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Ito integral is a martingale

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asked Apr 17 in BUS 4028F - Financial Economics by anonymous

How do you prove the Ito integral is a martingale and also that it's expectation is zero ?

1 Answer

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answered Apr 18 by Dumisani Ngwenza (140 points)

If $$\{g(t)\},$$ is deterministic, then we can use the fact that, the Ito-integral is normally distributed. Hence, using the properties of an Ito-integral on slide set 3, slide 13, we have $$X_t = \int_{0}^{t} g_s dW_s,$$, this means that there is no drift (easiest way to check is the fact that there is no dt term) and we can conclude that its expectation is 0. Moreover, if we can prove that $$\mathbb{E} [M_t] = M_0$$ (in this case we would have X(0) = 0), this will indicate that the Ito-integral has no drift process and hence meets the definition of a martingale (provided it meets certain criteria that needs to be made explicit).