= Solution
Use the usual stochastic-integrability assumptions implicit in the price <stochastic differential equation>: the coefficients have progressively measurable versions, the volatility is locally square integrable, and initial prices are finite deterministic positive numbers. A no-consumption <admissible trading strategy> consists of <predictable> risky share holdings $\pi$ for which the price <stochastic integral> exists and the <portfolio wealth> of the <self-financing portfolio> is
$$
X_t=x+\int_0^t\pi_s^T\,dS_s.
$$
The remainder $X_t-\pi_t^TS_t$ is held in the unit cash asset. Admissibility requires $X_t\geq-a$ throughout the horizon for some deterministic finite $a$; the more restrictive nonnegative-wealth convention also works. A finite-horizon <arbitrage> is such a strategy with $x=0$, $X_T\geq0$ almost surely and $P(X_T>0)>0$. The lower bound is essential to exclude doubling strategies.
Interpret the stated uniform ellipticity as $\sigma_t\sigma_t^T\geq\varepsilon I$ for a deterministic $\varepsilon>0$. If $|\mu_t|\leq K$, the <market price of risk> is bounded:
$$
\lambda_t=\sigma_t^{-1}\mu_t,\qquad |\lambda_t|^2=\mu_t^T(\sigma_t\sigma_t^T)^{-1}\mu_t\leq\frac{K^2}{\varepsilon}.
$$
The authoritative PDF puts both integrals inside the exponent. With that interpretation,
$$
Z_t=\exp\left(-\int_0^t\lambda_s^T\,dW_s-\frac12\int_0^t|\lambda_s|^2ds\right),\qquad dZ_t=-Z_t\lambda_t^T\,dW_t.
$$
On every finite horizon $T$, boundedness of $\lambda$ gives the <Novikov condition>:
$$
\mathbb E\exp\left(\frac12\int_0^T|\lambda_s|^2ds\right)\leq\exp\left(\frac{K^2T}{2\varepsilon}\right)<\infty.
$$
Hence the positive <stochastic exponential> $Z$ is a true <martingale> with $\mathbb EZ_T=1$. Define the equivalent probability measure $Q_T$ by $dQ_T=Z_T\,dP$. The <Girsanov theorem> says that $W_t^Q=W_t+\int_0^t\lambda_sds$ is a <Brownian motion> under $Q_T$, up to time $T$. Consequently
$$
dS_t=\operatorname{diag}(S_t)\sigma_t\,dW_t^Q,
$$
so the prices are <local martingales> under $Q_T$.
Every admissible zero-initial-capital <self-financing portfolio> has <portfolio wealth> which is therefore a $Q_T$-<local martingale> bounded below by $-a$. The shifted process $X+a$ is a <nonnegative local martingale>, hence a <supermartingale> by localization and the <Conditional Fatou lemma>. In particular $\mathbb E_{Q_T}X_T\leq0$. If the strategy were an <arbitrage>, nonnegativity of $X_T$ would force $X_T=0$ $Q_T$-almost surely. Equivalence of $P$ and $Q_T$ would then force $P(X_T>0)=0$, a contradiction. \b[The market has no admissible arbitrage on any finite horizon.] This proves the exclusion argument itself, rather than just asserting the <fundamental theorem of asset pricing>.
For the deflated price calculation, let $\sigma_{t,i}$ denote the column vector whose transpose is the $i$th row of $\sigma_t$. The <Itô product rule> includes the <quadratic covariation> term
$$
d[Z,S^i]_t=-Z_tS_t^i\sigma_{t,i}^T\lambda_t\,dt.
$$
Since $\sigma_{t,i}^T\lambda_t=\mu_t^i$, the full product equation is
$$
\begin{aligned}
d(Z_tS_t^i)&=Z_tS_t^i\left[(\mu_t^i-\sigma_{t,i}^T\lambda_t)dt+(\sigma_{t,i}-\lambda_t)^T\,dW_t\right]\\
&=Z_tS_t^i(\sigma_{t,i}-\lambda_t)^T\,dW_t.
\end{aligned}
$$
Localization makes the last <stochastic integral> a true <martingale> on each stopped interval, proving \b[each component of the deflated price vector is a local martingale under the original measure]. This is the <bounded-coefficient asset deflator> identity; bounded volatility has not yet been used.
If $\sigma$ is bounded, then $\sigma_{t,i}-\lambda_t$ is bounded as well. Solving the last linear <stochastic differential equation> gives
$$
Z_tS_t^i=S_0^i\exp\left(\int_0^t(\sigma_{s,i}-\lambda_s)^T\,dW_s-\frac12\int_0^t|\sigma_{s,i}-\lambda_s|^2ds\right).
$$
The <Novikov condition> now holds for this <stochastic exponential> on every finite horizon, because its bracket is bounded by a deterministic constant times the horizon. Therefore
$$
\boxed{\mathbb E[Z_tS_t^i\mid\mathcal F_s]=Z_sS_s^i\quad(0\leq s\leq t),\qquad\mathbb E[Z_tS_t^i]=S_0^i.}
$$
\b[The entire deflated price vector is a true martingale when volatility is bounded.] This finite-horizon argument does not assert uniform integrability as time tends to infinity, which is not required. Integrable random initial prices can also be handled by the conditional martingale argument; without any integrability assumption on initial prices, a true-martingale assertion would not be meaningful.
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