Abstract In this paper, we show that the spectral mapping theorem holds for ( m , n ) * {(m,n)^{*}} -paranormal operators. We also exhibit the self-adjointness of the Riesz idempotent E λ {E_{lambda}} of ( m , n ) * {(m,n)^{*}} -paranormal operators concerning for each isolated point λ of σ ( T ) {sigma(T)} . Moreover, we show Weyl’s theorem for ( m , n ) * {(m,n)^{*}} -paranormal operators and f ( T ) {f(T)} for every f ∈ ℋ ( σ ( T ) ) {finmathcal{H}(sigma(T))} . Furthermore, we investigate the class of totally ( m , n ) * {(m,n)^{*}} -paranormal operators and show that Weyl’s theorem holds for operators in this class.
{"title":"Riesz idempotent, spectral mapping theorem and Weyl's theorem for (m,n)*-paranormal operators","authors":"S. Ram, P. Dharmarha","doi":"10.2298/fil2110293d","DOIUrl":"https://doi.org/10.2298/fil2110293d","url":null,"abstract":"Abstract In this paper, we show that the spectral mapping theorem holds for ( m , n ) * {(m,n)^{*}} -paranormal operators. We also exhibit the self-adjointness of the Riesz idempotent E λ {E_{lambda}} of ( m , n ) * {(m,n)^{*}} -paranormal operators concerning for each isolated point λ of σ ( T ) {sigma(T)} . Moreover, we show Weyl’s theorem for ( m , n ) * {(m,n)^{*}} -paranormal operators and f ( T ) {f(T)} for every f ∈ ℋ ( σ ( T ) ) {finmathcal{H}(sigma(T))} . Furthermore, we investigate the class of totally ( m , n ) * {(m,n)^{*}} -paranormal operators and show that Weyl’s theorem holds for operators in this class.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45688118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we show that the spectral mapping theorem holds for (m,n)* {(m,n)^{*}} -paranormal operators. We also exhibit the self-adjointness of the Riesz idempotent Eλ {E_{lambda}} of (m,n)* {(m,n)^{*}} -paranormal operators concerning for each isolated point λ of σ(T) {sigma(T)} . Moreover, we show Weyl’s theorem for (m,n)* {(m,n)^{*}} -paranormal operators and f(T) {f(T)} for every f∈ℋ(σ(T)) {finmathcal{H}(sigma(T))} . Furthermore, we investigate the class of totally (m,n)* {(m,n)^{*}} -paranormal operators and show that Weyl’s theorem holds for operators in this class.
Abstract The aim of the present paper is to establish some strong stability results for solutions of stochastic differential equations driven by a fractional Brownian sheet with Hurst parameters H , H ′ ∈ ( 0 , 1 ) {H,H^{prime}in(0,1)} for which pathwise uniqueness holds.
{"title":"Stability results for stochastic differential equations driven by an additive fractional Brownian sheet","authors":"Oussama El Barrimi","doi":"10.1515/rose-2023-2013","DOIUrl":"https://doi.org/10.1515/rose-2023-2013","url":null,"abstract":"Abstract The aim of the present paper is to establish some strong stability results for solutions of stochastic differential equations driven by a fractional Brownian sheet with Hurst parameters H , H ′ ∈ ( 0 , 1 ) {H,H^{prime}in(0,1)} for which pathwise uniqueness holds.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"0 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41888113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Skew slash distribution is a distribution which considers both skewness and heavy tail. It is very useful in simulation studies and realistic in representing practical data due to its less peaks, especially in data sets that violate the assumption of normality. In this article, we propose a change-point detection procedure for skew slash distribution based on the modified information criterion (MIC). Meanwhile, we provide an estimation approach based on confidence distribution (CD) to measure the accuracy of change point location estimation. By comparing with the likelihood ratio test, the simulation results show that the MIC-based method performs better in terms of powers, the coverage probabilities and average lengths of confidence sets. In the end, we apply our proposed method to real data and locate the positions of the change points successfully.
{"title":"Modified information criterion for detecting changes in skew slash distribution","authors":"Mei Li, Yubin Tian, Wei Ning","doi":"10.1515/rose-2023-2011","DOIUrl":"https://doi.org/10.1515/rose-2023-2011","url":null,"abstract":"Abstract Skew slash distribution is a distribution which considers both skewness and heavy tail. It is very useful in simulation studies and realistic in representing practical data due to its less peaks, especially in data sets that violate the assumption of normality. In this article, we propose a change-point detection procedure for skew slash distribution based on the modified information criterion (MIC). Meanwhile, we provide an estimation approach based on confidence distribution (CD) to measure the accuracy of change point location estimation. By comparing with the likelihood ratio test, the simulation results show that the MIC-based method performs better in terms of powers, the coverage probabilities and average lengths of confidence sets. In the end, we apply our proposed method to real data and locate the positions of the change points successfully.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45238224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we analyze multidimensional generalized backward stochastic differential equations with jumps in a filtration that supports a Brownian motion and an independent integer-valued random measure. Under monotonicity and linear growth assumptions on the coefficients, we give the existence and uniqueness of 𝕃 2 {mathbb{L}^{2}} -solutions provided that the generators and the terminal condition satisfy some suitable integrability conditions.
{"title":"Generalized backward stochastic differential equations with jumps in a general filtration","authors":"Badr Elmansouri, M. El Otmani","doi":"10.1515/rose-2023-2007","DOIUrl":"https://doi.org/10.1515/rose-2023-2007","url":null,"abstract":"Abstract In this paper, we analyze multidimensional generalized backward stochastic differential equations with jumps in a filtration that supports a Brownian motion and an independent integer-valued random measure. Under monotonicity and linear growth assumptions on the coefficients, we give the existence and uniqueness of 𝕃 2 {mathbb{L}^{2}} -solutions provided that the generators and the terminal condition satisfy some suitable integrability conditions.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45373177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper deals with a class of delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also on the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is a divergence-type integral.
{"title":"Delay BSDEs driven by fractional Brownian motion","authors":"Sadibou Aidara, Ibrahima Sané","doi":"10.1515/rose-2023-2014","DOIUrl":"https://doi.org/10.1515/rose-2023-2014","url":null,"abstract":"Abstract This paper deals with a class of delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also on the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is a divergence-type integral.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46127830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper concerns a class of fractional impulsive neutral functional differential equations with an infinite delay driven by the Rosenblatt process. A set of sufficient conditions are established for the existence of new mild solutions using fixed point theory. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical result.
{"title":"Fractional neutral functional differential equations driven by the Rosenblatt process with an infinite delay","authors":"A. Lahmoudi, E. Lakhel","doi":"10.1515/rose-2023-2009","DOIUrl":"https://doi.org/10.1515/rose-2023-2009","url":null,"abstract":"Abstract This paper concerns a class of fractional impulsive neutral functional differential equations with an infinite delay driven by the Rosenblatt process. A set of sufficient conditions are established for the existence of new mild solutions using fixed point theory. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical result.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42399036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study a Schrödinger random operator where the potential is in terms of a continuous semimartingale. Our model is a generalization of the well-known case where the potential is the white-noise. Our approach is to analyze the random operator by means of its bilinear form. This allows us to construct an inverse operator using an explicit Green kernel. To characterize such homogeneous solutions we use certain stochastic equations in terms of stochastic integrals with respect to the semimartingale. An important tool that we use is the multi-dimensional Itô formula. Also, one important corollary of our results is that the operator has a discrete spectrum.
{"title":"A Schrödinger random operator with semimartingale potential","authors":"Jonathan Gutierrez-Pavón, Carlos G. Pacheco","doi":"10.1515/rose-2023-2008","DOIUrl":"https://doi.org/10.1515/rose-2023-2008","url":null,"abstract":"Abstract We study a Schrödinger random operator where the potential is in terms of a continuous semimartingale. Our model is a generalization of the well-known case where the potential is the white-noise. Our approach is to analyze the random operator by means of its bilinear form. This allows us to construct an inverse operator using an explicit Green kernel. To characterize such homogeneous solutions we use certain stochastic equations in terms of stochastic integrals with respect to the semimartingale. An important tool that we use is the multi-dimensional Itô formula. Also, one important corollary of our results is that the operator has a discrete spectrum.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41949629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The local time (LT) of a given stochastic process { X t : t ≥ 0 } {X_{t}:tgeq 0} is defined informally as L X ( t , x ) = ∫ 0 t δ x ( X s ) d s , L_{X}(t,x)=int_{0}^{t}delta_{x}(X_{s}),ds, where δ x delta_{x} denotes the Dirac function; actually, it counts the duration of the process’s stay at 𝑥 up to time 𝑡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ( u ) d B u X_{t}:=int_{0}^{t}f(u),dB_{u} for a given deterministic function f ∈ L 2 ( [ 0 , + ∞ [ ) fin L^{2}([0,+infty[) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments.
摘要给定随机过程{X t:t≥0}{X_{t}:tgeq 0}的局部时间(LT)被非正式地定义为L X≠(t, X)=∫0 t δ X≠(X s)∑ds, L_{X}(t, X)=int_{0}^{t}delta_{X}(X_{s})，ds，其中δ X delta_{X}表示Dirac函数;实际上，它会一直计算流程在端点处停留的时间𝑡。用近似方法研究了两类随机过程的LT过程的存在性和正则性。第一类是随机过程，定义为不定维纳积分X t:=∫0 t f _ (u) dB u X_{t}:=int_{0}^{t}f(u)，对于给定的确定性函数f∈l2([0，+∞[)fin L^{2}([0，+infty[))，dB_{u};第二类是l型过程，即平稳且具有独立增量的过程。
{"title":"On the local time of Gaussian and Lévy processes","authors":"Zineb Boudebane, Anis Rezgui","doi":"10.1515/rose-2023-2017","DOIUrl":"https://doi.org/10.1515/rose-2023-2017","url":null,"abstract":"Abstract The local time (LT) of a given stochastic process { X t : t ≥ 0 } {X_{t}:tgeq 0} is defined informally as L X ( t , x ) = ∫ 0 t δ x ( X s ) d s , L_{X}(t,x)=int_{0}^{t}delta_{x}(X_{s}),ds, where δ x delta_{x} denotes the Dirac function; actually, it counts the duration of the process’s stay at 𝑥 up to time 𝑡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ( u ) d B u X_{t}:=int_{0}^{t}f(u),dB_{u} for a given deterministic function f ∈ L 2 ( [ 0 , + ∞ [ ) fin L^{2}([0,+infty[) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48489332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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