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

  • Authors

    • Tcheick T. Kayembe Unikin
    • Pascal K. Mubenga Faculty of Science and Technology, University of Kinshasa, DR Congo
    • Emil K. Nawej Faculty of Science and Technology, University of Kinshasa, DR Congo
    • Ezechiel T. Tshisuaka Institute of Buildings and Public Works of Mbuji-Mayi, Department of Building and Public Works
    • Eugene M. Mbuyi Faculty of Science and Technology, University of Kinshasa, DR Congo
    2025-01-30
    https://doi.org/10.14419/2frsms38
  • FBSDE; Cash Flow Optimization; Financial Risk; Optimal Admissible Strategies and Numerical ‎Simulations‎.
  • 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.

  • References

    1. Merton, R. C. (1971). Optimum Consumption and Portfolio Rules in a Continuous-Time Model. Journal of Economic Theory, 3(4), 373-413. https://doi.org/10.1016/0022-0531(71)90038-X.
    2. Karatzas, I., & Shreve, S. E. (1998). Methods of Mathematical Finance. Springer-Verlag. https://doi.org/10.1007/b98840.
    3. Black, F., & Scholes, M. (1973). Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. https://doi.org/10.1086/260062.
    4. Fleming, W. H., & Soner, H. M. (2023). Controlled Markov Processes and Viscosity Solutions. Springer.
    5. Bismut, J. M. (1981). Stochastic Control: The Pontryagin Maximum Principle. MIT Press.
    6. Ekeland, I., & Taflin, E. (2022). A Stochastic Approach to Risk Management in Financial Markets. Journal of Financial Economics, 40(2), 127-148.
    7. Chiarella, C., & He, X. (2019). Dynamic Portfolio Allocation with Stochastic Control Methods. Quantitative Finance, 15(5), 823-836.
    8. Liu, J., & Pang, J. S. (2020). Optimal Control and Portfolio Selection: Theory and Applications. Financial Mathematics Journal, 22(1), 19-41.
    9. Ma, J., & Zhang, X. (2022). Stochastic Differential Equations in Finance: Risk Management and Pricing of Derivatives. Mathematical Finance Review, 45(2), 122-144.
    10. Devolder, M., Fouque, J. P., & Papanicolaou, G. (2023). Portfolio Optimization with Stochastic Dynamics: A Multi-Parameter Approach. Journal of Applied Probability and Stochastic Processes, 58(3), 289-307.
    11. Fouque, J. P., & Papanicolaou, G. (2011). Multi-scale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press. https://doi.org/10.1017/CBO9781139020534.
    12. Chiarella, C., & He, X. (2012). Financial Market Dynamics and Stochastic Models. Springer.
    13. Peng, S., & Wu, X. (2021). A New Approach to Stochastic Optimal Control with Applications to Finance. SIAM Journal on Control and Optimization, 59(4), 2839-2857.
    14. Driessen, J., & Laeven, R. (2022). Risk Management Strategies: Stochastic Control and Hedging in Derivatives Pricing. Risk Management, 23(6), 112-130.
    15. Duffie, D., & Epstein, L. G. (2021). Stochastic differential utility and asset pricing. Econometrica, 89(5), 1323-1348.
    16. Yong, J. (2020). Backward Stochastic Differential Equations in Financial Modelling. Annals of Finance, 16(4), 543-567.
    17. Glas, R., & Jain, S. (2023). Risk-Sensitive Portfolio Optimization with Backward Stochastic Differential Equations. Journal of Financial Mathematics, 28(4), 341-356.
    18. He, H., & Zhou, X. (2022). Dynamic hedging with stochastic volatility models. Journal of Quantitative Finance, 24(3), 215-232.
    19. Kayembe, T. T., Mubenga, P. K., & Mbuyi, E. M. (2024). Optimal Strategies for Investment and Consumption: Stochastic Analysis with Pontryagin's Principle under Economic Uncertainty. International Journal of Applied Mathematical Research, 13(2), 117-127. https://doi.org/10.14419/z8htty80
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  • How to Cite

    T. Kayembe, T., K. Mubenga , P. ., K. Nawej , E. ., T. Tshisuaka, E. . ., & M. Mbuyi , E. . (2025). Cash flow optimization in uncertain environments: forward backward stochastic ‎differential equation approach with pontryagin’s maximum principle‎. International Journal of Applied Mathematical Research, 14(1), 1-12. https://doi.org/10.14419/2frsms38