Editing Wiener process (section)

=== Integrated Brownian motion ===
The time-integral of the Wiener process
<math display="block">W^{(-1)}(t) := \int_0^t W(s) \, ds</math>
is called '''integrated Brownian motion''' or '''integrated Wiener process'''.  It arises in many applications and can be shown to have the distribution ''N''(0, ''t''<sup>3</sup>/3),<ref>{{Cite web|url=http://www.quantopia.net/interview-questions-vii-integrated-brownian-motion/|title=Interview Questions VII: Integrated Brownian Motion – Quantopia| website=www.quantopia.net| language=en-US| access-date=2017-05-14}}</ref> calculated using the fact that the covariance of the Wiener process is <math> t \wedge s = \min(t, s)</math>.<ref>Forum, [http://wilmott.com/messageview.cfm?catid=4&threadid=39502 "Variance of integrated Wiener process"], 2009.</ref>

For the general case of the process defined by 
<math display="block">V_f(t) = \int_0^t f'(s)W(s) \,ds=\int_0^t (f(t)-f(s))\,dW_s</math>
Then, for <math>a > 0</math>,
<math display="block">\operatorname{Var}(V_f(t))=\int_0^t (f(t)-f(s))^2 \,ds</math>
<math display="block">\operatorname{cov}(V_f(t+a),V_f(t))=\int_0^t (f(t+a)-f(s))(f(t)-f(s)) \,ds</math>
In fact, <math>V_f(t)</math> is always a zero mean normal random variable. This allows for simulation of <math>V_f(t+a)</math> given <math>V_f(t)</math> by taking
<math display="block">V_f(t+a)=A\cdot V_f(t) +B\cdot Z</math>
where ''Z'' is a standard normal variable and
<math display="block">A=\frac{\operatorname{cov}(V_f(t+a),V_f(t))}{\operatorname{Var}(V_f(t))}</math>
<math display="block">B^2=\operatorname{Var}(V_f(t+a))-A^2\operatorname{Var}(V_f(t))</math>
The case of <math>V_f(t)=W^{(-1)}(t)</math> corresponds to <math>f(t)=t</math>. All these results can be seen as direct consequences of [[Itô isometry]].
The ''n''-times-integrated Wiener process is a zero-mean normal variable with variance <math>\frac{t}{2n+1}\left ( \frac{t^n}{n!} \right )^2 </math>. This is given by the [[Cauchy formula for repeated integration]].