From Par(cle Filters to Malliavin Filtering with Applica(on to Target Tracking Sérgio Pequito Phd Student

Transcription

1 From Par(cle Filters to Malliavin Filtering with Applica(on to Target Tracking Sérgio Pequito Phd Student Final project of Detection, Estimation and Filtering course July 7th, 2010

2 Guidelines Mo(va(on Monte Carlo Methods Importance Sampling Control Variates Par(cle filters Malliavin Es(mator Example

3 Mo(va(on 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 the old variance to be smaller than 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 zero almost everywhere is

7 Control Variates Suppose we want to approximate Monte Carlo average using a simple 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 Par(cle filters

12 Bayes Theorem

13 Nonlinear Filtering State evolu(on discrete 4me stochas4c model (Markovian) Measurement equa(on Model predic(on given past measurements

14 Nonlinear Filtering Predic(on update using current measurement posterior likelihood dynamic prior

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

16 Par(cle Filtering Predic(on step: use the state update model Update step: with measurement, update the prior using Bayes rule:

17 Par(cle Filtering A par(cle filter itera(vely approximates the posterior pdf as a set: where: x k i is a point in the state space w ki is an importance weight associated with the point

18 Par(cle Filtering Implementa(on steps propose x 0 i and propagate par4cles compute likelihood of measurement w.r.t. each par4cle update par4cle weights based on likelihood normalize weights

19 Resampling Par(cle weights degenerate over (me measure of degeneracy: effec4ve sample size use normalized weights resample whenever new set of par4cles have same sta4s4cal proper4es

20 PF Flowchart Initialize PF Parameters Propose Initial Population, <x 0,w 0 > Propagate Particles using State Model, x k-1 x k Measuremen t z k Update Weights, w k-1 w k Weights degenerated? No Yes Resample

21 Malliavin Es(mator

22 Malliavin Es(mator Particle Filter Grid based method

23 Malliavin Es(mator

24 Malliavin Es(mator

25 Malliavin Es(mator 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 Es(mator 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 Es(mator where and The variance of is minimized for

28 Malliavin Es(mator For the numerical implementation one have to discretize the following By Euler scheme Moreover

29 Malliavin Es(mator

30 Example

31 Malliavin Es(mator are W.G.N.

32 N=1000 particles

33 N=20 particles Brownian Paths: runs!!!

34 Conclusion and Further Research Different SMC method with variance reduc(on (no need to calibra(on) 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 35