Optimal control of financial inclusion for women entrepreneurs with bifurcation, statistical and stochastic analysis

In this paper, we develop a comprehensive mathematical model incorporating market linkages, sociocultural interventions, social capital, and access to finance to examine the dynamics of women entrepreneurs. We extend the deterministic model by incorporating the random variations present in socioeconomic systems, in addition to constructiing and analyzing a stochastic nonstandard finite difference (NSFD) scheme. The methodology overcomes significant limitations of traditional numerical approaches by ensuring the preservation of positivity and boundedness for all step sizes. We compute key thresholds at which the system undergoes qualitative behavioral changes by using eigenvalue analysis, one-dimensional parameter sweeps, and two-dimensional stability surfaces. The absence of flip bifurcation verifies that the behaviour of the system is bounded and that the system does not exhibit period-doubling instabilities. The stochastic and deterministic simulations highlight the dynamic behavior of the suggested model. The impact of parameters, illustrated through radial, tornado, chord, and heat-map diagrams, shows that maintenance programs that lower the depreciation rate through regulatory change should be combined with initiatives that improve investment rates through financial literacy and digital infrastructure. Statistical validation shows the superiority of the nonstandard finite difference scheme and the effectiveness of control measures compared with the no-control case. By formulating a stochastic optimal control problem, we compare four policy cases with the no-control case, and the results show that the combination of all controls produces the highest access to finance. The NSFD scheme preserves positivity for all step sizes, whereas Euler-Maruyama generates economically non-feasible negative results for large step sizes. The findings indicate that, to prevent irregular losses in entrepreneurial capacity, effective interventions should prioritize stabilizing social networks and controlling financial access during turbulent periods, which underscores the importance of stable policy environments and consistent financial support mechanisms. In women's entrepreneurial development programs, our study offers a strong foundation for policy modeling, uncertainty quantification, and evidence-based decision-making.