Fractal Itô Calculus: Extensions and Applications
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In this paper, we first summarize fractal calculus and extend It\^{o} calculus to fractal sets, focusing on the integration and differentiation of stochastic processes within fractal structures. We compare Brownian motion on the real line with that on a ternary Cantor set, generalizing Ito's framework to accommodate fractal geometry complexities. We define the fractal Ito integral with respect to fractal Brownian motion and establish the fractal Ito lemma, forming a foundation for fractal stochastic differential equations. Applications span finance and physics, including modeling stock prices with a Fractal Black-Scholes equation and simulating particle movement and population dynamics on fractals.
