In advanced courses in stochastic processes, it is possible to define a stochastic integral with respect to a standard Brownian motion

\[ \int_{a}^{b} X_{s} d W_{s} \]

where \(\left(X_{s}\right)\) is a stochastic process and \(\left(W_{s}\right)\) is the Brownian motion, by a limit-taking process. These are used heavily in the area of mathematical finance. The building blocks are the following simple versions of stochastic integrals. Let \(x_{s}\) be a deterministic step process, with jumps at times \(t_{1}, t_{2}, \ldots, t_{n-1}\) and values

\[ x_{s}=\left\{\begin{array}{cc} x_{0} & \text { if } a \leq s

Let the stochastic integral \(\int_{a}^{b} x_{s} d W_{s}\) be defined as (the Riemann-Stieltjes integral)

\[ \sum_{i=0}^{n-1} x_{i}\left(W_{t_{i+1}}-W_{t_{i}}\right) \]

where \(t_{0}\) is taken to be \(a\) and \(t_{n}\) is \(b\). Note that \(\int_{a}^{b} x_{s} d W_{s}\) is a random variable. Find its mean and variance.