A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation (2205.00203v3)

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( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum {i=1}^{n}Y{i},\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\right) \right } {n=1}^{\infty} ] converges in distribution to $\tilde{L}{1}$, where $\tilde{L}{t}=(\tilde {\xi}{t},\tilde{\eta}{t},\tilde{\zeta}{t})$, $t\in [0,1]$, is a multidimensional nonlinear L\'{e}vy process with an uncertainty set $\Theta$ as a set of L\'{e}vy triplets. This nonlinear L\'{e}vy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) [ \left { \begin{array} [c]{l} \displaystyle \partial_{t}u(t,x,y,z)-\sup \limits_{(F_{\mu},q,Q)\in \Theta }\left { \int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu}(d\lambda)\right. \ \displaystyle \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. +\langle D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right } =0,\ \displaystyle u(0,x,y,z)=\phi(x,y,z),\ \ \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right. ] 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 L\'{e}vy-Khintchine representation formula to characterize $(\tilde{L}{t}){t\in [0,1]}$. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.