Cash flow optimization in uncertain environments: forward backward stochastic differential equation approach with pontryagin's maximum principle

Abstract

This article explores the application of Forward-Backward Stochastic Differential Equations ‎‎(FBSDEs) to cash flow optimization in uncertain financial environments. FBSDE provide a ‎rigorous framework for modeling investment and payment dynamics, enabling the maximization ‎of investor preferences while minimizing financial risks. The model considers a portfolio ‎composed of both risky and risk-free assets, incorporating constraints such as the balance between ‎discounted payments and accumulated premiums.‎ The analysis includes solving the optimization problem using the stochastic maximum principle ‎and Lagrange multipliers. Optimal admissible strategies are defined as stochastic processes ‎satisfying integrability conditions and backward differential equations. Numerical simulations ‎assess the impact of key parameters, such as initial wealth, discount rate, volatility, and risk ‎aversion, on investment and consumption decisions.‎ The results demonstrate that the FBSDE approach effectively captures complex dynamics and ‎facilitates the development of robust strategies under uncertainty. In conclusion, this article ‎highlights the potential of FBSDEs for portfolio management, financial product pricing, and ‎decision optimization in uncertain environments. Future research could expand this framework by ‎integrating exogenous factors, such as macroeconomic conditions, thereby broadening its ‎applicability and relevance‎.