Statistical Inferences on Non-stationary Increments Self-similar Stochastic Processes via Lamperti Transformations

Overview

The research goal is to develop an original method to fill some long-term existing gaps in statistical inferences on the self-similar stochastic processes. Self-similar processes are stochastic processes that exhibit the invariance of distribution under time and space. This family of processes are an essential tool to describe most of the financial instruments, such as stock prices, option prices, futures, and currency exchange rates, with extended applications in fields such as signal processing and teletraffic analysis. In the literature, there does not exist a general way to predict the paths of those instruments, in which the main difficulty lies in the estimation of the self-similarity parameter H, which is the key parameter driving the behavior of these financial phenomena. We introduce a novel method for estimating the self-similarity index and scaling parameters of a general H-self-similar process with either stationary or non-stationary increments. The estimation algorithm is developed based on a modified Lamperti transformation, which transforms H-self-similar processes to stationary ones. As an application, we show how to use this approach to estimate the self-similarity index and scaling parameters of fractional Brownian motion, sub-fractional Brownian motion, bi-fractional Brownian motion, and tri-fractional Brownian motion. Simulation study is performed to support the consistency of our estimators, with results consistent to that of other methods. Implementation in Python is also published on GitHub.

Competition history

• CSEF 2026 Mathematical Sciences (Senior Division) · Entry S-14-03

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