布朗運動積分(Brownian process integral)

• $\lbrace X(t),\ t \geq 0 \rbrace$ is Brownian process, $X(t) \sim N(0, \sigma^2 t)$.
• $Z(t)= \int_0^t X(s) ds$. is called integraded Brownian process (被積分函數為布朗運動，而積分算子為直線)。
• Note: Brownian process處處連續，但是均不可微分，以下僅具符號意義，而非真正的微分
  • $\frac{d Z(t)}{d t} = X(t) \Leftrightarrow Z(t) = Z(0)+ \int_0^t X(s) ds$.
  • $Z(t)$為不同實現值之sample path $X(s)$從$0$至$t$的總和，因為$X(s)$為一隨機分佈，因此$Z(t)$也為隨機分佈，而非定值。

Properties

期望值(Expectation)

• $X(t)$ is a Brownian process $\Rightarrow$ $X(t)$ is a Gaussian process.
• $Z(t)$ is sum of $X(s)$ $\Rightarrow$ $Z(t)$ is a Gaussian process.
• Expectation $E(Z(t)) = 0$.

$\begin{array}{rcl} \because Z(t) & = & \int_a^b X(s) ds \\ & = & \lim_{n \rightarrow \infty} \frac{b-a}{n} \sum_{k=1}^n X\left( a + \frac{b-a}{n}k \right). \\ \text{Let } \delta & = & \max_{1 \leq k \leq n}(t_k - t_{k-1}),\ t_{k-1} \leq d_k \leq t_k, \\ & & t_0 = a, t_0 \leq t_1 \leq \cdots \leq t_n = b, \\ \therefore E(Z(t)) & = & \lim_{\delta \rightarrow 0} \left( \sum_{k=1}^n E(X(d_i)) (t_k - t_{k-1}) \right) \\ & = & \int_a^b E(X(s))ds \\ & = & 0. \end{array}$

共變異數(Covariance)

• $0 \leq s \leq t$, $Cov(Z(s), Z(t)) = \sigma^2 \frac{s^2}{2} \left( t - \frac{s}{3}\right)$.
• If $t = s$, $Var(Z(t)) = \frac{\sigma^2 s^3}{3} = E^2(Z(t)) = E^2 \left(\int_0^t X(s) ds \right)$.
• If $t=1$, $\int_0^1X(s)ds \sim N \left(0, \frac{\sigma^2}{3} \right)$.

$\begin{array}{rcl} Cov(Z(s), Z(t) & = & E(Z(s) Z(t)) \\ & = & E \left( \int_0^s X(u)du \int_0^t X(v)dv \right) \\ & = & E \left( \int_0^s \int_0^t X(u)X(v) du dv \right) \\ (\because E(\cdot) \text{ is linear op.}) & = & \int_0^s \int_0^t E(X(u)X(v))dudv \\ & = & \int_0^s \int_0^t Cov(X(u), X(v)) dudv \\ & = & \int_0^s \int_0^t \sigma^2 \min(u,v) dudv. \end{array}$

• If $u < v$, $\int_0^t \min(u,v)dudv = \int_0^v u du$.
• If $u > v$, $\int_0^t \min(u,v)dudv = \int_0^t v du$.

$\begin{array}{rcl} \therefore Cov(Z(s), Z(t)) & = & \int_0^s \int_0^t \sigma^2 \min(u,v) dudv \\ & = & \sigma^2 \int_0^s \left( \int_0^v u du + \int_0^t v du \right)dv \\ & = & \sigma^2 \int_0^s \left( \frac{u^2}{2} \vert_0^v+ vu\vert_v^t \right)dv \\ & = & \sigma^2 \int_0^s \left(\frac{v^2}{2} + vt - v^2 \right)dv \\ & = & \sigma^2 \int_0^s \left(vt - \frac{v^2}{2} \right)dv \\ & = & \sigma^2 \frac{s^2}{2} \left(t - \frac{s}{3} \right). \end{array}$

Brownian process as integrated function

• $\lbrace X(t),\ t \geq 0 \rbrace$ is Brownian process, $X(t) \sim N(0, \sigma^2 t)$.
• $f(t)$ continuous on $[a,b]$, and $g(t)$ continuous on $[c,d]$.
• $F(t) = \int_a^b f(s)X(s)ds$, and $G(t) = \int_c^d g(s) X(s)ds$.

Properties

期望值(Expectation)

• $E\left( F(t) \right) = E\left( \int_a^b f(s)X(s) ds \right) = \int_a^b f(s)E(X(s))ds = 0$.
• $E(G(t)) = E\left( \int_c^d g(s)X(s) ds \right) = \int_c^d g(s)E(X(s))ds = 0$.
• 上述期望值算子可進積分符號內的原因是期望值為線性線子，且$E(f(s)) = f(s)$.

共變異數(Covariance)