A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation  

作　　者：Mingshang Hu Lianzi Jiang Gechun Liang Shige Peng 

机构地区：[1]Zhongtai Securities Institute for Financial Studies,Shandong University,Jinan 250100,China [2]College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao 266590,China [3]Department of Statistics,University of Warwick,Coventry CV47AL,UK [4]School of Mathematics,Shandong University,Jinan 250100,China

出　　处：《Probability, Uncertainty and Quantitative Risk》2023年第1期1-32,共32页概率、不确定性与定量风险（英文）

基　　金：supported by the National Key R&D Program of China(Grant No.2018YFA0703900);the National Natural Science Foundation of China(Grant No.11671231);the Qilu Young Scholars Program of Shandong University;supported by the Tian Yuan Projection of the National Natural Science Foundation of China(Grant Nos.11526205,11626247);the National Basic Research Program of China(973 Program)(Grant No.2007CB814900(Financial Risk)).

摘　　要：This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

关 键 词：Universal robust limit theorem Partial integro-differential equation Nonlinear Lévy process α-stable distribution Sublinear expectation 

分 类 号：O211.5[理学—概率论与数理统计]