随机微分方程的概率生成函数解法

Probability Generative Function Method to Solve Stochastic Differential Equation

讨论了当难以求出随机变量的分布函数时 ,如何研究随机变量的数学期望、方差、相关系数等数字特征的有关问题 ,利用概率生成函数与概率分布函数及相应的数字特征的关系 ,给出了概率生成函数为 gx( s) =∑∞k=0pksk时数学期望与方差的确定方法 ,并应用概率生成函数方法 ,证明了随机微分方程ddt Pk( t) =-λPk( t) +λPk- 1 ( t) ( k≥ 1)在边界条件 ddt P0 ( t) =-λP0 ( t) ,P0 ( 0 ) =1,Pk( 0 ) =0 ( k≥ 1)之下的解为 Pk( t) =1k!e-λt( λt) k ( k=0 ,1,2 ,… ) ,而随机微分方程ddt Pk( t) =-λk Pk( t) +λ( k -1) Pk- 1 ( t) ( k >1)在边界条件 ddt P1 ( t) =-λP1 ( t) ,P1 ( 0 ) =1,Pk( 0 ) =0 ( k>1)之下的解为 Pk( t) =e-λt( 1-e-λt) k- 1 .

It is discussed how to deal with numerical characteristics such as mathematical expectation,variance,correlation coefficient when distribution function of random variable can not be found.The relation between probability generative function and probability distribution function is used to give a method for finding mathematical expection and variance when probability generative function is g x(s)=∑∞k=0p ks k. It is proved that stochastic differential equation has the solution P k(t)=1/k! e -λt (λt) k(k=0,1,2,…) ,ifdd t P 0(t)=-λP 0(t),P 0(0)=1,P k(0)=0 k ≥1.And,dd t P k(t)=-λkP k(t)+λ(k-1)P k-1 (t)(k>1) has the solution P k(t)=e -λt (1-e -λt ) k-1 if dd t P 1(t)=-λP 1(t),P 1(0)=1,P k(0)=0(k>1).