[Paper Review] Levy Approximation of Impulsive Recurrent Process with Semi-Markov Switching

В. С. Королюк, Nikolaos Limnios+1 more|ArXiv.org|Oct 20, 2009

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TL;DR

This paper establishes the weak convergence of an impulsive recurrent process with semi-Markov switching to a Lévy process under a small parameter scaling, using semimartingale theory and singular perturbation of the compensating operator of the extended Markov renewal process. The key result is the convergence of the process to a Lévy process in the limit, under ergodicity and moment conditions on the jump distributions and switching mechanism.

ABSTRACT

In this paper, the weak convergence of impulsive recurrent process with semi-Markov switching in the scheme of Levy approximation is proved. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness.

Motivation & Objective

• To establish the weak convergence of a time-scaled impulsive recurrent process with semi-Markov switching to a Lévy process in the limit as the scaling parameter ε → 0.
• To analyze the asymptotic behavior of a coupled process (ξ⁰(t), x⁰(t)) where the dynamics are driven by impulsive jumps dependent on a semi-Markov switching process.
• To extend Lévy approximation theory to impulsive processes with state-dependent jump distributions and semi-Markov switching, beyond i.i.d. jump assumptions.
• To prove relative compactness of the semimartingale representation of the process family using compact containment and moment conditions.
• To apply singular perturbation techniques to the compensating operator of the extended Markov renewal process to derive the limiting generator.

Proposed method

• Formulate the impulsive process ξ⁰(t) as a sum of state- and regime-dependent jumps over a time-scaled renewal process with inter-jump times scaled by ε².
• Model the switching mechanism via a semi-Markov process with transition kernel Q(x, B, t) = P(x, B)Fₓ(t), defining a Markov renewal process (xₙ, τₙ).
• Define the compensating operator L⁰ of the extended Markov renewal process via L⁰φ = ε⁻²q(x)[∫P(x, dy)∫Gᵤₓ⁰(dz)φ(u+z, y) - φ(u, x)] for test functions φ.
• Apply singular perturbation analysis to the compensating operator L⁰φ⁰ = Lφ + θ⁰φ, decomposing the generator into leading-order terms involving drift, diffusion, and jump components.
• Use the asymptotic expansion φ⁰ = φ + εφ₁ + ε²φ₂ and solve the resulting hierarchy of equations to derive the limiting generator L.
• Establish convergence via Theorem 6.3 from [5], relying on compact containment, moment bounds, and relative compactness of quadratic variation.

Experimental results

Research questions

• RQ1 Under what conditions does the impulsive recurrent process with semi-Markov switching converge weakly to a Lévy process?
• RQ2 How does the dependence of jump distributions on the current state and switching regime affect the limiting behavior?
• RQ3 Can the compensating operator of the extended Markov renewal process be analyzed via singular perturbation to derive the limiting generator?
• RQ4 What role does the ergodicity of the switching semi-Markov process play in ensuring convergence to a Lévy process?
• RQ5 How do the moment and intensity conditions (L1–L3) on the jump distributions ensure relative compactness and convergence?

Key findings

• The process ξ⁰(t) converges weakly to a Lévy process ξ⁰(t) as ε → 0 under the stated conditions, including uniform ergodicity of the switching process.
• The limiting generator is derived as q⁻¹L = Π[(A(x) + C(x) + Gᵤₓ) + A₁(x)R̃₀A₁(x) - A₁²(x)]Π, where Π is the stationary projection.
• The compact containment condition holds due to uniform moment bounds on the jump distributions and initial state, ensuring relative compactness.
• The quadratic variation of the martingale component is relatively compact, a consequence of the moment and intensity conditions (L2, L3).
• The convergence is established via the combination of semimartingale tightness and singular perturbation of the compensating operator.
• The limiting process inherits characteristics from the averaged drift, diffusion, and jump components over the stationary distribution of the switching process.