Multi-Body Tracking

In June 2001, another particle breakthrough was achieved by Ballantyne, Chan, and Kouritzin using MIBR on the weakly interacting multi-target problem.  We were able to detect and track a small-unknown number (up to four) of simulated, weakly interacting bodies all evolving randomly over a sheet of water.  From the theoretical side this simply amounted to imbedding all the targets into a purely atomic Markov measure-valued process that becomes the signal.  Each particle then has a number of bodies in different locations.  On the practical side, we had to solve a whole host of new issues like ?x201c;when do we allow a particle to switch its number of targets??x201d; ?x201c;What is the best initial allotment of particles in terms of number of bodies and their positions??x201d; and ?x201c;How should the performance of the filter be judged??x201d;  These were all successfully answered and the filter worked remarkably well, given sufficient computing resources.  Its success attests to the efficiency of MIBR.  In the process of handling an unknown number of targets, we also solved the detection problem as a side effect.  Thus, we reduced the normal sequence of track detection, track association, tracking, and track removal to a single filtering problem on a larger space. 

We should have a working implementation of this problem with SERP in November 2002. It is expected to outperform MIBR and allow us to track up to five targets with careful solution of the stochastic control problem for selecting an optimal ?x3c1; resampling value.  SERP's resampling method allows us to consider potentially more efficient techniques for target count determination.

Gentil and Remillard obtained an efficient algorithm of detection of ships in using a sequence of noisy images.  We find the algorithm computing easily with the optimal filter.  We also present some simulations, in the case of one or two ships, showing that the prediction of the position of the ships is quite good.

Sensor Vibration Neutralization

One of Lockheed Martin Naval Electronics and Surveillance Systems' and Lockheed Martin Canada's most vital concerns is to improve the onboard performance of the Canadian Wescam sensor.  This sensor has been installed in several U.S. military aircraft and would be installed in many more if its performance were more satisfactory.  The most practical methods of improving performance are via software-implemented mathematical algorithms.  There are three separate problem areas in which nonlinear filtering can improve the performance of the Wescam filter.  In simplification, they are:  initialization, remote pod neutralization, and low observable tracking of vessels.  We have essentially provided our sponsors with their most promising methods of improving the low observable tracking performance of the Wescam sensor through our work on SERP and REST filters.  Our convolutional filter appears ideal for problems like remote pod movement neutralization.  This Wescam filter enhancement is part of the realistic and quite technical motivation for our previously mentioned Search and Rescue problem.

To explain the Pod Tracking problem further, we mention that the sensor must be mounted externally to be effective and this external sensor pod is subject to random vibrations and forces. Our goal is to neutralize the effect of the first sensor movement from the first sensor readings. This is done by a filtering algorithm and a second sensor in the craft, whose only task is to estimate the pod. Depending upon the choice for this second sensor different observation models are appropriate.  Due to the distances involved small errors in the pod location estimate have a dramatically amplified effect on the first sensor's readings.  Therefore, we wanted to avoid the extra approximations of particle or space-discretization methods and use IDEX.  Currently, we believe we have a workable explicit solution for the stochastic movement of the pod in terms of a small number of multiple Wiener integrals in the sense that the corresponding Stratonovich equation should adequately model the pod movement and the explicit solution is computer workable.  Most likely, we will have to modify our solution later to account for the pole behaviour more fully.  However, the explicit solution method has the highly desirable properties that the solution is forced to stay on the manifold even in the presence of numeric error and the evolution is in two dimensions (the dimension of the manifold), not three as direct particle or space discretization methods would do.

The mathematics behind this solution is reasonably involved but general enough to accommodate more general manifolds.  Basically, we prove such things as that there are explicit solutions involving the one and two-parameter stochastic integrals if and only if the vector fields corresponding to the columns of the dispersion coefficient in the Stratonovich equation are two step nilpotent and the ?x201c;drift?x201d; vector field satisfies some more general condition.  Then, we study the class of such explicit solutions and show that it is large and contains many suitable models for this problem.  In particular, we show that there are solutions that stay on the desired manifold and have the desired drift and diffusion properties on this manifold.  The work is being done by Kouritzin, Remillard, and Van Weelden.  Kouritzin and Wiersma are implementing the solution.

We are also talking with Lockheed about the possibility of combining the Pod Tracking problem with the Search and Rescue problem to better reflect reality and produce one large problem.

Communication Networks Applications

Our new corporate partner, Optovation of Ottawa, is paying us to investigate fibre optic signal properties.  In particular, they are building a product that tests the quality of optic signals at various frequencies, using a lot of proprietary hardware and software.  One of the most difficult problems left for them is the accurate simultaneous estimation of optical signal to noise ratios, peak powers, and bit rates for all the carrier frequencies on a fibre.  In the Ordered Multi-Target Tracking problem, we are solving a simplified version of this problem for them.  In early September 2002, Hailes and Kouritzin visited Optovation's lab to learn the characteristics of this problem and collect some data.  Since then, we have built a first model for their problem including the targets, their interactions, the optical noise, and the electrical noise.  The problem and model are naturally addressed by filtering theory.  However, due to the huge size of this problem (up to eighty interacting targets), we will have to make significant advances in multi-target tracking and applying REST to such tracking in order to help them.  Even with knowledge of problem specific simplifications that can be made, the problem seems too large for anything but possibly REST.  There is also a necessary parameter estimation that must be done to characterize the optical noise in this problem.  We may try to do this within REST or else using our recursive-combined algorithm.

Michael Kouritzin established a new class of models described by a nonlinear stochastic parabolic equation with arbitrary cádlág noise and arbitrary order elliptic operators.  The noise sources in these models include long-range dependent processes, heavy-tailed Lévy processes, and composite processes like iterated Brownian motion.  It is believed that these models are a reasonable starting point for modeling such events as the flow of information through a complicated communication network.  Filtering would then be used to determine more ?x201c;internal?x201d; quantities.

Mathematical Finance Applications 

We hope to find new corporate sponsors in the mathematical finance and communication industries.  Here, such data as a collection of stock prices or the number of jobs at various nodes of a network usually form the observation process, and long-range dependent or heavy-tailed models for observation processes are used for fidelity to reality.  Thus, the long-range dependent processes are not Markov.  However, as shown in a recent paper by M.L. Kleptsyna, A. Le Breton and M.C. Roubaud, entitled ?x201c;An elementary approach to filtering in systems with fractional Brownian observation noise?x201d;, there are still methods of constructing filters in certain long-range dependent observation settings.

Application of filtering and particle systems to mathematical finance is a growing area.  Hence, it seems natural to apply our algorithms to this area in hope of attract an industry partner.  In the volatility tracking problem we model and track stochastic volatility together with five unknown parameters using IDEX.  In fact, due to the nature of the model we can compute the maximum likelihood parameters from the filter.  Despite the large number of parameters to estimate, when applied with real stock data the model still performed very well.  We published some of the results in Kouritzin, Remillard and C. Chan (2001) and believe that this technique could benefit an industry partner.

When pricing options using natural random interest rate and stock models, the future price must be brought back to the current time using a random weighting.  Several books suggest constructing option prices using straight Monte Carlo methods.  However, this random weighting means that a resampled particle system like SERP should outperform straight Monte Carlo pricing.  We have applied SERP to a simple (single stock, bank account) model where the bank account interest rate depends on the stock price.  A minor modification of SERP is then used to price all pathspace and terminal options simultaneously by the creation of a ``pricing measure''.  The results were impressive, requiring far fewer particles than straight Monte Carlo.  We look forward to testing SERP on the multi-stock scenario.  In November and December, Jarett Hailes and Mike Kouritzin will meet with contacts from the finance sector in an attempt to secure a corporate partner and justify further work in this area.

Robotic Lighting Control Application

Acoustic Positioning Research (APR) was so impressed with our algorithms and the possibility of improving their product that they joined us as a corporate sponsor in November 2000.  Will Bauer, president of APR, suggested that including our algorithm into APR's product may increase the market for their product by fifty percent.  He has also agreed to pay a small royalty to our Centre on each sale of their enhanced product to reflect the fact that they will benefit from the previous work of the Centre.  The first problem is to direct lights and sound effects automatically based upon inaudible high frequency sound time measurements.  In the past, they have suffered from noisy measurements and poor prediction methods.  We now have simulations solving this initial one performer problem using MIBR, SERP, and REST.  The results are impressive.  The next large steps are to allow multiple performers and for parameters to be identified in the models.

The first APR problem was to predict the future position of a performer using timing measurements as described in the Performer Prediction problem.  This problem has now been solved effectively using SERP and APR is incorporating it into their full system.  Our solution can improve performance of the many systems that have been sold and are in the field.

Due to current inexpensive prices for video equipment, APR is now also interested in tracking and predicting performers using digital cameras also located on the perimeter of the stage.  A small pause in our collaboration was agreed upon due to the sabbatical of Kouritzin and the time required to incorporate the first solution.  However, both APR and PINTS are eager to restart collaboration on the second problem, striving to predict multiple performers using the smallest number of cameras, by the end of 2003.

Model Approximation and Robustness


Continuous and discrete Markov chain approximations can be important in predictive modeling and computationally efficient filtering for a number of reasons.  To dramatically increase the class of signals that can be filtered by our various methods, we have concerned ourselves with the Markov chain approximation of the extremely complex signals arising in multiple target tracking and other spatial filtering problems such as pollution or bacteria tracking within a sheet of water.  Then we can substitute the computer-tractable approximation for the real signal and rely on robustness results of, for example, Kushner or Bhatt and Karandikar  (1999, 2002) to justify our use of a simplified model.

With regard to constructing a general powerful method of approximating complicated signals with a Markov chain, Kouritzin and Long have made crucial innovations to advance the existing approximations of Ludwig Arnold, Peter Kotelenez, and Douglas Blount.  These innovations allow for driving noise sources, more computationally efficient implementations of novel Markov chain approximations, and a larger selection of spatial processes, which can be so approximated.  The innovations include enlarging the class of elliptic operators and the class of reaction functions that can be used in Markov chain approximations within stochastic reaction-diffusion equations, as well as improvements that allow for the use of much slower rates within these approximations to reduce the computational requirements.  Kouritzin and Long's alterations dramatically slow Markov chain state change rates, often yielding a one hundred-fold increase in the simulation speed over the previous version of the method.

In an attempt to attract Stantec or other potential industry partner with interests in environmental monitoring, the aforementioned innovations have been first incorporated into a stochastic reaction-diffusion equation motivated by pollution distribution.  (See the 1995 book by Kallianpur and Xiong for background information on this problem.)  Kouritzin and Long (2002) have established the convergence of Markov chain approximations to stochastic reaction diffusion equations in both the quenched and annealed senses.  Their first convergence result, a quenched law of large numbers, establishes convergence in probability of the Markov chains to the pathwise unique mild solution of the stochastic reaction diffusion equation for each fixed path of the driving Poisson measure source.  Their second result is the annealed approach that establishes convergence in probability of the Markov chains to the mild solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.  These results are vital for application of filtering theory to the pollution dispersion-tracking problem, as they can be combined with the robustness results of Kushner or Bhatt and Karandikar and the aforementioned particle filtering methods to create a computer workable algorithm.

In more detail, Kouritzin and Long considered the stochastic model of ground water pollution, which mathematically can be written with a stochastic reaction diffusion equation.  In the context of simulating the transport of a chemical or bacterial contaminant through a sheet of water, they extended a well-established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger class of reaction functions.  This work applies to Lockheed Martin's interest in detecting and classifying oil slicks and vessel traces or wakes.

A weighted L2 Hilbert space was chosen to symmetrize the elliptic operator in the stochastic reaction diffusion equation, and the existence of and convergence to pathwise unique mild solutions of the stochastic reaction-diffusion equation was considered.  The region [0,L1]  [0,L2] was divided into L1N  L2N cells, and the Markov chain approximation on these cells was analyzed as N .

The particles in cells evolve in time according to births and deaths from reaction, random walks from diffusion and drift, and some area dependent births from the Poisson noise sources.  In this stochastic particle model, the formalism allows for two kinds of randomness:  the external fluctuation coming from the Poisson driving sources and the internal fluctuation from the reaction and drift-diffusion on the particle level.  Independent standard Poisson processes defined on another probability were used to construct the Markov chains by the random time changes method.

In a second work on the stochastic model of water pollution, which mathematically can be written with a stochastic partial differential equation driven by Poisson measure noise,  Kouritzin, Long and Sun (2002) establish a more general annealed law of large numbers.  It shows convergence in probability for our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.  Our proof method of the main result is substantially different from the previous work Kouritzin and Long (2002) using the weak convergence method.  Here, we directly apply Cauchy criterion (convergence in probability) to our Markov chains and utilize the nice regularity of Green's function with a delicate iteration technique.  (The usual Gronwall's Lemma doesn't work in our case.)


The results of Kouritzin and Long (2002) and Kouritzin, Long and Sun (2002) have enabled the creation of prototype stochastic models of pollution. Methods for efficient simulation of the models have been implemented in a computer code.

Kouritzin, Long, Ballantyne, and H. Chan have created a simulation of a stochastic reaction-diffusion equation which can represent the transport of a pollutant or bacteria through a river or the leaching of a pollutant through a ground water system, including adsorption effects, all in a manner amenable to filtering. 

The stochastic models of the spread of pollution developed at MITACS-PINTS have the following general features:

  The models employ Markov chain approximations to nonlinear SPDEs representing stochastic reaction-diffusion equations.

  The equations include convective forces, as would be found in a flowing water system, and Poisson generating sources to model contamination from sites such as factories, storage ponds and agricultural facilities.

  The Markov chain approximations used in the models converge to the exact solution of the stochastic reaction-diffusion equation in both the annealed and the quenched senses.

  The approximations provide the basis for further work on applying filtering techniques to track the sources of contaminants given only imperfect, noise corrupted samples at a few locations.

  These concept proofs provide an effective foundation for incorporating novel filtering techniques into different models.  These techniques are also used to model other reactive flows, such as heat diffusion through a substance of varying heat capacity, or heat-activated internal reaction.


Lucic and Heunis (2002) study signal robustness in the extreme case of singular perturbations with the goal of characterizing the limiting nonlinear filter (if any) as the perturbation parameter tends to zero.  The signal arises from a singularly perturbed stochastic differential equation with a small parameter, in the case where the dynamics of the signal are conditioned by the observation process.  We show that the nonlinear filter is a solution of a particular measure-valued martingale problem, and then show that the limiting nonlinear filter exists and characterize it completely.  The approach is to use solvability of Poisson-type operator equations to construct a limiting measure-valued martingale problem, and use the uniqueness in law results of Lucic and Heunis (2001) to show that this limiting martingale problem is well posed and that its solution corresponds to the limiting nonlinear filter.

Kouritzin and Xiong (2002) study observation robustness to demonstrate the asymptotic correctness of the classical, non-instrumentable continuous-time observations via instrumentable coloured-noise approximations.  In particular, we consider the effect of the observations: , where  is an Ornstein-Uhlenbeck process, as .  In this case, the integrated observation noise converges to Brownian motion and we show that the filter also converges to the classical observation filter.  The non-integrated observations can be instrumented, so this result demonstrates that the classical observations are a natural idealized or limiting object.  Our result generalizes, in some manner, previous results by Kunita (1993), Mandal and Mandrekar (2000), Gawarecki and Mandrekar (2000), and Bhatt and Karandikar (2001).  Our method is to use the Kurtz-Xiong particle approach to derive a FKK-like filtering equation and uniqueness for this equation based upon observations , then we prove tightness for filter distributions, and identify a unique limit using the uniqueness result of Bhatt and Karandikar (1995).

Douglas Blount and Michael Kouritzin have derived Hölder continuity for processes related to the Zakai equation of filtering theory.  Blount and Kouritzin have obtained a criterion, which gives Hölder continuity results in Hilbert space for a class of solutions of stochastic evolution equations.  The class includes the superstable processes with critical binary branching and Ornstein-Uhlenbeck type SPDEs with a suitable eigenfunction expansion for the drift operator. It should also give regularity results for some types of SPDEs arising from filtering theory.  The resulting paper, ?x201c;Hölder continuity for spatial and path processes via spectral analysis?x201d;, appeared in Probability Theory and Related Fields and was the subject of an invited talk in a session on stochastic analysis of the AMS meeting held in January 2001 in New Orleans.

Filtering for Signals in Random Environments

Michael A. Kouritzin, Hongwei Long, and Wei Sun (2002) consider the nonlinear filtering problem in which the signal to be estimated is a (non-Markov) diffusion in a random environment that evolves in some Euclidean space.

Our motivation comes from the tracking problem of a dinghy lost at sea.  When the weather conditions are adverse, the motion of the dinghy will change dramatically due to drifting by random ocean surface wave propagation, rendering the Search and Rescue problem model inappropriate.  The dinghy itself would try to undergo the motion described by some SDE but the random wave formation would have the effect of adding random drift; hence, the signal process for the dinghy can be formally described as an Itô process whose drift term contains the gradient of W (W being a locally bounded random field in the Wiener space).  Of course, this does not really make sense as a normal SDE but is well defined through Dirichlet form or martingale problem theory.  We also allow correlation between the signal and the observation noise, which would be natural from dinghy occupant motion.

It is well known that the long-time behaviour of diffusion in a random environment is much different from that of an ordinary diffusion; note that the drift term  is not well defined, since W is a non-differentiable continuous function.  Diffusion in random medium is not a Markov process unless one fixes an instance of the random environment.  Consequently, it is impossible to construct, within the usual Itô's theory, a stochastic evolution equation for the filtering process of the diffusion in random medium based on noisy observations.  However, it can be constructed from the Dirichlet form theory in any finite dimension.


Although the signal itself is not Markov, it becomes Markov when a single occurrence of the random environment becomes fixed.  This results in two semigroups; one produced by the Dirichlet form on a (randomly) weighted L2-space and another on the Banach space of bounded measurable functions.  Assuming that the initial signal distribution satisfies a certain condition and we have access to the random environment, we show that there is a unique weak solution to the Zakai equation that remains in the L2-space and is expressible in terms of either semigroup. 

In the following, we remove the unrealistic assumption that we can ?x201c;see?x201d; the random environment.  We have chaos decomposition for our filtering process, namely we can represent the filtering process as a series in terms of multiple Wiener-Itô integrals, even in the case of correlated noise (i.e. when the observation noise is not independent of the signal).  We first obtain the expansion for each fixed random environment.  Then, we can use the classical filtering theory to obtain a Zakai equation for this fixed environment predictive model, from which we can express the unnormalized filtering process as the mild solution to the Zakai equation.  Further, we formulate the multiple Wiener integral representation for the unnormalized filtering process. Finally, we show that we can integrate this expansion term by term with respect to the distribution of the random environment.  We arrive at the desired unique multiple Wiener integral representation for the filtering process associated with the diffusion in the random environment.   No Zakai or Kushner-Stratonovich equation is possible.  Our representation formula in Kouritzin, Long and Sun (2002) provides the capability to simulate this filtering process on a computer.


Our practical simulation experience suggests that particle and space discretization methods of implementing filters often work better than chaos methods.  Therefore, it is important to come up with a particle system approximation for the conditional distribution of the signal in a random environment given only the observations.  The difficulty that arises is that we do not want to introduce an extremely large number of particles to account for the random environment.  Instead, we try to ?x201c;learn?x201d; the environment.  In particular, we construct a functional or white noise expansion for the random environment in the Dirichlet form, placing all randomness in the coefficients (and not the bases functions).  Then, we both truncate the expansion (after a very large number of terms) and employ combined filtering-parameter estimation algorithms to ?x201c;identify?x201d; the random environment coefficients while filtering or perform functional estimation to estimate all coefficients simultaneously.  Inasmuch as we have already discussed the former approach in (c), we only discuss the later here.

Kouritzin, Sun, and Xiong (2002) investigated combined functional environment estimation and filtering.  To justify the anticipation of a solution, we simplify the problem to one dimension and constrain the original signal to be [0,2x] by reflecting at the boundaries.  These models still make sense through Dirichlet form theory and can be thought of as a formal Skorohod stochastic differential equation.  Then, we replace the W gradient with its trigometric series.  In this manner, we are also replacingby some trigometric expansion.  (Clearly, since the gradient of a Brownian motion is not a square-integrable function we cannot expect that a trigometric series for is square summable but an infinite series expansion is still valid by deweighting the coefficients and using Sobolev spaces.)  Then, motivated by the method of moments, we assume h is 1-1, take the observation noise to be independent of the signal, and form a functional estimator of the form:

which, for appropriately chosen m n, can be shown to contain consistent (as n ) estimators for all the coefficients of our trigometric coefficients for the gradient of W.  Then, for each n we define an approximate signal and particle filter, show that the approximate signal converges to actual signal as n , and use the technique of Budhiraja and Kushner (2002) to prove that as n  the long-term error of this method converges to zero.

The drawbacks of this approach are:  1) In practice, you must fix n apriori; and 2) the amount of work between observations also increases to infinity as n .  We are also trying to derive an algorithm that will sample all frequencies infinitely often with the same number of calculations between observation so that over time we have a good estimator of all frequencies and n need not be fixed. The idea is to expand W (not ) and estimate the lowest frequencies first twice, then estimate the next lowest frequencies once, repeat both once, and then sample the third lowest frequency group, etc. Suppose we label the frequency groups of approximately the same size 1,2,3, ?x2026;  Then, the algorithm would use our functional estimation procedure on the groups in the following order:  1,1,2,1,1,2,3,1,1,2,1,1,2,3,4, ... The idea here is that the coefficients for W converge to zero as the frequencies increase, so lower frequency coefficients are more important, but they cannot be estimated properly without also estimating the higher coefficients.

l. Basic Filtering Theory


There is much interesting mathematics being researched at MITACS-PINTS on the basic mathematical equations of classical nonlinear filtering theory.  Such work is important for determining the expected properties of the often-difficult real world problems.  The first classical theoretical problem is that of uniqueness.  Uniqueness of the filtering equations provides an important tool in establishing convergence and rates of convergence for computer workable approximations.

In the simplest classical setting, where the signal and observation noise are independent, Szpirglas (1978) established pathwise uniqueness for the Kushner-Stratonovich and the Duncan-Mortensen-Zakai equations, which together with the Yamada-Watanabe argument also provide distributional uniqueness.  His results do not apply in the correlated or feedback case.  Lucic and Heunis (2001) consider uniqueness properties of the nonlinear filter equations, both normalized and unnormalized, for a nonlinear filtering problem in which the signal is conditioned by the observation process. This situation is typically encountered in stochastic feedback controlsystems, in which the output is fed back to the input.  Such feedback systems arise very commonly in the applications of nonlinear filtering to guidance and control problems in aerospace engineering.

Our particular interest is in the uniqueness in law property for the filter equations because of its applicability to the singular perturbation robustness problem discussed above. Although uniqueness in law for the nonlinear filter equations is well known in the case where the signal and observation noise are independent, it was not previously established for the case where the signal is conditioned by the observations, a basically new approach was utilized. We show uniqueness in law under rather mild restrictions for both the normalized and unnormalized filter equations. As a by-product we also get pathwise uniqueness for the unnormalized filter equation.  Our main tools were the uniqueness for measure-valued evolution results obtained by Bhatt and Kanadikar as well as the Yamada-Watanabe argument.  It should be mentioned that the nice uniqueness results of Kurtz and Ocone (1988) could not be used to deduce distributional uniqueness since Kurtz and Ocone's results only apply on probability spaces rich enough to contain the signal, observations and filter.


The uniqueness principle established by Kurtz and Ocone (1988) says that under very general conditions, any solution to the filtering equations on a rich enough probability space must be the conditional distribution of the signal given the observations (i.e. the desired distribution).  However, the conditions under which there are known solutions were far more severe.  It has been known for more than thirty years that there are solutions to the basic filtering equations under the mean-square finite energy condition and right continuity.  This is a much more stringent condition than those required by Kurtz and Ocone (1988) for uniqueness.  Hence, there was an open problem, that we just settled in Kouritzin and Long (2002), to determine existence (i.e. whether the conditional distributions satisfy the Duncan-Mortensen-Zakai and the Kushner-Stratonovich equations) under conditions as general as those used by Kurtz and Ocone for uniqueness.  Indeed, our conditions are milder that those used by Kurtz and Ocone, and demonstrate, for example, that the filtering equations continue to hold when the sensor function is linear and the signal has heavy tails like a Levy process.


When the signal-to-noise ratio goes to zero, Xiong (2002) derived a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces.  The approach follows from the framework established Xiong 1996.  The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.