Abstract We consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1 ) {Hin(frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.
{"title":"Malliavin calculus used to derive a stochastic maximum principle for system driven by fractional Brownian and standard Wiener motions with application","authors":"Tayeb Bouaziz, A. Chala","doi":"10.1515/rose-2020-2047","DOIUrl":"https://doi.org/10.1515/rose-2020-2047","url":null,"abstract":"Abstract We consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1 ) {Hin(frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"291 - 306"},"PeriodicalIF":0.4,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48690383","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 spectral equations for the minimax estimates of the parameters of some systems are obtained.
摘要得到了一些系统参数极大极小估计的谱方程。
{"title":"RAP-method (random perturbation method) for minimax G-filter","authors":"V. Girko, B. Shevchuk, L. Shevchuk","doi":"10.1515/rose-2020-2048","DOIUrl":"https://doi.org/10.1515/rose-2020-2048","url":null,"abstract":"Abstract The spectral equations for the minimax estimates of the parameters of some systems are obtained.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"307 - 312"},"PeriodicalIF":0.4,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2048","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44028464","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 article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.
{"title":"A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations","authors":"S. Singh, S. Ray","doi":"10.1515/rose-2020-2040","DOIUrl":"https://doi.org/10.1515/rose-2020-2040","url":null,"abstract":"Abstract In this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"209 - 216"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41625282","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 are interested in bounds on the large deviations probability and Berry–Esseen type inequalities for maximum likelihood estimator and Bayes estimator of the parameter appearing linearly in the drift of nonhomogeneous stochastic differential equation driven by fractional Brownian motion.
{"title":"Large deviations and Berry–Esseen inequalities for estimators in nonhomogeneous diffusion driven by fractional Brownian motion","authors":"Kouacou Tanoh, M. N’zi, A. F. Yodé","doi":"10.1515/rose-2020-2037","DOIUrl":"https://doi.org/10.1515/rose-2020-2037","url":null,"abstract":"Abstract We are interested in bounds on the large deviations probability and Berry–Esseen type inequalities for maximum likelihood estimator and Bayes estimator of the parameter appearing linearly in the drift of nonhomogeneous stochastic differential equation driven by fractional Brownian motion.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"183 - 196"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2037","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46998035","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 article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô’s formula for translation operators, and the uniqueness by an adapted form of “Monotonicity inequality”, proved earlier in the diffusion case. As a consequence, the solutions that we construct have the “translation invariance” property.
{"title":"Stochastic PDEs in 𝒮' for SDEs driven by Lévy noise","authors":"Suprio Bhar, R. Bhaskaran, Barun Sarkar","doi":"10.1515/rose-2020-2041","DOIUrl":"https://doi.org/10.1515/rose-2020-2041","url":null,"abstract":"Abstract In this article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô’s formula for translation operators, and the uniqueness by an adapted form of “Monotonicity inequality”, proved earlier in the diffusion case. As a consequence, the solutions that we construct have the “translation invariance” property.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"217 - 226"},"PeriodicalIF":0.4,"publicationDate":"2020-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47081649","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 Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes, which arise as approximations for open d-station queueing networks of various kinds. The data for such a process are a drift vector θ, a nonsingular d × d {dtimes d} covariance matrix Δ, and a d × d {dtimes d} reflection matrix R. The state space is the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motions, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions are formulated for some classes of reflection matrices and in two- and three-dimensional cases, but not more. In this work, we identify a new family of reflection matrices R for which the process is positive recurrent if and only if the drift θ ∈ Γ ̊ {thetainmathring{Gamma}} , where Γ ̊ {mathring{Gamma}} is the interior of the convex wedge generated by the opposite column vectors of R.
摘要反映布朗运动的半鞅（SRBM）是一个扩散过程，它是作为各种开放d站排队网络的近似而产生的。这种过程的数据是漂移向量θ、非奇异的d×d｛d times d｝协方差矩阵Δ和d×d｝反射矩阵R。状态空间是d维非负orthant，在其内部过程根据布朗运动演化，并以特定的方式反射到边界。一个标准问题是确定在什么条件下该过程是正循环的。对于某些类型的反射矩阵，在二维和三维情况下，给出了充要条件，但不是更多。在这项工作中，我们确定了一个新的反射矩阵族R，其过程是正递归的当且仅当漂移θ∈Γ。
{"title":"A new family of positive recurrent semimartingale reflecting Brownian motions in an orthant","authors":"Abdelhak Yaacoubi","doi":"10.1515/rose-2020-2036","DOIUrl":"https://doi.org/10.1515/rose-2020-2036","url":null,"abstract":"Abstract Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes, which arise as approximations for open d-station queueing networks of various kinds. The data for such a process are a drift vector θ, a nonsingular d × d {dtimes d} covariance matrix Δ, and a d × d {dtimes d} reflection matrix R. The state space is the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motions, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions are formulated for some classes of reflection matrices and in two- and three-dimensional cases, but not more. In this work, we identify a new family of reflection matrices R for which the process is positive recurrent if and only if the drift θ ∈ Γ ̊ {thetainmathring{Gamma}} , where Γ ̊ {mathring{Gamma}} is the interior of the convex wedge generated by the opposite column vectors of R.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"177 - 181"},"PeriodicalIF":0.4,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2036","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47161732","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 is devoted to derive a Freidlin–Wentzell type of the large deviation principle for stochastic differential equations with general delayed generator. We improve the result of Chi Mo and Jiaowan Luo [C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 2013, 202–210].
{"title":"Large deviations for stochastic differential equations with general delayed generator","authors":"C. Manga, A. Aman","doi":"10.1515/rose-2020-2039","DOIUrl":"https://doi.org/10.1515/rose-2020-2039","url":null,"abstract":"Abstract This paper is devoted to derive a Freidlin–Wentzell type of the large deviation principle for stochastic differential equations with general delayed generator. We improve the result of Chi Mo and Jiaowan Luo [C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 2013, 202–210].","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"197 - 207"},"PeriodicalIF":0.4,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2039","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43285780","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 propose a goodness-of-fit test based on the energy statistic for skew normality. Simulations indicate that the Type-I error of the proposed test can be controlled reasonably well for given nominal levels. Power comparisons to other existing methods under different settings show the advantage of the proposed test. Such a test is applied to two real data sets to illustrate the testing procedure.
{"title":"Goodness-of-fit test for skew normality based on energy statistics","authors":"Logan Opperman, Wei Ning","doi":"10.1515/rose-2020-2042","DOIUrl":"https://doi.org/10.1515/rose-2020-2042","url":null,"abstract":"Abstract In this paper, we propose a goodness-of-fit test based on the energy statistic for skew normality. Simulations indicate that the Type-I error of the proposed test can be controlled reasonably well for given nominal levels. Power comparisons to other existing methods under different settings show the advantage of the proposed test. Such a test is applied to two real data sets to illustrate the testing procedure.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"227 - 236"},"PeriodicalIF":0.4,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43028117","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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