[Paper Review] Properties of Switching Jump Diffusions: Maximum Principles and Harnack Inequalities

Xiaoshan Chen, Zhen-Qing Chen+2 more|arXiv (Cornell University)|Sep 30, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance14 references3 citations

TL;DR

This paper establishes maximum principles and Harnack inequalities for switching jump diffusion processes, which combine continuous dynamics, Lévy jumps, and regime switching governed by a Markovian switching process. By analyzing the associated non-local integro-differential systems, the authors derive key analytic properties essential for studying recurrence, ergodicity, and long-time behavior in hybrid systems with jumps and state-dependent switching.

ABSTRACT

This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.

Motivation & Objective

• To develop fundamental analytic tools—maximum principles and Harnack inequalities—for switching jump diffusion processes with non-local generators.
• To address the increased complexity arising from non-local operators and state-dependent switching, which distinguish these processes from standard diffusions and switching diffusions.
• To lay the theoretical groundwork for studying long-time behavior, including recurrence and ergodicity, in systems with random jumps and regime switching.
• To provide a rigorous framework for analyzing coupled systems of integro-partial differential equations arising from such hybrid processes.

Proposed method

• Analyzes the two-component Markov process $(X_t, ar{L}_t)$, where $X_t$ evolves as a jump diffusion and $ar{L}_t$ governs regime switching.
• Derives the generator of the process as a system of non-local integro-differential operators, incorporating drift, diffusion, jump, and switching components.
• Applies probabilistic potential theory and Green's operators to establish lower bounds on non-negative solutions of the associated elliptic systems.
• Uses a covering argument with balls of fixed radius to propagate local Harnack inequalities globally over compact sets.
• Employs a contradiction argument based on iterative point selection and growth control to prove boundedness and Harnack-type estimates.
• Establishes Harnack inequalities by comparing values of non-negative solutions at different points via transition probabilities and hitting time estimates.

Experimental results

Research questions

• RQ1 Under what conditions do switching jump diffusions satisfy a strong maximum principle?
• RQ2 How can Harnack inequalities be established for non-local, coupled systems arising from regime-switching jump diffusions?
• RQ3 What conditions ensure the boundedness and regularity of non-negative solutions to the associated elliptic systems?
• RQ4 How do the interplay between jumps, switching, and non-locality affect the analytic properties of the process?
• RQ5 Can the Harnack inequality be extended from local to global estimates on compact subsets of the state space?

Key findings

• A strong maximum principle holds for non-negative solutions of the elliptic system associated with switching jump diffusions, implying that if a solution achieves a non-negative maximum in the interior, it must be constant.
• Harnack inequalities are established for non-negative solutions, showing that the ratio of values at two points in a compact set is bounded by a universal constant depending on the geometry and parameters.
• The Harnack constant is shown to be uniform over compact subsets of the domain, with the bound depending on the number of covering balls and the ellipticity and jump parameters.
• The proof relies on a contradiction argument using iterative point selection and growth control, where the solution value is shown to grow exponentially unless bounded.
• The results are robust under state-dependent switching and non-local jump components, extending classical results from diffusion processes to hybrid jump-diffusion systems.
• The framework supports future analysis of recurrence, ergodicity, and long-run average control problems in systems with random jumps and regime switching.