The martingale problem for geometric stable-like processes

2024/12/24 by Iyer, Sarvesh Ravichandran
Mathematics · Economics, Econometrics and Finance · Earth and Planetary Sciences · #advanced mathematical theories #Stochastic processes and financial applications #Aquatic and Environmental Studies

paper · pdf · doi:10.48550/arxiv.2412.18677

Abstract

We prove that the martingale problem is well posed for pure-jump Lévy-type operators of the form (\mathcal Lf)(x) = ∫_\mathbb R^d ∖ \0\ (f(x+h)-f(x) - (∇ f(x) ⋅ h)1_{‖h‖ lt; 1})K(x,h) dh, where K(x,⋅) is a jump kernel of the form K(x,h) ∼ (l(‖h‖))/(‖h‖^d) for each x ∈ \mathbb R^d,‖h‖<1, and l is a positive function that is slowly varying at 0, under suitable assumptions on K. This includes jump kernels such as those of α-geometric stable processes, α∈ (0,2].

Related

• The martingale problem for a class of stable-like processes
• Uniqueness for the martingale problem associated with pure jump processes of variable order
• Construction and heat kernel estimates of general stable-like Markov processes
• Uniqueness of stable-like processes
• Martin kernels for Markov processes with jumps