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We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton-Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller [arXiv preprint arXiv:2408.02147 (2024)], where non-local path-dependent Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.
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This content will become publicly available on October 27, 2026
Non-local Hamilton–Jacobi–Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes
We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton-Jacobi-Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations.
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- Award ID(s):
- 2106077
- PAR ID:
- 10651773
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Stochastic Processes and their Applications
- Volume:
- 192
- Issue:
- C
- ISSN:
- 0304-4149
- Page Range / eLocation ID:
- 104813
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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