A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexi...A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.展开更多
In this paper we investigated theL 1 norm inequalities of theP square and the maximal functions of two-parameterB-valued strong martingales, which can be applied to characterizep-smoothness andq-convexity of Banach sp...In this paper we investigated theL 1 norm inequalities of theP square and the maximal functions of two-parameterB-valued strong martingales, which can be applied to characterizep-smoothness andq-convexity of Banach spaces.展开更多
In this paper we deal with the martingales in variable Lebesgue space over a probability space.We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditio...In this paper we deal with the martingales in variable Lebesgue space over a probability space.We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space.The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob’s maximal operators in martingale Lebesgue space with a variable exponent.In particular,we present two kinds of weak-type Doob’s maximal inequalities and some necessary and sufficient conditions for strong-type Doob’s maximal inequalities.Finally,we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1.展开更多
We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey sp...We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.展开更多
This paper studies the conditional version of Kolmogorov’s three-series theorem, and gets a new extention form of the conditional version. The results here present us an answer to the question when (or where) the con...This paper studies the conditional version of Kolmogorov’s three-series theorem, and gets a new extention form of the conditional version. The results here present us an answer to the question when (or where) the conditional version also provide necessary conditions for convergence in dependent cases. Furthermore, some new sufficient conditions are obtained.展开更多
A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the exis...A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.展开更多
This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended t...This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.展开更多
This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equati...This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels mar-tingales associated with Lévy processes.In either case,we obtain the optimality system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differen-tial equations.Moreover,the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.展开更多
“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always...“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always of the same gender as the member making the selection. This model for evolution is isomorphic to the classic Pólya’s urn. The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, then replacing it and adding a new ball of the same color. It is well known that for Pólya’s urn, the proportion of red balls in the urn is a martingale. It follows that for a consanguinity, the proportion of the membership that is male is a martingale. Furthermore, being bounded, this martingale converges to a limit. For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is also a well-known second-degree martingale from which the variance of the limiting distribution can be deduced. What the author discovered, in the process of solving his own examination problem, is that a similar martingale exists also for Pólya’s urn, even though in this case the number of red balls is the sum of random variables that are not independent. This new martingale can be used to calculate the variance of the limiting distribution. Traditionally, the probability that r red balls will be drawn from Pólya’s urn in n trials is derived by a rather tricky argument involving conditional probability. This article uses an obvious but overlooked simpler approach. Pólya’s formula for the probability that m male members will be chosen in n weeks is derived, without any mention of conditional probability, by an elementary counting argument, and its limit is shown to be a beta distribution.展开更多
A Berry–Esseen bound is obtained for self-normalized martingales under the assumption of finite moments.The bound coincides with the classical Berry–Esseenboundforstandardizedmartingales.Anexampleisgiventoshowtheopt...A Berry–Esseen bound is obtained for self-normalized martingales under the assumption of finite moments.The bound coincides with the classical Berry–Esseenboundforstandardizedmartingales.Anexampleisgiventoshowtheoptimality of the bound.Applications to Student’s statistic and autoregressive process are also discussed.展开更多
In this paper, we prove that the process of product variation of a two-parameter smooth martingale admits an ∞ modification, which can be constructed as the quasi-sure limit of sum of the corresponding product variat...In this paper, we prove that the process of product variation of a two-parameter smooth martingale admits an ∞ modification, which can be constructed as the quasi-sure limit of sum of the corresponding product variation.展开更多
We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob’s inequality in terms of(p,r)-capacity for two parameter smooth martingales.
In this paper, we prove that under the F<sub>4</sub> condition, any L log<sup>+</sup> L bounded two-parameter Banach space valued martingale converges almost surely to an integrable Banach spac...In this paper, we prove that under the F<sub>4</sub> condition, any L log<sup>+</sup> L bounded two-parameter Banach space valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if the F<sub>4</sub> condition is replaced by the weaker local F<sub>4</sub> condition.展开更多
In this paper, we study the constants in a version of Rosenthal’s inequality for locally square integrable martingales. We prove that the order of growth rates of the constants is the same as in the case of discrete ...In this paper, we study the constants in a version of Rosenthal’s inequality for locally square integrable martingales. We prove that the order of growth rates of the constants is the same as in the case of discrete time martingales.展开更多
基金Supported by the Scientific Research Foundation of Hubei Province (D200613001)the National Natural Science Foundation of China (10371093)
文摘A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.
基金Supported by the National Natural Science Foundation of China
文摘In this paper we investigated theL 1 norm inequalities of theP square and the maximal functions of two-parameterB-valued strong martingales, which can be applied to characterizep-smoothness andq-convexity of Banach spaces.
文摘In this paper we deal with the martingales in variable Lebesgue space over a probability space.We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space.The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob’s maximal operators in martingale Lebesgue space with a variable exponent.In particular,we present two kinds of weak-type Doob’s maximal inequalities and some necessary and sufficient conditions for strong-type Doob’s maximal inequalities.Finally,we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1.
文摘We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.
基金Supported by the Key Science and Technology Project of Ministry of Education(207407) NSF of Anhui Educational Bureau(2006kj251B)the Special Project Grants of AnhuiNormal University (2006xzx08)
文摘This paper studies the conditional version of Kolmogorov’s three-series theorem, and gets a new extention form of the conditional version. The results here present us an answer to the question when (or where) the conditional version also provide necessary conditions for convergence in dependent cases. Furthermore, some new sufficient conditions are obtained.
文摘A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.
基金supported by Science Engineering Research Board(SERB),DST,GovtYSS Project F.No:YSS/2014/000447 dated 20.11.2015UGC,New Delhi,for providing BSR fellowship for the year 2015.
文摘This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.
基金supported by the Key Projects of Natural Science Foundation of Zhejiang Province of China(no.Z22A013952)the National Natural Science Foundation of China(no.11871121)supported by the Natural Science Foundation of Zhejiang Province of China(no.LY21A010001).
文摘This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels mar-tingales associated with Lévy processes.In either case,we obtain the optimality system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differen-tial equations.Moreover,the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.
文摘“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always of the same gender as the member making the selection. This model for evolution is isomorphic to the classic Pólya’s urn. The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, then replacing it and adding a new ball of the same color. It is well known that for Pólya’s urn, the proportion of red balls in the urn is a martingale. It follows that for a consanguinity, the proportion of the membership that is male is a martingale. Furthermore, being bounded, this martingale converges to a limit. For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is also a well-known second-degree martingale from which the variance of the limiting distribution can be deduced. What the author discovered, in the process of solving his own examination problem, is that a similar martingale exists also for Pólya’s urn, even though in this case the number of red balls is the sum of random variables that are not independent. This new martingale can be used to calculate the variance of the limiting distribution. Traditionally, the probability that r red balls will be drawn from Pólya’s urn in n trials is derived by a rather tricky argument involving conditional probability. This article uses an obvious but overlooked simpler approach. Pólya’s formula for the probability that m male members will be chosen in n weeks is derived, without any mention of conditional probability, by an elementary counting argument, and its limit is shown to be a beta distribution.
文摘A Berry–Esseen bound is obtained for self-normalized martingales under the assumption of finite moments.The bound coincides with the classical Berry–Esseenboundforstandardizedmartingales.Anexampleisgiventoshowtheoptimality of the bound.Applications to Student’s statistic and autoregressive process are also discussed.
文摘In this paper, we prove that the process of product variation of a two-parameter smooth martingale admits an ∞ modification, which can be constructed as the quasi-sure limit of sum of the corresponding product variation.
文摘We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob’s inequality in terms of(p,r)-capacity for two parameter smooth martingales.
基金Project supported by the National Natural Science Foundation of Chinathe State Education Commission Ph. D. Station Foundation
文摘In this paper, we prove that under the F<sub>4</sub> condition, any L log<sup>+</sup> L bounded two-parameter Banach space valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if the F<sub>4</sub> condition is replaced by the weaker local F<sub>4</sub> condition.
基金the National Natural Science Foundation of China (No.10571176)
文摘In this paper, we study the constants in a version of Rosenthal’s inequality for locally square integrable martingales. We prove that the order of growth rates of the constants is the same as in the case of discrete time martingales.
