# Variance of a Wiener process

Problem statement: a continuous wiener process $w(t)$ with unit incremental variance and $w(0)=0$ is given, and then we check the wiener process at every $h$ seconds, $h>0$ is a positive number. If $\left\vert w(kh)\right\vert \geq d$ with $k$ is a positive integer and $d$ is a positive number, then $w(kh)$ is reset to zero. The process continues until $t$ goes to infinite. We want to calculate the variance $V=\lim_{T\rightarrow \infty }\frac{1}{T}E\int_{0}^{T}w^{2}\left( t\right) dt$

One paper gives the result $V=d^{2}\left( \frac{1}{6}+\frac{5}{6}\frac{h}{h+d^{2}}\right)$ but no details. I want to know how to get this result or the result is correct or not?

Thanks very much for any helps!

-
Is it possible to provide a reference to the paper? –  Sasha Apr 13 '12 at 5:05
Thanks for your interest. This is the reference -- K. J. Astrom, B. M. Bernhardsson "Comparison of Riemann and Lebesgue sampling for first order stochastic systems", Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, USA, December 2002. Please refer to section 3 subsection "Approximate Lebesgue Sampling". –  Xiangyu Meng Apr 13 '12 at 5:21
I don't know if it is correct, but there are a couple of straightforward approaches. One is to observe that the sum over resets is a geometric number of truncated normals, and another is to use the renewal theorem to find limiting distribution of $w^2$ –  mike Apr 13 '12 at 12:17
Thanks for your comments. Can you give some details? Since I am an Engineering student, I am not familiar with the approaches you mentioned. –  Xiangyu Meng Apr 13 '12 at 15:47