Exercise 13.8 Let where {z(t)} is a standard Brownian motion. Let n = inf{t > 0 :

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Exercise 13.8 Let

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where {z(t)} is a standard Brownian motion. Let τn = inf{t > 0 : ez2(t) = n}. Prove that τn → ∞ as n → ∞
and that {M(t)} is a local martingale with the localizing sequence {τn}. Also, prove that


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hence {M(t)} is not a martingale.

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