[Paper Review] Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

Gerardo Barrera, Juan Carlos Pardo|arXiv (Cornell University)|Jan 1, 2020

Stochastic processes and financial applications36 references10 citations

TL;DR

This paper establishes the cut-off phenomenon for d-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes under the total variation distance. Under conditions including finite log-moments and regularity of the Lévy process, it proves the existence of a cut-off time with a profile function in reversible cases and conditions under which it exists in non-reversible cases, extending results from the Brownian motion case to jump-diffusion processes with complex jump structures.

ABSTRACT

In this paper, we study the cut-off phenomenon under the total variation distance of d-dimensional Ornstein-Uhlenbeck processes which are driven by Lévy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined.

Motivation & Objective

• To investigate the cut-off phenomenon in d-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes.
• To determine conditions under which the cut-off phenomenon occurs under the total variation distance.
• To establish the existence of a profile function in reversible cases and sufficient conditions for its existence in non-reversible cases.
• To analyze the cut-off behavior for superposition and average processes of such OUL processes.
• To extend known results from Brownian motion-driven SDEs to Lévy-driven SDEs with jump components.

Proposed method

• Analyzes the SDE dX(ε)t = −QX(ε)t dt + √ε dξt for t > 0, with X(ε)0 = x0 ∈ Rd \ {0}, where ξ is a d-dimensional Lévy process.
• Imposes conditions on the Lévy process: finite log-moment and regularity to ensure convergence and cut-off behavior.
• Uses linearization techniques and coupling methods adapted to Lévy noise, differing from Brownian motion approaches due to lack of Gaussian structure.
• Applies change-of-variable and convolution inequalities for total variation distance to compare laws of processes.
• Introduces an 'invariance-type' condition to prove profile cut-off without explicit computation of total variation distances.
• Studies superposition and average processes, proving profile cut-off for superposition via self-decomposable distributions and for average processes under stability assumptions.

Experimental results

Research questions

• RQ1 Under what conditions does the cut-off phenomenon occur for Lévy-driven Ornstein-Uhlenbeck processes under total variation distance?
• RQ2 When does a profile function exist for the cut-off phenomenon in reversible and non-reversible cases?
• RQ3 How does the addition of Lévy jumps affect the cut-off behavior compared to the Brownian motion case?
• RQ4 Can profile cut-off be established for superposition and average processes of OUL processes?
• RQ5 What role does the limiting distribution’s symmetry and self-decomposability play in profile cut-off?

Key findings

• The cut-off phenomenon occurs for OUL processes driven by Lévy processes under the total variation distance when the Lévy process has finite log-moments and satisfies regularity conditions.
• A profile function exists for the cut-off phenomenon in the reversible case, and may exist in non-reversible cases under an invariance-type condition on the limiting distribution.
• For symmetric unidimensional limiting distributions, the profile cut-off condition is fully satisfied.
• Profile cut-off is established for the superposition process, with the profile function linked to a self-decomposable distribution.
• For the average process under stable Lévy noise, profile cut-off is proven with an explicit profile function, cut-off time, and window.
• The results extend the cut-off theory beyond Brownian motion to Lévy-driven SDEs, highlighting the impact of jump components on convergence dynamics.