ON THE L P -CONVERGENCE OF A GIRSANOV THEOREM BASED PARTICLE FILTER. Eric Moulines. Center of Applied Mathematics École Polytechnique, France

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1 O TH L P -COVRGC OF A GIRSAOV THORM BASD PARTICL FILTR Simo Särä Det. of lectrical ngineering and Automation Aalto University Finland ric Moulines Center of Alied Mathematics École Polytechnique France ABSTRACT We analyze the L -convergence of a reviously roosed Girsanov theorem based article filter for discretely observed stochastic differential equation (SD) models. We rove the convergence of the algorithm with the number of articles tending to infinity by requiring a moment condition and a ste-wise initial condition boundedness for the stochastic exonential rocess giving the lielihood ratio of the SDs. The ractical imlications of the condition are illustrated with an Ornstein Uhlenbec model and with a non-linear Bene s model. Index Terms Girsanov theorem article filter convergence stochastic differential equation 1. ITRODUCTIO In this article we analyze the L -convergence of the Girsanov theorem based article filter introduced in 1 3]. The article filter is concerned with the classical roblem 4] of discretely observed stochastic differential equations of the form dx(t) = f(x(t) t) dt + L(t) db(t) y ρ (y x ) where ρ (y x ) is the conditional robability density (here w.r.t. the Lebesgue measure) of the measurement y R d given the state X(t ) = x R n. We assume that the vector of standard Brownian motions B(t) R n and that L(t) is invertible but this can be relaxed 3]. Although there exists a wide range of L -convergence results for article filters (see e.g. 5 13] and references therein) the main difficulty in alying these results to the resent filter is that unlie in many other cases the imortance weights cannot be assumed to be oint-wise bounded. Therefore we base our analysis on the recently roosed more general moment conditions on the weights 14 15]. Furthermore although here we only consider the convergence of the filtering measures at the measurement times the article filter method 1 3] actually roduces samles of the full aths of the osterior rocess. ven though this limits Thans to Academy of Finland for funding. (1) the ossible choices of imortance rocesses to those which are absolutely continuous with resect to the dynamic model rocess it also enables the ossibility estimate the values of functionals of the aths. 2. TH PARTICL FILTR The article filter introduced in 1 3] is based on the classical Girsanov theorem 16] which gives an exression for the lielihood ratio between an SD and its driving Brownian motion. From the theorem it is also ossible to derive an exression for the stochastic exonential rocess Z(t) giving the lielihood ratio between two SDs driven by the same Brownian motion (see 3]) which can be used for imortance samling of the SDs in article filtering. The resulting article filter algorithm is the following. Algorithm 1 (Girsanov theorem based article filter). Given a set of Monte Carlo samles {x (i) 1 : i = 1... } and the new measurement y a single ste of the filter is: 1. Simulate indeendent realizations of the imortance rocess from t = t 1 to t = t : ds (i) (t) = g(s (i) t) dt + L(t) db (i) (t) S (i) (t 1 ) = x (i) 1. (2) 2. Simulate the corresonding log-lielihood ratios dλ (i) (t) = h T (S (i) (t) t) L 1 (t)] T db (i) (t) 1 2 ht (S (i) (t) t) ( L(t) L T (t) ) 1 h(s (i) (t) t) dt where we have defined (3) h(s t) = f(s t) g(s t) (4) from t = t 1 to t = t with Λ (i) (t 1 ) = 0 and set x (i) = S(i) (t ) { } = ex Λ (i) (t ). z (i) (5) ote that the realizations of Brownian motions must be the same as in simulation of the imortance rocesses.

2 3. For each i comute w (i) and normalize them: 4. Resamle { x (i) w (i) w(i) = z(i) = w (i) ρ (y x (i) ) (6) n j=1 w(j) } to obtain {x(i). (7) : i = 1... }. 3. MASUR-THORTICAL ITRPRTATIO Let us denote the set of bounded Borel-measurable functions φ < by B(R n ). If Q is the transition ernel of a Marov rocess we denote Q(x) = φ(y) Q(x dy). We denote the transition ernel from X(t 1 ) = x 1 to X(t ) defined by the SD in (1) as Q which is usually intractable to write down exlicitly. We write η for the conditional (filtering) measure of X(t ) given the observations y 1... y. The Bayesian filter in a test function form can then be written as 8] α = η 1 (Q (φ ρ )) η = α α (1) (8) where α is an unnormalized measure and ρ (y x ) is considered as a function of x. To account for the imortance rocess it is convenient to rewrite the equations into the following equivalent form (cf. 15]) α = η 1 (Π (φ w )) η = α α (1) (9) where Π is the transition ernel of the Marov rocess defined by the imortance rocess SD for the transition from S(t 1 ) = x 1 to S(t ). In the above dislay we have the weight function w (x 1 x ) = ρ (y x ) dq dπ (x 1 x ) (10) where dq /dπ is a Radon iodym derivative. The advantage of this formulation is that the article filter in Algorithm 1 can be seen as a direct Monte Carlo aroximation of quations (9) as follows: 1. The simulation of the imortance rocess in (2) can be seen as drawing samles from the measure η 1 (Π ) where η 1 is the -article aroximation of the filtering distribution from the time ste Combining with (3) and (5) leads to the aroximation 3. In (6) and (7) we form the aroximation η =. (12) (1) 4. Resamling ste forms a new measure η from η. 4. L P COVRGC THORY The main convergence theorem is the following. Theorem 2. Assume that 1. The measurement model density is bounded ρ (y x ) D <. 2. The transition ernels of the SDs are Feller. 3. The imortance weights and the measurement model density satisfy the inequality su x 1 Π ( w )(x 1 ) (13) for some constants < for all = 1... M that is it bounds uniformly for all starting oints x The resamling algorithm satisfies (e.g. 8]): η ] ĉ φ 2 for some constant ĉ indeendent of. (14) Then for some set of constants c for all = 1... M indeendent of for all φ B(R n ) we have η η ] c φ. (15) 2 We start the roof of the above with the following lemma. Lemma 3. Let 2 and {ξ i : i = 1... } be conditionally indeendent random variables given a sigma-algebra G such that ξ i G] <. Then we have ] ( ) 2 ξ i ξ i G] G C ξ i G] 2 (16) where C is indeendent of. Proof. Follows from Theorem 2.12 in 17]. Lemma 4. Assume that we have η 1 η 1 ] c 1 φ 2 for some constant c 1 indeendent of. Then (17) = φ( x (i) ) w (x (i) 1 x(i) ). (11) η ] c φ 2 for some constant c indeendent of. (18)

3 Proof. By using Minowsi s inequality we get η ] 1 = (1) α (1) ] 1 For the first term above we get (1) α ] 1 α (1) + α (1) α ] 1 α (1). (19) (1) ] α (1) φ α (1) α (1) (1) ]. (20) For the second term we get α (1) α ] 1 α (1) = α (1) α ]. (21) We also have α ] 1 ] ] 1 G 1 G 1 + ] G 1 α ] 1 (22) where G 1 denotes the sigma-algebra generated by the articles {x (i) 0: 1 : i = 1... }. For the first term here we get by using Lemma 3 and assumtion 3: G 1 ] ] G 1 ( C ) (i) φ( x ) z(i) ρ (y x (i) ) ] /2 2 G 1 C φ /2. (23) The second term gives by using the induction assumtion ] G 1 α ] η = 1 (Q (φ ρ )) η 1 (Q (φ ρ )) ] (24) c Q (φ ρ ) /2 c D φ /2 where we have used G 1] = η 1 (Q (φ ρ )). The result follows by substituting (23) and (24) into (22) then the result to (20) and (21) (to former with φ = 1) and by finally using (19). Lemma 5. Assume that we have η ] c φ 2 for some constant c indeendent of. Then η η ] c φ 2 for some constant c indeendent of. (25) (26) Proof. The result follows from η η ] 1 η η ] 1 + η η ] 1 together with the assumtion (14). Proof of Theorem 2. The result follows by combining Lemmas 4 and 5 together with a simle induction argument similarly to 15]. 5. SURIG TH ASSUMPTIO 3 Let us now discuss what the condition that Π ( w )(x 1 ) uniformly for all starting oints x 1 actually means and how it can be checed in ractice. If the Lebesgue densities of Π and Q exist and are π and q resectively then the condition is equivalent to the following being true regardless of x 1 : ] ρ (y x ) q (x x 1 ) π (x x 1) dx. π (x x 1 ) (27) This will certainly be true if we can ensure that the unnormalized weights in the bracets above are uniformly bounded in both variables x and x 1. However we cannot generally ensure that. One way to roceed is to exlicitly chec that the condition above is true for the transition densities of the dynamic model and imortance rocess SDs. However for non-linear SDs the comutation of the densities is usually intractable (they are solutions of the Foer Planc Kolmogorov artial differential equation). Still sometimes analytical or numerical analysis is ossible. We have Π ( w ) D Π (dq /dπ ] ) and thus we can also attemt to ensure that Π (dq /dπ ] ) regardless of the starting oint x 1. It is worth noting that this gives a sufficient condition for the convergence but Π ( w ) might be true even when Π (dq /dπ ] ) Ẽ is not due to aearance of the otentially regularizing function ρ. xlicitly written the latter condition is (recall (4)) t ( x 1 ex h T (S(t) t) L 1 (t)] T db(t) t 1 t h T (S(t) t) ( L(t) L T (t) ) ] 1 h(s(t) t) dt 2 t 1 Ẽ Ẽ (28)

4 which is related to so called oviov s conditions for martingales (with = 1) and the moments of the lielihood ratio considered in 18]. These conditions essentially say that rovided that t x 1 ex (c h T (S(t) t) ( L(t) L T (t) ) 1 t 1 )] (29) h(s(t) t) dt < for a suitably chosen constant c then the moment is bounded. However these conditions do not say anything about the boundedness in the initial conditions (i.e. x 1 ). We can also ut bac the measurement model into the condition (28) which leads to the condition ( t x 1 ex h T (S(t) t)l 1 (t)] T db(t) t 1 t h T (S(t) t) ( L(t) L T (t) ) ) 1 h(s(t) t) dt (30) 2 t 1 ] ρ (y S(t )) Ẽ. 6. XAMPL: ORSTI UHLBCK MODL In this section we illustrate the condition (27) discussed in the revious section by exlicitly analyzing its imlications on the following Ornstein Uhlenbec model: dx(t) = a X(t) dt + q 1/2 db(t) ρ(y x ) = 1 ex ( (y X(t )) 2 ) 2πR 2 R with an imortance distribution of the form (31) ds(t) = b S(t) dt + q 1/2 db(t). (32) In the above dislays a b q and R are ositive constants. We now obtain that the condition Π ( w ) < is satisfied if by selecting the ranges of the arameters suitably. Figure 1 shows the ranges of a and b when these conditions are met with R = 1 and R = 1/10 when the other arameters are q = 1 t = 1 and = XAMPL: O-LIAR B S MODL We now consider the non-linear model dx(t) = tanh(x(t)) dt + db(t) ( (y θ(x(t ))) 2 ρ(y X(t )) = 1 2πR ex 2 R ) (33) where θ( ) is a non-linear function with an imortance distribution of the form ds(t) = b dt + db(t). (34) Fig. 1: Ranges of Ornstein Uhlenbec model arameters where the condition (27) is met (the gray area) with R = 1 (left) and R = 1/10 (right). In both the figures the lower right forbidden art is the result of the initial condition deendence and the uer left art deends on both and the variance R of the measurement noise. The above ind of imortance distribution tyically arises when we use an extended Kalman filter (KF) unscented Kalman filter (UKF) or a similar method to form the imortance distribution 3]. By using the closed-form transition density for the SD in (33) 1 19] it is easy to show that the ratio between the SD transition densities is bounded both in x and x 1 and thus the article filter converges regardless of the value of b. It is also easy to show that the oviov conditions are also satisfied due to boundedness of the drifts in both of the SDs. 8. COCLUSIO AD DISCUSSIO In this article we have roved that the Girsanov theorem based article filter roosed in 1 3] converges in L sense rovided that a moment condition is satisfied by the lielihood ratio rocess and if it is bounded with resect to the ste-wise initial condition. It is worth noting that the results also imly the almost sure convergence of the emirical filtering measure due to a Borel Cantelli argument (see e.g. 15]). Although we have required that the moments are bounded for any x 1 in fact they only need to be bounded given G 1 which might oen u chance to relax the initial condition boundedness requirement. In this article we have also comletely ignored the discretization error caused by numerical integration of the SDs which certainly affects convergence. However more detailed analysis of the effect of this error is left as a future wor. 9. RFRCS 1] S. Särä Recursive Bayesian Inference on Stochastic Differential quations Doctoral dissertation Helsini University of Technology Deartment of lectrical and Communications ngineering 2006.

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