Let Q be the Q-matrix of an irreducible, positive recurrent Markov process on a countable state space. We show that, under a number of conditions, the stationary distributions of the n × n north-west corner augme...Let Q be the Q-matrix of an irreducible, positive recurrent Markov process on a countable state space. We show that, under a number of conditions, the stationary distributions of the n × n north-west corner augmentations of Q converge in total variation to the stationary distribution of the process. Two conditions guaranteeing such convergence include exponential ergodicity and stochastic monotonicity of the process. The same also holds for processes dominated by a stochastically monotone Markov process. In addition, we shall show that finite perturbations of stochastically monotone processes may be viewed as being dominated by a stochastically monotone process, thus extending the scope of these results to a larger class of processes. Consequently, the augmentation method provides an attractive, intuitive method for approximating the stationary distributions of a large class of Markov processes on countably infinite state spaces from a finite amount of known information.展开更多
Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, a...Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, assuming that the training examples are input in a stochastic way. The monotonicity of the error function in the iteration and the boundedness of the weight are both guaranteed. We also present a numerical experiment to support our results.展开更多
The criteria on separation cutoff for birth and death chains were obtained by Diaconis and Saloff-Coste in 2006. These criteria are involving all eigenvalues. In this paper, we obtain the explicit criterion, which dep...The criteria on separation cutoff for birth and death chains were obtained by Diaconis and Saloff-Coste in 2006. These criteria are involving all eigenvalues. In this paper, we obtain the explicit criterion, which depends only on the birth and death rates. Furthermore, we present two ways to estimate moments of the fastest strong stationary time and then give another but equivalent criterion explicitly.展开更多
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator g satisfies a weak s...This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator g satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable y, and a stochastic-Lipschitz condition in the state variable z. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.展开更多
This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we o...This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we obtain an L^(P)-result(p>2),including the existence and uniqueness of the p-th power integrable solution,a p-th power estimate,and a related continuous dependence property of the solution on the coefficients,for coupled linear FBSDEs in the monotonicity framework over large time intervals.In order to get rid of the stubborn constraint commonly existing in the literature,i.e.,the Lipschitz constant of σ with respect to z is very small,we introduce a linear transformation to overcome the difficulty on small intervals,and then"splice"the L^(P)-results obtained on many small intervals to yield the desired one on a large interval.展开更多
In this paper we consider one dimensional mean-field backward stochastic differential equations(BSDEs)under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the s...In this paper we consider one dimensional mean-field backward stochastic differential equations(BSDEs)under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the solution(Y,Z)but also on the law PY of Y.The first part of the paper is devoted to the existence and uniqueness of solutions in Lp,1<p≤2,where the monotonicity conditions are satisfied.Next,we show that if the generator/is uniformly continuous in(μ,y,z),uniformly with respect to(t,ω) and if the terminal valueξbelongs to Lp(Ω,F,P)with 1<p≤2,the mean-field BSDE has a unique Lp solution.展开更多
A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properti...A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properties for the extended birth_death processes with catastrophes are obtained.展开更多
In this paper,the authors establish the existence and uniqueness theorem of L^(p)(1<p≤2)solutions for multidimensional backward doubly stochastic differential equations(BDSDEs for short)under the p-order globally(...In this paper,the authors establish the existence and uniqueness theorem of L^(p)(1<p≤2)solutions for multidimensional backward doubly stochastic differential equations(BDSDEs for short)under the p-order globally(locally)weak monotonicity conditions.Comparison theorem of L^(p) solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.展开更多
This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity condition...This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity conditions and the unified approach-to ensure the existence and uniqueness of solutions of fully coupled linear FBSDEs. The authors show that the first method (the method of continuation under monotonicity conditions) can be deduced as a special case of the second method (the unified approach). An example is given to illustrate it in linear FBSDEs case. And then, a linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs can not be dealt with by the the method of continuation under monotonicity conditions and the unified approach directly. As a powerful supplement to the the method of continuation under monotonicity conditions and the unified approach, linear transformation method overall develops the well-posedness theory of fully coupled linear forward-backward stochastic differential equations which have potential applications in optimal control and partial differential equation theory.展开更多
In this article, we are interested in least squares estimator for a class of pathdependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribu...In this article, we are interested in least squares estimator for a class of pathdependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown parameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of this article lie in three aspects:(i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no longer works;(ii) We take the advantage of linear interpolation with respect to the discrete-time observations to approximate the functional solution;(iii) Our model is more applicable and practice as we are dealing with SDEs with irregular coefficients (for example, Holder continuous) and pathdistribution dependent.展开更多
文摘Let Q be the Q-matrix of an irreducible, positive recurrent Markov process on a countable state space. We show that, under a number of conditions, the stationary distributions of the n × n north-west corner augmentations of Q converge in total variation to the stationary distribution of the process. Two conditions guaranteeing such convergence include exponential ergodicity and stochastic monotonicity of the process. The same also holds for processes dominated by a stochastically monotone Markov process. In addition, we shall show that finite perturbations of stochastically monotone processes may be viewed as being dominated by a stochastically monotone process, thus extending the scope of these results to a larger class of processes. Consequently, the augmentation method provides an attractive, intuitive method for approximating the stationary distributions of a large class of Markov processes on countably infinite state spaces from a finite amount of known information.
基金Partly supported by the National Natural Science Foundation of China,and the Basic Research Program of the Committee of ScienceTechnology and Industry of National Defense of China.
文摘Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, assuming that the training examples are input in a stochastic way. The monotonicity of the error function in the iteration and the boundedness of the weight are both guaranteed. We also present a numerical experiment to support our results.
基金Acknowledgements This work was supported in part by 985 Project, 973 Project （No. 2011CB808000）, the National Natural Science Foundation of China （Grant No. 11131003）, the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20100003110005）, and the Fundamental Research Funds for the Central Universities.
文摘The criteria on separation cutoff for birth and death chains were obtained by Diaconis and Saloff-Coste in 2006. These criteria are involving all eigenvalues. In this paper, we obtain the explicit criterion, which depends only on the birth and death rates. Furthermore, we present two ways to estimate moments of the fastest strong stationary time and then give another but equivalent criterion explicitly.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No.2017XKZD11)the National Natural Science Foundation of China(Grant No.12171471).
文摘This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator g satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable y, and a stochastic-Lipschitz condition in the state variable z. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11871310,12271304 and 11971262)the Natural Science Foundation of Shandong Province(Grant No.ZR2020MA014)。
文摘This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we obtain an L^(P)-result(p>2),including the existence and uniqueness of the p-th power integrable solution,a p-th power estimate,and a related continuous dependence property of the solution on the coefficients,for coupled linear FBSDEs in the monotonicity framework over large time intervals.In order to get rid of the stubborn constraint commonly existing in the literature,i.e.,the Lipschitz constant of σ with respect to z is very small,we introduce a linear transformation to overcome the difficulty on small intervals,and then"splice"the L^(P)-results obtained on many small intervals to yield the desired one on a large interval.
基金supported in part by the NSFC(11222110,11871037)Shandong Province(JQ201202)+1 种基金NSFC-RS(11661130148,NA150344)111 Project(B12023)。
文摘In this paper we consider one dimensional mean-field backward stochastic differential equations(BSDEs)under weak assumptions on the coefficient.Unlike[3],the generator of our mean-field BSDEs depends not only on the solution(Y,Z)but also on the law PY of Y.The first part of the paper is devoted to the existence and uniqueness of solutions in Lp,1<p≤2,where the monotonicity conditions are satisfied.Next,we show that if the generator/is uniformly continuous in(μ,y,z),uniformly with respect to(t,ω) and if the terminal valueξbelongs to Lp(Ω,F,P)with 1<p≤2,the mean-field BSDE has a unique Lp solution.
文摘A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properties for the extended birth_death processes with catastrophes are obtained.
基金supported by the National Natural Science Foundation of China(No.11601509).
文摘In this paper,the authors establish the existence and uniqueness theorem of L^(p)(1<p≤2)solutions for multidimensional backward doubly stochastic differential equations(BDSDEs for short)under the p-order globally(locally)weak monotonicity conditions.Comparison theorem of L^(p) solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.
基金supported by the National Natural Science Foundation of China under Grant No.61573217the National High-Level Personnel of Special Support Programthe Chang Jiang Scholar Program of Chinese Education Ministry
文摘This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity conditions and the unified approach-to ensure the existence and uniqueness of solutions of fully coupled linear FBSDEs. The authors show that the first method (the method of continuation under monotonicity conditions) can be deduced as a special case of the second method (the unified approach). An example is given to illustrate it in linear FBSDEs case. And then, a linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs can not be dealt with by the the method of continuation under monotonicity conditions and the unified approach directly. As a powerful supplement to the the method of continuation under monotonicity conditions and the unified approach, linear transformation method overall develops the well-posedness theory of fully coupled linear forward-backward stochastic differential equations which have potential applications in optimal control and partial differential equation theory.
文摘In this article, we are interested in least squares estimator for a class of pathdependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown parameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of this article lie in three aspects:(i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no longer works;(ii) We take the advantage of linear interpolation with respect to the discrete-time observations to approximate the functional solution;(iii) Our model is more applicable and practice as we are dealing with SDEs with irregular coefficients (for example, Holder continuous) and pathdistribution dependent.
