# Onsager--Machlup functional as the density across a mesh of discrete points

It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint density of the diffusion along two paths, evaluated at a partition of the time interval, also converges to the to the Onsager--Machlup functional as the mesh of the partition vanishes.

Formally, let $$X_t$$ be an $$n$$-dimensional diffusion process satisfying the following Itō SDE over $$[0,1]$$:

$$dX_t = f(X_t)\,dt + dW_t,$$

where $$W_t$$ is an $$n$$-dimensional Wiener process and $$f$$ is of class $$C^2_b$$, i.e., it is twice differentiable and the function and all its derivatives up to order 2 are bounded. For simplicity, assume that $$X_0=0$$. Additionally, let $$\phi,\varphi$$ be paths in the Cameron--Martin space, i.e., absolutely continuous with $$\phi(0)=0$$ and $$\dot\phi\in L^2([0,1])$$. We know that $$\lim_{\epsilon\downarrow 0} \frac{ P(\sup_{t\in[0,1]} \lVert\phi(t)-X_t \rVert<\epsilon) }{ P(\sup_{t\in[0,1]} \lVert\varphi(t)-X_t \rVert<\epsilon) } = \exp\Big(J(\phi) - J(\varphi)\Big),$$ where $$J$$ is the Onsager--Machlup functional $$J(\phi):= -\frac{1}{2} \int_0^1\lVert\dot\phi(t) - f\big(\phi(t)\big)\rVert^2 dt -\frac{1}{2} \int_0^1\operatorname{div}f\big(\phi(t)\big)\, dt.$$

My question: can we obtain the same ratio by dividing the densities along the paths, that is $$\lim_{N\to\infty} \prod_{i=1}^N \frac{ p\Big(\varphi(\tfrac{i-1}{N});\tfrac1N,\varphi(\tfrac iN)\Big) }{ p\Big(\phi(\tfrac{i-1}{N});\tfrac1N,\phi(\tfrac iN)\Big) } = \exp\Big(J(\phi) - J(\varphi)\Big),$$ where $$p(x_t; \delta, x_{t+\delta})$$ is the transition density?

I have proved that this is true when the SDE is discretized using the trapezoidal scheme [1]. We also found out that if the Euler--Maruyama scheme is used, the ratio of densities converges to a different limit, where we have only the quadratic term of the OM functional, without the drift divergence. We generally understood that this means that the Euler--Maruyama scheme cannot be used to compute the probability of paths. This spurred interesting discussions into the use of discretizations and the correct approximations or functionals to use for estimating paths [2, 3], and I've been questioned whether the OM functional is the correct result for the general case. If it is, this would simplify some analyses as it is usually simpler to approximate its integral than densities of nonlinear SDEs.

• Pet peeve: Itô was a cultured man and knew well that the standard Hepburn romanization of his name is Itō. Still, according to several people who knew him personally, he strongly preferred the spelling Itô, which is used in pretty much every article he authored. At the end of the day, I think the bearer of a name gets to decide its spelling... Commented May 6, 2022 at 14:48

In preparing the material for asking the question, I understood it better and ended out figuring out an answer it. As answering your own question is encouraged in MathOverflow, I'm posting it here. The proof relies on the fact that, using Levy's modulus of continuity, when restricting $$W_t$$ to pass between a series of hoops at discrete time points it is contained within a larger tube. As both the hoop diameter and density of the mesh vanish, $$\sup_{t\in[0,1]}|W_t|\to 0$$. Under this double limit the standard derivation of the Onsager--Machlup functional still holds. Together with the fact that, for each fixed partition, the normalized hoop probability converges to the density, we have that the double limit is equal to the iterated limit, which is the limit of densities as the partition mesh vanishes. $$\newcommand{\limsup}{\operatorname*{lim\,sup}} \newcommand{\reals}{{\rm I\!R}} \newcommand{\naturals}{\mathbb{N}} \newcommand{\parti}{\mathcal{P}} \newcommand{\xeballi}{% \max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon % } \newcommand{\weballi}{% \max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon % } \newcommand{\xeiball}{% \max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon_i % } \newcommand{\weiball}{% \max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon_i % }$$

In what follows, let $$\{\parti_i\}_{i=1}^\infty$$ be a sequence of partitions $$\parti_i:=\{\tau_{k,i}\}_{k=0}^{N_i}$$ of the unit interval whose mesh $$\bar\delta_i$$ vanishes, i.e.,

\begin{align} \tau_{0,i}&=0,& \tau_{N_i,i}&= 1,& \tau_{k,i}&<\tau_{k+1,i},\\ \end{align} \begin{align} \delta_{k,i} &:= \tau_{k,i} - \tau_{k-1,i},& \bar\delta_i &:= \max_{0< k\leq N_i} \delta_{k,i}, & \lim_{i\to\infty}\bar\delta_i &= 0. \end{align}

Lemma 1: For all $$i \in\naturals$$ and $$\phi:[0,1]\to\reals^n$$,

$$\lim_{\epsilon\to 0} \frac{ P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon) }{ \mu(\{x\in\reals^n:|x|<\epsilon\})^{N_i} } = \prod_{k=1}^{N_i} p\big(\phi(\tau_{k-1,i});\delta_{ki}, \phi(\tau_{ki})\big),$$ where $$\mu$$ is the Lebesgue measure.

This is a corollary of the Lebesgue differentiation theorem and the fact that $$X_\tau$$ admits a continuous probability density function.

Lemma 2: For all $$\phi:[0,1]\to\reals^n$$ in the Cameron--Martin space,

$$\lim_{\substack{\epsilon\to 0 \\ i\to\infty}} \frac{ P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon) }{ P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon) } = \exp\big(J(\phi)\big).$$

This is follows from the same arguments that are used to prove the Onsager--Machlup functional as the fictitious density of tubes around $$\phi$$, which will be detailed shortly.

Corollary 3: For all $$\phi:[0,1]\to\reals^n$$ in the Cameron--Martin space,

$$\lim_{i\to\infty} \prod_{k=1}^{N_i} \frac{ p\big(\phi(\tau_{k-1,i});\delta_{ki}, \phi(\tau_{ki})\big) }{(2\pi)^{-\frac n2}(\delta_{ki})^{-\frac12}} = \exp\big(J(\phi)\big).$$

Proof: Applying Lemma 1 to both the $$X_t$$ and $$W_t$$ processes, we have that, for all $$i\in naturals$$,

$$\lim_{\epsilon\to 0} \frac{ P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon) }{ P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon) } = \prod_{k=1}^{N_i} \frac{ p\big(\phi(\tau_{k-1,i});\delta_{ki}, \phi(\tau_{ki})\big) }{(2\pi)^{-\frac n2}(\delta_{ki})^{-\frac12}}.$$

As the double limit of Lemma 2 exists, and the single limit for each fixed $$i\in\naturals$$ exists and is finite, we have that

$$$$\tag*{\blacksquare} \lim_{\substack{\epsilon\to 0 \\ i\to\infty}} \frac{ P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon) }{ P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon) } = \lim_{i\to\infty} \lim_{\epsilon\to 0} \frac{ P(\max_{\tau\in\mathcal{P}_i} |X_\tau-\phi(\tau)|<\epsilon) }{ P(\max_{\tau\in\mathcal{P}_i} |W_\tau|<\epsilon) }.$$$$

Now only Lemma 2 remains to be proved. For that, we will the following lemma, which is a consequence of Levy's modulus of continuity theorem.

Lemma 4: There exist $$c\in\reals$$ such that, for all sufficiently large $$i$$ and almost all outcomes in the event $$\{\weballi\}$$,

$$|W_t| < \epsilon + c\sqrt{\bar \delta_i\log(1/\delta_i)}, \qquad\forall t\in[0,1].$$

The proof of Lemma 2 now follows essentially the same steps as the derivation of the Onsager--Machlup functional (see, e.g., Capitaine, 2000).

Proof of Lemma 2: As we are dealing with a double limit, let $$\{\epsilon_i\}_{i=1}^\infty$$ be a sequence of vanishing hoop diameters $$\epsilon_i\to 0$$, indexed by the same $$i$$ as the partitions. Applying the Girsanov transformation,

$$\begin{multline} \frac{\xeiball}{\weiball} = \exp\big(J(\phi)\big) E\Bigg[\exp\Big( -\int_0^1\dot\phi_t^T\,dW_t \\ -\frac12\int_0^1|f(\phi_t+W_t)|^2\,dt + \frac12\int_0^1|f(\phi_t)|^2\,dt \\ +\int_0^1 f(\phi_t+W_t)^T d\phi_t - \int_0^1 f(\phi_t)^T d\phi_t \\ +\int_0^1 f(\phi_t+W_t)^T d\phi_t + +\frac12 \int_0^1\operatorname{div}f(\phi_t)\,dt \Big)\Bigg|\weiball\Bigg] \end{multline}$$

It now suffices to prove that, for all $$c\in\reals$$,

$$$$\tag{1} \limsup_{i\to\infty} E\Bigg[\exp\Big(c \int_0^1\dot\phi_t^T\,dW_t \Big)\Bigg|\weiball\Bigg]\leq 1,$$$$

$$$$\tag{2} \limsup_{i\to\infty} E\Bigg[\exp\Big( c\frac12\int_0^1|f(\phi_t+W_t)|^2\,dt - c\frac12\int_0^1|f(\phi_t)|^2\,dt \Big)\Bigg|\weiball\Bigg]\leq 1,$$$$

$$$$\tag{3} \limsup_{i\to\infty} E\Bigg[\exp\Big( c\int_0^1 f(\phi_t+W_t)^T d\phi_t - c\int_0^1 f(\phi_t)^T d\phi_t \Big)\Bigg|\weiball\Bigg]\leq 1,$$$$

$$$$\tag{4} \limsup_{i\to\infty} E\Bigg[\exp\Big( c\int_0^1 f(\phi_t+W_t)^T dW_t +c\frac12 \int_0^1\operatorname{div}f(\phi_t)\,dt \Big)\Bigg|\weiball\Bigg]\leq 1,$$$$

Eqs. (2) and (3) are easy to verify, as $$f$$ is assumed to be Lipschitz and bounded. Eq. (4) follows from the application of the stochastic Stokes' theorem. Eq. (1) is proved by Shepp and Zeitouni (1992) for conditioning on the events $$\{\sup_{t\in[0,1]}|W_t|<\epsilon\}$$. However, their proof holds for the events $$\{\weiball\}$$, as they are symmetric and convex, and Lemma 4 implies that the width of the bounding tubes vanish as $$i\to\infty$$.