= Solution
On a fixed finite horizon, a <self-financing strategy> consists of predictable <stock> holdings $h^i$, integrable against the price processes, and a <bank account> holding $h^0$. Its wealth and gains satisfy
$$
X_t=h_t^0B_t+\sum_{i=1}^dh_t^iS_t^i,\qquad dX_t=h_t^0dB_t+\sum_{i=1}^dh_t^idS_t^i.
$$
An <admissible trading strategy> has discounted wealth $X_t/B_t$ bounded below by a deterministic constant throughout the horizon. With the bounded <short rate> in this part, an undiscounted deterministic lower bound is equivalent after adjusting the constant. An <arbitrage> is an admissible zero-initial-cost strategy with $X_T\ge0$ almost surely and $\mathbb P(X_T>0)>0$.
Put $\mathbf1=(1,\ldots,1)^\top$ and define the <market price of risk>
$$
\lambda_t=\sigma_t^{-1}(\mu_t-r_t\mathbf1).
$$
The bounded coefficients and bounded inverse make $\lambda$ bounded. Define the <stochastic exponential>
$$
L_t=\exp\left(-\int_0^t\lambda_s\cdot dW_s-\frac12\int_0^t\lVert\lambda_s\rVert^2ds\right).
$$
The <Novikov condition> states that $\mathbb E\exp(\tfrac12\int_0^T\lVert\lambda_s\rVert^2ds)<\infty$ makes this exponential a true <martingale> with expectation one. Boundedness verifies the condition. Therefore $dQ=L_TdP$ defines an equivalent probability measure. The <Girsanov theorem> states that
$$
W_t^Q=W_t+\int_0^t\lambda_sds
$$
is a $d$-dimensional <Brownian motion> under $Q$. Substituting into the asset equations gives
$$
\frac{dS_t^i}{S_t^i}=r_tdt+\sum_j\sigma_t^{ij}dW_t^{Q,j},\qquad
\frac{d(S_t^i/B_t)}{S_t^i/B_t}=\sum_j\sigma_t^{ij}dW_t^{Q,j}.
$$
Thus $Q$ is an <equivalent local martingale measure>. The sufficient direction of the <fundamental theorem of asset pricing> says that such a measure excludes <arbitrage> among admissible <self-financing strategies>. Here the reason can be seen directly: $X/B$ is a <local martingale> under $Q$, and its lower bound makes it a <supermartingale>. Consequently $X_0=0$ implies $\mathbb E_Q[X_T/B_T]\le0$, incompatible with the nonnegative terminal payoff of an <arbitrage>. Therefore \b[the bounded-inverse market has no <arbitrage>].
For the requested <law of one price> counterexample, take a broader continuous-time local-martingale model on $[0,T]$. Let $X$ be a <Brownian motion>, let $\tau=\inf\{u\ge0:X_u=-1\}$, and set $q(t)=t/(T-t)$ for $t<T$. Define
$$
\boxed{B_t=1,\qquad S_t=2+X_{q(t)\wedge\tau}\ (t<T),\qquad S_T=1.}
$$
The given hitting-time fact gives $\tau<\infty$ almost surely. The corresponding physical hitting time is $T\tau/(1+\tau)<T$, so $S$ reaches one and then stays one. In particular it is continuous at maturity and $S_t\ge1$.
For a rigorous closed-horizon localization, stop additionally when $S$ hits $n+2$, or at $T$ if it never does. The stopped process is <Brownian motion> stopped on first exiting $(-1,n)$ and subjected to the deterministic clock; it is a bounded <martingale>, including at $T$. The stopping times increase to $T$, since every Brownian path on the finite random interval $[0,\tau]$ has a finite maximum. Thus $S$ is a positive <local martingale> on the whole horizon. With the time-changed Brownian filtration, the original probability measure is an <equivalent local martingale measure>. The admissible-wealth <supermartingale> argument above proves \b[no admissible <arbitrage>].
Nevertheless $S_T=B_T=1$ while $S_0=2\ne B_0=1$, so \b[the <law of one price> fails]. Shorting one share and buying two units of cash has zero initial cost and terminal gain one, but its wealth is $2-S_t$. It has no deterministic lower bound: for every $n$, the probability of <Brownian motion> hitting $n$ before $-1$ is $1/(n+1)>0$, obtained by optional stopping of the bounded stopped <Brownian motion>. That apparently profitable strategy is therefore inadmissible. The counterexample deliberately uses a <strict local martingale>; the bounded-coefficient class in the first part has true discounted asset <martingales> and does satisfy the <law of one price>.
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