= Solution
Set $N_t=\int_0^tZ_s^{-1}\,dZ_s$, the <stochastic logarithm> of $Z$. On each finite horizon, a strictly positive continuous path has its reciprocal bounded; stopping at levels of $Z$, $Z^{-1}$ and $[Z]$ rigorously localizes this <stochastic integral>. Thus $N$ is a zero-start <continuous local martingale>, with
$$
[N]_t=\int_0^tZ_s^{-2}\,d[Z]_s.
$$
The <Itô formula> for the ordinary logarithm gives
$$
\log Z_t=\log Z_0+N_t-\frac12[N]_t.
$$
Accordingly, with the exponential convention printed in this question,
$$
\boxed{M_t=\log Z_0+\int_0^t\frac{dZ_s}{Z_s},\qquad Z_t=\exp\!\left(M_t-\frac12[M]_t\right).}
$$
The initial constant in $M$ contributes no <quadratic variation>. Under the standard normalized <Doléans-Dade exponential> convention, which subtracts the driver's initial value in the exponent, the same formula is written $Z=Z_0\mathcal E(N)$.
For uniqueness, suppose another continuous driver $L$ gives the printed exponential. Evaluating at zero forces $L_0=\log Z_0$. Applying the <Itô formula> to its exponential gives $dZ_t=Z_t\,dL_t$. Since $Z_t>0$, <associativity of stochastic integration> yields $dL_t=Z_t^{-1}dZ_t$. Thus $L_t=L_0+N_t=M_t$, up to <indistinguishability of stochastic processes>.
There is an initial-integrability qualification if “<local martingale>” is used with the usual integrable-initial-value definition. The displayed $M$ is a <continuous local martingale> when $\log Z_0\in L^1$, in particular when $Z_0$ is a positive deterministic number. An arbitrary positive integrable $Z_0$ does not ensure this. For example, take an $\mathcal F_0$-measurable positive integer $W$ with $\mathbb P(W=n)=6/(\pi^2n^2)$ and put $Z_t=e^{-W}$ for all $t$. This is a bounded strictly positive <martingale>, but any representing driver must have $M_0=-W$, which is not integrable. Thus the unqualified claim is false under that definition. For general $Z_0$, the always-valid local-<martingale> statement is the zero-start representation \b[$Z=Z_0\mathcal E(N)$]; the printed representation additionally requires the stated integrability or the convention that allows arbitrary finite initial constants.
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