Consider a one-dimensional stochastic differential equation with jumps
\(dX\left(t\right)=a\left(X\left(t\right)\right)dt+\sum_{k=1}^{m}{b}_{k}\left(X\left(t-\right)\right)d{Z}_{k}\left(t\right),\)
where Zk, k ∈ {1, 2, . . . ,m}, are independent centered Lévy processes with finite second moments. We prove that if the coefficient a(x) has a certain power asymptotics as x ⟶ ∞ and the coefficients bk, k ∈ {1, 2, . . . ,m}, satisfy certain growth condition, then the solution X(t) has the same asymptotics as the solution of the ordinary differential equation dx(t) = a(x(t))dt as t ⟶ ∞ a.s.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 11, pp. 1570–1584, November, 2023. DOI: https://doi.org/10.3842/umzh.v75i11.7684.
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Yuskovych, V. On the Asymptotics of Solutions of Stochastic Differential Equations with Jumps. Ukr Math J 75, 1778–1795 (2024). https://doi.org/10.1007/s11253-024-02291-1
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DOI: https://doi.org/10.1007/s11253-024-02291-1