Stochastic Integral basics

Question

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?

Answer 1

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.

Numerically it could be approximated by ordinary quadrature methods, given a Wiener trajectory (which you could approximate by cumulatively summing approximate increments on small intervals). This would converge in pretty much any sense you like. By contrast, numerical methods for approximating stochastic integrals will not converge in any pathwise sense. The important matter for that case is that these methods do converge in the mean square sense.