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Math Help - Problems understanding Riemann integral with Martingales

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    Problems understanding Riemann integral with Martingales

    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 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.
    Last edited by chopet; June 1st 2008 at 08:01 AM.
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