Pub Date : 2019-07-11DOI: 10.1080/17442508.2019.1641093
Wolfgang Bock, Sascha Desmettre, José Luís da Silva
In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case.
{"title":"Integral representation of generalized grey Brownian motion.","authors":"Wolfgang Bock, Sascha Desmettre, José Luís da Silva","doi":"10.1080/17442508.2019.1641093","DOIUrl":"10.1080/17442508.2019.1641093","url":null,"abstract":"<p><p>In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case.</p>","PeriodicalId":93054,"journal":{"name":"Stochastics (Abingdon, England : 2005)","volume":"92 4","pages":"552-565"},"PeriodicalIF":0.0,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7455069/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38388724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-12eCollection Date: 2020-01-01DOI: 10.1080/17442508.2019.1626859
Christel Geiss, Alexander Steinicke
We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a local Lipschitz condition in the Z and U variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value ξ and its Malliavin derivative . Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in U. BSDEs of the latter type find use in exponential utility maximization.
{"title":"Existence, uniqueness and Malliavin differentiability of Lévy-driven BSDEs with locally Lipschitz driver.","authors":"Christel Geiss, Alexander Steinicke","doi":"10.1080/17442508.2019.1626859","DOIUrl":"https://doi.org/10.1080/17442508.2019.1626859","url":null,"abstract":"<p><p>We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a local Lipschitz condition in the <i>Z</i> and <i>U</i> variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value <i>ξ</i> and its Malliavin derivative <math><mi>D</mi> <mi>ξ</mi></math> . Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in <i>U</i>. BSDEs of the latter type find use in exponential utility maximization.</p>","PeriodicalId":93054,"journal":{"name":"Stochastics (Abingdon, England : 2005)","volume":"92 3","pages":"418-453"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/17442508.2019.1626859","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38151618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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