Develop efficient numerical schemes for high-dimensional stochastic Hamilton–Jacobi–Bellman equations

Develop efficient numerical schemes for viscosity solutions of stochastic Hamilton–Jacobi–Bellman equations in high-dimensional state spaces, including the additional backward non-Markovian setting induced by random coefficients.

Background

The paper studies stochastic optimal control problems with two independent Brownian motions and coefficients that depend non-Markovianly on one of them. Their dynamic programming equation is a second-order backward stochastic partial differential equation, whose numerical treatment is substantially more difficult than that of a classical deterministic Hamilton–Jacobi–Bellman equation.

The authors emphasize that even in the classical Markovian case, efficient numerical schemes for viscosity solutions remain unresolved when the state dimension is high. The non-Markovian backward structure of the stochastic Hamilton–Jacobi–Bellman equation introduces further numerical difficulties, motivating alternative duality and rough-path formulations.

References

For instance, efficient numerical schemes for viscosity solutions are open challenges even in the classical Markovian case (in this case, the SHJB equation becomes a classical PDE), when the dimension of the state space $d$ is high.

— Duality for Stochastic Control with non-Markovian Random Coefficients  (2609.05101 - Bank et al., 4 Sep 2026) in Section 1, paragraph “Motivations”