Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation \begin{equation} dX_{t} = f(X_{t})dt + dW_{t}, \end{equation} where $f \in C_{b}^{2}(R)$ is a bounded, twice continuously differentiable function with bounded derivatives. Y. Takahashi and S. Watanabe (Springer Lecture Notes in Math. 851: 432–463, 1980) have shown that for any function $\phi \in C^{2}([0,T])$ \begin{equation*} P\bigl( \sup_{ 0 \leq t \leq T } \lvert X_{t} - \phi(t) \rvert < \epsilon \bigr) \underset{\epsilon \rightarrow 0}{\sim} e^{-\frac{1}{2} \int_{0}^{T} \mathcal{L}(\phi,\phi') dt } \end{equation*} where \begin{align*} \mathcal{L}(\phi,\phi') = \bigl( \phi'(t) - f( \phi(t) ) \bigr)^{2} + f'(\phi(t)) \end{align*} is the Onsager-Machlup function. A minimization of $\mathcal{L}(\phi,\phi')$ using the Euler-Lagrange equation will yield the most probable path.

I was wondering if it is possible to transfer (some of) the boundedness conditions on $f$ to $\phi$? For instance, Dürr and Bach (Commun. Math. Phys. 60: 153–170, 1978) seem to suggest that the theorem is true when $f \in C^{2}(R)$ and $\phi \in C_{b}^{2}([0,T])$.

Additional question (added later):

What if we introduce a "finite boundary" at which $f$ diverges? That is, what if $X_{t} \in \ ]0,\infty[$ almost surely and $f \in C^{2}(]0,\infty[)$ such that $\lvert f(x) \rvert, \lvert f'(x) \rvert, \lvert f''(x) \rvert \rightarrow \infty$ as $x \rightarrow 0^{+}$. Would the theorem still hold for $\phi \in C_{b}^{2}([0,T])$?