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

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.