Two Optimization Problems of a Continuous-in-Time Financial Model

In the previous papers of Frenod & Menard & Safa [1] and Frenod & Safa [2] we used the continuous-in-time financial model developed by Frenod & Chakkour [3], which describes working of loan and repayment, in an optimal control theory framework to effectively conduct project objectives. The goal was to determine the optimal loan scheme taking into account the objectives of the project, the income and the spending. In this article, we enrich this continuous-in-time financial model to take into account possibility of saving. Then we propose two new optimization problems involving this model. The first one consists in finding a loan scheme minimizing the cost of the loan for a project, together with the time to achieve its objectives. The second problem consists, from given loan, saving and withdrawal schemes, to finding optimal variants of them. We have built a mathematical framework for the optimal control problem which consists to solving a constraint optimization problem. We have used Simplex algorithm to solve the first optimization problem and the quadratic programming for the second one. The simulations’ results are consistent with the theoretical part and show the performance and stability of our approach.