Pub Date : 2023-11-14DOI: 10.1080/17442508.2023.2280693
Haide Gou
AbstractIn this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.Keywords: Evolution equationsdelaynonlocal problemsmonotone iterative techniqueC0-semigroupS-asymptotically ω-periodic mild solutionsMathematics Subject Classifications: 35R1235K9047D06 Data availability statementMy manuscript has no associate data.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis research was supported by the National Natural Science Foundation of China [No. 12061062], Science Research Project for Colleges and Universities of Gansu Province [No. 2022A-010].
{"title":"Monotone iterative technique for evolution equations with delay and nonlocal conditions in ordered Banach space","authors":"Haide Gou","doi":"10.1080/17442508.2023.2280693","DOIUrl":"https://doi.org/10.1080/17442508.2023.2280693","url":null,"abstract":"AbstractIn this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.Keywords: Evolution equationsdelaynonlocal problemsmonotone iterative techniqueC0-semigroupS-asymptotically ω-periodic mild solutionsMathematics Subject Classifications: 35R1235K9047D06 Data availability statementMy manuscript has no associate data.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis research was supported by the National Natural Science Foundation of China [No. 12061062], Science Research Project for Colleges and Universities of Gansu Province [No. 2022A-010].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134900685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1080/17442508.2023.2279316
Zhang Chen, Li Yang
AbstractIn this paper, we consider the well-posedness for the anticipated backward stochastic Schrödinger equation in a bounded domain or the whole space Rd, which is associated to a stochastic control problem of nonlinear Schrödinger equations with time delay effect. The approach to establish the existence and uniqueness of adapted solutions are mainly based on the complex Itô's formula, the Galerkin's approximation method and the martingale representation theorem.Keywords: Stochastic Schrödinger equationanticipated backward stochastic differential equationswell-posednessGalerkin's approximation2010 Mathematics Subject Classifications: 60H0560H1560G99 AcknowledgmentsThe authors would like to thank the referees for a careful reading of the manuscript and for a number of useful comments and suggestions for improving the paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by NSFC [grant numbers 11471190 and 11971260].
{"title":"Well-posedness for anticipated backward stochastic Schrödinger equations","authors":"Zhang Chen, Li Yang","doi":"10.1080/17442508.2023.2279316","DOIUrl":"https://doi.org/10.1080/17442508.2023.2279316","url":null,"abstract":"AbstractIn this paper, we consider the well-posedness for the anticipated backward stochastic Schrödinger equation in a bounded domain or the whole space Rd, which is associated to a stochastic control problem of nonlinear Schrödinger equations with time delay effect. The approach to establish the existence and uniqueness of adapted solutions are mainly based on the complex Itô's formula, the Galerkin's approximation method and the martingale representation theorem.Keywords: Stochastic Schrödinger equationanticipated backward stochastic differential equationswell-posednessGalerkin's approximation2010 Mathematics Subject Classifications: 60H0560H1560G99 AcknowledgmentsThe authors would like to thank the referees for a careful reading of the manuscript and for a number of useful comments and suggestions for improving the paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by NSFC [grant numbers 11471190 and 11971260].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":" 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-04DOI: 10.1080/17442508.2023.2263606
Houlin Zhou, Chao Lu, Xuejun Wang
AbstractIn this paper, we mainly study the complete f-moment convergence for sums of asymptotically almost negatively associated (AANA, for short) random variables and provide an application. A general result on complete f-moment convergence for arrays of rowwise AANA random variables is obtained. We also give an application to nonparametric regression models based on AANA errors by using the result on complete f-moment convergence that we have established. A sufficient and necessary moment condition for the complete consistency is presented. Finally, a numerical simulation is provided to verify the validity of the theoretical result.Keywords: Asymptotically almost negatively associated random variablescomplete f-moment convergencecomplete convergencenonparametric regression modelcomplete consistencyMathematical subject classifications: 60F1562G05 AcknowledgmentsThe authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and valuable suggestions which helped in improving an earlier version of this paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingSupported by the National Social Science Foundation of China (22BTJ059).
{"title":"Complete <i>f</i> -moment convergence for sums of asymptotically almost negatively associated random variables with statistical applications","authors":"Houlin Zhou, Chao Lu, Xuejun Wang","doi":"10.1080/17442508.2023.2263606","DOIUrl":"https://doi.org/10.1080/17442508.2023.2263606","url":null,"abstract":"AbstractIn this paper, we mainly study the complete f-moment convergence for sums of asymptotically almost negatively associated (AANA, for short) random variables and provide an application. A general result on complete f-moment convergence for arrays of rowwise AANA random variables is obtained. We also give an application to nonparametric regression models based on AANA errors by using the result on complete f-moment convergence that we have established. A sufficient and necessary moment condition for the complete consistency is presented. Finally, a numerical simulation is provided to verify the validity of the theoretical result.Keywords: Asymptotically almost negatively associated random variablescomplete f-moment convergencecomplete convergencenonparametric regression modelcomplete consistencyMathematical subject classifications: 60F1562G05 AcknowledgmentsThe authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and valuable suggestions which helped in improving an earlier version of this paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingSupported by the National Social Science Foundation of China (22BTJ059).","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135592411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-04DOI: 10.1080/17442508.2023.2262666
M. Perninge
We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional Lp-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.
{"title":"Infinite horizon impulse control of stochastic functional differential equations driven by Lévy processes","authors":"M. Perninge","doi":"10.1080/17442508.2023.2262666","DOIUrl":"https://doi.org/10.1080/17442508.2023.2262666","url":null,"abstract":"We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional Lp-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135548083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-25DOI: 10.1080/17442508.2023.2259534
Qinjing Qiu, Reiichiro Kawai
AbstractWe establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.Keywords: Jump-diffusion processestime-state dependent jump ratePicard iterationpartial integro-differential equationsfirst exit times2020 Mathematics Subject Classifications: 91B3060G5165M1565N15 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by JSPS Grants-in-Aid for Scientific Research 20K22301 and 21K03347.
{"title":"A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes","authors":"Qinjing Qiu, Reiichiro Kawai","doi":"10.1080/17442508.2023.2259534","DOIUrl":"https://doi.org/10.1080/17442508.2023.2259534","url":null,"abstract":"AbstractWe establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.Keywords: Jump-diffusion processestime-state dependent jump ratePicard iterationpartial integro-differential equationsfirst exit times2020 Mathematics Subject Classifications: 91B3060G5165M1565N15 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by JSPS Grants-in-Aid for Scientific Research 20K22301 and 21K03347.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135814888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-16DOI: 10.1080/17442508.2023.2258250
Bin Pei, Yong Xu, Min Han
AbstractWe prove the validity of averaging principles for two-time-scale neutral stochastic delay partial differential equations (NSDPDEs) driven by fractional Brownian motions (fBms) under two-time-scale formulation. Firstly, in the sense of mean-square convergence, we obtain not only the averaging principles for NSDPDEs involving two-time-scale Markov switching with a single weakly recurrent class but also for the case of two-time-scale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for NSDPDEs driven by fBms with random delay modulated by two-time-scale Markovian switching are established. We proved that there is a limit process in which the fast changing noise is averaged out. The limit process is substantially simpler than that of the original full fast–slow system.Keywords: Averaging principlesneutral stochastic delay partial differential equationsrandom delayfractional Brownian motionstwo-time-scale Markov switching2010 Mathematics Subject Classifications: Primary: 60G22Secondary: 60H15 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingPei's work was partially supported by National Natural Science Foundation of China (NSFC) [grant number 12172285], NSFC-Chongqing [grant number cstc2021jcyj-msxmX0296], Shaanxi Fundamental Science Research Project for Mathematics and Physics [grant number 22JSQ027], Fundamental Research Funds for the Central Universities, Young Talent Fund of the University Association for Science and Technology in Shaanxi, China. Xu's work was partially supported by NSFC [grant number 12072264], and NSFC Key International (Regional) Joint Research Program [grant number 12120101002].
{"title":"Averaging principles for two-time-scale neutral stochastic delay partial differential equations driven by fractional Brownian motions","authors":"Bin Pei, Yong Xu, Min Han","doi":"10.1080/17442508.2023.2258250","DOIUrl":"https://doi.org/10.1080/17442508.2023.2258250","url":null,"abstract":"AbstractWe prove the validity of averaging principles for two-time-scale neutral stochastic delay partial differential equations (NSDPDEs) driven by fractional Brownian motions (fBms) under two-time-scale formulation. Firstly, in the sense of mean-square convergence, we obtain not only the averaging principles for NSDPDEs involving two-time-scale Markov switching with a single weakly recurrent class but also for the case of two-time-scale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for NSDPDEs driven by fBms with random delay modulated by two-time-scale Markovian switching are established. We proved that there is a limit process in which the fast changing noise is averaged out. The limit process is substantially simpler than that of the original full fast–slow system.Keywords: Averaging principlesneutral stochastic delay partial differential equationsrandom delayfractional Brownian motionstwo-time-scale Markov switching2010 Mathematics Subject Classifications: Primary: 60G22Secondary: 60H15 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingPei's work was partially supported by National Natural Science Foundation of China (NSFC) [grant number 12172285], NSFC-Chongqing [grant number cstc2021jcyj-msxmX0296], Shaanxi Fundamental Science Research Project for Mathematics and Physics [grant number 22JSQ027], Fundamental Research Funds for the Central Universities, Young Talent Fund of the University Association for Science and Technology in Shaanxi, China. Xu's work was partially supported by NSFC [grant number 12072264], and NSFC Key International (Regional) Joint Research Program [grant number 12120101002].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135308312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-13DOI: 10.1080/17442508.2023.2256922
Ilie Grigorescu, Min Kang
AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"Asymptotics and criticality for a space-dependent branching process","authors":"Ilie Grigorescu, Min Kang","doi":"10.1080/17442508.2023.2256922","DOIUrl":"https://doi.org/10.1080/17442508.2023.2256922","url":null,"abstract":"AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-13DOI: 10.1080/17442508.2023.2258248
Mohamed Fkirine, Said Hadd
AbstractThis paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.Keywords: Semilinear stochastic equationsunbounded nonlinear perturbationHilbert spacesemigroupequations with delays Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"Solving stochastic equations with unbounded nonlinear perturbations","authors":"Mohamed Fkirine, Said Hadd","doi":"10.1080/17442508.2023.2258248","DOIUrl":"https://doi.org/10.1080/17442508.2023.2258248","url":null,"abstract":"AbstractThis paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.Keywords: Semilinear stochastic equationsunbounded nonlinear perturbationHilbert spacesemigroupequations with delays Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-11DOI: 10.1080/17442508.2023.2255340
Zhiyan Shi, Xiaoyu Zhu
The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we study the generalized asymptotic equipartition property in the form of moving average for N bifurcating Markov chains indexed by an N-branch Cayley tree, which is a special case of Markov Urandom fields. Firstly, we construct a class of random variables containing a parameter with means of 1, and establish a strong limit theorem for the moving average of multivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we present the strong law of large numbers for the frequencies of occurrence of states of the moving average, as well as the generalized asymptotic equipartition property for N bifurcating Markov chains indexed by an N-branch Cayley tree. As corollaries, we also generalize some known results.
{"title":"The asymptotic equipartition property for a special Markov random field","authors":"Zhiyan Shi, Xiaoyu Zhu","doi":"10.1080/17442508.2023.2255340","DOIUrl":"https://doi.org/10.1080/17442508.2023.2255340","url":null,"abstract":"The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we study the generalized asymptotic equipartition property in the form of moving average for N bifurcating Markov chains indexed by an N-branch Cayley tree, which is a special case of Markov Urandom fields. Firstly, we construct a class of random variables containing a parameter with means of 1, and establish a strong limit theorem for the moving average of multivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we present the strong law of large numbers for the frequencies of occurrence of states of the moving average, as well as the generalized asymptotic equipartition property for N bifurcating Markov chains indexed by an N-branch Cayley tree. As corollaries, we also generalize some known results.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135980388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-11DOI: 10.1080/17442508.2023.2256506
Gaofeng Zong
We study the mean-field type stochastic control problem where the dynamics is governed by a general Lévy process with moments of all orders. For this, we introduce the power jump processes and the related Teugels martingales and give the Malliavin derivative with respect to Teugels martingales. We derive necessary and sufficient conditions for optimality of our control problem in the form of a mean-field stochastic maximum principle.
{"title":"Malliavin derivative of Teugels martingales and mean-field type stochastic maximum principle","authors":"Gaofeng Zong","doi":"10.1080/17442508.2023.2256506","DOIUrl":"https://doi.org/10.1080/17442508.2023.2256506","url":null,"abstract":"We study the mean-field type stochastic control problem where the dynamics is governed by a general Lévy process with moments of all orders. For this, we introduce the power jump processes and the related Teugels martingales and give the Malliavin derivative with respect to Teugels martingales. We derive necessary and sufficient conditions for optimality of our control problem in the form of a mean-field stochastic maximum principle.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135980703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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