CFL-type relation between $Δt$ and $h$.