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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.13729 |
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| _version_ | 1866916217001869312 |
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| author | Mytnik, Leonid Weinberger, Johanna |
| author_facet | Mytnik, Leonid Weinberger, Johanna |
| contents | We consider the one-dimensional stochastic differential equation
\begin{equation*}
X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0,
\end{equation*} where $μ$ is a finite measure of Kato class $K_η$ with $η\in (0,α-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $α$-stable process with $α\in (1,2)$. We derive weak and strong well posedness for this equation when $η\leqα-1$ and $η< α-1$, respectively, and show that the condition $η\leq α-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $α$-stable processes with $α\in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13729 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes Mytnik, Leonid Weinberger, Johanna Probability 60H10, 60G52 We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0, \end{equation*} where $μ$ is a finite measure of Kato class $K_η$ with $η\in (0,α-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $α$-stable process with $α\in (1,2)$. We derive weak and strong well posedness for this equation when $η\leqα-1$ and $η< α-1$, respectively, and show that the condition $η\leq α-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $α$-stable processes with $α\in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation. |
| title | Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes |
| topic | Probability 60H10, 60G52 |
| url | https://arxiv.org/abs/2404.13729 |