= Solution
Here the <bank account> is continuous and of finite variation, but for $t>0$ its rate is $r_t=t^{-1/2}$. This is integrable at zero, so the account itself is well defined. The stock satisfies $dS_t=S_tdW_t$. The previous continuity hypothesis on the rate no longer holds at the initial time.
Suppose a positive normalized <state-price density> existed; even a <local martingale deflator> would suffice for a contradiction. Then $ZB$ would be a continuous <local martingale> by the <Brownian martingale representation theorem>. Dividing by the positive continuous account gives continuous $Z$, with $Z_0=1$, and
$$
dZ_t=-Z_tt^{-1/2}dt+\eta_tdW_t,
\qquad \int_0^t\eta_s^2ds<\infty\quad\text{locally}.
$$
The <Itô product rule> for $ZS$ forces its drift to vanish, so $\eta_t=Z_tt^{-1/2}$ for almost every positive time. Continuity and $Z_0=1$ imply that each path has a positive interval on which $Z_t\ge1/2$. But then
$$
\int_0^\delta\eta_t^2dt=\int_0^\delta\frac{Z_t^2}{t}dt\ge\frac14\int_0^\delta\frac{dt}{t}=\infty,
$$
contradicting the local square-integrability required for the <stochastic integral>. Thus
$$
\boxed{\text{there is no positive normalized local deflator, hence no state-price density.}}
$$
This is the <singular initial market-price-of-risk obstruction>: the necessary market price of risk is $-t^{-1/2}$, whose squared integral diverges at zero. On an interval starting at a strictly positive time this particular obstruction disappears; the initial-time normalization is essential.
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