A nonlinear system related to investment under uncertainty solved using the fractional pseudo-Newton method
A. Torres-Hernandez, F. Brambila-Paz, J. J. Brambila
TL;DR
The paper addresses solving a nonlinear algebraic system arising from investment under uncertainty, specifically a Dixit-Pindyck model with income dynamics $dI=(\mu dt+\sigma dz)I$ and a project value $V(I)=AI^{b}+BI^{a}+\frac{I}{l-\mu}-\frac{c}{l}$. It introduces a one-step fractional pseudo-Newton method, updating via $x_{i+1}=x_i-P_{\epsilon,\beta}(x_i) f(x_i)$ with a regularized fractional-derivative matrix $P_{\epsilon,\beta}(x_i)$ and an index $\alpha\in\mathbb{R}\setminus\mathbb{Z}$, applicable to one or several variables. The authors illustrate the approach on the Dixit-Pindyck two-variable reduction, demonstrating convergence from nonlocal initial guesses and the ability to obtain multiple solutions from a single seed. This work extends fractional-calculus-based iterative schemes to nonlinear algebraic systems in economics and engineering, offering a robust tool for investment-under-uncertainty analyses and related applications.
Abstract
A nonlinear algebraic equation system of two variables is numerically solved, which is derived from a nonlinear algebraic equation system of four variables, that corresponds to a mathematical model related to investment under conditions of uncertainty. The theory of investment under uncertainty scenarios proposes a model to determine when a producer must expand or close, depending on his income. The system mentioned above is solved using a fractional iterative method, valid for one and several variables, that uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find solutions of nonlinear systems.