Viscosity Solution Approach Research Articles

Explicit Solution to an Optimal Switching Problem in the Two‐Regime Case

This paper considers the problem of determining the optimal sequence of stopping times for a diffusion process subject to regime switching decisions. This is motivated in the economics literature by the investment problem under uncertainty for a multi-activity firm involving opening and closing decisions. We use a viscosity solutions approach combined with the smooth-fit property, and explicitly solve the problem in the two-regime case when the state process is of geometric Brownian nature. The results of our analysis take several qualitatively different forms, depending on model parameter values.

Numerical solution to the optimal birth feedback control of a population dynamics: viscosity solution approach

AbstractThis paper is concerned with the optimal birth control of a McKendrick‐type age‐structured population dynamic system. We use the dynamic programming approach in our investigation. The Hamilton–Jacobi–Bellman equation satisfied by the value function is derived. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal birth feedback control is found explicitly through the value function. A finite difference scheme is designed to obtain the numerical solution of the optimal birth feedback control. The validity of the optimality of the obtained control is verified numerically by comparing with different controls under the same constraint. All the data utilized in the computation are taken from the census of the population of China in 1989. Copyright © 2005 John Wiley & Sons, Ltd.

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach

A class of nonlinear integro-differential Cauchy problems is studied by means of the viscosity solutions approach. In view of financial applications, we are interested in continuous initial data with exponential growth at infinity. Existence and uniqueness of solution is obtained through Perron's method, via a comparison principle; besides, a first order regularity result is given. This extension of the standard theory of viscosity solutions allows to price derivatives in jump--diffusion markets with correlated assets, even in the presence of a large investor, by means of the PDE's approach. In particular, derivatives may be perfectly hedged in a completed market.

APPROXIMATION OF DIFFERENTIAL GAMES WITH MAXIMUM COST AND INFINITE HORIZON

We consider an integral form of the Isaacs equations associated to differential games with L∞ criterion, for the characterization of their value functions. We prove that upper and lower values are the lowest super-solution and the largest element of a special set of sub-solutions, of the dynamic programming equation. This is an alternative to the viscosity solutions approach, without requiring any regularity assumption on the value functions.For finite horizon approximations, we propose a scheme in terms of an infinitesimal operator defined over the set of Lipschitz continuous functions. The images of this operator can be characterized classically in terms of viscosity solutions.We illustrate these results on a example, which values functions can be determined analytically.

Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Levy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Levy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

Second Order Hamilton--Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control

The paper is concerned with fully nonlinear second order Hamilton--Jacobi--Bellman--Isaacs equations of elliptic type in separable Hilbert spaces which have unbounded first and second order terms. The viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved. A stochastic optimal control problem driven by a parabolic stochastic PDE with control of Dirichlet type on the boundary is considered. It is proved that the value function of this problem is the unique viscosity solution of the associated Hamilton--Jacobi--Bellman equation.

Asset allocation with time variation in expected returns

This paper analyzes the consumption investment problem of a risk averse investor in continuous time when there are several asset classes. The classic paper in this area is due to Merton who solved the problem when the returns were assumed to be stationary. We assume that there is time variation in the expected returns on the different assets and that this time variation arises from movements in the underlying state variables. We formulate the investor's decision as a problem in optimal stochastic control. Our work extends the paper by Brennan et al. (1997) to incorporate a different interest rate process. In addition we investigate the impact of transaction costs on the stock. We employ a viscosity solution approach to the problem and to guarantee a solution we need to impose strong assumptions.

Infinite-Dimensional Hamilton–Jacobi Equations and Dirichlet Boundary Control Problems of Parabolic Type

The paper is concerned with an infinite-dimensional Hamilton–Jacobi equation. This equation is related to boundary control problems of Dirichlet type for semilinear parabolic systems. The viscosity solution approach is adapted to the equation under consideration, using the properties of fractional powers of generators of analytic semigroups. An existence-and-uniqueness result for such problem is obtained.

Deterministic Near-Optimal Controls. Part II: Dynamic Programming and Viscosity Solution Approach

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal controls, for systems governed by deterministic ordinary differential equations, and uses dynamic programming to study the near-optimality. Since nonsmoothness is inherent in this subject, the viscosity solution approach is employed to investigate the problem. The dynamic programming equation is derived in terms of ϵ-superdifferential/subdifferential. The relationships among the adjoint functions, the value functions, and the Hamiltonian along near-optimal trajectories are revealed. Verification theorems with which near-optimal feedback controls can be constructed are obtained.

Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach

A stochastic optimal switching and impulse control problem in a finite horizon is studied. The continuity of the value function, which is by no means trivial, is proved. The Bellman dynamic programming principle is shown to be valid for such a problem. Moroever, the value function is characterized as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.

A Viscosity Solution Approach to the Asymptotic Analysis of Queueing Systems

We consider a system of several interconnected queues (with a single class of customers) and model the state (Xt) of the system as a jump Markov process. The problem of interest is to estimate the large deviations behavior of the rescaled system Xt = eXt/s, corresponding to large time and large excursions of the original (unscaled) system. The techniques employed are those of the theory of viscosity solutions to Hamilton-Jacobi equations. From the point of view of large deviation theory, the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty, since an abrupt change in the jump measure occurs. From the point of view of viscosity solutions, the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations. Much of the paper is devoted to proving uniqueness of viscosity solutions for these equations, and these sections are of independent interest. While our use of test functions in proving the uniqueness is an adaptation of the usual technique, the construction of the test functions themselves via the Legendre transform is new. We obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem, which suggests the correct integrand in the large deviation rate functional. Since it is the treatment of the effects due to the boundaries that is novel, we devote the majority of the paper to the detailed development of a simple two-dimensional system that exhibits all the essential new features. However, the arguments may be applied to nueueing systems that are considerably more general, and we attempt to indicate this generality as well.