This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.

Abstract We study the well-posedness of Hamilton–Jacobi–Bellman equations on subsets $${\mathbb {R}}^d$$ R d in a context without boundary conditions. The Hamiltonian is given as supremum over two parts: an internal depending external control vari...

We consider the optimal control of stochastic delayed systems with jumps, in which both the state and controls can depend on the past history of the system, for a value function which depends on the initial path of the process. We derive the Hamilton-Jacobi-Bellman equation and the associated verification theorem and prove a necessary and a sufficient maximum principles for such problems. Expli...

This paper investigates the optimal mean-variance reinsurance-investment problem for an insurer with a common shock dependence under two kinds of popular premium principles: variance principle and expected value principle. We formulate optimization within game theoretic framework derive closed-form expressions equilibrium strategy function different principles by solving extended Hamilton-Jacob...

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics is governed by a general system of stochastic functional differential equations with a bounded memory. An infinite-dimensional HJB equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.

An optimal stopping problem for stochastic differential equations with random coefficients is considered. The dynamic programming principle leads to a Hamiltion–Jacobi–Bellman equation, which, for the current case, is a backward stochastic partial differential variational inequality (BSPDVI, for short) for the value function. Well-posedness of such a BSPDVI is established, and a verification th...

a random walk on a lattice is one of the most fundamental models in probability theory. when the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (rwre). the basic questions such as the law of large numbers (lln), the central limit theorem (clt), and the large deviation principle (ldp) are no...

Bellman showed how classical mechanics can be obtained from Hamilton’s principle by dynamic programming. If we add noise in a particular way, we obtain Schrödinger’s equation and many other results from the elementary theory of quantum mechanics. Some new results can also be obtained, and there are important consequences for the philosophy of science.