[Paper Review] Weak convergence of stochastic integrals driven by continuous-time random walks

Meredith N. Burr|arXiv (Cornell University)|Oct 2, 2011

Fractional Differential Equations Solutions17 references3 citations

TL;DR

This paper establishes weak convergence of stochastic integrals driven by scaled continuous-time random walks (CTRWs) to those driven by limit processes, specifically α-stable Lévy motion and time-changed Brownian motion. Using Kurtz and Protter's theorem on weak convergence of stochastic integrals, it proves convergence under uniform tightness of the integrators, verifying a conjecture on convergence to Itô integrals of Brownian motion.

ABSTRACT

Brownian motion is a well-known model for normal diffusion, but not all physical phenomena behave according to a Brownian motion. Many phenomena exhibit irregular diffusive behavior, called anomalous diffusion. Examples of anomalous diffusion have been observed in physics, hydrology, biology, and finance, among many other fields. Continuous-time random walks (CTRWs), introduced by Montroll and Weiss, serve as models for anomalous diffusion. CTRWs generalize the usual random walk model by allowing random waiting times between successive random jumps. Under certain conditions on the jumps and waiting times, scaled CTRWs can be shown to converge in distribution to a limit process M(t) in the cadlag space D[0,infinity) with the Skorohod J_1 or M_1 topology. An interesting question is whether stochastic integrals driven by the scaled CTRWs X^n(t) converge in distribution to a stochastic integral driven by the CTRW limit process M(t). We prove weak convergence of the stochastic integrals driven by CTRWs for certain classes of CTRWs, when the CTRW limit process is an alpha-stable Levy motion and when the CTRW limit process is a time-changed Brownian motion.

Motivation & Objective

• To establish conditions under which stochastic integrals driven by scaled CTRWs converge weakly to those driven by their limit processes.
• To verify a conjecture by Germano et al. regarding weak convergence of stochastic integrals from scaled CTRWs to Itô integrals of Brownian motion.
• To extend the theoretical foundation for modeling anomalous diffusion using CTRW-based stochastic processes.
• To provide a rigorous weak convergence framework for stochastic integration in the context of non-Markovian, heavy-tailed, and time-changed processes.

Proposed method

• Applies Kurtz and Protter’s theorem on weak convergence of stochastic integrals, requiring uniform tightness of the integrator processes.
• Uses the Skorohod J1 topology on the space of càdlàg functions D[0,∞) to analyze convergence in distribution.
• Establishes convergence of the CTRW process X^n(t) to a limit process M(t), either an α-stable Lévy motion or a time-changed Brownian motion.
• Employs the continuous-mapping theorem and J1-continuity of operations like quadratic variation and exponential function to derive convergence of related processes.
• Analyzes the logarithmic expansion of the Doléans-Dade exponential to control the convergence of the stochastic exponential of the CTRW process.
• Relies on scaling limits of CTRWs derived from Meerschaert and Scheffler’s framework, where waiting times and jumps are regularly varying and in the domain of attraction of stable laws.

Experimental results

Research questions

• RQ1 Under what conditions does the stochastic integral ∫₀ᵗ Hⁿ(s−) dXⁿ(s) converge weakly to ∫₀ᵗ H(s−) dM(s) as n→∞?
• RQ2 Does the stochastic integral of a scaled CTRW converge in distribution to the Itô integral of Brownian motion, as conjectured by Germano et al.?
• RQ3 Can weak convergence of stochastic integrals be established when the CTRW limit process is an α-stable Lévy motion?
• RQ4 Can weak convergence be proven when the limit process is a time-changed Brownian motion driven by a subordinator?

Key findings

• The stochastic integral driven by scaled CTRWs converges weakly to the stochastic integral driven by the α-stable Lévy motion limit process under appropriate scaling and uniform tightness.
• The conjecture by Germano et al. is verified: ∫₀ᵗ Xⁿ(s−) dXⁿ(s) ⇒ ∫₀ᵗ B(s) dB(s) in distribution as n→∞.
• For time-changed Brownian motion limits, weak convergence of stochastic integrals holds when the time change is a subordinator and the CTRW scaling satisfies domain of attraction conditions.
• The convergence relies on the uniform tightness of the CTRW processes Xⁿ, which ensures applicability of Kurtz and Protter’s weak convergence theorem.
• The convergence of the Doléans-Dade exponential exp{Zⁿ − ½[Zⁿ,Zⁿ]} to exp{B(E) − ½E} is established via J1-continuity and the continuous-mapping theorem.
• The logarithmic correction term in the exponential converges to zero in probability, confirming the convergence of the normalized stochastic exponential to the limit process.