布朗运动和方差 W is a Brownian motion 专题 Finance / 金融 难度 L4 来源 QuantQuestion 收藏 标记掌握 个人笔记 题目详情 设 WWW 为布朗运动，且 s<ts<ts<t。求 Var(Ws+Wt).\operatorname{Var}(W_s+W_t).Var(Ws+Wt). 英文原题 If WWW is a Brownian motion and s<ts < ts<t , find var(Ws+Wt)\operatorname {var}\left(W_{s} + W_{t}\right)var(Ws+Wt) 解析 对布朗运动 Var(Wu)=u\operatorname{Var}(W_u)=uVar(Wu)=u，且 Cov(Ws,Wt)=min(s,t)=s\operatorname{Cov}(W_s,W_t)=\min(s,t)=sCov(Ws,Wt)=min(s,t)=s（因 s<ts<ts<t）。 因此 Var(Ws+Wt)=Var(Ws)+Var(Wt)+2Cov(Ws,Wt)=s+t+2s=t+3s.\operatorname{Var}(W_s+W_t)=\operatorname{Var}(W_s)+\operatorname{Var}(W_t)+2\operatorname{Cov}(W_s,W_t) =s+t+2s=t+3s.Var(Ws+Wt)=Var(Ws)+Var(Wt)+2Cov(Ws,Wt)=s+t+2s=t+3s. 即 Var(Ws+Wt)=t+3s.\boxed{\operatorname{Var}(W_s+W_t)=t+3s}.Var(Ws+Wt)=t+3s. 英文解析 Var(Ws+Wt)=Var(Ws)+Var(Wt)+2Cov(Ws,Wt)=s+t+2s=t+3s.\operatorname{Var}(W_s+W_t)=\operatorname{Var}(W_s)+\operatorname{Var}(W_t)+2\operatorname{Cov}(W_s,W_t) =s+t+2s=t+3s.Var(Ws+Wt)=Var(Ws)+Var(Wt)+2Cov(Ws,Wt)=s+t+2s=t+3s. Var(Ws+Wt)=t+3s.\boxed{\operatorname{Var}(W_s+W_t)=t+3s}.Var(Ws+Wt)=t+3s.