论文标题
在标准性期望下的非线性Lévy过程的通用稳健限制定理
A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation
论文作者
论文摘要
本文确立了在透明性期望框架下的普遍强大限制定理。在时刻和一致性条件下，我们表明，（1,2）$ in I.I.D. $α\。序列\ [\ left \ {\ left（\ frac {1} {\ sqrt {n}}} \ sum_ {i = 1}^{n} x_ {i}，\ frac {1} {n} {n} \ sum sum _ {i = 1}^{n} y_ {i}，\ frac {1} {\ sqrt [α] {n}}} \ sum_ {i = 1}^{n}^{n} Z__________________________________________ {i} \ right） $ \ tilde {l} _ {1} $，其中$ \ tilde {l} _ {t} =（\ tilde配ξ_{t}，\tildeη_{t}，\tildeζ_{t} {t}） $θ$作为一组Lévy三胞胎。这个非线性的lévy过程的特征是完全非线性，可能是退化的部分间隙分化方程（pide）\ [\ [\ left \ {\ oken {array} [c] {c] {l} \ displaystyle \ displayStyle \ partial_ partial_ partial_ { θ} \ left \ {\ int _ {\ mathbb {r}^{d}}Δ_λu（t，x，x，x，y，z）f_μ（dλ）\ right。 \\ \ displayStyle \ text {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \} \ left。 +\ langle d_ {y} u（t，x，y，z），q \ rangle+\ frac {1} {2} {2} tr [d_ {x}^{2}^{2} u（t，x，x，x，x，y，z）\ right \ right \} \ forall（t，x，y，z）\ in \ lbrack 0,1] \ times \ mathbb {r}^{3d}，\ end {array} \ right。 \]带有$Δ_λU（t，x，y，z）：= u（t，x，x，y，z+λ）-U（t，x，x，y，z） - \ langle d_ {z {z} u（t，t，x，x，y，z），λ\ rangle $。要构建极限过程$（\ tilde {l} _ {t}）_ {t \ in \ lbrack0,1]} $，我们基于均匀期望空间的紧密度和弱紧凑性的概念开发了一种新颖的弱收敛方法。我们进一步证明了一种新型的lévy-khintchine表示公式来表征$（\ tilde {l} _ {t}）_ {t \ in [0,1]} $。作为副产品，我们还提供了一种概率方法来证明上述完全非线性退化派迪的存在。
This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $α\in(1,2)$, the i.i.d. sequence \[ \left \{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\sqrt[α]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \} _{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where $\tilde{L}_{t}=(\tilde ξ_{t},\tildeη_{t},\tildeζ_{t})$, $t\in [0,1]$, is a multidimensional nonlinear Lévy process with an uncertainty set $Θ$ as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l} \displaystyle \partial_{t}u(t,x,y,z)-\sup \limits_{(F_μ,q,Q)\in Θ}\left \{ \int_{\mathbb{R}^{d}}δ_λu(t,x,y,z)F_μ(dλ)\right. \\ \displaystyle \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. +\langle D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\ \displaystyle u(0,x,y,z)=ϕ(x,y,z),\ \ \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right. \] with $δ_λu(t,x,y,z):=u(t,x,y,z+λ)-u(t,x,y,z)-\langle D_{z}u(t,x,y,z),λ\rangle$. 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 $(\tilde{L}_{t})_{t\in [0,1]}$. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.
