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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 $X_0=0$.

It is known that for paths $\phi,\varphi$ in the Cameron--Martin space, i.e., absolutely continuous with $\phi(0)=0$ and $\dot\phi\in L^2([0,1])$, $$ \lim_{\epsilon\downarrow 0} \frac{ P(\sup_{t\in[0,1]} \lVert\phi(t)-X_t \rVert<\epsilon) }{ P(\sup_{t\in[0,1]} \lVert\varphi(t)-X_t \rVert<\epsilon) } = \exp\Big(J(\phi) - J(\varphi)\Big), $$ where $J$ is the Onsager--Machlup functional $$ J(\phi):= -\frac{1}{2} \int_0^1\lVert\dot\phi(t) - f\big(\phi(t)\big)\rVert^2 dt -\frac{1}{2} \int_0^1\operatorname{div}f\big(\phi(t)\big)\, dt. $$ This is used by many to define the "most probable" path of diffusion processes (Dürr and Bach, 1978) and to obtain maximum a posteriori state paths (Zeitouni and Dembo, 1987; Aihara and Bagchi, 1999a,b). The exponential of the Onsager--Machlup functional is also described by Takahashi (1981) to be "an ideal density with respect to a fictitious uniform measure", justifying its use as a fictitious density for the purpose of comparing the "probability" of individual paths.


My question: for a path $\phi$ outside the Cameron--Martin space and $\varphi$ in the Cameron--Martin space, does $$ \lim_{\epsilon\downarrow 0} \frac{ P(\sup_{t\in[0,1]} \lVert\phi(t)-X_t \rVert<\epsilon) }{ P(\sup_{t\in[0,1]} \lVert\varphi(t)-X_t \rVert<\epsilon) } = 0, $$ i.e., does the probability of an $\epsilon$-ball centered paths outside the Cameron--Martin space decays much faster than those centered in paths in the Cameron--Martin space? This seems to be a requirement (and an implicit assumption) for using the Onsager--Machlup function for maximum a posteriori estimation of paths of diffusions (my thesis topic).


I've attempted to prove this by taking slightly larger tubes around piecewise-linear interpolations of $\phi$ (which belong to the Cameron--Martin space) and converge uniformly to $\phi$. However, I have problems because some of the bounds I obtained depend on $\lVert\dot\phi_i\rVert$ which explodes. I've also searched extensively and failed to find any mention to this issue. The fact that the functional $J$ includes this quadratic term which depends on $\dot\phi$ seems to imply that the answer is yes.

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In terms of terminology, there is nothing wrong with thinking of the density only over path in H^1. So the answer to your question is not necessary for the interpretation as posterior density (the MAP estimator anyway lives in H^1, and even smoother). But the question itself is interesting, and I will comment on it separately. –  ofer zeitouni Mar 20 at 10:27
1  
If you take as norm the L^2 norm, I believe the answer is ``yes''. This is because you can use the Karhunen-Loeve expansion and bring the question to a question about (shifted) ellipsoids; computations for the shifted case (as in wisdom.weizmann.ac.il/~zeitouni/pdf/addendum.pdf for the unshifted case) seem to give that. Did you try that route? –  ofer zeitouni Mar 20 at 10:29
    
I believe that if the answer to my question is "no", then we would need a strong argument for not considering the paths there, besides not having an expression for the fictitious density. Whenever reading about the OM functional, it seems almost implied that the "most probable path" is in that space. It seems, additionally, that the Cameron--Martin space is the largest space for which the density is finite as outside it the expression of the OM functional would be ill defined. –  Dimas Abreu Dutra Mar 20 at 17:22
    
If the answer is "yes" for the $L^2$ norm, then it would suffice for my purposes. The fact that for finite-dimensional diffusions the fictitious density is independent of the norm used (as far as I know) seems to imply that if it could also be true for the uniform norm. I will try to work a proof with this route you mentioned. –  Dimas Abreu Dutra Mar 20 at 17:39
    
This is not quite true. The OM functional is not derived for arbitrary norms (especially in the multi dimensional case). See for example ``Onsager-Machlup functional for some smooth norms on Wiener space'' by M. Capitaine (PTRF 1995) and also wisdom.weizmann.ac.il/~zeitouni/pdf/omrev2.pdf –  ofer zeitouni Mar 20 at 22:20

1 Answer 1

up vote 7 down vote accepted
+100

Here's a proof of the statement for $f=0$, so that $X=W$ is a Wiener process. (The proof with general $f$ is a bit more involved, and I give this further below). I'll base the proof on the following simple result. Here, I am using $B_\epsilon=\left\{\omega\in\mathcal{W}^n\colon\sup_{t\in[0,1]}\lVert\omega(t)\rVert\le\epsilon\right\}$ for the $\epsilon$-ball in Wiener space.

Lemma 1: For any continuous $\gamma\colon[0,1]\to\mathbb{R}^n$ and $\epsilon > 0$ $$ \mathbb{P}\left(W\in B_\epsilon+\gamma\right)\le\mathbb{P}\left(W\in B_\epsilon\right). $$

Proof: As the Wiener measure is Gaussian, it is log-concave. As $B_\epsilon$ is convex then its indicator function is log-concave and, as the convolution of log-concave functions is log-concave, this implies that $\gamma\mapsto\mathbb{P}(W\in B_\epsilon+\gamma)$ is log-concave. As it is also symmetric this gives $$ \mathbb{P}\left(W\in B_\epsilon\right)\ge\sqrt{\mathbb{P}\left(W\in B_\epsilon+\gamma\right)\mathbb{P}\left(W\in B_\epsilon-\gamma\right)}=\mathbb{P}\left(W\in B_\epsilon+\gamma\right). $$ QED

Choosing any smooth $\gamma\colon[0,1]\to\mathbb{R}^n$ with $\gamma(0)=0$ then, using a Girsanov transform in the usual way, $$ \begin{align} &\mathbb{P}\left(W\in B_\epsilon+\phi\right) = \mathbb{P}\left(W-\gamma\in B_\epsilon+\phi-\gamma\right)\\ &\qquad=\mathbb{E}\left[\exp\left(-\int_0^1\dot\gamma dW-\frac12\int_0^1\dot\gamma^2dt\right)\Bigg\vert W\in B_\epsilon+\phi-\gamma\right]\mathbb{P}(W\in B_\epsilon+\phi-\gamma)\\ &\qquad\le\mathbb{E}\left[\exp\left(\int_0^1\ddot\gamma W dt-\dot\gamma(1)W(1)-\frac12\int_0^1\dot\gamma^2dt\right)\Bigg\vert W\in B_\epsilon+\phi-\gamma\right]\mathbb{P}(W\in B_\epsilon) \end{align} $$ The last inequality here is using the lemma, together with integration by parts in the exponent. So, letting $\epsilon$ go to zero and using uniform convergence in the expectation, $$ \begin{align} \limsup_{\epsilon\to0}\frac{\mathbb{P}(W\in B_\epsilon+\phi)}{\mathbb{P}(W\in B_\epsilon)} &\le\exp\left(\int_0^1\ddot\gamma (\phi-\gamma) dt-\dot\gamma(1)(\phi(1)-\gamma(1))-\frac12\int_0^1\dot\gamma^2dt\right)\\ &=\exp\left(\frac12\int_0^1\dot\gamma^2dt-\int_0^1\dot\gamma\dot\phi dt\right). \end{align} $$ In the last line, the derivative $\dot\phi$ is understood in the sense of distributions. Now, if $\phi$ is not in Cameron--Martin space, then if we let $\gamma$ be smooth approximations to $\phi$ (e.g., convolve with a smooth bump function), the right hand side tends to zero giving the result.


Here's a proof of the general case, based on the following lemma which generalizes Lemma 1 above (a proof is given further below).

Lemma 2: Suppose that $X$ satisfies an SDE of the form $dX=g(X_t,t)dt+dW$, $X_0=0$, where $g\colon\mathbb{R}^n\times[0,1]\to\mathbb{R}^n$. is (jointly) continuous and $g(x,t)$ has (jointly) continuous first and second order derivatives with respect to $x$. Then, for any continuous path $\phi\colon[0,1]\to\mathbb{R}^d$, $$ \limsup_{\epsilon\to0}\frac{\mathbb{P}(X\in B_\epsilon+\phi)}{\mathbb{P}(W\in B_\epsilon)}\le K$$ where $K$ is some increasing function of $\lVert Dg(\phi,\cdot)\rVert$ and $\lVert D^2g(\phi,\cdot)\rVert$.

Let's use this to prove the result in the question. For any smooth $\gamma\colon[0,1]\to\mathbb{R}^n$, define $Y$ by $Y_0=0$ and $dY=f(Y+\gamma)dt+dW$. We see that $X-\gamma$ satisfies the same SDE as $Y$, but with $W-\gamma$ in place of $W$. Applying a Girsanov transformation as above, $$ \frac{\mathbb{P}(X\in B_\epsilon+\phi)}{\mathbb{P}(Y\in B_\epsilon+\phi-\gamma)}=\frac{\mathbb{P}(X-\gamma\in B_\epsilon+\phi-\gamma)}{\mathbb{P}(Y\in B_\epsilon+\phi-\gamma)}=\mathbb{E}\left[\exp\left(-\int_0^1\left(\dot\gamma dW_t+\frac12\dot\gamma^2dt\right)\right)\Bigg\vert Y\in B_\epsilon+\phi-\gamma\right]=\mathbb{E}\left[\exp\left(-\int_0^1\left(\dot\gamma dY_t-\dot\gamma f(Y_t+\gamma)dt+\frac12\dot\gamma^2dt\right)\right)\Bigg\vert Y\in B_\epsilon+\phi-\gamma\right]. $$ Taking limits as $\epsilon$ goes to 0 and applying Lemma 2, $$ \limsup_{\epsilon\to0} \frac{\mathbb{P}(X\in B_\epsilon+\phi)}{\mathbb{P}(W\in B_\epsilon)}\le K\exp\left(\int_0^1\left(\frac12\dot\gamma^2-\dot\gamma\dot\phi-\dot\gamma f(\phi)\right)dt\right). $$ As above, we use integration by parts to take the limit, and $\dot\phi$ is the derivative in the sense of distributions. The term $K$ is the right hand side of the inequality in Lemma 2, evaluated with $g(x,t)=f(x+\gamma_t,t)$ evaluated along the path $\phi-\gamma$. This is the same as the right hand side of the inequality evaluated with $g(x,t)=f(x)$ along the path $\phi$, so is independent of $\gamma$. Letting $\gamma$ be smooth approximations to $\phi$, the right hand side of the above inequality can be made arbitrarily close to 0, giving the required result.


Proof of Lemma 2:

Note that the SDE can be expressed as $W_t=X_t-\int_0^tg(X_s,s)ds$. Let us define the path $\psi_t=\phi_t-\int_0^tg(\phi_s,s)ds$. Expanding $g(X_s,s)$ as a power series to second order in $X-\phi$ gives $$ W_t-\psi_t=X_t-\phi_t-\int_0^t\left(A_s(X_s-\phi_s)+u_s\right)ds, $$ where $A_s=Dg(\phi_s,s)$ and $u_s$ is of second order in $X_s-\phi_s$. Choosing a fixed $L > \lVert D^2g(\phi_t,t)\rVert/2$ then $\lVert u_s\rVert\le L\lVert X_s-\phi_s\rVert^2\le L\epsilon^2$ for $X\in B_\epsilon+\phi$ and small enough $\epsilon$. Now, define $S$ to be set of all paths $\omega$ of the form $$ \omega_t=y_t-\int_0^t(A_sy_s+v_s)ds $$ for some $y\in B_\epsilon$ and $v\in B_{L\epsilon^2}$. From the above, we have $W\in S+\psi$ whenever $X\in B_\epsilon+\phi$. As $S$ is convex and symmetric, using log-concavity as in the proof of Lemma 1, $\mathbb{P}(W\in S+\psi)\le\mathbb{P}(W\in S)$.

Now, define a process $Y$ by $$ dY_t=dW_t+AY_t,\ \ Y_0=0. $$ Suppose that $\omega\in S$, $y\in B_\epsilon$ and $v\in B_{L\epsilon^2}$ are as above and $W_t=\omega_t$ then, we have $$ d\lVert Y_t-y_t\rVert/dt\le\lVert A(Y_t-y_t)-v_t\rVert\le a\lVert Y_t-y_t\rVert+L\epsilon^2 $$ where $a=\sup_t\lVert A_t\rVert$. From this we can bound $Y_t-y_t$ by $L\epsilon^2a^{-1}(e^{at}-1)$, which I will write as $M\epsilon^2$. So, $Y\in B_{\epsilon+M\epsilon^2}$, and we have $\mathbb{P}(W\in S)\le\mathbb{P}(Y\in B_{\epsilon+M\epsilon^2})$.

Next, define the process $$ dZ = dW - aZdt,\ \ Z_0=0. $$ This has drift $-a\lVert Z\rVert$ in the radial direction, whereas $Y$ has radial drift bounded below by $-a\lVert Y\rVert$. By comparing these processes we have $\mathbb{P}(Y\in B_{\epsilon+M\epsilon^2})\le\mathbb{P}(Z\in B_{\epsilon+M\epsilon^2})$.

Now, applying the Onsager--Machlup functional to the simple case with $f(x)=-ax$ and $\phi=0$ gives $$ \lim_{\epsilon\to0}\frac{\mathbb{P}(Z\in B_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B_{\epsilon+M\epsilon^2})}=\exp(a). $$ We can also use the standard result that $\mathbb{P}(W\in B_\epsilon)\sim\exp(-c/\epsilon)$ in the limit $\epsilon\to0$ (where $c$ depends only on the dimension $n$) to get $$ \lim_{\epsilon\to0}\frac{\mathbb{P}(W\in B_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B_{\epsilon})}=\exp(cM). $$

Putting the above results together gives the following sequence of inequalities, which should be understood in the asymptotic limit $\epsilon\to0$. $$ \begin{align} \frac{\mathbb{P}(X\in B_\epsilon+\phi)}{\mathbb{P}(W\in B_\epsilon)} &\le \frac{\mathbb{P}(W\in S+\psi)}{\mathbb{P}(W\in B_\epsilon)} \le \frac{\mathbb{P}(W\in S)}{\mathbb{P}(W\in B_\epsilon)}\\ &\le \frac{\mathbb{P}(Y\in B_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B_\epsilon)} \le \frac{\mathbb{P}(Z\in B_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B_\epsilon)}\\ &\le \exp(a)\frac{\mathbb{P}(W\in B_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B_\epsilon)} \le\exp(a+cM). \end{align} $$ QED

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Very clever solution, in particular that the Girsanov transformation is performed with the smooth mollification of $\phi$ and integration by parts can be used. I believe that this can be adapted to $f\in C^2_b$, as the Girsanov transformation is used in a similar way. I'll post here when I make more progress. –  Dimas Abreu Dutra Mar 27 at 1:34
    
Including the drift term $f(X)$ for $X$, you can use the same idea, but there are additional terms in the Girsanov transform. Although not trivial, you should be able to handle them with standard methods. –  George Lowther Mar 27 at 1:47
    
I managed to prove that for bounded variation functions outside the Cameron--Martin space the fictitious density is zero. I'm writing the proof down and will reference it here whenever I'm finished. –  Dimas Abreu Dutra Apr 3 at 18:13
    
I also have a proof of the general case, and have started to add it. It should also extend to a more general result where $f$ is also allowed to depend non-differentiably on time. –  George Lowther Apr 4 at 2:03

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