# Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this:

\begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*}

where dW(s) represents standard wiener increments. Then we say that the first integral is our classic Riemann integral and the second integral is our stochastic integral which is then defined and so on.

I was wondering, how I would compute:

\begin{equation*} Y_t = \int_0^t W(s) ds? \end{equation*}

Would that still be a classic Riemann Integral? How would I compute it? Would my interpretation that it is 'the area under a realised path of a wiener process' be correct?

For (almost) every fixed $\omega \in \Omega$ it is a just a Riemann integral, because you are integrating a continuous function over a compact interval. So overall what you have is a random variable, and your interpretation of this variable is correct (although it is of course signed area). The resulting random variable doesn't really have a simple representation to my knowledge.
• Thanks. So just to be very clear. Since $Y_t$ is a random variable I can't really compute $Y_t$, I can only do so for fixed $\omega$, i.e. given a trajectory?! And then I guess it would make more sense to ask for something like $E[Y_t]$ which should be 0 I guess since the wiener process is a martingale starting at 0? Commented May 20, 2015 at 16:14
• $E[Y_t]$ is zero, but you need a Fubini theorem for that (after which you can use the martingale property).