Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.     

On the Bellman Equation for Infinite Horizon Problems with Unbounded Cost Functional

We study a class of infinite horizon control problems for nonlinear systems, which includes the Linear Quadratic (LQ) problem, using the Dynamic Programming approach. Sufficient conditions for the regularity of the value function are given. The value function is compared with sub- and supersolutions of the Bellman equation and a uniqueness theorem is proved for this equation among locally Lipschitz functions bounded below. As an application it is shown that an optimal control for the LQ problem is nearly optimal for a large class of small unbounded nonlinear and nonquadratic pertubations of the same problem. Accepted 8 October 1998     

A Characterization of the Existence of Solutions for Hamilton—Jacobi Equations in Ergodic Control Problems with Applications

We give a characterization of the existence of bounded solutions for Hamilton—Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton—Jacobi equations. Accepted 1 December 1999     

Variational Solutions of Coupled Hamilton—Jacobi Equations

In this note we study variational solutions of weakly coupled Hamilton—Jacobi equations in the case where the Hamiltonians are convex. More precisely, we build the variational solution by an approximation scheme. Accepted 24 April 1998     

A Class of Hamilton—Jacobi Equations with Unbounded Coefficients in Hilbert Spaces

We establish existence and comparison theorems for a class of Hamilton—Jacobi equations. The class of Hamilton—Jacobi equations includes and is broader than those studied in [8] We apply the existence and uniqueness results to characterizing the value functions associated with the optimal control of systems governed by partial differential equations of parabolic type. Accepted 11 May 2001. Online publication 5 October 2001.     

An Auxiliary Equation for the Bellman Equation in a One-Dimensional Ergodic Control

In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First, we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.} \amsclass{49L20, 35G20, 93E20.} Accepted 11 September 2000. Online publication 16 January 2001.     

Hamilton—Jacobi Equations and Distance Functions on Riemannian Manifolds

   Abstract. The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities.     

Radon—Nikodym Theorem in L ∞

We prove that for any given set function F which satisfies F(∪ A _i ) =sup _i F(A _i ) and F(A)=-∈fty if meas (A)=0 , there must exist a measurable function g so that F(A) = ess sup_ y ∈ A g(y) . Two proofs of this result are given. Then a Riesz representation theorem for ``linear' operators on L ^∈fty is proved and used to establish the existence of Green's function for first-order partial differential equations. In the special case u _t +H(u,Du)=0 , Green's function is explicitly found, giving the extended Lax formula for such equations. Accepted 20 March 2000. Online publication 7 July 2000.     

Equivalence between Nonlinear H ∞ Control Problems and Existence of Viscosity Solutions of Hamilton—Jacobi—Isaacs Equations rid="*" id="*" This work was written while the author was visiting the University of California at Santa Barbara. He was partially supported by Consiglio Nazionale delle Ricerche of Italy.

In this paper we extend to completely general nonlinear systems the result stating that the suboptimal control problem is solved if and only if the corresponding Hamilton—Jacobi—Isaacs (HJI) equation has a nonnegative (super)solution. This is well known for linear systems, using the Riccati equation instead of the HJI equation. We do this using the theory of differential games and viscosity solutions. Accepted 14 February 1997