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Outline. 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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  1. Outline • 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

  2. I. Formulation of FilteringProblem • 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

  3. 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 .

  4. 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 .

  5. II. 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

  6. Reference probability measure: • Under :and areindependent, is a standard Brownianmotion. • Kallianpur-Striebel formula (Bayes formula):

  7. 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?

  8. Define • is a martingale under is a martingale under (also ) • is a martingale under

  9. Letbe a refining partition of [0,T] • Equi-continuity via uniform integrability is sum of a and a martingale under , i.e

  10. 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

  11. III.Filtering Model for Reflecting Diffusions • Signal: reflecting diffusions in rectangular region D

  12. The associated diffusion generator • is symmetric on • The observation : • is defined on

  13. IV. Wong-Zakai Approximation to DMZ Equation • has a density which solves where

  14. Let (unitary transformation) then satisfies the following SPDE: where defined on

  15. Kushner-Huang’s wide-band observation noise approximation where is stationary , bounded, and -mixing, converges to in distribution

  16. 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.

  17. V. 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

  18. : 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

  19. Step 4: Particle balance equation where are independent Poisson processes defined on another probability space

  20. Construction of Markov Chains (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

  21. Then satisfies • Compare with our previous equation for • To get mild formulation for and via semigroups

  22. Define a product probability space (for annealed result) • From we can construct a unique probability measure defined on for each

  23. VI. 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

  24. Proof 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

  25. VII. Simulation for Fish Problem

  26. Fish Model • 2-dimensional fish motion model (in a tank ) • Observation: • To estimate: • In our simulation: • Panel size : pixel, fish size : pixel,

  27. SIMULATION

  28. VIII. 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

  29. SIMULATION

  30. Pollution Tracking

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