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

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.