Definition Of Viscosity Solution Research Articles

Discrete-to-Continuum Convergence of Charged Particles in 1D with Annihilation

We consider a system of charged particles moving on the real line driven by electrostatic interactions. Since we consider charges of both signs, collisions might occur in finite time. Upon collision, some of the colliding particles are effectively removed from the system (annihilation). The two applications we have in mind are vortices and dislocations in metals. In this paper we achieve two goals. First, we develop a rigorous solution concept for the interacting particle system with annihilation. The main innovation here is to provide a careful management of the annihilation of groups of more than two particles, and we show that the definition is consistent by proving existence, uniqueness, and continuous dependence on initial data. The proof relies on a detailed analysis of ODE trajectories close to collision, and a reparametrization of vectors in terms of the moments of their elements. Second, we pass to the many-particle limit (discrete-to-continuum), and recover the expected limiting equation for the particle density. Due to the singular interactions and the annihilation rule, standard proof techniques of discrete-to-continuum limits do not apply. In particular, the framework of measures seems unfit. Instead, we use the one-dimensional feature that both the particle system and the limiting PDE can be characterized in terms of Hamilton–Jacobi equations. While our proof follows a standard limit procedure for such equations, the novelty with respect to existing results lies in allowing for stronger singularities in the particle system by exploiting the freedom of choice in the definition of viscosity solutions.

Nonconcave Utility Maximization with Portfolio Bounds

The problems of nonconcave utility maximization appear in many areas of finance and economics, such as in behavioral economics, incentive schemes, aspiration utility, and goal-reaching problems. Existing literature solves these problems using the concavification principle. We provide a framework for solving nonconcave utility maximization problems, where the concavification principle may not hold, and the utility functions can be discontinuous. We find that adding portfolio bounds can offer distinct economic insights and implications consistent with existing empirical findings. Theoretically, by introducing a new definition of viscosity solution, we show that a monotone, stable, and consistent finite difference scheme converges to the value functions of the nonconcave utility maximization problems. This paper was accepted by Agostino Capponi, finance. Funding: M. Dai acknowledges financial support from the National Natural Science Foundation of China [Grants 12071333 and 11671292] and the SingaporeMinistry of Education [Grants R-146-000-243-114, R-146-000-306-114, R-146-000-311-114, and R-703-000-032-112]. X. Wan is supported by the National Natural Science Foundation of China [Grants 71850010, 72171109, 71972131, and 72271157]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2021.4228 .

Non-Concave Utility Maximization without the Concavification Principle

The problems of non-concave utility maximization appear in many areas of finance and economics, such as in behavior economics, incentive schemes, aspiration utility, and goal-reaching problems. Existing literature solves these problems using the concavification principle. We provide a framework for solving non-concave utility maximization problems, where the concavification principle may not hold and the utility functions can be discontinuous. In particular, we find that adding bounded portfolio constraints, which makes the concavification principle invalid, can significantly affect economic insights in the existing literature. Theoretically, we give a new definition of viscosity solution and show that a monotone, stable, and consistent finite difference scheme converges to the solution of the utility maximization problem.

A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

Near-Optimal Control of Stochastic Recursive Systems Via Viscosity Solution

In this paper, we study the near-optimal control for systems governed by forward–backward stochastic differential equations via dynamic programming principle. Since the nonsmoothness is inherent in this field, the viscosity solution approach is employed to investigate the relationships among the value function, the adjoint equations along near-optimal trajectories. Unlike the classical case, the definition of viscosity solution contains a perturbation factor, through which the illusory differentiability conditions on the value function are dispensed properly. Moreover, we establish new relationships between variational equations and adjoint equations. As an application, a kind of stochastic recursive near-optimal control problem is given to illustrate our theoretical results.

Approximation schemes for solving disturbed control problems with non-terminal time and state constraints

A finite difference scheme for finding viscosity solutions of specifically stated problems for Hamilton–Jacobi equations is proposed. Solutions of such problems satisfy two differential inequalities involved into the definition of viscosity solutions. The specifics is that the inequalities must hold only in the region where solutions are less than a given function or greater than another given function. From the point of view of optimal control theory and the theory of differential games, these results can be applied to finding value functions in problems with nonterminal payoff functionals and state constraints.

Invariant Solutions of Differential Games and Hamilton--Jacobi--Isaacs Equations for Time-Measurable Hamiltonians

We characterize invariance of time-varying domains with respect to differential games with time-measurable dynamics. We deduce from this result a new definition of viscosity solutions to some first order Hamilton--Jacobi equations with time-measurable Hamiltonians.

Semicontinuous viscosity solutions for Hamilton-Jacobi equations with a degenerate coefficient

The Hamilton-Jacobi equation with concave Hamiltonians and degenerate coefficients is treated. A new definition of viscosity solutions is proposed to deal with lower semicontinuous solutions. Under the setting, the comparison principle for lower semicontinuous viscosity solutions is established. The justification of our definition in applications is shown.

Uniqueness of viscosity solutions for monotone systems of fully nonlinear PDES under Dirichlet condition

is a given function, where L2 is a bounded open set of R” and S” is the set of symmetric matrices of order n. Several authors have studied some systems of PDEs via viscosity solution methods; e.g. [l-5]. Recently, Ishii and Koike [6] have pointed out that these systems have a monotone structure and have obtained uniqueness results under the assumption of monotonicity on F. Moreover, Ishii [7] has shown that Perron’s method gives the existence result under this monotonicity hypothesis. On the other hand, it is known that, in general, boundary conditions should be considered in the viscosity sense especially in the case when F is degenerate elliptic. In the case of single PDEs Ishii [8] first pointed out this fact and studied Dirichlet boundary value problems in the viscosity sense. We also refer to [9-131. The aim of this paper is to obtain some uniqueness results for viscosity solutions of (1.1) under Dirichlet condition in the viscosity sense. By using a perturbation technique in [3] we will also weaken a uniform continuity assumption on F. This paper is organized as follows. In Section 2 we recall the definition of viscosity solutions and give an equivalent definition. In Section 3 we state our main results and give some remarks. In Section 4 we prove the main results. In the final section, as a modification of main results, we will treat a so-called switching game with Dirichlet data in the viscosity sense.

Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable hamiltonians

The objective of this paper is to extend the idea of viscosity solution for nonlinear first order Hamilton-Jacobi equations, U, + H(t, x, D.,u) = 0, with time-continuous H to time-measurable hamiltonians. We are primarily motivated by the fact that control problems, for which the value function is the viscosity solution of a Hamilton-Jacobi equation, should not be restric- ted to continuity in time. In Barron and Jensen [ 1 ] we were confronted with a linear Hamilton-Jacobi equation arising in the proof of the Pon- tryagin which had time-measurable coeflicients. The existing theory of viscosity solutions did not apply to even this case. Moreover, important models of controlled first order HamiltonJacobi equations require time- measurable controls. For example, models of controlled traffic flow or queueing processes might be of this type. We are about then to extend the definition of viscosity solution to “generalized viscosity solution” applying to time-measurable hamiltonians. In this paper we give the definition and derive the corresponding uni- queness result with an implicit domain of dependence consequent. The fun- damental technique used in the proof is the so-called “blow up” method which has also been used elsewhere [S], although not in the generality discussed here. This method effectively freezes the point (to, x0) under consideration and essentially removes it from the problem. Of course t, is

Viscosity Solutions for the Monotone Control Problem

The value function, V, of the optimal control problem in which the controls must be monotone nondecreasing is a $W^{1,\infty } $ generalized solution of the quasi-variational inequality (QVI) $\max \{ LV,V_y \} = 0$. Here $LV = V_t + f \cdot \nabla _x V + h$ and f, h the dynamics of the control problem. We extend the Crandall, Evans, Lions definition of viscosity solution for nonlinear first-order p.d.e.s. to systems like (QVI) and prove that V is the unique viscosity solution of (QVI). Further, V is the smallest function satisfying the inequalities $LV \leqq 0$, $V_y \leqq 0$.