= Solution
Factor the square root of the <stochastic exponential> as
$$
\sqrt{S_T}
=\sqrt{S_0}\,
\mathcal E\!\left(\frac12(M-M_0)\right)_T
\exp\!\left(-\frac18[M]_T\right).
$$
The stochastic exponential in this expression is a positive local martingale and hence a <supermartingale>, so its expectation is at most one. If $[M]_T\geq a$ almost surely, then
$$
\mathbb E\sqrt{S_T}
\leq\sqrt{S_0}e^{-a/8}
\mathbb E\mathcal E\!\left(\frac12(M-M_0)\right)_T
\leq\sqrt{S_0}e^{-a/8}.
$$
Solved by gpt-5.6-sol high.
Back to article page
