Consistent Finite Difference Scheme Research Articles

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.

Convergence of Finite Difference Schemes to the Aleksandrov Solution of the Monge-Ampère Equation

We present a technique for proving convergence to the Aleksandrov solution of the Monge-Ampere equation of a stable and consistent finite difference scheme. We also require a notion of discrete convexity with a stability property and a local equicontinuity property for bounded sequences.

On a Stable and Consistent Finite Difference Scheme for a Time-Dependent Schrodinger Wave Equation in a Finitely Low Potential Well

In this paper, a stable and consistent criterion to an explicit finite difference scheme for a time-dependent Schrodinger wave equation (TDSWE) was presented. This paper is a departure from the well-established time independent Schrodinger Wave Equation (SWE). To develop the stability criterion for the scheme, the Fourier series method of von Newmann was adopted, while in establishing the consistency property, the concise definition of the consistent scheme was applied. This research is carried out for a particular case of a finitely low potential well.

On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions

A new explicit conditionally consistent finite difference scheme for one-dimensional third-order linear pseudoparabolic equation with nonlocal conditions is constructed. The stability of the finite difference scheme is investigated by analysing a nonlinear eigenvalue problem. The stability conditions are stated and stability regions are described. Some numerical experiments are presented in order to validate theoretical results.

Propagation of graphs in two-dimensional inhomogeneous media

We consider the propagation of a front in a two-dimensional striated medium. In the case of a front that can be written as a graph over the real line we are led to an Eikonal equation for the height function which contains a possibly discontinuous coefficient. We prove existence and uniqueness of a viscosity solution in the sense of Ishii. Furthermore, we obtain an O ( h ) -error bound for a monotone and consistent finite difference scheme. We conclude with some numerical tests.

Implicit solution of the incompressible Navier-Stokes equations on a non-staggered grid

This paper presents an implicit procedure for the solution of the incompressible Navier-Stokes equations in primitive variables. The time dependent momentum equations are solved implicitly for the velocity field using the approximate factorization technique. The continuity equation is satisfied at each time step through the solution of a Poisson type equation for the static pressure. A consistent finite-difference scheme which satisfies the compatibility condition using a non-staggered grid is used in the finite difference approximation of the static pressure Poisson equation. The results of the numerical computations in a driven cavity, which are presented for the history of the residues, at several Reynolds numbers using different computational parameters, demonstrate the excellent convergence characteristics of the solution procedure. A stability analysis is conducted for the non-iterative phase and numerical results are presented for a given set of compational parameters. Numerical results obtained for the steady state static pressure in the driven cavity are presented for the first time at Re =1000 using a non-staggered grid, and the steady state velocity, vorticity, and pressure contours at Re = 100, 400, and 1000, are compared with the computational results of other investigators. Additional results using a curvilinear grid are also presented for the flow in a cascade of circular airfoils at Re = 1000.

On the Construction of Consistent Finite Difference Schemes with Certain Invariant Subspaces

On the Construction of Consistent Finite Difference Schemes with Certain Invariant Subspaces