Abstract
In this article a notion of viscosity solutions is introduced for second-order path-dependent Hamilton–Jacobi–Bellman (PHJB) equations associated with optimal control problems for path-dependent stochastic differential equations. We identify the value functional of optimal control problems as unique viscosity solution to the associated PHJB equations. We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions and satisfies a stability property. Applications to backward stochastic Hamilton–Jacobi–Bellman equations are also given.
Funding Statement
The author was partially supported by the National Natural Science Foundation of China (Grant No. 11401474), Shaanxi Natural Science Foundation (Grant No. 2021JM-083) and the Fundamental Research Funds for the Central Universities (Grant No. 2452022374).
Acknowledgments
The author would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.
Citation
Jianjun Zhou. "Viscosity solutions to second order path-dependent Hamilton–Jacobi–Bellman equations and applications." Ann. Appl. Probab. 33 (6B) 5564 - 5612, December 2023. https://doi.org/10.1214/23-AAP1954
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