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

Mingshang Hu; Lianzi Jiang; Gechun Liang; Shige Peng

doi:10.3934/puqr.2023001

Article Contents

2023, Volume 8, Issue 1: 1-32. Doi: 10.3934/puqr.2023001

This issue Previous Article Next Article Mean-field BSDEs with jumps and dual representation for global risk measures

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

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 CV4 7AL, UK
4.
School of Mathematics, Shandong University, Jinan 250100, China

Corresponding author: Lianzi Jiang
Corresponding author: Lianzi Jiang

Received: April 29, 2022

Accepted: September 13, 2022

Early access: November 9, 2022

Published online: November 9, 2022

We thank two referees for their constructive and helpful comments, which help to improve the presentation. Hu is supported by the National Key R&D Program of China (Grant No. 2018YFA0703900), the National Natural Science Foundation of China (Grant No. 11671231), and the Qilu Young Scholars Program of Shandong University. Peng is supported by the Tian Yuan Projection of the National Natural Science Foundation of China (Grant Nos. 11526205 and 11626247) and the National Basic Research Program of China (973 Program) (Grant No. 2007CB814900 (Financial Risk))..

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Abstract

This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for$ \alpha \in(1,2) $, the i.i.d. sequence

　　　　　　　　$ \left\{ {\left( {\dfrac{1}{{\sqrt n }} \displaystyle\sum\limits_{i = 1}^n {{X_i}} ,\dfrac{1}{n} \displaystyle\sum\limits_{i = 1}^n {{Y_i}} ,\dfrac{1}{{\sqrt[\alpha ]{n}}} \displaystyle\sum\limits_{i = 1}^n {{Z_i}} } \right)} \right\}_{n = 1}^\infty $

converges in distribution to$ \tilde{L}_{1} $, where$ \tilde{L}_{t}=(\tilde {\xi}_{t},\tilde{\eta}_{t},\tilde{\zeta}_{t}) $,$ t\in \lbrack0,1] $, is a multidimensional nonlinear Lévy process with an uncertainty set$ \Theta $as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE)

　　　　$\begin{aligned}[b]\left \{ \begin{array} {l} \partial_{t}u(t,x,y,z)-\sup \limits_{(F_{\mu},q,Q)\in \Theta }\left \{ \displaystyle\int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu} ({\rm{d}}\lambda)\right. \\ \qquad\left. +\langle D_{y}u(t,x,y,z),q\rangle+\dfrac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\ u(0,x,y,z)=\phi(x,y,z),\quad \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right.\end{aligned}$

with$ \delta_{\lambda}u(t,x,y,z):=u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle D_{z}u(t,x,y,z),\lambda \rangle $. To construct the limit process$ (\tilde {L}_{t})_{t\in \lbrack0,1]} $, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Lévy-Khintchine representation formula to characterize$ (\tilde{L}_{t})_{t\in \lbrack0,1]} $. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

Keywords:

Mathematics Subject Classification: 60F05, 60G51, 60G52, 60G65, 45K05.

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Figure 1. The three sets $ {\Re},\mathfrak{F}_{0} $ and $ \mathfrak{F}_{1} $.

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