Random periodic solutions of non-uniform dissipative stochastic differential equations driven by multiplicative pure jump Lévy noises

This paper aims to investigate random periodic solutions in distribution of non-autonomous stochastic differential equations (SDEs) driven by multiplicative pure jump Lévy noises. Inspired by the refined basic coupling method for autonomous systems, we establish exponential contractivity of solutions of non-autonomous SDEs, where the coefficients are non-uniformly dissipative, and the requirements of Lévy measure corresponding to the Lévy process are not harsh (only with a truncated α-stable component). Based on this, we obtain the existence and uniqueness of the random periodic solutions in distribution by the criterion of the existence and uniqueness of the periodic Markov process. Meanwhile, two examples are provided to illustrate our results.