[Paper Review] On the small-time behaviour of Lévy-type processes

Victoria Knopova, René L. Schilling|arXiv (Cornell University)|Oct 1, 2013

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper establishes Chung-type laws of the iterated logarithm (LIL) for a class of one-dimensional Feller and Lévy-type processes, characterizing small-time path behavior via the symbol of the infinitesimal generator. The key result provides almost sure liminf and limsup asymptotics for the supremum and increment of the process near zero, with norming functions derived from the symbol, generalizing classical LIL results for Lévy processes.

ABSTRACT

We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.

Motivation & Objective

• To extend classical Chung-type laws of the iterated logarithm (LIL) from Lévy processes to a broader class of Feller processes with state-dependent characteristics.
• To characterize the small-time behavior of pure-jump Feller processes through the symbol of their infinitesimal generator, which generalizes the characteristic exponent in the Lévy case.
• To derive almost sure liminf and limsup results for the supremum and increment of the process near time zero, using norming functions based on the symbol.
• To establish criteria for the almost sure positivity or explosion of the liminf of |X_t - x| / w(t,x), depending on the growth of the symbol.
• To generalize existing LIL results for Lévy processes by incorporating state-dependent dynamics through variable-coefficient pseudo-differential generators.

Proposed method

• The authors use the symbol $ p(x, ho) $ of the infinitesimal generator of a Feller process as the central tool to define norming functions $ u^{-1}(x, t / /log|\ ext{log} frac{1}{t}|) $, $ v(t,x) $, and $ w(t,x) $, which govern the small-time scaling of the process.
• They derive bounds on the characteristic function $ ho o ho(x, ho) $ using the Lévy-Khintchine representation for the symbol, particularly focusing on the jump measure and drift components.
• The proof technique involves estimating the characteristic function $ ho o ho(x, ho) $ via exponential moments and applying Fatou’s lemma and the Borel-Cantelli lemma to derive almost sure convergence results.
• The authors use a contradiction argument based on the real part of the characteristic function: assuming the liminf is zero leads to a contradiction with the exponential decay of the characteristic function under the symbol's growth condition.
• They apply a key inequality involving $ ext{Re}(e^{i heta}) o 1 $ as $ heta o 0 $, combined with the lower bound $ | ext{Re}(z)| o 1 $, to derive a contradiction when assuming $ ext{liminf} |X_t - x| / w(t,x) = 0 $.
• The proof distinguishes two cases: when $ ext{liminf}_{t o 0} t g(1/w(x,t)) = c(x) > 0 $, leading to $ ext{liminf} |X_t - x| / w(t,x) > 0 $, and when it diverges to infinity, implying $ ext{liminf} = ext{infty} $.

Experimental results

Research questions

• RQ1 What is the small-time asymptotic behavior of the supremum $ ext{sup}_{0 o s o t} |X_s - x| $ for a Feller process near time zero?
• RQ2 How can the symbol of the infinitesimal generator be used to define norming functions that yield almost sure LIL-type results for Feller processes?
• RQ3 Under what conditions on the symbol does the liminf of $ |X_t - x| / w(t,x) $ remain positive or diverge to infinity almost surely?
• RQ4 How do the results generalize classical LIL results for Lévy processes to the non-homogeneous setting of Feller processes?
• RQ5 What is the role of the function $ g( ho) $, derived from the symbol, in determining the growth rate of the norming function $ w(t,x) $ in the LIL?

Key findings

• For a Feller process with symbol $ p(x, ho) $, the liminf of $ rac{ ext{sup}_{0 o s o t} |X_s - x|}{u^{-1}(x, t / ext{log}|\text{log} frac{1}{t}|)} $ is almost surely equal to a positive constant $ C(x) $, where $ u $ is derived from the symbol.
• The limsup of $ rac{ ext{sup}_{s o t} |X_s - x|}{v(t,x)} $ is either 0 or $ + ext{infty} $ almost surely, depending on the growth of the symbol.
• The liminf of $ rac{|X_t - x|}{w(t,x)} $ is almost surely positive or infinite, depending on whether $ ext{liminf}_{t o 0} t g(1/w(x,t)) $ is positive or infinite, where $ g $ is a function related to the symbol.
• In the symmetric Lévy case, if $ ext{liminf}_{t o 0} t ho(1/w(t)) = c > 0 $, then $ ext{liminf} |X_t| / w(t) > 0 $ almost surely; if the limit is $ ext{infty} $, then the liminf is $ ext{infty} $ almost surely.
• The proof shows that assuming $ ext{liminf} |X_t - x| / w(t,x) = 0 $ leads to a contradiction with the exponential decay of the characteristic function, implying the liminf must be positive.
• When $ ext{liminf}_{t o 0} t g(1/w(x,t)) = c(x) > 0 $, the liminf of $ |X_t - x| / w(t,x) $ is bounded away from zero almost surely, and the constant $ ext{inf}_x c(x) > 0 $ implies $ ext{inf}_x ext{liminf} |X_t - x| / w(t,x) > 0 $.