2022/08/25 by Victoria Knopova, Knopova, Victoria, Yana Mokanu +1
Mathematics · Engineering · Economics, Econometrics and Finance · #advanced mathematical theories #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2208.12056
In this note we prove some sufficient conditions for ergodicity of a Levy-type process, such that on the test functions the generator of the respective semigroup is of the form Lf(x) = a(x)f'(x) + ∫ℝ ( f(x+u)-f(x)- ∇ f(x)⋅ u \mathbbI|u|≤ 1 ) ν(x,du), f∈ C∞2(ℝ). Here ν(x,du) is a Levy-type kernel and a(⋅): ℝ→ ℝ. We consider the case when the tails are of polynomial decay as well as the case when the decay is (sub)-exponential. For the proof the Foster-Lyapunov approach is used.