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  4. A New Approach for Solving a Control Stochastic Problem Driven by a Diffusion Process with Jumps

A New Approach for Solving a Control Stochastic Problem Driven by a Diffusion Process with Jumps

MMC.
2026;
Volume 13, Number 1
: pp. 77–90
Received: August 13, 2025
Revised: January 28, 2026
Accepted: January 31, 2026

Oulghazi H. A New Approach for Solving a Control Stochastic Problem Driven by a Diffusion Process with Jumps.  Mathematical Modeling and Computing. Vol. 13, No. 1, pp. 77–90 (2026)

Authors:
  1. H. Oulghazi
1
IMIA Laboratory, A2MSDS Team, Department of Mathematics, Faculty of Sciences and Technics, Moulay Ismail University of Meknes

In this paper, we focus on the numerical solution of high-dimensional stochastic optimal control problems, whose system states are modeled as jump-diffusion processes.  Through the maximum principle and deep neural networks, we restate the original control problem as a variational problem, and we introduce specialized algorithms to solve this new formulation.  The algorithms and the various architectures employed have been introduced.  The mean-variance portfolio selection problem in a financial market consisting of two kinds of assets in a jump-diffusion process setting validates the effectiveness of proposed algorithms.

forward-backward stochastic differential equations with jumps
stochastic maximum principle
deep neural networks
optimal stochastic control
jump diffusions
mean-variance portfolio selection problem
  1. Merton R. C.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.  The Review of Economics and Statistics.  51 (3), 247–257 (1969).
  2. Zariphopoulou T.  Consumption–Investment Models with Constraints.  SIAM Journal on Control and Optimization.  32 (1), 59–85 (1994).
  3. Oksendal B., Sulem A.  Optimal consumption and portfolio with both fixed and proportional transaction costs.  SIAM Journal on Control and Optimization.  40 (6), 1765–1790 (2002).
  4. Haussmann U. G.  A Stochastic Maximum Principle for Optimal Control of Diffusions.  John Wiley & Sons, Inc. (1986).
  5. Bellman R.  Dynamic programming and stochastic control processes.  Information and Control.  1 (3), 228–239 (1958).
  6. Kushner H. J.  Numerical methods for stochastic control problems in continuous time.  SIAM Journal on Control and Optimization.  28 (5), 999–1048 (1990). 
  7. Goodfellow I., Bengio Y., Courville A.  Deep Learning.  MIT Press (2016).
  8. Jiequn H., Weinan E.  Deep learning approximation for stochastic control problems.  Preprint arXiv:1611.07422 (2016).
  9. Weinan E., Jiequn H., Arnulf J.  Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations.  Communications in Mathematics and Statistics.  5 (4), 349–380 (2017).
  10. Ji S., Peng S., Peng Y., Zhang X.  Three algorithms for solving high-dimensional fully coupled FBSDEs through deep learning.  IEEE Intelligent Systems.  35 (3), 71–84 (2020).
  11. Ji S., Peng S., Peng Y., Zhang X.  Solving stochastic optimal control problem via stochastic maximum principle with deep learning method.  Journal of Scientific Computing.  93 (1), 30 (2022).
  12. Germain M., Pham H., Warin X.  Neural networks-based algorithms for stochastic control and PDEs in finance.  Preprint arXiv:2101.08068 (2021).
  13. Carmona R., Laurière M.  Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II –- the finite horizon case.  The Annals of Applied Probability.  32 (6), 4065–4105 (2022).
  14. Hu R., Laurière M.  Recent developments in machine learning methods for stochastic control and games.  Preprint arXiv:2303.10257 (2023).
  15. Andersson K., Andersson A., Oosterlee C. W.  Convergence of a robust deep FBSDE method for stochastic control.  SIAM Journal on Scientific Computing.  45 (1), A226–A255 (2023).
  16. Boltyanski V. G., Gamkrelidze R. V., Mishchenko E. F., Pontryagin L. S.  The maximum principle in the theory of optimal processes of control.  IFAC Proceedings Volumes.  1 (1), 464–469 (1960).
  17. Peng S.  A general stochastic maximum principle for optimal control problems.  SIAM Journal on Control and Optimization.  28 (4), 966–979 (1990).
  18. Tang S., Li X.  Necessary conditions for optimal control of stochastic systems with random jumps.  SIAM Journal on Control and Optimization.  32 (5), 1447–1475 (1994).
  19. Framstad N. C., Øksendal B., Sulem A.  Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance.  Journal of Optimization Theory and Applications.  121, 77–98 (2004).
  20. Meng Q., Dong Y., Shen Y., Tang S.  Optimal controls of stochastic differential equations with jumps and random coefficients: stochastic Hamilton–Jacobi–Bellman equations with jumps.  Applied Mathematics & Optimization.  87 (1), 3 (2023).
  21. Øksendal B., Sulem A.  Applied Stochastic Control of Jump Diffusions.  Springer Berlin, Heidelberg (2007).
  22. Han J., Long J.  Convergence of the deep BSDE method for coupled FBSDEs.  Probability, Uncertainty and Quantitative Risk.  5 (1), 5 (2020).
  23. Agram N., Øksendal B.  Deep learning for conditional McKean–Vlasov jump diffusions.  Systems & Control Letters.  188, 105815 (2024).
  24. Kingma D. P., Ba J.  Adam: A method for stochastic optimization.  Preprint arXiv:1412.6980 (2015).
  25. Hornik K., Stinchcombe M., White H.  Multilayer feedforward networks are universal approximators.  Neural Networks.  2 (5), 359–366 (1989).
  26. Liu D. C., Nocedal J.  On the limited memory BFGS method for large scale optimization.  Mathematical Programming.  45 (1), 503–528 (1989).