General path integrals and stable SDEs

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
S. Baguley, L. Doering, A. Kyprianou
{"title":"General path integrals and stable SDEs","authors":"S. Baguley, L. Doering, A. Kyprianou","doi":"10.4171/jems/1331","DOIUrl":null,"url":null,"abstract":"The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \\[ \n{\\rm d}Z_t=\\sigma(Z_{t-}){\\rm d} X_t \n\\]driven by a (symmetric) $\\alpha$-stable Levy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $\\alpha\\in(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jems/1331","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 4

Abstract

The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \[ {\rm d}Z_t=\sigma(Z_{t-}){\rm d} X_t \]driven by a (symmetric) $\alpha$-stable Levy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $\alpha\in(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.
一般路径积分与稳定sde
由布朗运动驱动的一维随机微分方程理论是经典的,几十年来已经得到了很大的理解。对于有跳跃的随机微分方程,我们的认识仍然不完整,甚至一些最基本的问题也只是部分理解。本文本着经典Engelbert-Schmidt时变方法的精神,研究了由(对称)$\alpha$ -稳定Levy过程驱动的\[ {\rm d}Z_t=\sigma(Z_{t-}){\rm d} X_t \]弱解的存在唯一性。推广并补全了Zanzotto的结果,得到了$\alpha\in(0,1)$弱解存在唯一性的完整刻画。我们的方法不是基于经典的随机微积分论证,而是基于马尔可夫过程的一般理论。在最小假设条件下,证明了路径积分有限的积分检验。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信