2023/09/11 by Zhenxin Liu, Jun Ma, Liu, Zhenxin +1 · 1 citation
Economics, Econometrics and Finance · #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2309.05411
openalex publication_date 2023/09/11 · openalex created_date 2023/09/13 · openalex updated_date 2026/07/28
In this paper, we prove the existence and uniqueness of solutions as well as ergodicity for McKean-Vlasov SDEs under Lyapunov conditions, in which the Lyapunov functions are defined on \mathbb Rd× \mathcal P2(\mathbb Rd), i.e. the Lyapunov functions depend not only on space variable but also on distribution variable. It is reasonable and natural to consider distribution-dependent Lyapunov functions since the coefficients depends on distribution variable. We apply the martingale representation theorem and a modified Yamada-Watanabe theorem to obtain the existence and uniqueness of solutions. Furthermore, the Krylov-Bogolioubov theorem is used to get ergodicity since it is valid by linearity of the corresponding Fokker-Planck equations on \mathbb Rd× \mathcal P2(\mathbb Rd). In particular, if the Lyapunov function depends only on space variable, we obtain exponential ergodicity for semigroup Pt^* under Wasserstein quasi-distance. Finally, we give some examples to illustrate our theoretical results.