= Solution
Let $Q$ be the given <risk-neutral measure>, and write $B_t=B_0e^{rt}$. Take $V$ to be classical on $t<T$ and continuous at its prescribed terminal value. By <Itô formula> and the <pricing equation for a local volatility model>,
$$
\begin{aligned}
dV(t,S_t)&=\left[V_t+rS_tV_S+\tfrac12\sigma(S_t)^2S_t^2V_{SS}\right]dt+\sigma(S_t)S_tV_S\,d\widehat W_t\\
&=rV(t,S_t)dt+\sigma(S_t)S_tV_S\,d\widehat W_t,
\end{aligned}
$$
so
$$
d\left(\frac{\xi_t}{B_t}\right)=\frac{\sigma(S_t)S_tV_S(t,S_t)}{B_t}\,d\widehat W_t.
$$
The discounted added asset is a <local martingale>, as is $S_t/B_t$, while $B_t/B_t=1$. Consequently $Q$ is an <equivalent local martingale measure> for all three assets.
Here is the <arbitrage> argument, including the usual <admissible trading strategy> condition. If $X$ is the discounted wealth of a <self-financing strategy>, it is a <stochastic integral> against the discounted asset prices, plus its initial value, and hence a <local martingale>. Assume $X_t\geq-c$ for a fixed $c$, as required for an <admissible trading strategy>. For a localizing sequence $\tau_n$, the <stopped process> $X^{\tau_n}$ is a <martingale>. Apply the conditional <Fatou lemma> to $X_{t\wedge\tau_n}+c\geq0$: for $s\leq t$, continuity and the stopped <martingale> identity give $\mathbb E_Q[X_t+c\mid\mathcal F_s]\leq X_s+c$. Thus $X$ is a <supermartingale>. An <arbitrage> would have $X_0=0$, $X_T\geq0$ and positive <probability> of $X_T>0$. Equivalence of $P,Q$ makes that <probability> positive under $Q$, contradicting $\mathbb E_Q[X_T]\leq0$. Therefore \b[the augmented market has no admissible <arbitrage>].
Nonnegativity of $V$ gives the needed price process, but does not by itself make its discounted <local martingale> a true <martingale>. If $V$ is bounded, the latter follows by bounded <localization>, and its price is also $B_t\mathbb E_Q[g(S_T)/B_T\mid\mathcal F_t]$. The <arbitrage> proof above applies to the stated nonnegative classical solution without assuming that additional bound.
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