This paper gives comparison principles for first-order PDEs of the Hamilton-JacobiBellman type that arise in the problem of filtering under unknown disturbances with set-membership bounds on the uncertainty. The exact solutions of this problem, given in set-theoretic terms as “information sets,” are expressed as level sets to the solutions of some specific types of the HJB equation. But these s...

The study of the separability of the Hamilton–Jacobi and the corresponding scalar field equations in a curved spacetime has a long history. Robertson [1] and Eisenhart [2] discussed general conditions for such a separability in spaces which admit a complete set of mutually orthogonal families of hypersurfaces. An important class of 4-dimensional separable spacetimes, including several type D me...

Wavelets, which have many good properties such as time/freqency localization and compact support, are considered for solving the Hamilton-Jacobi-Bellman (HJB) equation as appears in optimal control problems. Specifically, we propose a Successive Wavelet Collocation Algorithm (SWCA) that uses interpolating wavelets in a collocation scheme to iteratively solve the Generalized-Hamilton-Jacobi-Bell...

Under usual assumptions on the Hamiltonian, we prove that any viscosity solution of the corresponding Hamilton-Jacobi equation on the manifold M is locally semiconcave and C loc outside the closure of its singular set (which is nowhere dense in M). Moreover, we prove that, under additional assumptions and in low dimension, any viscosity solution of that Hamilton-Jacobi equation satisfies a gene...

We consider Hamilton–Jacobi equations which characterize optimal controlled partial differential equations of the following types: the Allen–Cahn equation, the Cahn–Hilliard equation, a nonlinear Fokker–Planck equation, and aVlasov–Fokker–Planck equation. In each of the examples, the optimal control problem and its associated cost functional can be derived as limit from a microscopically define...

A π,z J(z) = lim t>0,t→0 e −αt E z,π [J(z(t))] − J(z) t Lemma 1. Let J ∈ J satisfy J(0, a, b) = 0. Let τ = inf{t : J(z t) = 0}. E τ 0 e −αt e −π(z)/r a b π(z) + A π,z J(z) dt = J π (z 0) − J(z 0) Proof: Define H π J(z) = e −π(z)/r a b π(z) + A π,z J(z) Lemma 6 later verifies that this definition is in agreement with our previous definition (in [2]) provided J ∈ J. Let τ be a stopping time of th...

In a stationary case and for any potential, we solve the three-dimensional quantum Hamilton-Jacobi equation in terms of the solutions of the corresponding Schrödinger equation. Then, in the case of separated variables, by requiring that the conjugate momentum be invariant under any linear transformation of the solutions of the Schrödinger equation used in the reduced action, we clearly identify...

In this paper we provide a maximum norm analysis of an overlapping Schwarz method on non-matching grids for evolutionary HJB equation with nonlinear source terms with the mixed boundary conditions and a general elliptic operator. Moreover, an asymptotic behavior in uniform norm is established.

We study an optimal investment and risk control problem for an insurer under stochastic factor. The insurer allocates his wealth across a riskless bond and a risky asset whose drift and volatility depend on a factor process. The risk process is modeled by a jump-diffusion with state-dependent jump measure. By maximizing the expected power utility of the terminal wealth, we characterize the opti...

In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the erg...