stochastic processes – How to understand m-fold integrated Wiener process?


Consider a stochastic process

$$X(t) = sum_{j=1}^m theta_j frac{t^{j-1}}{(j-1)!} + sqrt{b} Z(t)$$

where

$$Z(t) = int_0^t frac{(t-u)^{m-1}}{(m-1)!},dW(u)$$

where $W(t)$ is a Wiener process. Here, $Z(t)$ is called a $m$-fold integrated Wiener process.

My question is that how to understand the process defined here? For example, what does it “look” like? What are some “commonly known” (e.g. $MR(p), AR(q)$) models that can be described in this form?


Some of my attempts: I have computed some special case. When $m=1$,
$$ X(t) = theta_1 + sqrt{b} Z(t) = theta_1 + sqrt{b} int_0^t 1,dW(u) = theta_1 + sqrt{b} W(t)$$

This is easy to understand: a constant level plus a scaled Wiener process.

When $m=2$,
begin{align*}
X(t) &= theta_1 + theta_2 t + sqrt{b} int_0^t (t-u) ,dW(u) \
&= theta_1 + theta_2 t + sqrt{b} left{ t W(t) – int_0^t u,dW(u)right}\
&= theta_1 + theta_2 t + sqrt{b} t W(t) – sqrt{b}mathcal{N}left(0, frac{t^3}{3}right)
end{align*}

This is not so obvious to me what it is.