Finite Difference Methods for the Hamilton–Jacobi–Bellman Equations Arising in Regime Switching Utility Maximization

AbstractFor solving the regime switching utility maximization, Fu et al. (Eur J Oper Res 233:184–192, 2014) derive a framework that reduce the coupled Hamilton–Jacobi–Bellman (HJB) equations into a sequence of decoupled HJB equations through introducing a functional operator. The aim of this paper is to develop the iterative finite difference methods (FDMs) with iteration policy to the sequence of decoupled HJB equations derived by Fu et al. (2014). The convergence of the approach is proved and in the proof a number of difficulties are overcome, which are caused by the errors from the iterative FDMs and the policy iterations. Numerical comparisons are made to show that it takes less time to solve the sequence of decoupled HJB equations than the coupled ones.

Download Full-text

ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202503002660 ◽

2003 ◽

Vol 13 (05) ◽

pp. 613-644 ◽

Cited By ~ 32

Author(s):

ESPEN ROBSTAD JAKOBSEN

Keyword(s):

Finite Difference ◽

Rate Of Convergence ◽

Optimal Stopping ◽

Initial Conditions ◽

Finite Difference Methods ◽

Approximation Schemes ◽

Dynamic Programming Principle ◽

Bellman Equations ◽

Difficult Time ◽

Difference Methods

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

Download Full-text

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Numerical Algorithms ◽

10.1007/s11075-021-01159-w ◽

2021 ◽

Author(s):

Ying Ma ◽

Jia Yin

Keyword(s):

Dirac Equation ◽

Finite Difference ◽

Error Estimates ◽

Finite Difference Methods ◽

Difference Methods

Download Full-text

Biharmonic navigation using radial basis functions

Robotica ◽

10.1017/s0263574721000709 ◽

2021 ◽

pp. 1-12

Author(s):

Xu-Qian Fan ◽

Wenyong Gong

Keyword(s):

Finite Difference ◽

Boundary Conditions ◽

Radial Basis Functions ◽

Real World ◽

Biharmonic Equation ◽

Finite Difference Methods ◽

Basis Functions ◽

Large Size ◽

Radial Basis ◽

Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

Download Full-text

Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽

10.3390/math9030206 ◽

2021 ◽

Vol 9 (3) ◽

pp. 206

Author(s):

María Consuelo Casabán ◽

Rafael Company ◽

Lucas Jódar

Keyword(s):

Stochastic Process ◽

Finite Difference ◽

Coming Out ◽

Finite Difference Methods ◽

Diffusion Reaction ◽

Finite Difference Schemes ◽

Finite Degree ◽

Mean Square Stability ◽

Reaction Models ◽

Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

Download Full-text

Policy iteration for Hamilton–Jacobi–Bellman equations with control constraints

Computational Optimization and Applications ◽

10.1007/s10589-021-00278-3 ◽

2021 ◽

Author(s):

Sudeep Kundu ◽

Karl Kunisch

Keyword(s):

Linear Equations ◽

Hjb Equation ◽

Policy Iteration ◽

Optimal Feedback Control ◽

Iteration Step ◽

Control Constraints ◽

Hjb Equations ◽

Bellman Equations ◽

Hamilton Jacobi Bellman ◽

Feedback Control Theory

AbstractPolicy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.

Download Full-text