The stochastic Hamilton–Jacobi–Bellman equation on Jacobi structures

The paper’s Theorem 4.1 (stochastic contact HJ formalism) establishes the equivalence between (i) the lifted Hamiltonian semimartingale condition and (ii) a pair of stochastic contact HJB equations, and it reduces them to the single scalar Stratonovich identity ∂tS dt + h(j1S) · δXt = 0. The candidate solution proves the same equivalence by a chain-rule computation combined with a pointwise decomposition along im j1S into horizontal, vertical, and Reeb components. This mirrors the paper’s argument, which uses the same chain rule, the vertical-lift operator, and Lemma 4.5 to pass from the integral identity to the HJB equations. The only substantive issue in the candidate solution is a sign slip in the preliminary identity ιΛ♯(α) dη = −(α − α(R)η), which conflicts with the paper’s contact identities and leads to a sign-flipped coordinate expression for Vh; however, the subsequent key decomposition (⋆) and the resulting HJB system and scalar identity match the paper’s result. Hence both are correct; the proofs are substantially the same up to minor sign conventions and presentation differences. See Theorem 4.1 and its proof, Lemma 4.5, and the contact vector-field formulas (Proposition 4.18/Example 4.19) in the paper .