A Numerical Method for Solving Stochastic Optimal Control Problems with Linear Control

Chavanasporn, Walailuck; Ewald, Christian‐Oliver

doi:10.1007/s10614-011-9263-1

Cited by 3 publications

(2 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [19], the authors examined an alternative method of deriving numerical solutions to continuous-time finite-horizon SOC problems. The authors in [20] introduced a numerical method to solve SOC problems, which are linear in the control. In [21], the authors investigated stochastic optimal strategy for unknown linear discrete-time system quadratic zero-sum games in input-output form with communication imperfections.…”

Section: Introductionmentioning

confidence: 99%

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Kafash

Delavarkhalafi

Karbassi

2015

Asian Journal of Control

View full text Add to dashboard Cite

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.

show abstract

Section: Introductionmentioning

confidence: 99%

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Kafash

Delavarkhalafi

Karbassi

2015

Asian Journal of Control

View full text Add to dashboard Cite

show abstract

“…Munk discussed about the applicability of the approach to financial control problems and alternative approaches by using the Kushner's method [30], [31]. Chavanasporn and Ewald introduced a numerical method to solve stochastic optimal control problems, which are linear in the control [32]. The aim of this research is to develop a numerical method based on block pulse functions (BPFs) to solve the general optimal control of stochastic Volterra integral equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Method For Solving Optimal Control Problem Of Stochastic Volterra Integral Equations Using Block Pulse Functions

Saffarzadeh¹,

Delavarkhalafi²,

Nikoueinezhad³

2014

J. Math. Computer Sci.

View full text Add to dashboard Cite

In this paper, a numerical method for solving a general optimal control of systems is presented. These systems governed by stochastic Volterra integral equations. This method is based on block pulse functions. By using the properties of block pulse functions and associated operational matrices, optimal control problem is converted to an optimization problem and will be solved via mathematical programming techniques. The error estimations and associated theorems have been provided. Finally, some numerical examples are presented to show the validity and efficiency of the proposed method.

show abstract

Numerical solution of Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems using fractional-order Legendre collocation method

Nikooeinejad,

Heydari

2023

Journal of Vibration and Control

View full text Add to dashboard Cite

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton’s portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton’s portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation.

show abstract

A Numerical Method for Solving Stochastic Optimal Control Problems with Linear Control

Cited by 3 publications

References 9 publications

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Numerical Method For Solving Optimal Control Problem Of Stochastic Volterra Integral Equations Using Block Pulse Functions

Numerical solution of Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems using fractional-order Legendre collocation method

Contact Info

Product

Resources

About