 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-202/4/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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} .

 New to topics? Read the docs here!