布朗运动在不同时刻的相关系数

Correlation between Brownian

What is the correlation between 2 Brownian motions at time $t_1$ and $t_2$ ? Assume that they are jointly normal with correlation?

解析

对同一个布朗运动 $W_t$ ，

\operatorname{Cov}(W_{t_1},W_{t_2})=\min(t_1,t_2),\quad \operatorname{Var}(W_{t_i})=t_i.

因此相关系数为

\boxed{\operatorname{Corr}(W_{t_1},W_{t_2})=\frac{\min(t_1,t_2)}{\sqrt{t_1t_2}}= \sqrt{\frac{\min(t_1,t_2)}{\max(t_1,t_2)}}}.