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If $\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]=\sqrt{S_t}\exp (-{\int }_t^T{f}_t(u)du)$ and $d{f}_t(T)=A_t(T)dt+B_t(T)d{W}_t$, the stochastic Fubini theorem and Itô formula give $f_t(t)={\sigma }_t^2/8$ and $A_t(T)=B_t(T)({\int }_t^T{B}_t(u)du-{\sigma }_t/2)$. The final term comes from the product cross-variation. Continuity extends the drift equality to the specified continuous versions.

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.

For fixed maturity $T$, define $b_t^T={\int }_t^T\sigma (t,u)du$ and $I_t^T={\int }_t^{T}f(t,u)du$. The stochastic Fubini theorem and the moving lower endpoint giveThe factor $1/2$ is the integral over one of the two triangles in the square $[t,T{]}^2$. Applying the Itô formula to $P(t,T)=e^{-I_t^T}$, its quadratic-variation correction cancels that factor:Set $D_t=e^{-{\int }_0^t{r}_{s}ds}$, the reciprocal of the continuous-time bank account. The Itô product rule then givesIf $|\sigma |\le M$, then $|b_t^T|\le MT$ on this finite horizon. The Novikov condition holds, so this stochastic exponential is a true martingale, not merely a local martingale. The discounted price is thereforeThe authoritative PDF discounts to $t$ in this part. The TeX transcription's upper endpoint $T$ would include future short rates and is incorrect here.