Can someone please help me with the following text:

For Riemann Integral, the limit of sum of rectangles will be the same regardless of the heights of the rectangles. i.e. The lower and upper integrals are the same.

(This I am familiar).

For stochastic (random) environments, this is not true. Suppose $\displaystyle f(W_t)$ is a function of random variable $\displaystyle W_t$ and we are interested in calculating:

$\displaystyle \int^T_{t_0} f(W_s) dW_s$

still following...just a definition anyway

$\displaystyle f(W_{t_i})( W_{t_i} - W_{t_{i-1}})$ ---- Eq 1

is generally different from

$\displaystyle f(W_{t_{i-1}})( W_{t_i} - W_{t_{i-1}})$ ----Eq 2

are they saying the lower and upper integrals are not necessarily the same?

Proof:

Let W be a martingale. The expectation of the term in Eq 2, conditional on information at time $\displaystyle t_{i-1}$ will vanish. This is the case, because by definition, future increments of a martingale will be unrelated to the current information set.

lost here... I thought conditional expectation of the future value is the current value. Why does it vanish?

On the other hand, the same conditional expectation of the term in Eq 1 will in general be non-zero.

Hence, Riemann integrals in stochastic environments fail.