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
Let N, N1, N2 be simple point processes on a LCCB space (E,) such that N=N1+N2, and p() be a measurable function with 0<p()<1 on (E,). Then any two of the following statements yield another two:(Ⅰ) N is a Poiss...Let N, N1, N2 be simple point processes on a LCCB space (E,) such that N=N1+N2, and p() be a measurable function with 0<p()<1 on (E,). Then any two of the following statements yield another two:(Ⅰ) N is a Poisson process;(Ⅱ) N1 is the p( )thinning of N, N2 is the (1-p())-thinning of N;(Ⅲ) N1 and N2 are independent;(Ⅳ) N1, N2 are Poisson processes with respect to a filtration {F(A), A}, whereF(A)={N1(B), N2(B), B, BA},i.e., for each bounded set A, N1(A) and N2(A) are Poisson variables, independent of F(A ).Indeed, only the fact, (Ⅱ)+(Ⅲ)(Ⅳ)+(Ⅰ), is new.展开更多
For the two classes of stochastic processes, namely, martingale difference sequences withconstant conditional variances and processes with independent increments, each square-inte-grable functional of the process has ...For the two classes of stochastic processes, namely, martingale difference sequences withconstant conditional variances and processes with independent increments, each square-inte-grable functional of the process has been shown to have chaos decomposition if and only ifthe process has the property of predictable representation. The definition of chaos is thesame as P. A. Meyer’s, that is polynomial functional in discrete parameter case and ortho-gonal stochastic multiple integral in continuous parameter case. The proofs mainly rely onthe necessary and sufficient conditions for the property of predictable representation forthese two classes of processes, obtained previously by the authors.展开更多
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 ,
文摘Let N, N1, N2 be simple point processes on a LCCB space (E,) such that N=N1+N2, and p() be a measurable function with 0<p()<1 on (E,). Then any two of the following statements yield another two:(Ⅰ) N is a Poisson process;(Ⅱ) N1 is the p( )thinning of N, N2 is the (1-p())-thinning of N;(Ⅲ) N1 and N2 are independent;(Ⅳ) N1, N2 are Poisson processes with respect to a filtration {F(A), A}, whereF(A)={N1(B), N2(B), B, BA},i.e., for each bounded set A, N1(A) and N2(A) are Poisson variables, independent of F(A ).Indeed, only the fact, (Ⅱ)+(Ⅲ)(Ⅳ)+(Ⅰ), is new.
基金Supported by the National Natural Science Foundation of China.
文摘For the two classes of stochastic processes, namely, martingale difference sequences withconstant conditional variances and processes with independent increments, each square-inte-grable functional of the process has been shown to have chaos decomposition if and only ifthe process has the property of predictable representation. The definition of chaos is thesame as P. A. Meyer’s, that is polynomial functional in discrete parameter case and ortho-gonal stochastic multiple integral in continuous parameter case. The proofs mainly rely onthe necessary and sufficient conditions for the property of predictable representation forthese two classes of processes, obtained previously by the authors.
