Some Large Deviations Asymptotics in Small Noise Filtering Problems

In this work we study certain large deviation asymptotics for nonlinear filtering problems with small signal and observation noise. As the noise in the signal and observation processes vanish, the filtering problem can formally be replaced by a variational problem and one may approximate the filtering estimates (namely suitable conditional probabilities or expectations) by solutions of certain deterministic optimization problems. However due to randomness there will be occasional large deviations of the true nonlinear filter estimates from the variational problem solutions. The main goal of this work is to investigate the probabilities of such deviations by establishing a suitable large deviation principle. Large deviations and related asymptotic problems in the context of small noise nonlinear filtering have been investigated, under different settings, in many works [15, 13, 2, 16, 21, 3, 24, 18, 19, 11, 22, 1]. We summarize the main results of these works and their relation to the asymptotic questions considered in the current work at the end of this section.

In order to describe our results precisely, we begin by introducing the filtering model that we study. We consider a signal process $X^{\varepsilon}$ given as the solution of the $d$-dimensional stochastic differential equation (SDE)

and an $m$-dimensional observation process $Y^{\varepsilon}$ governed by the equation

on some probability space $(\bar{\mathnormal{\Omega}},\bar{{\mathcal{F}}},\bar{{\mathbb{P}}})$. Here $\varepsilon\in(0,\infty)$ is a small parameter, $T\in(0,\infty)$ is some given finite time horizon, $W$ and $B$ are mutually independent standard Brownian motions in ${\mathbb{R}}^{k}$ and ${\mathbb{R}}^{m}$ respectively, $x_{0}\in{\mathbb{R}}^{d}$ is known deterministic initial condition of the signal, and the functions $b,\sigma$ and $h$ are required to satisfy the following condition.

Note that under Assumption 1 there is a unique pathwise solution of (1.1) and the solution is a stochastic process with sample paths in ${\mathcal{C}}_{d}$ (the space of continuous functions from $[0,T]$ to ${\mathbb{R}}^{d}$ equipped with the uniform metric).

The filtering problem is concerned with the computation of the conditional expectations of the form

where ${\mathcal{Y}}^{\varepsilon}_{T}\doteq\sigma\{Y^{\varepsilon}(s):0\leq s\leq T\}$ and $\phi:{\mathcal{C}}_{d}\to{\mathbb{R}}$ is a suitable map. The stochastic process with values in the space of probability measures on ${\mathcal{C}}_{d}$, given by

is usually referred to as the nonlinear filter.

In this work we are interested in the study of the asymptotic behavior of the nonlinear filter as $\varepsilon\to 0$. Denote by $\xi^{*}\in{\mathcal{C}}_{d}$ the unique solution of

It can be shown that, under additional conditions (see discussion in Section 2), that, as $\varepsilon\to 0$,

weakly. In particular for Borel subsets $A$ of ${\mathcal{C}}_{d}$ whose closure does not contain $\xi^{*}$ one will have $\bar{\mathbb{P}}\left[X^{\varepsilon}\in A\mid{\mathcal{Y}}^{\varepsilon}_{T}\right]\to 0$ in probability as $\varepsilon\to 0$. It is of interest to study the rate of decay of conditional probabilities of such non-typical state trajectories. As a special case of the results of the current paper (see Corollary 4.2) it will follow that for every real continuous and bounded function $\phi$ on ${\mathcal{C}}_{d}$, denoting

where $\stackrel{{\scriptstyle\bar{\mathbb{P}}}}{{\longrightarrow}}$ denotes convergence in probability under $\bar{\mathbb{P}}$, and $J$ is the rate function on ${\mathcal{C}}_{d}$ associated with the large deviation principle for $\{X^{\varepsilon}\}_{\varepsilon>0}$ (see Section 2). From this convergence it follows using standard arguments (see e.g. [6, Theorem 1.8]), that, for all Borel subsets $A$ of ${\mathcal{C}}_{d}$)

where for real random variables $Z^{\varepsilon}$ and a constant $\alpha\in{\mathbb{R}}$ we say $\overline{\lim}^{\bar{\mathbb{P}}}_{\varepsilon\to 0}Z^{\varepsilon}\leq\alpha$ [ resp. $\underline{\lim}^{\bar{\mathbb{P}}}_{\varepsilon\to 0}Z^{\varepsilon}\geq\alpha$] if $(Z^{\varepsilon}-\alpha)^{+}$ [resp. $(\alpha-Z^{\varepsilon})^{+}$] converges to $0$ in $\bar{\mathbb{P}}$-probability, and for a set $A$, $A^{o}$ and $\bar{A}$ denote its interior and closure respectively..

Thus the convergence in (1.7) gives information on rate of decays of conditional probabilities of non-typical state trajectories. Formally, denoting the infimum in the above display as $S(\xi^{*},A)$, we can write approximations for conditional probabilities:

However, due to stochastic fluctuations, one may find that for some ‘rogue’ observation trajectories the conditional probabilities on the left side of (1.9) are quite different from the deterministic approximation on the right side of (1.9). In order to quantify the probabilities of observing such rogue observation trajectories that cause deviations from the bounds in (1.8), a natural approach is to study a large deviation principle for ${\mathbb{R}}$ valued random variables $\{-\varepsilon^{2}\log U^{\varepsilon}[\phi]\}$ whose typical (law of large numbers) behavior is described by the right side of (1.7). Establishing such a large deviation principle is the goal of this work. Such a result gives information on decay rates of probabilities of the form

for suitable sets $A\in{\mathcal{B}}({\mathcal{C}}_{d})$ and $\delta>0$. Our main result is Theorem 2.1 which gives a large deviation principle for $\{-\varepsilon^{2}\log U^{\varepsilon}[\phi]\}$, for every continuous and bounded function $\phi$ on ${\mathcal{C}}_{d}$ with a rate funcion defined by the variational formula in (2.16)-(2.17).

In Section 2 we will introduce a continuous map $\hat{\Lambda}^{\varepsilon}_{T}:{\mathcal{C}}_{m}\to{\mathcal{P}}({\mathcal{C}}_{d})$ such that $\hat{\Lambda}^{\varepsilon}_{T}(Y^{\varepsilon})=\bar{\mathbb{P}}(X^{\varepsilon}\in\cdot\mid{\mathcal{Y}}^{\varepsilon}_{T})$ a.s. In [15], Hijab established, under conditions (that include boundedness and smoothness of various coefficients functions), a large deviation principle for the collection of probability measures (on ${\mathcal{C}}_{d}$) $\{\hat{\Lambda}^{\varepsilon}_{T}(y)\}_{\varepsilon>0}$ (with speed $\varepsilon^{-2}$), for a fixed $y$ in ${\mathcal{C}}^{1}_{m}$, with rate function $\hat{I}_{y}:{\mathcal{C}}_{d}\to[0,\infty]$ given by

In a related direction, Hijab’s Ph.D. dissertation [14], studied asymptotics of the unnormalized conditional density and established, under conditions, an asymptotic formula of the form

where $q^{\varepsilon}(x,t)$ denotes the solution of the Zakai equation associated with the nonlinear filter (cf. [17]). The deterministic function $W(x,t)$ coincides with Mortensen’s (deterministic) minimum energy estimate [20] which is given as solution of a certain minimization problem related to the objective function $I_{y}(\eta)$. Results of Hijab were extended to random initial conditions in [16], once again assuming boundedness and smoothness of coefficients. In related work, the problem of constructing observers for dynamical systems as limits of stochastic nonlinear filters is studied in [2]. Heunis[13] studies a somewhat different asymptotic problem for small noise nonlinear filters. Specifically, it is shown in [13], that for every $\phi\in C_{b}({\mathcal{C}}_{d})$, $w\in{\mathcal{C}}_{m}$, and for any $\eta\in{\mathcal{C}}_{d}$ for which the map defined in (2.13) has a unique minimizer (at say $\eta^{*}$),

This result and its connection to our work are further discussed in Section 2. In particular the statement in (1.5) follows readily on using similar ideas as in [13]. The work of Pardoux and Zeitouni[21] considers a one dimensional nonlinear filtering problem where the observation noise is small while the signal noise is $O(1)$ (specifically, the term $\varepsilon\sigma(X^{\varepsilon}(t))$ in (1.1) is replaced by $1$). In this case the conditional distribution of $X(T)$ given ${\mathcal{Y}}^{\varepsilon}_{T}$ converges a.s. to a Dirac measure $\delta_{X(T)}$ as $\varepsilon\to 0$. The paper [21] proves a quenched LDP for this conditional distribution (regarded as a collection of probability measures on ${\mathcal{C}}_{d}$ parametrized by $X(T)(\omega)$) in ${\mathcal{C}}_{d}$. In a somewhat different direction, in a sequence of papers [24, 19, 18], the authors have studied asymptotics of the filtering problem under a small signal to noise ratio limit, under various types of model settings. In this case the nonlinear filter converges to the unconditional law of the signal and the authors establish large deviation principles characterizing probabilities of deviation of the filter from the above deterministic law. An analogous result in a correlated signal-observation noise case was studied in [3]. Finally, yet another type of large deviation problem in the context of nonlinear filtering (with correlated signal-observation noise) when the observation noise is $O(1)$ and the signal noise and drift are suitably small has been considered in a series of papers [11, 22, 1].

The closest connections of the current work are with [15] and [13]. Specifically, the asymptotic statements in (1.7) and (1.8) which follow as a special case of our results (see Corollary 4.2) is analogous to results in [15], except that instead of a fixed observation path we consider the actual observation process $Y^{\varepsilon}$ (also we make substantially weaker assumption on coefficients than [15]). However our main interest is in a large deviation principle for the convergence to the deterministic limit in (1.7) , thus roughly speaking we are interested in quantifying the probability of deviations from the convergence statement in [15] (when a fixed observation path is replaced with the observation process $Y^{\varepsilon}$). This large deviation result, given in Theorem 2.1, is the main contribution of our work.

Recall that $X^{\varepsilon}$ has sample paths in ${\mathcal{C}}_{d}$. Similarly, the processes $Y^{\varepsilon},W,B$ have sample paths in ${\mathcal{C}}_{m},{\mathcal{C}}_{k},{\mathcal{C}}_{m}$ respectively. It will be convenient to formulate the filtering problem on suitable path spaces. Denote, for $p\in{\mathbb{N}}$, the standard Wiener measure on $({\mathcal{C}}_{p},{\mathcal{B}}({\mathcal{C}}_{p}))$ as ${\mathcal{W}}_{p}$ and the Wiener measure with variance parameter $\varepsilon^{2}$ as ${\mathcal{W}}_{p}^{\varepsilon}$. Denote the canonical coordinate process on $({\mathcal{C}}_{k},{\mathcal{B}}({\mathcal{C}}_{k}))$ as $\{\gamma(t):0\leq t\leq T\}$ and consider the SDE on the probability space $({\mathcal{C}}_{k},{\mathcal{B}}({\mathcal{C}}_{k}),{\mathcal{W}}_{k})$,

From Assumption 1, the above SDE has a unique strong solution with sample paths in ${\mathcal{C}}_{d}$.

Consider the map ${\mathcal{C}}_{k}\to\mathnormal{\Omega}_{x}\doteq{\mathcal{C}}_{d}\times{\mathcal{C}}_{k}$ defined as $\omega\mapsto(x^{\varepsilon}(\omega),\gamma(\omega))$ and let

Next, let $\mathnormal{\Omega}_{y}\doteq{\mathcal{C}}_{m}$ and consider the probability space

Abusing notation, denote the coordinate maps on the above probability space as $\xi,\gamma,\zeta$, namely

We will frequently write $\xi(\omega)(s)$ as $\xi(s)$ for $(\omega,s)\in\mathnormal{\Omega}\times[0,T]$. Similar notational shorthand will be followed for other coordinate maps.

Note that, under ${\mathbb{Q}}^{\varepsilon}$, $\xi(0)=x_{0}$, $\gamma$ and $\varepsilon^{-1}\zeta$ are independent standard Brownian motions in ${\mathbb{R}}^{k}$ and ${\mathbb{R}}^{m}$ respectively and

Define, for ${\mathbb{Q}}^{\varepsilon}$ a.e. $\omega=((\omega_{1},\omega_{2}),\omega_{3})$, for $t\in[0,T]$,

Note that, since under ${\mathbb{Q}}^{\varepsilon}$, $\varepsilon^{-1}\zeta$ is a standard Brownian martingale with respect to the filtration ${\mathcal{F}}_{t}^{0}\doteq\sigma\{\gamma(s),\xi(s),\zeta(s):0\leq s\leq t\}$, the first integral in the exponent is well-defined as an Itô integral. From the independence of $\xi$ and $\zeta$ under ${\mathbb{Q}}^{\varepsilon}$ and Assumption 1 it follows that $L^{\varepsilon}_{t}$ is a $\{{\mathcal{F}}^{0}_{t}\}$-martingale under ${\mathbb{Q}}^{\varepsilon}$. Define a probability measure ${\mathbb{P}}^{\varepsilon}$ on $(\mathnormal{\Omega},{\mathcal{F}})$ as

Note that, by Girsanov’s theorem, under ${\mathbb{P}}^{\varepsilon}$

is a standard $m$-dimensional Brownian motion which is independent of $(\xi,\gamma)$. Rewriting the above equation as

we see that

Next, for $\varepsilon>0$, define $\Gamma^{\varepsilon}_{T}:{\mathcal{C}}_{m}\to{\mathcal{M}}({\mathcal{C}}_{d})$ as

The maps are well defined ${\mathbb{P}}^{\varepsilon}$-a.s. and using results of [8, 9, 10], one can obtain versions of these maps (denoted as $\hat{\Gamma}^{\varepsilon}_{T}$) which are continuous on ${\mathcal{C}}_{m}$. Also, define $\Lambda^{\varepsilon}_{T}:{\mathcal{C}}_{m}\to{\mathcal{P}}({\mathcal{C}}_{d})$ as

Once again, for each $\varepsilon>0$, this map is well defined ${\mathbb{P}}^{\varepsilon}$-a.s. and a continuous version of the map exists (which we denote as $\hat{\Lambda}^{\varepsilon}_{T}$) from [8, 9, 10] . Write, for $f\in\mbox{BM}({\mathcal{C}}_{d})$

Then with $(X^{\varepsilon},Y^{\varepsilon})$ as in (1.1)-(1.2), for $\phi\in\mbox{BM}({\mathcal{C}}_{d})$

Also,

where ${\mathbb{E}}_{{\mathbb{P}}^{\varepsilon}}$ denotes the expectation under the probability measure ${\mathbb{P}}^{\varepsilon}$, and

Let for $\xi_{0}\in{\mathcal{C}}_{d}$,

where ${\mathcal{U}}(\xi_{0})$ is the collection of all $\varphi$ in ${\mathcal{L}}^{2}_{k}$ such that

Note that, by Assumption 1, for every $\varphi\in{\mathcal{L}}^{2}_{k}$ there is a unique solution of (2.10). By classical results of Freidlin and Wentzell (see e.g. [6, Theorem 10.6]) the collection $\{X^{\varepsilon}\}$ of ${\mathcal{C}}_{d}$ valued random variables satisfies a LDP with rate function $J$ and speed $\varepsilon^{-2}$, namely, for all $F\in C_{b}({\mathcal{C}}_{d})$

where we denote the first coordinate process on $\mathnormal{\Omega}_{x}$ by $\hat{\xi}$, i.e. $\hat{\xi}(\omega)=\omega_{1}$ for $\omega=(\omega_{1},\omega_{2})\in\mathnormal{\Omega}_{x}={\mathcal{C}}_{d}\times{\mathcal{C}}_{k}$. In [13] it is shown that for every $w\in{\mathcal{C}}_{m}$, and a given $\eta\in{\mathcal{C}}_{d}$, the probability measure

weakly, if the map

attains its infimum over ${\mathcal{C}}_{d}$ uniquely at $\eta^{*}$, where recall that $\hat{\Lambda}^{\varepsilon}_{T}$ is the continuous version of $\Lambda^{\varepsilon}_{T}$. We remark that [13] assumes in addition to (1) that $h$ and $b$ are bounded, but an examination of the proof shows (see calculations in Section 5) that these conditons can be replaced by linear growth conditions that are implied by Assumption 1 .

Recall the function $\xi^{*}\in{\mathcal{C}}_{d}$ from (1.4). Then using similar ideas as in [13], under Assumption 1, and assuming in addition that either $\sigma\sigma^{\dagger}$ is positive definite or $h$ is a one-to-one function, it follows that

weakly, as $\varepsilon\to 0$. This is a consequence of the fact that when $\eta=\xi^{*}$ the map in (2.13) achieves its minimum (which is $0$) uniquely at $\xi^{*}$.

As a consequence of the results of the current paper (see Corollary 4.2) one can show the Laplace asymptotic formula in (1.7). Recall from the discussion in the Introduction that the convergence in (1.7) gives information on asymptotics of conditional probabilities of non-typical state trajectories. In order to quantify the decay rate of probabilities of observing rare observation trajectories that cause deviations from the deterministic variational quantity in (1.7), we will establish a large deviation principle for $\{-\varepsilon^{2}\log U^{\varepsilon}[\phi]\}$ defined in (1.6).

We now present the rate function associated with this LDP.

Define the map $H:{\mathcal{C}}_{d}\times{\mathcal{C}}_{d}\times{\mathcal{L}}^{2}_{m}\to{\mathbb{R}}_{+}$ as

Also, for $\varphi\in{\mathcal{L}}^{2}_{k}$, let $\xi_{0}^{\varphi}$ be given as the unique solution of (2.10).

We now introduce the rate function that will govern the large deviation asymptotics of $-\varepsilon^{2}\log U^{\varepsilon}[\phi]$.

Fix $\phi\in C_{b}({\mathcal{C}}_{d})$ and define $I^{\phi}:{\mathbb{R}}\to[0,\infty]$ as

where ${\mathcal{S}}(z)$ is the collection of all $(\varphi,\psi)$ in ${\mathcal{L}}^{2}_{k}\times{\mathcal{L}}^{2}_{m}$ such that

The following is the main result of the work.

Fix $\phi\in C_{b}({\mathcal{C}}_{d})$. Recall the functional $U^{\varepsilon}[\phi]$ from (1.6). From (2.6), note that one can write $U^{\varepsilon}[\phi]$ as

whose distribution under $\bar{\mathbb{P}}$ is same as the distribution of $\Lambda^{\varepsilon}_{T}\left(\exp\{-\varepsilon^{-2}\phi(\cdot)\},\zeta\right)$ under ${\mathbb{P}}^{\varepsilon}$. Let

Using this equality of laws and the equivalence between Large deviation principles and Laplace principles (see e.g. [6, Theorems 1.5 and 1.8]), in order to prove Theorem 2.1 it suffices to show that, $I^{\phi}$ has compact sub-level sets, i.e.,

and for every $G\in C_{b}({\mathbb{R}})$

The proof of the identity in (3.2) will use a variational representation for nonnegative functionals of Brownian motions given by Boué and Dupuis[4]. We now use this representation to give a variational formula for the left side of the above equation. Let ${\mathcal{F}}_{t}$ denote the ${\mathbb{P}}^{\varepsilon}$-completion of ${\mathcal{F}}^{0}_{t}$ and denote by ${\mathcal{A}}^{k}$ [resp. ${\mathcal{A}}^{m}$] the collection of all $\{{\mathcal{F}}_{t}\}$-progressively measurable ${\mathbb{R}}^{k}$ [resp. ${\mathbb{R}}^{m}$] valued processes $g$ such that for some $M=M(g)\in(0,\infty)$

For $(u,v)\in{\mathcal{A}}^{k}\times{\mathcal{A}}^{m}$, let $\xi^{u}$ be given as the unique solution of the SDE on $(\mathnormal{\Omega},{\mathcal{F}},\{{\mathcal{F}}_{t}\},{\mathbb{P}}^{\varepsilon})$:

Also define

Occasionally, to emphasize the dependence of above processes on $\varepsilon$ we will write $(\xi^{u},\zeta^{u,v})$ as $(\xi^{\varepsilon,u},\zeta^{\varepsilon,u,v})$.

Now let

When clear from context we will drop $(u,v,\phi)$ from the notation in $\bar{V}^{\varepsilon,u,v}[\phi]$ and simply write $\bar{V}^{\varepsilon}$. Then it follows from [4] (cf. [6, Theorems 3.17]) that

For $M\in(0,\infty)$, let

We equip, $S_{M}$ with the weak topology under which $(\varphi_{n},\psi_{n})\to(\varphi,\psi)$ as $n\to\infty$ if and only if for all $(f,g)\in{\mathcal{L}}^{2}_{k}\times{\mathcal{L}}^{2}_{m}$

as $n\to\infty$. This topology can be metrized so that $S_{M}$ is a compact metric space.

Recall $\phi\in C_{b}({\mathcal{C}}_{d})$ in the statement of Theorem 2.1. For $(\varphi,\psi)\in{\mathcal{L}}^{2}_{k}\times{\mathcal{L}}^{2}_{m}$ define

Note that with this notation ${\mathcal{S}}(z)$ (introduced below (2.16)) is the collection of all $(\varphi,\psi)$ in ${\mathcal{L}}^{2}_{k}\times{\mathcal{L}}^{2}_{m}$ such that $V_{0}^{\varphi,\psi}[\phi]=z$.

The following lemma will be the key to the proof of Theorem 2.1.