# Math Help - Problems understanding Riemann integral with Martingales

1. ## Problems understanding Riemann integral with Martingales

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 $f(W_t)$ is a function of random variable $W_t$ and we are interested in calculating:

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

still following...just a definition anyway

$f(W_{t_i})( W_{t_i} - W_{t_{i-1}})$ ---- Eq 1
is generally different from
$f(W_{t_{i-1}})( W_{t_i} - W_{t_{i-1}})$ ----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 $t_{i-1}$ 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.