Analytical and Numerical Solution to a Novel Partial Differential Equation of the Optimal Investment–Consumption Combination Problem

In this paper, we obtain the analytical solution to a novel partial differential equation of the optimal investment–consumption combination problem under the condition that the random fluctuation of stock return is completely correlated with the random fluctuation of interest rate, and the random fluctuation of stock return is completely correlated with the random fluctuation of income growth rate. We also explore the numerical solution to the model in the same equation form and conclude that the numerical operation of the model can only be carried out when the random fluctuations between interest rate, stock return, unit commodity price growth rate and income growth rate satisfy a certain condition. Meanwhile, we investigate the impact of different parameters on the investment–consumption combination under a given set of basic parameters and plot the corresponding figures. We incorporate the inflation factors commonly present in the market into the model and have extended the form of the investor’s utility function, allowing investor to have an unequal asset utility function and consumption utility function, making the model more widely applicable.