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Let $\{W_t\}$ be a standard Wiener process on a probability space $(\Omega, \mathcal{F},P)$. Let $\mathcal{F}^W$ be the natural filtration generated by $\{W_t\}$.

Let $\{\theta_t\}$ be an $\mathcal{F}_t^W$-adapted stochastic process satisfying $$ E_P\left[ \exp\left(\int_0^t \frac{\theta_s^2}{2} ds\right) \right]<\infty , \forall t\geq 0, $$ Then, the process $$ \varepsilon_t = \exp\left(\int_0^t\theta_s dW_s - \frac{1}{2}\int_0^t\theta^2_s ds\right) $$ is an $\mathcal{F}^W_t$-martingale (because Novikov's condition is satisfied) and it represents the Radon-Nikodym derivative $dQ/dP$ for some $Q$ that is equivalent to $P$. We also have that $\{X_t\}$ defined by $X_t=\int_0^t dW_s -\int_0^t\theta_sds $ is a standard Wiener process on $(\Omega, \mathcal{F}^W, Q)$.

My question: Take the natural filtration generated by $\{X_t\}$, $\mathcal{F}^X$. What is the relationship between $\mathcal{F}^W$and $\mathcal{F}^X$? Do they coincide?

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