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Supercritical SDEs driven by multiplicative stable-like Lévy processes
Transactions of the American Mathematical Society ( IF 1.3 ) Pub Date : 2021-08-23 , DOI: 10.1090/tran/8343
Zhen-Qing Chen , Xicheng Zhang , Guohuan Zhao

Abstract:In this paper, we study the following time-dependent stochastic differential equation (SDE) in $\mathbb {R}^d$: \begin{equation*} \mathrm {d} X_{t}= \sigma (t, X_{t-}) \mathrm {d} Z_t + b(t, X_{t})\mathrm {d} t, \quad X_{0}=x\in \mathbb {R}^d, \end{equation*} where $Z$ is a $d$-dimensional non-degenerate $\alpha$-stable-like process with $\alpha \in (0,2)$, and uniform in $t\geqslant 0$, $x\mapsto \sigma (t, x): \mathbb {R}^d\to \mathbb {R}^d\otimes \mathbb {R}^d$ is $\beta$-order Hölder continuous and uniformly elliptic with $\beta \in ( (1-\alpha )^+ , 1)$, and $x\mapsto b(t, x)$ is $\beta$-order Hölder continuous. The Lévy measure of the Lévy process $Z$ can be anisotropic or singular with respect to the Lebesgue measure on $\mathbb {R}^d$ and its support can be a proper subset of $\mathbb {R}^d$. We show in this paper that for every starting point $x \in \mathbb {R}^d$, the above SDE has a unique weak solution. We further show that the above SDE has a unique strong solution if $x\mapsto \sigma (t, x)$ is Lipschitz continuous and $x\mapsto b(t, x)$ is $\beta$-order Hölder continuous with $\beta \in (1-\alpha /2,1)$. When $\sigma (t, x)=\mathbb {I}_{d\times d}$, the $d\times d$ identity matrix, and $Z$ is an arbitrary non-degenerate $\alpha$-stable process with $0<\alpha <1$, our strong well-posedness result in particular gives an affirmative answer to the open problem in a paper by Priola.


中文翻译:

由乘法稳定类 Lévy 过程驱动的超临界 SDE

摘要:在本文中,我们研究了 $\mathbb {R}^d$ 中的以下瞬态随机微分方程 (SDE): \begin{equation*} \mathrm {d} X_{t}= \sigma (t , X_{t-}) \mathrm {d} Z_t + b(t, X_{t})\mathrm {d} t, \quad X_{0}=x\in \mathbb {R}^d, \end {equation*} 其中 $Z$ 是一个 $d$ 维非退化 $\alpha$-stable-like 过程,具有 $\alpha\in (0,2)$,并且在 $t\geqslant 0$ 中是一致的, $x\mapsto \sigma (t, x): \mathbb {R}^d\to \mathbb {R}^d\otimes \mathbb {R}^d$ 是 $\beta$-阶 Hölder 连续一致椭圆与 $\beta \in ( (1-\alpha )^+ , 1)$ 和 $x\mapsto b(t, x)$ 是 $\beta$-order Hölder 连续的。Lévy 过程 $Z$ 的 Lévy 测度相对于 $\mathbb {R}^d$ 上的 Lebesgue 测度可以是各向异性的或奇异的,并且它的支持可以是 $\mathbb {R}^d$ 的适当子集。我们在本文中表明,对于每个起点 $x \in \mathbb {R}^d$,上述 SDE 都有一个唯一的弱解。我们进一步证明,如果 $x\mapsto \sigma (t, x)$ 是 Lipschitz 连续的并且 $x\mapsto b(t, x)$ 是 $\beta$-order Hölder 连续$\beta \in (1-\alpha /2,1)$。当$\sigma(t, x)=\mathbb {I}_{d\times d}$,$d\times d$单位矩阵,而$Z$是任意非退化$\alpha$-stable在 $0<\alpha <1$ 的过程中,我们的强适定性结果特别为 Priola 的一篇论文中的开放问题给出了肯定的答案。x)$ 是 $\beta$-order Hölder 与 $\beta \in (1-\alpha /2,1)$ 连续。当$\sigma(t, x)=\mathbb {I}_{d\times d}$,$d\times d$单位矩阵,而$Z$是任意非退化$\alpha$-stable在 $0<\alpha <1$ 的过程中,我们的强适定性结果特别为 Priola 的一篇论文中的开放问题给出了肯定的答案。x)$ 是 $\beta$-order Hölder 与 $\beta \in (1-\alpha /2,1)$ 连续。当$\sigma(t, x)=\mathbb {I}_{d\times d}$,$d\times d$单位矩阵,而$Z$是任意非退化$\alpha$-stable在 $0<\alpha <1$ 的过程中,我们的强适定性结果特别为 Priola 的一篇论文中的开放问题给出了肯定的答案。
更新日期:2021-10-21
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