Customer portfolio model driven by continuous-time Markov chains: An l₂ Lagrangian regularization method

This paper provides a solution to the customer portfolio for a given fixed desired expected rate of return under constraints based. We restrict the solution to a class of finite, ergodic, controllable continuous-time, finite-state Markov chains. We propose a regularized Lagrange method for the portfolio representation that ensures the strong convexity of the objective function and the existence of a unique solution of the portfolio. The solution is obtained by using the standard Lagrange method introducing the positive parameters θ and δ, and the Lagrange vector-multipliers μ₀ and μ₁ for the equality and inequality constraints, respectively, and forming the Lagrangian. We prove that if the ratio_δ^θn tends to zero, then the solution of the original portfolio converges to a unique solution withⁿ the minimal weighted norm. We introduce a recurrent procedure based on the projection-gradient method for finding the extremal points of the portfolio. In addition, we prove the convergence of the method. A numerical example validates the effectiveness of the regularized portfolio Lagrange method.