= Solution
Use positive initial asset prices and the usual augmentation of the <natural Brownian filtration>. Define the <market price of risk>
$$
\boxed{\lambda_t=\frac{\mu_t-r_t}{\sigma_t}.}
$$
Continuity and strict positivity of $\sigma$ imply $\int_0^T\lambda_t^2dt<\infty$ on every finite horizon <almost surely>, because each path has a positive minimum of $\sigma$ there. Hence the strictly positive <stochastic exponential>
$$
\boxed{Y_t=\exp\!\left(-\int_0^tr_sds-\int_0^t\lambda_s\,dW_s-\frac12\int_0^t\lambda_s^2ds\right)}
$$
is defined and has $Y_0=1$ and $dY_t=-Y_t(r_tdt+\lambda_tdW_t)$. The <Itô product rule> gives
$$
d(Y_tB_t)=-Y_tB_t\lambda_t\,dW_t,\qquad d(Y_tS_t)=Y_tS_t(\sigma_t-\lambda_t)\,dW_t,
$$
so it is a <local martingale deflator>.
For uniqueness, let $\widetilde Y$ be any normalized strictly positive <local martingale deflator>. The <Brownian martingale representation theorem> makes $B\widetilde Y$ a continuous <local martingale> with a <Brownian motion> integral representation. Dividing by $B$ therefore gives $d\widetilde Y=-r\widetilde Ydt+\eta dW$. The vanishing <drift> of $S\widetilde Y$ requires $\widetilde Y(\mu-r)+\eta\sigma=0$, so $\eta=-\widetilde Y\lambda$. This scalar linear <stochastic differential equation> has exactly the exponential solution above, proving \b[the normalized <local martingale deflator> is unique.]
The assumptions do not make $\lambda$ deterministically bounded: $\sigma$ need not be bounded away from zero uniformly over outcomes. Thus a true <martingale> density or <Novikov condition> is not inferred here; the required conclusion is local.
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