2. 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

3. I. Formulation of Filtering Problem 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

4. 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.

5. 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.

6. 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

7. Reference probability measure: Under : and are independent, is a standard Brownian motion. Kallianpur-Striebel formula (Bayes formula):

8. 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?

9. Define is a martingale under is a martingale under (also ) is a martingale under

10. Let be a refining partition of [0,T] Equi-continuity via uniform integrability is sum of a and a martingale under, i.e

11. 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

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

13. The associated diffusion generator is symmetric on The observation : is defined on

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

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

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

17. 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.

18. 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

19. : 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

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

21. 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

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

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

24. 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

25. 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

26. VII. Simulation for Fish Problem

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

28. SIMULATION

29. 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

30. SIMULATION

31. Pollution Tracking