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

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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.

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  • $\begingroup$ @zeitouni Thank you very much! Would you also happen to know the answer to my additional question (that I added just now)? $\endgroup$ – tot Dec 10 '14 at 10:10
  • $\begingroup$ well, if $\phi$ can take the value 0 then you have a problem. Did you mean to exclude that with your notation $C_b^2(]0,\infty[)$? $\endgroup$ – ofer zeitouni Dec 10 '14 at 12:17
  • $\begingroup$ Yes, I did. So $\phi$ is defined, $C^{2}$, and bounded with bounded derivatives (up to second order) on $]0,\infty[$, that is, on $(0,\infty)$. $\endgroup$ – tot Dec 10 '14 at 12:27
  • $\begingroup$ Your notation is still not satisfactory. What do you mean that $\phi$ is bounded ON $(0,\infty)$? Normally such a statement means $\sup_{t\in (0,\infty)} |\phi(t)|<\infty$. But I suspect this is not what you mean, rather you mean that $\inf_{t\in [0,T]} |\phi(t)|>0$. You should clarify what you mean. $\endgroup$ – ofer zeitouni Dec 10 '14 at 12:45
  • $\begingroup$ Sorry, I got things mixed up. Obviously, I mean $\phi \in C_{b}^{2}([0,\infty[)$ such that $\rvert \phi(t) \lvert > 0$ for all $t \geq 0$. I would also be happy if I had to require $\rvert \phi(t) \lvert > \delta$ for all $t \geq 0$ and some small $\delta > 0$. $\endgroup$ – tot Dec 10 '14 at 13:04

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