= Solution
Assume the usual positive initial asset values. The coefficients determine a unique normalized <local martingale deflator>. The <state-price density and local deflator distinction> matters here: calling it a <state-price density> uses the local convention, while true expectation pricing needs an additional qualification, addressed below.
Let $ZB$ and $ZS$ be <local martingales>, with $Z_0=1$. The <Brownian martingale representation theorem> says that every <local martingale> in the usual <natural Brownian filtration> is continuous and is a <stochastic integral> against $W$. Applying it to $ZB$, and dividing by the positive finite-variation account, forces
$$
dZ_t=-r_tZ_tdt+\eta_tdW_t
$$
for a locally square-integrable predictable $\eta$. The <Itô product rule> for $ZS$ has drift
$$
S_t\{Z_t(\mu_t-r_t)+\eta_t\sigma_t\}dt.
$$
It must vanish. Since $S,Z,\sigma$ are positive, this yields
$$
\boxed{\lambda_t=\frac{\mu_t-r_t}{\sigma_t},\qquad
\eta_t=-Z_t\lambda_t,\qquad dZ_t=-Z_t(r_tdt+\lambda_tdW_t).}
$$
Conversely this drift choice makes both $ZB$ and $ZS$ <local martingales>. Continuity and strict positivity of $\sigma$ make $\lambda$ bounded along each path on every finite time interval, so its pathwise square integral is finite. The unique linear SDE solution is
$$
\boxed{Z_t=\exp\left(-\int_0^t r_sds-\int_0^t\lambda_sdW_s-\frac12\int_0^t\lambda_s^2ds\right)>0.}
$$
Uniqueness follows from the forced drift and diffusion coefficients and uniqueness of this linear <stochastic differential equation>.
\b[Continuity alone does not make the density a true <martingale>.] For a true <equivalent martingale measure>, the <stochastic exponential> $D_t=Z_tB_t/B_0$ must have expectation one, for example under the <Novikov condition> on each horizon. True <martingale> pricing of all desired deflated payoffs also requires the relevant integrability. These stronger conclusions do not follow just from pathwise continuity.
An explicit counterexample to the stronger reading uses a three-dimensional <Bessel process> $R$ with $R_0=1$ and $dR=dW+R^{-1}dt$, which is a positive strong solution in the Brownian filtration. Take $B=1$ and $S=R$. Then $r=0$, $\mu=R^{-2}$ and $\sigma=R^{-1}$ are continuous and $\sigma>0$. The unique local candidate is $Z=R^{-1}$, with $ZS=1$. The <reciprocal three-dimensional Bessel strict local martingale> is not a true <martingale>. To verify the loss of expectation, use the standard Bessel transition density
$$
p_t(1,y)=\frac{y}{\sqrt{2\pi t}}\left(e^{-(y-1)^2/(2t)}-e^{-(y+1)^2/(2t)}\right),\qquad y>0.
$$
Integrating $p_t(1,y)/y$ gives $\mathbb E[R_t^{-1}]=2\Phi(1/\sqrt t)-1<1$ for $t>0$, where $\Phi$ is the <standard normal cumulative distribution function>. A true pricing density for the constant bank account would have expectation one. Hence the printed hypotheses establish the local deflator statement, while a true-density reading needs an additional condition and is false as stated.
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