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 $\phi \in C\_{b}^{2}([0,T])$?



Onsager-Machlup function and most probable path of a diffusion process
Certainly, if $\phi\in C\_b$ then you only care about the values of $f$ at in a neighborhood of points in the range of $\phi$. In particular, the behavior of $f$ at infinity is irrelevant.


