Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 4

For $S_0>0$, the stochastic exponential solution stays strictly positive. Put $U_t=\sqrt{S_t}$. The Itô formula givesThus $U$ is a nonnegative local supermartingale. To justify the true supermartingale property, stop where $U$ or its stochastic integral exceeds successive bounds. The stopped Itô formula gives $\mathbb{E}[U_{T\wedge {\tau }_m}\mid {\mathcal{F}}_t]\le U_{t\wedge {\tau }_m}$ for $T\ge t$. The conditional Fatou lemma and ${\tau }_m\uparrow \infty$ yieldIn particular $\mathbb{E}{U}_T\le U_0<\infty$. The case $S_0=0$ is the identically zero process. This is the square-root stock supermartingale.

The coefficient is $\boxed{k=1/8}$, but independence alone does not make the printed right-hand side ${\mathcal{F}}_t$-measurable. The future variance integral need not be known at time $t$. This is a genuine missing information assumption in the PDF.

The Itô formula or explicit stochastic exponential givesIf the volatility path is fixed at time zero and independent of $W$, conditioning on that path makes the first factor an exponential of a centered Gaussian variable with the compensating half-variance, so its conditional expectation is one. More generally, this conditioning works when enlarging the filtration by the entire independent volatility path preserves the Brownian property. For the usual joint filtration of Brownian history and independent volatility history, the valid conditional square-root price under independent volatility isWhen the integral is already ${\mathcal{F}}_t$-measurable, the outer conditional expectation can be removed, giving the intended printed formula. In particular this holds for deterministic volatility or an independent path disclosed initially.

For a counterexample to the unqualified printed assertion, take an independent fair Bernoulli variable $A\in \{0,1\}$, disclose it at time $1$, and letWith the filtration generated by the Brownian history and this disclosure, $W$ is Brownian and $\sigma$ is bounded, continuous, adapted, and independent of $W$. At $t=0,T=2$, the variance integral is $A/3$, while ${\mathcal{F}}_0$ is trivial. Direct Gaussian conditioning givesa constant. The proposed factor $e^{-A/24}$ is random, so cannot equal that conditional expectation. The formula requires knowledge of future integrated variance; independence by itself is insufficient.

This part's conditional-expectation representation is its own hypothesis; it does not need the incorrect unrestricted claim in part (b). Fix $T$, and writeThe assumed representation makes $M(T)$ a martingale. The stochastic Fubini theorem givesApply the Itô formula to $e^{-I}$ and use the equation for $U$ from part (a), including the cross-variation. The result isUniqueness of the continuous semimartingale decomposition makes the drift vanish. Initially this is a $dtd\mathbb{P}$ statement; the assumed continuity in time and maturity extends it to the continuous versions simultaneously. Let $T\downarrow t$ to obtainSubstitute back and differentiate the maturity integrals using their continuous integrands:This is the forward drift restriction for square-root stock claims. The $-{\sigma }_t/2$ term comes from the product cross-variation and must be retained. For the uninformative zero-stock case, the representation does not identify $f$; as usual a positive initial stock price is understood.