Temporal Fluctuation Scaling and Temporal Theil Scaling in Financial Time Series

Fluctuation scaling, an emergent property in complex systems, is expressed by the relationship Ξ 2 ∼ M 1 α TFS $$\varXi _{2}\sim M_{1}^{\alpha _{TFS}}$$ , connecting the variance ( Ξ 2 $$\varXi _{2}$$ ) and mean ( M 1 $$M_{1}$$ ) from empirical data. Utilizing the path integral formalism by H. Kleinert, we explore the origin and temporal evolution of the temporal fluctuation scaling exponent, denoted as α TFS ( t ) $$\alpha _{TFS}(t)$$ . Thus, by introducing a non-linear term in the cumulant generating function, ℋ ( n ) ( p , t ; γ ) $$\mathcal {H}^{(n)}(p,t;\gamma )$$ , where n denotes the moment order, we create a model allowing arbitrary evolution of probability distribution moments. Thence, the temporal fluctuation scaling is then described through a linear combination of ℋ ( n ) ( p , t ; γ ) $$\mathcal {H}^{(n)}(p,t;\gamma )$$ with n ∈ 1 , 2 $$n\in {1,2}$$ , providing an analytical expression for the evolution of α TFS ( t ) $$\alpha _{TFS}(t)$$ . Additionally, a power-law relation, termed temporal Theil scaling, is established between the Theil index T and the mean M 1 ( t ) $$M_{1}(t)$$ , being similar to the Ginzburg-Landau theory’s order parameter-temperature relation. Finally, the proposed approach is validated across diverse financial time series with daily frequency.