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To demonstrate Ito Integrals are martingales for simple processes, let P = {t0,t1,...,tn} be a partition of [0,T]. Assume we have a filtration Ft, a Brownian motion Wt and an adapted process D(t) ie Ft measurable.

we want to show I(t) = $\int_0^T D(u) dW_t$ is a martingale.

Let 0 <= s <= t <= T be given. And assume that s and t are in different subintervals of the partition P ie. 0<= t_l <= s <= t_(l+1) <= t_k <= t <= t_(k+1) <= T

Split the into integral into I(t) = sum from j=0 to j=l-1 (D(t_j)(W_(t_(j+1)) - W_(t_j)) + D(t_l)(W_(t_(l+1))-W_(t_l)) + sum from j=l+1 to j=k-1 (D(t_j)(W_(t_(j+1)) - W_(t_j)) + D(t_k)(W_(t))-W_(t_k))

Note that 0<= t_(j+1) <= s, E[D(t_j)(W_(t_(j+1)) - W_(t_j)) | F_s] = D(t_j)(W_(t_(j+1)) - W_(t_j)) as W_t_(j+1)) - W_t_(j)) is F_s measurable

Note that t_l <= s <= t_(l+1), E[D(t_l)(W_(t_(l+1)) - W_(t_l)) | F_s] = D(t_l) E[(W_(t_(l+1))| F_s] - W_(t_l))] (linearity)

= D(t_l) E[(W_s) - W_(t_l))] (Martingale Propoerty of BM)

Note for s <= t_j,

E[D(t_j)(W_(t_(j+1)) - W_(t_j)) | F_s] = E[E[D(t_j)(W_(t_(j+1)) - W_(t_j)) | F_t_j] | F_s] (Tower Property)

= E[D(t_j) E[W_(t_(j+1)) | F_t_j] - W_(t_j)) | F_s] (linearity)

= E[D(t_j) E[W_(t_(j)) - W_(t_j)) | F_s] (Martingale property of BM)

=0

Therefore E[I(t)| F_s] = E[Sum of four components above | F_s]

= sum from(j=0 to j=l-1) of D(t_j)(W_(t_(j+1)) - W_(t_j)) + D(t_l) ((W_s) - W_(t_l)) +0 +0 = I(s) therefore a martingale.

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