Path-dependent Hamilton-Jacobi equations with super-quadratic growth in the gradient and the vanishing viscosity method. (English) Zbl 1492.60183
Summary: The nonexponential Schilder-type theorem in [J. Backhoff-Veraguas et al., Ann. Appl. Probab. 30, No. 3, 1321–1367 (2020; Zbl 1472.60091)] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails a non-Markovian counterpart to the vanishing viscosity method. We show uniqueness of maximal subsolutions for path-dependent viscous Hamilton-Jacobi equations related to convex super-quadratic backward stochastic differential equations. We establish well-posedness for the Hamilton-Jacobi-Bellman equation associated to a Bolza problem of the calculus of variations with path-dependent terminal cost. In particular, uniqueness among lower semicontinuous solutions holds, and state constraints are admitted.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 35F21 | Hamilton-Jacobi equations |
| 35K10 | Second-order parabolic equations |
| 60F10 | Large deviations |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 49J52 | Nonsmooth analysis |
Keywords:
large deviations; vanishing viscosity method; path-dependent partial differential equations; minimax solutions; dini solutions; calculus of variations; optimal control; state constraints; backward stochastic differential equations; nonsmooth analysisCitations:
Zbl 1472.60091References:
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