The answer is yes.
Proof:
Theorem (Radon-Nikodym)
Let $(\Omega, \mathcal{F})$ be a measurable space. Let $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ be two $\sigma$-finite measures. Let $\widetilde{\mathbb{P}}$ be absolutely continuous w.r.t. $\mathbb{P}$ (i.e. $\widetilde{\mathbb{P}}\ll\mathbb{P}$). THEN: $(\exists)$ measurable function $f:\Omega\to[0;+\infty)$ such that:
$$
\widetilde{\mathbb{P}}(A)=\int_A f(\omega)d\mathbb{P}(\omega), (\forall)A\in\mathcal{F}.
$$
$f$ is unique up to indistinguishability, i.e. if there is another $g$ with the same properties as above, then $f=g, \mathbb{P}-a.s.$ (or $\mathbb{P}$-a.e.).
Note that if $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ are equivalent measures (denoted by $\mathbb{P}\approx\widetilde{\mathbb{P}}$), then $\widetilde{\mathbb{P}}\ll\mathbb{P}$ and $\mathbb{P}\ll\widetilde{\mathbb{P}}$.
Let now $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=(\mathcal{F}_t)_{t\geq0}$ is the filtration. We use the Radon-Nikodym theorem to prove the next proposition:
Proposition. Let $\mathbb{P}\approx\widetilde{\mathbb{P}}$ be two equivalent probability measures on $(\Omega, \mathcal{F}_T)$, a measurable space from the notation above. THEN, $(\exists)$ a strictly positive $(\mathbb{P}, \mathbb{F})$-martingale $(L_t)_{t\geq 0}$ such that
$$
\widetilde{\mathbb{P}}(A)=\int_A L_t(\omega)d\mathbb{P}(\omega), (\forall) A\in\mathcal{F}_t, (\forall) t\leq T
$$
with the properties that:
- $\mathbb{E}_{\widetilde{\mathbb{P}}}[X]=\mathbb{E}_{\mathbb{P}}[L_tX]$, for all $\mathcal{F}_t$-measurable, non-negative, random variables $X$, when $t\leq T$.
- $L_0 = 1$
- $\mathbb{E}_{\mathbb{P}}[L_t]=1, (\forall) t\leq T$.
Proof:
We know from the Radon-Nikodym theorem above that since $\mathbb{P}\approx\widetilde{\mathbb{P}}$ on $(\Omega, \mathcal{F}_T)$, then there must exist a non-negative, $\mathcal{F}_T$-measurable random variable $Z$ with the property that
$$
\widetilde{\mathbb{P}}(A)=\int_AZ(\omega)d\mathbb{P}(\omega), (\forall)A\in\mathcal{F}_T
$$
Since we have already assumed that $\widetilde{\mathbb{P}}$ is a probability measure, we have that:
$$
\widetilde{\mathbb{P}}(\Omega)=1=\int_{\Omega}Z(\omega)d\mathbb{P}(\omega)=\mathbb{E}_{\mathbb{P}}[Z].
$$
Since we now know that $\mathbb{E}_{\mathbb{P}}[Z]=1$, we can apply (Steve Shreve, Stochastic Calculus for Finance II - Continuous Models, p. 33, Theorem 1.6.1) to reach the conclusion that for any wandom variable $X$ that is a non-negative and $\mathcal{F}_T$-measurable we have:
$$
\mathbb{E}_{\widetilde{\mathbb{P}}}[X]=\mathbb{E}_{\mathbb{P}}[ZX].
$$
In particular, for $X=1$ this leads to:
$$
\mathbb{E}_{\mathbb{P}}[Z]=1.
$$
Let us define $L_t=\mathbb{E}_{\mathbb{P}}[Z|\mathcal{F}_t]$. Clearly, $(L_t)_{t\geq 0}$ is a $(\mathbb{P}, \mathbb{F})$-martingale because for all $s\leq t$:
$$
\mathbb{E}_{\mathbb{P}}[L_t|\mathcal{F}_s]=\mathbb{E}_{\mathbb{P}}[\mathbb{E}_{\mathbb{P}}[Z|\mathcal{F}_t]|\mathcal{F}_s]=
\mathbb{E}_{\mathbb{P}}[L_t|\mathcal{F}_s]=L_s,
$$
where the first equality is from the definition of $L_t$, the second inequality is due to the tower law, and the third equality is due to the definition of $L_s$. Taking expectation in the above we get the property that $\mathbb{E}_{\mathbb{P}}[L_t]=1, (\forall)t\leq T$. If we take $\mathcal{F}_0=\{\emptyset, \Omega\}$, as is usual, then $L_0$ is deterministic and $L_0=1$. This proves items (2.) and (3.) of the proposition.
We can then use (Steve Shreve, Stochastic Calculus for Finance II - Continuous Models, p. 211, Lemma 5.2.1) to prove item (1.) of the proposition, namely that:
$$
\mathbb{E}_{\widetilde{\mathbb{P}}}[X]=\mathbb{E}_{\mathbb{P}}[L_tX],\text{ for all } \mathcal{F}_t\text{-measurable, non-negative, random variables }X,\text{ when }t\leq T.
$$
In the above, let us substitute $1_A$ for $X$ and T for t. This proves immediately the rest of the proposition.
Note that from this answer, $L_t$ is $\mathbb{P}$-a.s. non-negative.
Also note that we can take $Z$ to be strictly positive since the two measures are equivalent. Therefore, we can also take a version of $L_t$ that is strictly positive and this changes nothing. We will consider in what follows that we use such $L_t$.$$\Box $$
We have constructed above the Radom-Nikodym derivative process $(L_t)_{t\geq 0}$ , which is a $(\mathbb{P}, \mathbb{F})$-martingale. Because $\mathbb{F}$ is generated by $(W_t)_{t\in[0;T]}$ we can apply the martingale representation theorem $\Rightarrow (\exists) (\psi_t)_{t\geq 0}$ an $\mathbb{F}$-measurable process s.t.:
$$
L_t=1+\int_0^t \psi_udW_u.
$$
or, alternatively, that:
$$
dL_t=\psi_tdW_t, L_0=1.
$$
Since the Radon-Nikodym derivarive process is strictly positive, using Ito lemma we get:
$$
d\log(L_t)=\frac{1}{L_t}dL_t-\frac{1}{2}\frac{1}{L^2_t}d\langle L \rangle_t=\frac{\psi_t}{L_t}dW_t-\frac{1}{2}\frac{\psi^2_t}{L^2_t}dt
$$
Since $L_t$ is strictly positive, we can simplify things a bit by introducing
$$
\Theta_t=-\frac{\psi_t}{L_t}.
$$
This is also an $\mathbb{F}$-adapted process. With this notation, by integrating the result of the application of the Ito lemma and exponentiating we get:
$$
L_t=e^{-\int_0^t\Theta_udu-\frac{1}{2}\int_0^t\Theta^2_u dWu}.
$$
The result also rests on the uniqueness (up to indistinguishability) of the Radon-Nikodym derivative (in the R-N theorem).
So yes, all changes of measure must be of this form.
