Determining optimal consumption and bequest policies is complex but essential aspect of retirement planning. It can balance the immediate need for consumption with the desire to leave a legacy, managing risks and economic activities more sustainably and efficiently. One of the key challenges for such optimal retirement planning is the stochasticity in the retiree’s income, which can include uncertain pension payments, fluctuating investment returns, or unexpected windfalls or losses. This paper proposes a robust Monte Carlo-based-dynamic programming technique to approximate the deterministic equivalent of the intractable stochastic utility bequest-consumption problem. The utility function is modelled using constant relative risk aversion (CRRA), and a weighting function modulates the bequest utility. This stochastic approach allows for a more realistic representation of the uncertainties faced by retirees. The dynamic programming approach involves recursive value functions that consider the maximum expected utility from the current period to the end of the planning horizon, given the state variables of wealth and income. This paper contributes to retirement planning by offering a sophisticated tool for optimizing consumption and bequest strategies in uncertain future income and survival probabilities.

This study compares the price dynamics of a Caputo fractional order delay differential cobweb model with existing cobweb models that have conformable fractional derivatives, Caputo fractional derivatives, and nonsingular kernel fractional derivatives to determine the effect of the time delay parameter on commodity price, besides the positivity of the solution that was investigated. In contrast to previous research, the stability study shows the practical usefulness of our model since the literature’s various time intervals showed short‐term equilibrium price convergence but long‐term price divergence. The results showed that the time gap between supply and demand accounts for the noise associated with the fractional‐order time delay differential cobweb regarding convergence (or divergence), which is nonexistent in the literature models. It is observed in the paper that lower fractional order goes with a higher time delay, while we have the reverse for high fractional order. This depicts the realities of price behaviour in connection with the law of demand and supply of commodities that take a finite period for them to be ready for distribution to the market. It is, therefore, recommended that price adjustment models be modeled using fractional delay differential equations. The numerical simulations were done using MATLAB.

This research investigates the stability and occurrence of Hopf bifurcation in a credit risk contagion model, which includes distributed delay, using the chain trick method. The model is a generalized version of those previously examined. The model is an expanded version of those previously studied. Comparative analysis showed that unlike earlier models, which only used the nonlinear resistance coefficient to determine the rate of credit risk infection, the credit risk contagion rate is also affected by the weight given to past behaviors of credit risk participants. Therefore, it is recommended to model the transmission of credit risk contagion using dispersed delays.

Retirement planning requires strategies that balance immediate needs with long-term financial stability, especially as retirees face challenges like income uncertainty, market volatility, and unreliable traditional investments. Renewable energy—particularly solar—has emerged as a viable alternative, offering reduced energy costs and alignment with sustainability goals. However, adoption is hindered by high upfront costs and fluctuating market conditions. This paper presents a novel Monte-Carlo-based optimization framework integrated with dynamic programming (DP) to address these issues. The framework simulates diverse economic and energy market scenarios to support informed retirement decisions. It compares outcomes of investing solely in annuities with those from a hybrid strategy combining annuities and solar energy. Despite uncertainty in solar returns, findings show that retirees who diversify enjoy greater financial security. By merging Monte Carlo simulations with the flexibility of DP, this approach enables smarter investment planning, optimizes spending, and strengthens long-term resilience. It offers the dual benefit of financial sustainability and environmental responsibility—a transformative shift for retirement planning.