[Paper Review] Self-similar Gaussian Markov processes

Benedict Bauer, Stefan Gerhold|arXiv (Cornell University)|Aug 7, 2020

Stochastic processes and financial applications16 references4 citations

TL;DR

This paper establishes that all centered self-similar Gaussian Markov processes are scaled versions of a two-parameter family of processes defined by $ X^{H,c}_t = t^{2H+c} W(t^{-2H-2c}) $, where $ W $ is Brownian motion and $ c \in [-\infty, -H] $. The key contribution is a complete characterization of such processes via their covariance structure, enabling simple verification of Markovianity or non-Markovianity for self-similar Gaussian processes.

ABSTRACT

We characterize all multi-dimensional real self-similar Gaussian Markov processes. Three types of covariance matrix functions occur: white-noise type functions, covariances that can be expressed by continuous matrix semigroups, and covariances based on non-continuous solutions of Cauchy's functional equation. Characterizing the latter requires us to develop some results on the representation theory of non-continuous matrix semigroups, which are presented in a companion paper. In dimension one, besides white noise, the self-similar Gaussian Markov processes reduce to a two-parameter family of time-changed Brownian motions. This observation simplifies several proofs of non-Markovianity of concrete processes found in the literature.

Motivation & Objective

• To fully characterize the class of centered, self-similar Gaussian Markov processes.
• To determine whether a given self-similar Gaussian process is Markovian by analyzing its covariance function.
• To provide a unified framework that includes Brownian motion and other self-similar processes as special cases.
• To prove that fractional Brownian motion and related variants are non-Markovian using this characterization.
• To establish that a self-similar Gaussian process with asymptotically stationary increments is not a semimartingale.

Proposed method

• Define a two-parameter family $ X^{H,c}_t = t^{2H+c} W(t^{-2H-2c}) $, where $ W $ is standard Brownian motion, to generate self-similar Gaussian Markov processes.
• Use self-similarity and the Markov property to derive a functional equation for the covariance function $ R(s,t) $, leading to $ R(s,1)R(1,1) = R(s/t,1)R(t,1) $.
• Transform the covariance function using $ g(x) = R(e^{-x},1)/R(1,1) $, which satisfies $ g(x+y) = g(x)g(y) $, reducing the problem to solving a multiplicative functional equation.
• Apply Ostrowski’s theorem on additive functions to classify solutions of the functional equation, distinguishing between exponential-type solutions and pathological ones.
• Use the limit $ c \to -\infty $ to define a degenerate process with covariance $ R_{H,-\∞}(s,t) = \begin{cases} t^{2H} & s=t \\ 0 & s\neq t \end{cases} $, corresponding to a white noise-type process.
• In the appendix, use stochastic integral representations and limit analysis to show that a certain self-similar Gaussian process with asymptotically stationary increments is not a semimartingale.

Experimental results

Research questions

• RQ1 What is the complete class of centered, self-similar Gaussian Markov processes?
• RQ2 Can the Markov property of a self-similar Gaussian process be determined solely from its covariance function?
• RQ3 Why is fractional Brownian motion not Markovian, and can this be shown uniformly across variants?
• RQ4 What is the role of additive functions and Cauchy’s functional equation in characterizing such processes?
• RQ5 Is a self-similar Gaussian process with asymptotically stationary increments necessarily a semimartingale?

Key findings

• Any centered, self-similar Gaussian Markov process is distributed as a constant multiple of $ X^{H,c} $ for some $ H>0 $ and $ c \in [-\infty, -H] $.
• The covariance function of such a process is $ R(s,t) = R(1,1) R_{H,c}(s,t) $, where $ R_{H,c}(s,t) = (s \vee t)^{2H+c} (s \wedge t)^{-c} $ for $ s \wedge t > 0 $.
• The limiting case $ c \to -\infty $ yields a process with covariance $ R_{H,-\infty}(s,t) = t^{2H} $ if $ s=t $, and 0 otherwise, which is Markov and self-similar.
• The characterization allows immediate verification of non-Markovianity for processes like fractional Brownian motion, which do not match the $ R_{H,c} $ form.
• A self-similar Gaussian process with asymptotically stationary increments is not a semimartingale, as shown via limit analysis of quadratic variation.
• The proof relies on functional equations and properties of additive functions, with Ostrowski’s theorem on Cauchy’s equation as a key tool.