Stochastic self-similarity and stationarity: Novel perspectives for heterogeneous and multifractal processes

We introduce novel perspectives on stochastic self-similarity and stationarity, addressing limitations and extensions of classical definitions when applied to heterogeneous and multifractal processes. We propose a new notion of stochastic self-similarity encompassing processes with random parameters and establish a corresponding Lamperti transformation. The concept of stationarity in marginal distributions is introduced and connected with self-similarity in marginal distributions, particularly relevant for processes with time-dependent Hurst exponent. Our results provide a comprehensive overview of definitions of self-similarity and stationarity accompanied by illustrative examples and explore relationships between the different notions through Lamperti transformations.