Abstract This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1 2 {frac{1}{2}} . We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.
{"title":"Backward doubly stochastic differential equations driven by fractional Brownian motion with stochastic integral-Lipschitz coefficients","authors":"Assane Ndiaye, Sadibou Aidara, A. B. Sow","doi":"10.1515/rose-2023-2024","DOIUrl":"https://doi.org/10.1515/rose-2023-2024","url":null,"abstract":"Abstract This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1 2 {frac{1}{2}} . We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139438616","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 the existence, uniqueness and stability of solutions of stochastic Volterra integral equations perturbed by some random processes. The obtained results extend, generalize and enrich the theory of stochastic Volterra integral equations in literature. Lastly, for illustration, we give an example that agrees with the theoretical analysis.
{"title":"On Ulam type of stability for stochastic integral equations with Volterra noise","authors":"Sheila A. Bishop, Samuel A. Iyase","doi":"10.1515/rose-2023-2026","DOIUrl":"https://doi.org/10.1515/rose-2023-2026","url":null,"abstract":"Abstract This paper concerns the existence, uniqueness and stability of solutions of stochastic Volterra integral equations perturbed by some random processes. The obtained results extend, generalize and enrich the theory of stochastic Volterra integral equations in literature. Lastly, for illustration, we give an example that agrees with the theoretical analysis.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229510","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 work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory.
{"title":"Existence results for some stochastic functional integrodifferential systems driven by Rosenblatt process","authors":"Amadou Diop, Mamadou Abdoul Diop, Khalil Ezzinbi, Essozimna Kpizim","doi":"10.1515/rose-2023-2020","DOIUrl":"https://doi.org/10.1515/rose-2023-2020","url":null,"abstract":"Abstract This work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136226889","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 consider the problem of drift parameter estimation in a stochastic differential equation driven by fractional Brownian motion with Hurst parameter H∈(12,1) {Hin(frac{1}{2},1)} and small diffusion. The technique that we used is the trajectory fitting method. Strong consistency and asymptotic distribution of the estimator are established as a small diffusion coefficient goes to zero.
{"title":"Trajectory fitting estimation for stochastic differential equations driven by fractional Brownian motion","authors":"Hector Araya, John Barrera","doi":"10.1515/rose-2023-2018","DOIUrl":"https://doi.org/10.1515/rose-2023-2018","url":null,"abstract":"Abstract We consider the problem of drift parameter estimation in a stochastic differential equation driven by fractional Brownian motion with Hurst parameter <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {Hin(frac{1}{2},1)} and small diffusion. The technique that we used is the trajectory fitting method. Strong consistency and asymptotic distribution of the estimator are established as a small diffusion coefficient goes to zero.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234220","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 a noise driven by a multivariate point process μ with predictable compensator ν, we prove existence and uniqueness of the reflected backward stochastic differential equation’s solution with a lower obstacle (ξt)t∈[0,T] {(xi_{t})_{tin[0,T]}} which is assumed to be a right upper-semicontinuous, but not necessarily right-continuous process, and a Lipschitz driver f . The result is established by using the Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.
摘要在具有可预测补偿器ν的多元点过程μ驱动的噪声中，证明了具有下障碍(ξ t) t∈[0,t] {(xi_{t})_{t In [0, t]}}的反射后向随机微分方程解的存在唯一性，该方程被假设为右上半连续过程，但不一定是右连续过程，并具有Lipschitz驱动器f。利用可选强(但不一定是正确连续)超鞅的Mertens分解、Gal ' chouk和Lenglart对Itô公式的适当推广以及最优停止理论中的一些工具，建立了该结果。给出了这类方程的一个比较定理。
{"title":"Existence and uniqueness for reflected BSDE with multivariate point process and right upper semicontinuous obstacle","authors":"Baadi, Brahim, Marzougue, Mohamed","doi":"10.1515/rose-2023-2019","DOIUrl":"https://doi.org/10.1515/rose-2023-2019","url":null,"abstract":"Abstract In a noise driven by a multivariate point process μ with predictable compensator ν, we prove existence and uniqueness of the reflected backward stochastic differential equation’s solution with a lower obstacle <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> {(xi_{t})_{tin[0,T]}} which is assumed to be a right upper-semicontinuous, but not necessarily right-continuous process, and a Lipschitz driver f . The result is established by using the Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233034","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 the present paper, we provide some existence results for the Darboux problem of partial fractional random differential equations in Fréchet spaces with an application of a generalization of the classical Darbo fixed point theorem and the concept of measure of noncompactness.
{"title":"Random differential hyperbolic equations of fractional order in Fréchet spaces","authors":"Mohamed Helal","doi":"10.1515/rose-2023-2021","DOIUrl":"https://doi.org/10.1515/rose-2023-2021","url":null,"abstract":"Abstract In the present paper, we provide some existence results for the Darboux problem of partial fractional random differential equations in Fréchet spaces with an application of a generalization of the classical Darbo fixed point theorem and the concept of measure of noncompactness.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234213","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 prove the existence result for a mild solution of a fractional stochastic evolution inclusion involving the Caputo derivative in the Hilbert space driven by a fractional Brownian motion with the Hurst parameter H>12 {H>frac{1}{2}} . The results are obtained by using fractional calculation, operator semigroups and the fixed point theorem for multivalued mappings.
{"title":"Stochastic fractional differential inclusion driven by fractional Brownian motion","authors":"Rahma Yasmina Moulay Hachemi, Toufik Guendouzi","doi":"10.1515/rose-2023-2012","DOIUrl":"https://doi.org/10.1515/rose-2023-2012","url":null,"abstract":"Abstract In this paper, we prove the existence result for a mild solution of a fractional stochastic evolution inclusion involving the Caputo derivative in the Hilbert space driven by a fractional Brownian motion with the Hurst parameter <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>H</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> {H>frac{1}{2}} . The results are obtained by using fractional calculation, operator semigroups and the fixed point theorem for multivalued mappings.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234216","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 introduce a generalization of the two-parameter double Lindley distribution (TPDLD) of Kumar and Jose [C. S. Kumar and R. Jose, A new generalization to Laplace distribution, J. Math. Comput. 31 2020, 8–32], namely the generalized double Lindley distribution (GDLD) along with its location-scale extension (EGDLD). Then we discuss the estimation of parameters of the EGDLD by the maximum likelihood estimation procedure. Next, we illustrate this estimation procedure with the help of certain real life data sets, and a simulation study is carried out to examine the performance of various estimators of the parameters of the distribution.
{"title":"Generalized double Lindley distribution: A new model for weather and financial data","authors":"C. Satheesh Kumar, Rosmi Jose","doi":"10.1515/rose-2023-2015","DOIUrl":"https://doi.org/10.1515/rose-2023-2015","url":null,"abstract":"Abstract In this paper, we introduce a generalization of the two-parameter double Lindley distribution (TPDLD) of Kumar and Jose [C. S. Kumar and R. Jose, A new generalization to Laplace distribution, J. Math. Comput. 31 2020, 8–32], namely the generalized double Lindley distribution (GDLD) along with its location-scale extension (EGDLD). Then we discuss the estimation of parameters of the EGDLD by the maximum likelihood estimation procedure. Next, we illustrate this estimation procedure with the help of certain real life data sets, and a simulation study is carried out to examine the performance of various estimators of the parameters of the distribution.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48384955","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 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":null,"pages":null},"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}
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