In this paper we analyze a Boltzmann type mean field game model for knowledge growth, which was proposed by Lucas and Moll [14]. We discuss the underlying mathematical model, which consists of a coupled system of a Boltzmann type equation for the agent density and a Hamilton-Jacobi-Bellman equation for the optimal strategy. We study the analytic features of each equation separately and show loc...

Abstract. The Symplectic Pontryagin method was introduced in a previous paper. This work shows that this method is applicable under less restrictive assumptions. Existence of solutions to the Symplectic Pontryagin scheme are shown to exist without the previous assumption on a bounded gradient of the discrete dual variable. The convergence proof uses the representation of solutions to a Hamilton...

This paper derives the optimal debt ratio and dividend payment strategies for an insurance company. Taking into account the impact of reinsurance policies and claims from the credit derivatives, the surplus process is stochastic that is jointly determined by the reinsurance strategies, debt levels, and unanticipated shocks. The objective is to maximize the total expected discounted utility of d...

We consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment. In particular, we consider an investment project that yields a payoff at a rate that depends on its installed capacity level and on a random economic indicator such as, for instance, the price of the project’s output commodity. We mode...

The paper is concerned with fully nonlinear second order Hamilton{Jacobi{Bellman{ Isaacs equations of elliptic type in separable Hilbert spaces which have unbounded rst and second order terms. The viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions is proved. A stochastic optimal control problem driven by a paraboli...

Abstract This paper investigates a feature tracking control method for visual servoing (VS) manipulators adaptive dynamic programming (ADP)-based the unknown dynamics. The major superiority of ADP-based optimal lies in that problem is converted to error with cost function. Moreover, an neural network observer developed approximate entire uncertainties, which are utilized construct improved By e...

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...

The illiquid market model HJB equations and characterization of the solution Power utility functions and numerical results

In this paper we present a version of the balancing technique for nonlinear systems which are dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system. We discuss an input output approach that allows us to apply the theory of balancing based upon Hankel singular value analysis. In order to do that we prove that the available storage and ...

Using the Hamilton-Jacobi-Bellman equation, we derive both a Keynes-Ramsey rule and a closed form solution for an optimal consumption-investment problem with labor income. The utility function is unbounded and uncertainty stems from a Poisson process. Our results can be derived because of the proofs presented in the accompanying paper by Sennewald (2006). Additional examples are given which hig...