We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we...We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we obtain the stochastic maximum principle as the necessary condition for an optimal control,and we also prove its sufficiency under proper conditions.The stochastic linear quadratic problem in this setting is also discussed.展开更多
The article first studies the fully coupled Forward-Backward Stochastic Differential Equations （FBSDEs） with the continuous local martingale. The article is mainly divided into two parts. In the first part, it consi...The article first studies the fully coupled Forward-Backward Stochastic Differential Equations （FBSDEs） with the continuous local martingale. The article is mainly divided into two parts. In the first part, it considers Backward Stochastic Differential Equations （BSDEs） with the continuous local martingale. Then, on the basis of it, in the second part it considers the fully coupled FBSDEs with the continuous local martingale. It is proved that their solutions exist and are unique under the monotonicity conditions.展开更多
In this paper, we consider the strong approximation for locally square-integrable martingales. In our results, the limit process may be a process with jumps. This is an extension of the former results.
Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a numbe...Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.展开更多
This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates,and also from the random ...This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates,and also from the random delays of the incubation and immunity periods.Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state,and also for the disease to remain permanently in the system over time.Moreover,the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined,and the statistical characteristics of the distribution are given mumerically.The results of this study show that the disease will persist and become permanent in the system,regardless of(1)whether the noises are from the discase transmission rate and/or from the natural death rates or(2)whether the delays in the system are constant or random for individuals in the system.Furthermore,it is shown that"weak"noise is associated with the existence of an endemic stationary distribution for the disease,while"strong"noise is associated with extinction of the population over time.Numerical simulation examples for Plasnodiurr vitar malaria are given.展开更多
Backward stochastic differential equations (BSDE) are discussed in many papers. However, in those papers, only Brownian motion and Poisson process are considered. In this paper, we consider BSDE driven by continuous l...Backward stochastic differential equations (BSDE) are discussed in many papers. However, in those papers, only Brownian motion and Poisson process are considered. In this paper, we consider BSDE driven by continuous local martingales and random measures.展开更多
By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of ...By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of the following conditions is satisfied,(2) Suppose the Levy's measure of X may be written as v(dt,ds) = Ft(dx) dV(t) and there is a σ-finite measure G such tnat ,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The authors are also grateful to the two anonymous referees for their valuable comments.J.Song is partially supported by Shandong University(Grant No.11140089963041)the National Natural Science Foundation of China(Grant No.12071256).
文摘We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we obtain the stochastic maximum principle as the necessary condition for an optimal control,and we also prove its sufficiency under proper conditions.The stochastic linear quadratic problem in this setting is also discussed.
文摘The article first studies the fully coupled Forward-Backward Stochastic Differential Equations （FBSDEs） with the continuous local martingale. The article is mainly divided into two parts. In the first part, it considers Backward Stochastic Differential Equations （BSDEs） with the continuous local martingale. Then, on the basis of it, in the second part it considers the fully coupled FBSDEs with the continuous local martingale. It is proved that their solutions exist and are unique under the monotonicity conditions.
基金Supported by National Natural Science Foundation of China (Grant No. 10871177)Specialized Research Fund for the Doctor Program of Higher Education (Grant No. 20090101110020)
文摘In this paper, we consider the strong approximation for locally square-integrable martingales. In our results, the limit process may be a process with jumps. This is an extension of the former results.
基金supported by the National Natural Science Foundation of China (No.10271091,10571139)
文摘Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.
文摘This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates,and also from the random delays of the incubation and immunity periods.Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state,and also for the disease to remain permanently in the system over time.Moreover,the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined,and the statistical characteristics of the distribution are given mumerically.The results of this study show that the disease will persist and become permanent in the system,regardless of(1)whether the noises are from the discase transmission rate and/or from the natural death rates or(2)whether the delays in the system are constant or random for individuals in the system.Furthermore,it is shown that"weak"noise is associated with the existence of an endemic stationary distribution for the disease,while"strong"noise is associated with extinction of the population over time.Numerical simulation examples for Plasnodiurr vitar malaria are given.
文摘Backward stochastic differential equations (BSDE) are discussed in many papers. However, in those papers, only Brownian motion and Poisson process are considered. In this paper, we consider BSDE driven by continuous local martingales and random measures.
文摘By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PⅡ). Let X = {Xt, t ≥ 0} be a PII with Ext=0,V(t)=Ext2<∞and limt∞V(t)=∞ If one of the following conditions is satisfied,(2) Suppose the Levy's measure of X may be written as v(dt,ds) = Ft(dx) dV(t) and there is a σ-finite measure G such tnat ,
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.
