Partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations and applications to stochastic control
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916584304410624 |
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| author | Federico, Salvatore Ferrari, Giorgio Rosestolato, Mauro |
| author_facet | Federico, Salvatore Ferrari, Giorgio Rosestolato, Mauro |
| contents | In this note, we demonstrate that a locally semiconvex viscosity supersolution to a possibly degenerate fully nonlinear elliptic Hamilton-Jacobi-Bellman (HJB) equation is differentiable along the directions spanned by the range of the coefficient associated with the second-order term. The proof leverages techniques from convex analysis combined with a contradiction argument. This result has significant implications for various stationary stochastic control problems. In the context of drift-control problems, it provides a pathway to construct a candidate optimal feedback control in the classical sense and establish a verification theorem. Furthermore, in optimal stopping and impulse control problems, when the second-order term is nondegenerate, the value function of the problem is shown to be differentiable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations and applications to stochastic control Federico, Salvatore Ferrari, Giorgio Rosestolato, Mauro Optimization and Control Analysis of PDEs 35D40, 26B25, 35R35, 49L25, 49J40, 60G40 In this note, we demonstrate that a locally semiconvex viscosity supersolution to a possibly degenerate fully nonlinear elliptic Hamilton-Jacobi-Bellman (HJB) equation is differentiable along the directions spanned by the range of the coefficient associated with the second-order term. The proof leverages techniques from convex analysis combined with a contradiction argument. This result has significant implications for various stationary stochastic control problems. In the context of drift-control problems, it provides a pathway to construct a candidate optimal feedback control in the classical sense and establish a verification theorem. Furthermore, in optimal stopping and impulse control problems, when the second-order term is nondegenerate, the value function of the problem is shown to be differentiable. |
| title | Partial regularity of semiconvex viscosity supersolutions to fully nonlinear elliptic HJB equations and applications to stochastic control |
| topic | Optimization and Control Analysis of PDEs 35D40, 26B25, 35R35, 49L25, 49J40, 60G40 |
| url | https://arxiv.org/abs/2501.15285 |