A Stochastic Model for the Variation of Fourier Series Expansions with Time Delay Arising in Financial Market Price Changes

Abstract

In this paper, we derive a closed-form solution for the Stochastic Delay Differential Equation (SDDE). We formulated and proved theorems using Fourier series coefficients, which provided exact conditions for asset proce returns in three scenarios : linear, quadratic, and cubic functions. These price functions were utilized as the drift, representing the return rate in the SDDE
solution, resulting in three distinct solutions. We empirically evaluated these solutions to analyze the periodic impact of delay on each asset price function, revealing that an increase in the delay parameter reduces the value of time-varying asset investments.
Finally, our comparison of the asset values indicated that return rates following a linear trend offer the highest precision.