OutlineOutline Formulation of Filtering Problem General Conditions for Filtering Equation Filtering Model for Reflecting Diffusions Wong-Zakai Approximation to DMZ Equation Construction of Markov Chains Laws of Large Numbers Simulation for Fish Problem Concluding Remarks

I. Formulation of FilteringI. Formulation of Filtering ProblemProblem

We require a predictive model for (signal observations).• Signal is a valued measurable Markov

process with weak generator

where is a complete separable metric space

is the transition semigroup (on )

• Define• Let

Let

: weak generator of with domain

• is measure-determining if is bp-dense in

(Kallianpur & Karandikar, 1984)

• The observation are :

where is a measurable function and is a Brownian motion independent of .

• Optimal filter=random measure

with

Kushner (1967) got a stochastic evolution equation for

Fujisaki, Kallianpur, and Kunita (1972) established it rigorously under

(old)

Kurtz and Ocone (1988) wondered if this condition

could be weakened .

II. General Conditions for Filtering EquationII. General Conditions for Filtering Equation

K & L prove that if (new) then FKK equation

where is the innovation process

• The new condition is more general, allowing and with - stable distributions with

• No right continuity of or filtration

Reference probability measure:

Under : and are independent,

is a standard Brownian motion.

Kallianpur-Striebel formula (Bayes formula):

Under the new condition, satisfies the Duncan-Mortensen-Zakai (DMZ) equation

Ocone (1984) gave a direct derivation of DMZ equation under finite energy condition

• just measurable, not right continuous; no stochastic calculus. How would you establish DMZ equation?

Define

is a martingale under

is a martingale under (also )

is a martingale under

Let be a refining partition of [0,T] Equi-continuity via uniform integrability

is sum of a and a martingale under ,

i.e

Then is a zero

mean martingale

Using martingale representation, stopping arguments, Doob’s optional sampling theorem to identify

FKK equation can be derived by Ito’s formula, integration by parts and the DMZ equation

III.Filtering Model for Reflecting III.Filtering Model for Reflecting DiffusionsDiffusions

Signal: reflecting diffusions in rectangular region D

The associated diffusion generator

is symmetric on

The observation :

• is defined on

IV. Wong-Zakai Approximation to DMZ IV. Wong-Zakai Approximation to DMZ EquationEquation

has a density which solves

where

Let (unitary transformation)

then satisfies the following SPDE:

where

defined on

Kushner-Huang’s wide-band observation noise approximation

where is stationary , bounded, and -mixing,

converges to in distribution

Find numerical solutions to the random PDE by replacing with and adding correction term,

Kushner or Bhatt-Kallianpur-Karandikar’s robustness

result can handle this part: the approximate

filter converges to optimal filter.

V. Construction of Markov ChainsV. Construction of Markov Chains Use stochastic particle method developed by Kurtz (1971),

Arnold and Theodosopulu (1980), Kotelenez (1986, 1988), Blount (1991, 1994, 1996), Kouritzin and Long (2001).

Step 1: divide the region D into cells Step 2: construct discretized operator via (discretized) Dirichlet form.

where is the potential term in

• : number of particles in cell k at time t

• Step 3: particles evolve in cells according to

(i) births and deaths from reaction:

at rate

(ii) random walks from diffusion-drift

at rate

where is the positive (or negative) part of

• Step 4: Particle balance equation

where are independent Poisson processes defined on another probability space

Construction of Markov Chains Construction of Markov Chains (cont.)(cont.)

is an inhomogeneous Markov chain

via random time changes

Step 5: the approximate Markov process is given by

where denotes mass of each particle

Then satisfies

Compare with our previous equation for

To get mild formulation for and via semigroups

Define a product probability space (for annealed result)

• From we can construct a unique probability measure

defined on for each

VI. Laws of Large NumbersVI. Laws of Large Numbers

• The quenched (under ) and annealed (under ) laws of large numbers ( ):

Quenched approach: fixing the sample path of observation process

Annealed approach: considering the observation process as a random medium for Markov chains

~P

Proof IdeasProof Ideas

Quadratic variation for mart. in

Martingale technique, semigroup theory, basic inequalities to get uniform estimate

Ito’s formula, Trotter-Kato, dominated convergence and Gronwall inequality

VII. Simulation for Fish ProblemVII. Simulation for Fish Problem

Fish ModelFish Model

2-dimensional fish motion model (in a tank )

Observation: To estimate:

In our simulation:

Panel size : pixel, fish size : pixel,

SIMULATIONSIMULATION

VIII. Concluding RemarksVIII. Concluding Remarks

Find implementable approximate solutions to filtering equations.

Our method differs from previous ones

such as Monte Carlo method (using Markov

chains to approximate signals, Kushner 1977), interacting particle method (Del Moral, 1997), weighted particle method (Kurtz and Xiong, 1999, analyze), and branching particle method (Kouritzin, 2000)

Future work: i)weakly interacting multi-target

ii) infinite dimensional signal

SIMULATIONSIMULATION

Pollution TrackingPollution Tracking