Numerical solution of competitive advertising problem with a stochastic differential game approach using a combined Chelyshkov collocation with policy iteration method

In the field of advertising‎, ‎there are always situations in which individuals or companies promote their products in order to find retrieval opportunities and attract customers in a competitive environment‎. ‎Several goals are followed in this paper‎. ‎First‎, ‎the historical development of applications of differential games in modeling strategic situations in competitive advertising is mentioned‎. ‎We then introduce the problem in a duopoly market under the influence of uncertainty in the framework of a stochastic differential game‎. ‎Finding the equilibrium strategy for this problem requires solving a nonlinear partial differential equations system also known as the Hamilton-Jacoby equation‎. ‎Another purpose of this paper is to propose an efficient and appropriate computational method for solving the Hamilton-Jacobi partial differential equations‎. ‎The proposed method for solving the problem is a combination of collocation methods by the derivative operator matrix based on Chelyshkov polynomials and policy iteration method‎. ‎The advantage of using the policy iteration method is that at each step‎, ‎instead of finding the solution to a nonlinear partial differential equation‎, ‎it is sufficient to solve a sequence of linear partial differential equations systems‎. ‎The convergence of the proposed method is provided in detail‎. ‎Finally‎, ‎we solve the corresponding Hamilton-Jacobi equations system by the proposed iterative algorithm‎.