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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
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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
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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 .
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• 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 .
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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
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Reference probability measure:
Under : and are independent,
is a standard Brownian motion.
Kallianpur-Striebel formula (Bayes formula):
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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?
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Define
is a martingale under
is a martingale under (also )
is a martingale under
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Let be a refining partition of [0,T] Equi-continuity via uniform integrability
is sum of a and a martingale under ,
i.e
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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
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III.Filtering Model for Reflecting III.Filtering Model for Reflecting DiffusionsDiffusions
Signal: reflecting diffusions in rectangular region D
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The associated diffusion generator
is symmetric on
The observation :
• is defined on
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IV. Wong-Zakai Approximation to DMZ IV. Wong-Zakai Approximation to DMZ EquationEquation
has a density which solves
where
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Let (unitary transformation)
then satisfies the following SPDE:
where
defined on
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Kushner-Huang’s wide-band observation noise approximation
where is stationary , bounded, and -mixing,
converges to in distribution
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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.
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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
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• : 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
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• Step 4: Particle balance equation
where are independent Poisson processes defined on another probability space
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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
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Then satisfies
Compare with our previous equation for
To get mild formulation for and via semigroups
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Define a product probability space (for annealed result)
• From we can construct a unique probability measure
defined on for each
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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
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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
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VII. Simulation for Fish ProblemVII. Simulation for Fish Problem
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Fish ModelFish Model
2-dimensional fish motion model (in a tank )
Observation: To estimate:
In our simulation:
Panel size : pixel, fish size : pixel,
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SIMULATIONSIMULATION
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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
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SIMULATIONSIMULATION
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Pollution TrackingPollution Tracking