On Asymptotics of Solutions of Stochastic Differential Equations with Jumps

2023/11/21 by Viktor Yuskovych, Yuskovych, Viktor
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2311.12422

openalex publication_date 2023/11/21 · openalex created_date 2023/11/24 · openalex updated_date 2026/07/28

Abstract

Consider a one-dimensional stochastic differential equation with jumps \mathrm d X(t) = a(X(t))\mathrm d t + ∑_{k = 1}^m b_k(X(t-))\mathrm d Z_k(t), where Z_k, k ∈ \1, 2, ..., m\ are independent centered Lévy processes with finite second moments. We prove that if coefficient a(x) has certain power asymptotics as x → ∞ and coefficients b_k, k ∈ \1, 2, ..., m\, satisfy certain growth condition then a solution X(t) has the same asymptotics as a solution of \mathrm d x(t) = a(x(t))\mathrm d t as t → ∞ a.s.

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