1. Sérgio Pequito Phd Student
From Particle Filters to Malliavin Filtering with Application to Target Tracking
Sérgio Pequito
Phd Student
Final project of Detection, Estimation and Filtering course
July 7th, 2010

2. Guidelines Motivation Monte Carlo Methods Particle filters
Importance Sampling
Control Variates
Particle filters
Malliavin Estimator
Example

3. Motivation - Barely relevant sampling

4. Monte Carlo Methods

5. Importance sampling
Importance sampling involves a change of probability measure.
Instead of taking X from a distribution with pdf , we instead
take it from a different distribution with pdf
where is the Radon-Nikodym derivative.

6. Importance sampling We want the new variance to be smaller than
the old variance
How to achieve this? By making small where is large,
and making large when is small.
Small large relative to so more random samples in
region where is large
Particularly important for rare event simulation where is
zero almost everywhere

7. Control Variates Suppose we want to approximate using a simple
Monte Carlo average
If there is another payoff for which we know ,
can use to reduce error in
How? By defining a new estimator
Unbiased since

8. Control Variates For a single sample For a average of N samples
To minimize this, the optimum value for is

9. Control Variates The resulting variance is
Where is the correlation between and .
The challenge is the choose a good which is well
correlated with the covariance, and hence the
optimal , can be estimated from the data.

10. Remarks Importance sampling – very useful for applications with rare
events, but again needs to be fine-tuned for each application
Control variates – easy to implement and can be very effective
But requires careful choice of control variate in each case
Overall, a tradeoff between simplicity and generality on one
hand, and efficiency and programming effort on the other.

11. Particle filters

12. Bayes’ Theorem

13. Nonlinear Filtering State evolution Measurement equation
discrete time stochastic model (Markovian)
Measurement equation
Model prediction given past measurements

14. Nonlinear Filtering Prediction update using current measurement
posterior
likelihood
dynamic prior

15. Particle Filtering represent state as a pdf
sample the state pdf as a set of particles and associated weights
propagate particle values according to model
update weights based on measurement
P(x) x tk
tk+1 t
actual state value
measured state value
state particle value
state pdf (belief)
actual state trajectory
estimated state trajectory
particle propagation
particle weight x

16. Particle Filtering Prediction step: use the state update model
Update step: with measurement, update the prior using Bayes’ rule:

17. Particle Filtering
A particle filter iteratively approximates the posterior pdf as a set:
where:
xki is a point in the state space
wki is an importance weight associated with the point

18. Particle Filtering Implementation steps
propose x0i and propagate particles
compute likelihood of measurement w.r.t. each particle
update particle weights based on likelihood
normalize weights

19. Resampling Particle weights degenerate over time
measure of degeneracy: effective sample size
resample whenever
new set of particles have same statistical properties
use normalized weights

20. PF Flowchart Initialize PF Parameters
Propose Initial Population , <x0,w0>
Propagate Particles using State Model , xk-1xk
Measurement zk
Update Weights, wk-1wk
Weights degenerated? No
Yes
Resample

21. Malliavin Estimator

22. Malliavin Estimator
Particle Filter
“Grid based method”

23. Malliavin Estimator

24. Malliavin Estimator

25. Malliavin Estimator
If Hörmander hypothesis is satisfied then density exists and is smooth
Use Malliavin calculus to develop an expression for H(X,1) that can be
Simulated. So we get a Monte Carlo for (with variance reduction)
Where are independent Euler approximations of
For us then (pdf of X)

26. Malliavin Estimator
The same simulated paths give good estimates for densities at any point.
That is, one can compute the density over the whole real line with the
same number of paths.
Let a localization function and a parameter and
a “control variate”.
The density function
where

27. Malliavin Estimator
where
and
The variance of is minimized for

28. Malliavin Estimator
For the numerical implementation one have to discretize the following
By “Euler” scheme
Moreover

29. Malliavin Estimator

30. Example

31. Malliavin Estimator
are W.G.N.

32. N=1000 particles

33. N=20 particles
Brownian Paths: 50
1000 runs!!!

34. Conclusion and Further Research
Different SMC method with variance reduction (no need to calibration)
Possibility to extend to nonlinear case and extend to higher dimensions

35. References
B. Bouchard, I. Ekeland and N. Touzi (2002). On the Malliavin approach to Monte Carlo approximation
of conditional expectations.
B. Bouchard and N. Touzi (2002), Discrete time approximation and Monte-Carlo simulation of Backward
Stochastic Differential Equations
E. Fournié, J.M.Larsy, J. Lebouchoux, P.L. Lions, N. Touzi (1999). Applications of Malliavin calculus to
Monte Carlo methods in Finance I
E. Fournié, J.M.Larsy, J. Lebouchoux, P.L. Lions (2001). Applications of Malliavin calculus to Monte
Carlo methods in Finance II
D. Nualart (1995). The Malliavin Calculus and Related Topics.
Doucet, A.; De Freitas, N.; Gordon, N.J. (2001). Sequential Monte Carlo Methods in Practice
Doucet, A.; Johansen, A.M.; (2008). A tutorial on particle filtering and smoothing: fifteen years later
Arulampalam, M.S.; Maskell, S.; Gordon, N.; Clapp, T.; (2002). A tutorial on particle filters for online
nonlinear/non-Gaussian Bayesian tracking