关于 $\int_0^T W(t)dt$

Stochastic calculus for brownian

How fresh is your stochastic calculus? What can you tell me about $\int_{0}^{T}w(t)dt$ , where $w(t)$ is a standard Brownian motion?

解析

$W(t)$ 路径几乎处处连续，因此 $\int_0^T W(t)dt$ 是逐路径的 Riemann 积分。

它是正态随机变量，且可用积分分部把它写成 Itô 积分：

\int_0^T W(t)dt=\int_0^T (T-s)\,dW_s.

因此

\boxed{\mathbb{E}\left[\int_0^T W(t)dt\right]=0},

\boxed{\operatorname{Var}\left(\int_0^T W(t)dt\right)=\int_0^T (T-s)^2ds=\frac{T^3}{3}}.