Existence and Uniqueness of Viscosity Solutions for Stochastic Hamilton--Jacobi--Bellman Equations
Abstract
We study stochastic Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with random coefficients and possible degenerateness. The viscosity solution is reformulated by using adapted smooth random test fields that touch on stopped space--time neighborhoods almost surely. The state coefficients are predictable, with no continuity assumption on their dependence on the underlying Brownian path. Under linear growth of locally Lipschitz drift and diffusion coefficients, polynomial growth of the running and terminal cost coefficients, and a nonempty separable metric space-valued control set, we show that the value function is the unique viscosity solution in this formulation among adapted continuous fields with square-integrable random polynomial growth bounds. The comparison proof constructs adapted fields from controlled trajectories and spatially shaken conditional values, using a weak semimartingale decomposition and regularization of its positive finite-variation part. In particular, the uniqueness allows for diffusion coefficients being random, degenerate, and depending on time, state, and control, whereas earlier uniqueness results of Qiu [SIAM J. Control Optim. 56 (2018), pp. 3708--3730] and Qiu and Zhang [Ann. Appl. Probab. 33 (2023), pp. 889--930] were obtained under additional structural restrictions on the diffusion coefficient, such as determinism combined with either superparabolicity or independence of the state and controls.
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BibTeX
@misc{qiu2026existence,
title = {{Existence and Uniqueness of Viscosity Solutions for Stochastic Hamilton--Jacobi--Bellman Equations}},
author = {Jinniao Qiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10806},
eprint = {2610.10806},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10806},
}
RIS
TY - UNPB
TI - Existence and Uniqueness of Viscosity Solutions for Stochastic Hamilton--Jacobi--Bellman Equations
AU - Jinniao Qiu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10806
UR - https://arxiv.org/abs/2610.10806
AB - We study stochastic Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with random coefficients and possible degenerateness. The viscosity solution is reformulated by using adapted smooth random test fields that touch on stopped space--time neighborhoods almost surely. The state coefficients are predictable, with no continuity assumption on their dependence on the underlying Brownian path. Under linear growth of locally Lipschitz drift and diffusion coefficients, polynomial growth of the running and terminal cost coefficients, and a nonempty separable metric space-valued control set, we show that the value function is the unique viscosity solution in this formulation among adapted continuous fields with square-integrable random polynomial growth bounds. The comparison proof constructs adapted fields from controlled trajectories and spatially shaken conditional values, using a weak semimartingale decomposition and regularization of its positive finite-variation part. In particular, the uniqueness allows for diffusion coefficients being random, degenerate, and depending on time, state, and control, whereas earlier uniqueness results of Qiu [SIAM J. Control Optim. 56 (2018), pp. 3708--3730] and Qiu and Zhang [Ann. Appl. Probab. 33 (2023), pp. 889--930] were obtained under additional structural restrictions on the diffusion coefficient, such as determinism combined with either superparabolicity or independence of the state and controls.
ER -