Properties of switching jump diffusions: Maximum principles and Harnack inequalities

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.

Recurrence and Ergodicity for A Class of Regime-Switching Jump Diffusions

“…Repeat this procedure. As pointed out in [6], since j∈M q ij (x) = 0 on R d for every i ∈ M, the resulting process (X(t), Λ(t)) is a strong Markov process with lifetime ζ = ∞. It is easy to check that the law of (X(t), Λ(t)) solves the martingale problem for G, C 2 b (R d ) so it is the desired switched jump-diffusion.…”

“…It follows from (4.6) that P φ(x) = f (x, l) for x ∈ E. Using the Harnack inequality for Gharmonic functions (see Remark 2.1(c) and [6,Theorem 4.7]), there exists a positive constant κ E such that P φ(x) ≤ κ E P φ(y) for all x, y ∈ E, φ ∈ B(E). Note that κ E is independent of φ ∈ B(E).…”

“…Define u(y, j) = P y,j (σ E < τ G ) for (y, j) ∈ R d × M. Then u(y, j) = E y,j 1 {X(τ G\E )∈E} . Hence u(y, j) is a G-harmonic function in G \ E × M. By Remark 2.1(c) and [6, Theorem 3.4], u(y, j) > 0 for all (y, j) ∈ (G \ E) × M. Let H be a domain such that D ⊂ H ⊂ H ⊂ G. Define v(y, j) = E y,j u (X(τ H ), Λ(τ H )) for (y, j) ∈ H × M.Again by Remark 2.1(c) and[6, Theorem 3.4], v(y, j) > 0 for (y, j) ∈ H × M. By Remark 2.1(c) and the Harnack inequality[6, Theorem 4.7], there is a constant δ 1 ∈ (0, 1) such that inf (y,j)∈D×M…”

“…Suppose that ν(D, l) = 0. Let x 0 ∈ D and r ∈ (0, 1) such that B(x 0 , 2r) ∈ D. By Remark 2.1(c) and [6,Proposition 4.1], there exists a constant c 1 > 0 such that…”

Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties

“…Continuing on the effort of studying regime-switching diffusions, Chen et al (2018a) obtained maximum principle and Harnack inequalities for switching jump diffusions using mainly probabilistic arguments, and Chen el al. (2018b) proceeded further to obtain recurrence and ergodicity of switching jump diffusions.…”