Analytical Solution on Stochastic Systems to Assess the Wealth Function of Periodic Corporate Investors

Abstract

This paper investigates a system of stochastic differential equations (SDEs) to evaluate the wealth dynamics of corporate investors, focusing on disparities between linear periodic and quadratic periodic return models. Using Ito’s lemma, we derived analytical solutions for two distinct investment scenarios, enabling closed-form expressions for wealth valuation over time. The models incorporate key stochastic parameters such as intrinsic growth rate (μ), interest rate (r), and volatility (σ), with periodic functions defined as μ(t)=μ0+μ1cos⁡(ωt)\mu(t) = \mu_0 + \mu_1 \cos(\omega t)μ(t)=μ0​+μ1​cos(ωt) for linear periodic returns and (ωt)  for quadratic periodic returns. The analysis revealed that (i) An increase in the intrinsic growth rate (e.g., from μ = 0.05 to μ = 0.10) results in up to a 45% increase in final wealth values, (ii) When the interest rate rises from r = 0.02 to r = 0.08, the expected wealth declines by approximately 20%, while reducing r to 0.01 leads to a 12% increase in investor wealth, (iii) increasing volatility frm σ = 0.10 to σ = 0.30 decreases expected wealth by over 35% and that under periodic volatility , wealth becomes increasingly sensitive, with fluctuations up to ±25% observed compared to constant volatility models.Results, presented in tabular form, show that the second investment strategy—governed by quadratic periodic returns—consistently yields higher wealth accumulation, with final wealth values exceeding the linear model by 18–27% depending on parameter configurations. This affirms the strategic advantage of incorporating nonlinear periodic trends in return modeling. Overall, the study quantitatively underscores how variations in financial market parameters significantly influence independent investor wealth in stochastic environments