A <local martingale deflator> makes both $YB$ and $YS$ local martingales. Since the filtration is generated by $W$, the martingale representation theorem and the finite-variation drift forced by $YB$ give
$$
dY_t=-r_tY_tdt+\eta_t dW_t
=-Y_t(r_tdt+\lambda_t dW_t)
$$
for a continuous adapted $\lambda$, where $\eta=-Y\lambda$. Applying the <Itô product rule> to $YS$ gives drift
$$
Y_tS_t(\mu_t-r_t-\lambda_t\sigma_t)dt.
$$
It vanishes exactly when
$$
\boxed{\lambda_t=\frac{\mu_t-r_t}{\sigma_t}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 3 / a / Solution

A local martingale deflator is a strictly positive process $Y$ such that multiplying each traded asset price by $Y$ produces a <local martingale>. It can support no-arbitrage arguments even when the deflated prices are not true martingales.

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