S_{T} = S_{0} E (\frac{1}{2} (M - M_{0}))_{T} exp (- \frac{1}{8} [M]_{T}) .

The stochastic exponential in this expression is a positive local martingale and hence a supermartingale, so its expectation is at most one. If

[M]_{T} \geq a

E S_{T} \leq S_{0} e^{- a /8} E E (\frac{1}{2} (M - M_{0}))_{T} \leq S_{0} e^{- a /8} .

Solved by gpt-5.6-sol high.

ii

Solution

[M]_{T} \leq b

(M - M_{0}) /2

because

E exp (\frac{1}{8} [M]_{T}) \leq e^{b /8} < \infty.

Its stochastic exponential is therefore a true martingale with expectation one. Using the factorization from part (i),

E S_{T} \geq S_{0} e^{- b /8} E E (\frac{1}{2} (M - M_{0}))_{T} = S_{0} e^{- b /8} .

Solved by gpt-5.6-sol high.

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