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Path continuity of the nonlinear filter
Statistics & Probability Letters 54 (2001) 75 – 78
Path continuity of the nonlinear lter Abhay G. Bhatt ∗ , Rajeeva L. Karandikar Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi 110016, India Received July 2000; received in revised form March 2001
Abstract We consider the nonlinear ltering model with signal and observation noise independent, and show that in case the signal is continuous in probability, the lter admits a version whose paths are continuous. The analysis is based on c 2001 Elsevier Science expressing the nonlinear lter as a Wiener functional via the Kallianpur–Striebel Bayes formula. B.V. All rights reserved MSC: primary 60G35; 62M20; secondary 60G17; 60G44 Keywords: Nonlinear ltering; Path continuity
1. Introduction Consider the nonlinear ltering model t Yt = h(Xs ) ds + Wt ; 06t6T; 0
(1.1)
where X is the signal process, assumed to take values in a complete separable metric space E and having r.c.l.l. paths, the observation noise W is assumed to be an Rk valued Brownian motion independent of the signal X , h is a measurable function and Y is the observation process. The optimal lter t is given by t ; f = E[f(Xt )|FtY ];
∀f ∈ Cb (E):
(1.2)
Here Cb (E) is the class of bounded continuous functions on E, the processes X and W are de ned on a probability space (; F; P) and FtY = {Ys : 06s6t} is the observation - eld. The function h is assumed to satisfy T |h(Xs )|2 ds¡∞ a:s: [P]: 0
∗
Corresponding author. Tel.: +91-11-651-6200; fax: +91-11-685-6779. E-mail address: [email protected] (A.G. Bhatt).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 0 6 8 - 2
(1.3)
76
A.G. Bhatt, R.L. Karandikar / Statistics & Probability Letters 54 (2001) 75 – 78
It is well known that t admits an r.c.l.l. version (Yor, 1977). When X is a Markov process then ( t ) satis es the FKK equation and it follows that, for f in the domain of the generator of the Markov process, f; t admits a continuous version. (Here, for a measure on E; f; stands for f d.) Thus, when the domain of the generator of X is rich enough (for example when X is a nite dimensional diIusion), it can be shown that ( t ) admits a continuous version. In this article, we show that this result holds in general. We do not assume that the signal is Markov. Assuming only that X is continuous in probability we show that lter ( t ) admits a continuous version. The only condition imposed on h is (1.3). This is achieved via the Kallianpur–Striebel Bayes formula which gives an explicit representation of the lter. Thus, we do not use the measure valued equations of ltering, namely the FKK and Zakai equations. Similar arguments have been used in Bhatt et al. (1999) to obtain robustness of the lter. Let P(E) denote the space of probability measures on E and let M+ (E) denote the space of positive nite measures on E. These spaces will be equipped with the respective Skorokhod topologies of weak convergence; viz., n converges weakly to if and only if f; n → f; for every f ∈ Cb (E). Let 0 = C([0; T ]; Rk ), F0 be the Borel - eld on 0 and Q be the Wiener measure on (0 ; F0 ). Let Y˜ ˆ F; ˆ P) ˆ where the be the co-ordinate process on 0 . Let X˜ be a process de ned on some probability space (; ˜ F; ˆ F; ˜ P) ˆ P) ˜ = (; ˆ ⊗ (0 ; F0 ; Q). Let law of X˜ is the same as the law of X . Let (; k k t t 1 i 0 i ˜ 0 i 2 ˆ ! ) = exp h (X s (!)) ˆ d Y˜ s (! ) − (h (X˜ s (!))) ˆ ds : (1.4) qt (!; 2 0 0 i=1
i=1
We choose and x a version of qt (!; ˆ !0 ) that is continuous in t for all (!; ˆ !0 ). De ne F by ˆ !); ˆ t (!; ˆ !0 ) d P( ˆ ∀f ∈ Cb (E): Ft (!0 ); f = f(X˜ t (!))q
(1.5)
Also, let H be de ned by Ht (!0 ); f =
Ft (!0 ) : Ft (!0 ); 1
(1.6)
Then, E[f(Xt )|FtY ] =
Ft (Y ); f = Ht (Y ) a:s: P: Ft (Y ); 1
(1.7)
This is the Kallianpur–Striebel formula. See (Kallianpur and Karandikar, 1988, appendix). In view of (1.7), we de ne the conditional distribution t of Xt , given FtY , under P by t ; f(!) = Ht (Y (!)); f:
(1.8)
We will show that de ned by (1.8) is continuous a.s. 2. Continuity of the lter Lemma 2.1. Let 0 ˆ !); qt (!; ˆ !0 ) d P( ˆ t (! ) =
06t6T:
Then t is continuous a.s. Q and further 0 0 Q ! : inf t (! ) ¿ 0 = 1: 06t6T
(2.1)
A.G. Bhatt, R.L. Karandikar / Statistics & Probability Letters 54 (2001) 75 – 78
77
˜ F; ˜ P) ˜ where Proof. Note that (qt ; Gt ) is a martingale on (; Gt = {(X˜ s ; Y˜ s ): 06s6t}: This follows from the independence of X˜ and Y˜ . See (Kallianpur, 1980). As a consequence, (t ; Ft ) is a martingale on (0 ; F0 ; Q) where Ft = {Y˜ s : 06s6t}. Since Y˜ is a Brownian motion, this implies that t admits a continuous modi cation ( t ). Let N = {!0 ∈ 0 : r (!0 ) = r (!0 ) for some rational r}: Then Q(N ) = 0. Let Ht = {Y˜ s : 06s6t}: Note that ˆ !); ˆ !0 )) d P( ˆ 06t6T nt = (n ∧ qt (!; is a continuous process and hence (Ht )-predictable. Further, t is the pointwise limit of nt as n tends to ∞ and hence is also (Ht )-predictable. Fix a (Ht )-stopping time . Let n (!0 ) = 2−n [2n (!0 ) + 1]. (here, [x] denotes the integer part of x). Note that n (!0 ) is rational and hence for !0 ∈ N ,
n (!0 )
(!0 ) =
n (!0 )
(!0 ):
(2.2)
Fix !0 ∈ N . Using (2.2) and Fatou’s lemma we conclude that
(!0 ) (!
0
(!0 )
) = lim
n (!0 )
= lim
n (!0 )
n n
= lim inf ¿
(!0 ) q
n
lim inf n q
= =
q
(!0 ) (!
(!0 ) (!
0
0
n (!0 )
ˆ !) (!0 ; !) ˆ d P( ˆ
n (!0 )
ˆ !) (!0 ; !) ˆ d P( ˆ
ˆ !) ; !) ˆ d P( ˆ
):
Thus, ¿ a.s. Q. By Fubini and the de nition of it follows that EQ [ ] = 1. Also, ( t ) is a mean one-continuous martingale and hence EQ [ ] = 1: These observations give us Q( = ) = 1
for all stopping times :
(2.3)
Since and are predictable processes, (2.3) implies that Q(t = t for all t) = 1
(2.4)
see (MQetivier, 1982, Theorem 14:2). The last part follows from the fact that t is a continuous martingale and Q(T ¿ 0) = 1. We are now ready to prove our main result. Theorem 2.2. Assume that the signal X is continuous in probability. Then a.s. the paths of (Ft ) are continuous.
78
A.G. Bhatt, R.L. Karandikar / Statistics & Probability Letters 54 (2001) 75 – 78
Proof. Let N1 = {!0 : t (!0 ) is not continuous in t}. By the previous lemma, Q(N1 ) = 0. Fix tn → t. Then for f bounded continuous on E, f(Xtn ) → f(Xt ) in Pˆ probability: Also, for !0 ∈ N1 , qtn (!0 ; !) ˆ → qt (!0 ; !) ˆ and
ˆ !) qtn (!0 ; !) ˆ d P( ˆ →
∀!ˆ
ˆ !): qt (!0 ; !) ˆ d P( ˆ
ˆ Thus, for !0 ∈ N1 ; {qtn (!0 ; ·): n¿1} is P-uniformly integrable. Since f is bounded, it now follows that ˆ {f(Xtn )qtn (!0 ; ·): n¿1} is P–uniformly integrable: Thus, Ftn (!0 ); f → Ft (!0 ); f for all !0 ∈ N1 : Since this holds for all bounded continuous f, the result follows. The above theorem now gives the path continuity of the lter mentioned in Section 1. Theorem 2.3. Consider the 7ltering model (1:1). Assume that the function h satis7es (1:3). Further; assume that the signal process is continuous in probability. Then the paths of the optimal 7lter t de7ned by (1:8) are continuous a.s. P. Proof. Note that t (!0 ) = Ft (!0 ); 1 and hence t (!) =
Ft (Y (!)) : t (Y (!))
Since the law P ◦ Y −1 of Y is mutually absolutely continuous with respect to the Wiener measure Q, the Q-a.s. path continuity of Ft , t (proved above) implies that Ft (Y (!)) and t (Y (!)) are both P-a.s. continuous. Now (2.1) implies the required result. References Bhatt, A.G., Kallianpur, G., Karandikar, R.L., 1999. Robustness of the optimal lter. Stoch. Processes Appl. 81, 247–254. Kallianpur, G., 1980. Stochastic Filtering Theory. Springer, Berlin. Kallianpur, G., Karandikar, R.L., 1988. White Noise Theory of Prediction, Filtering and Smoothing. Gordon and Breach, New York. MQetivier, M., 1982. Semimartingales. Walter de Gruyter, Berlin. Yor, M., 1977. Sur les thQeories du ltrage et de la prQediction. SQeminaire de ProbabilitQes XI LNM, Vol. 581. Springer, Berlin, pp. 257–297.
Path continuity of the nonlinear lter Abhay G. Bhatt ∗ , Rajeeva L. Karandikar Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi 110016, India Received July 2000; received in revised form March 2001
Abstract We consider the nonlinear ltering model with signal and observation noise independent, and show that in case the signal is continuous in probability, the lter admits a version whose paths are continuous. The analysis is based on c 2001 Elsevier Science expressing the nonlinear lter as a Wiener functional via the Kallianpur–Striebel Bayes formula. B.V. All rights reserved MSC: primary 60G35; 62M20; secondary 60G17; 60G44 Keywords: Nonlinear ltering; Path continuity
1. Introduction Consider the nonlinear ltering model t Yt = h(Xs ) ds + Wt ; 06t6T; 0
(1.1)
where X is the signal process, assumed to take values in a complete separable metric space E and having r.c.l.l. paths, the observation noise W is assumed to be an Rk valued Brownian motion independent of the signal X , h is a measurable function and Y is the observation process. The optimal lter t is given by t ; f = E[f(Xt )|FtY ];
∀f ∈ Cb (E):
(1.2)
Here Cb (E) is the class of bounded continuous functions on E, the processes X and W are de ned on a probability space (; F; P) and FtY = {Ys : 06s6t} is the observation - eld. The function h is assumed to satisfy T |h(Xs )|2 ds¡∞ a:s: [P]: 0
∗
Corresponding author. Tel.: +91-11-651-6200; fax: +91-11-685-6779. E-mail address: [email protected] (A.G. Bhatt).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 0 6 8 - 2
(1.3)
76
A.G. Bhatt, R.L. Karandikar / Statistics & Probability Letters 54 (2001) 75 – 78
It is well known that t admits an r.c.l.l. version (Yor, 1977). When X is a Markov process then ( t ) satis es the FKK equation and it follows that, for f in the domain of the generator of the Markov process, f; t admits a continuous version. (Here, for a measure on E; f; stands for f d.) Thus, when the domain of the generator of X is rich enough (for example when X is a nite dimensional diIusion), it can be shown that ( t ) admits a continuous version. In this article, we show that this result holds in general. We do not assume that the signal is Markov. Assuming only that X is continuous in probability we show that lter ( t ) admits a continuous version. The only condition imposed on h is (1.3). This is achieved via the Kallianpur–Striebel Bayes formula which gives an explicit representation of the lter. Thus, we do not use the measure valued equations of ltering, namely the FKK and Zakai equations. Similar arguments have been used in Bhatt et al. (1999) to obtain robustness of the lter. Let P(E) denote the space of probability measures on E and let M+ (E) denote the space of positive nite measures on E. These spaces will be equipped with the respective Skorokhod topologies of weak convergence; viz., n converges weakly to if and only if f; n → f; for every f ∈ Cb (E). Let 0 = C([0; T ]; Rk ), F0 be the Borel - eld on 0 and Q be the Wiener measure on (0 ; F0 ). Let Y˜ ˆ F; ˆ P) ˆ where the be the co-ordinate process on 0 . Let X˜ be a process de ned on some probability space (; ˜ F; ˆ F; ˜ P) ˆ P) ˜ = (; ˆ ⊗ (0 ; F0 ; Q). Let law of X˜ is the same as the law of X . Let (; k k t t 1 i 0 i ˜ 0 i 2 ˆ ! ) = exp h (X s (!)) ˆ d Y˜ s (! ) − (h (X˜ s (!))) ˆ ds : (1.4) qt (!; 2 0 0 i=1
i=1
We choose and x a version of qt (!; ˆ !0 ) that is continuous in t for all (!; ˆ !0 ). De ne F by ˆ !); ˆ t (!; ˆ !0 ) d P( ˆ ∀f ∈ Cb (E): Ft (!0 ); f = f(X˜ t (!))q
(1.5)
Also, let H be de ned by Ht (!0 ); f =
Ft (!0 ) : Ft (!0 ); 1
(1.6)
Then, E[f(Xt )|FtY ] =
Ft (Y ); f = Ht (Y ) a:s: P: Ft (Y ); 1
(1.7)
This is the Kallianpur–Striebel formula. See (Kallianpur and Karandikar, 1988, appendix). In view of (1.7), we de ne the conditional distribution t of Xt , given FtY , under P by t ; f(!) = Ht (Y (!)); f:
(1.8)
We will show that de ned by (1.8) is continuous a.s. 2. Continuity of the lter Lemma 2.1. Let 0 ˆ !); qt (!; ˆ !0 ) d P( ˆ t (! ) =
06t6T:
Then t is continuous a.s. Q and further 0 0 Q ! : inf t (! ) ¿ 0 = 1: 06t6T
(2.1)
A.G. Bhatt, R.L. Karandikar / Statistics & Probability Letters 54 (2001) 75 – 78
77
˜ F; ˜ P) ˜ where Proof. Note that (qt ; Gt ) is a martingale on (; Gt = {(X˜ s ; Y˜ s ): 06s6t}: This follows from the independence of X˜ and Y˜ . See (Kallianpur, 1980). As a consequence, (t ; Ft ) is a martingale on (0 ; F0 ; Q) where Ft = {Y˜ s : 06s6t}. Since Y˜ is a Brownian motion, this implies that t admits a continuous modi cation ( t ). Let N = {!0 ∈ 0 : r (!0 ) = r (!0 ) for some rational r}: Then Q(N ) = 0. Let Ht = {Y˜ s : 06s6t}: Note that ˆ !); ˆ !0 )) d P( ˆ 06t6T nt = (n ∧ qt (!; is a continuous process and hence (Ht )-predictable. Further, t is the pointwise limit of nt as n tends to ∞ and hence is also (Ht )-predictable. Fix a (Ht )-stopping time . Let n (!0 ) = 2−n [2n (!0 ) + 1]. (here, [x] denotes the integer part of x). Note that n (!0 ) is rational and hence for !0 ∈ N ,
n (!0 )
(!0 ) =
n (!0 )
(!0 ):
(2.2)
Fix !0 ∈ N . Using (2.2) and Fatou’s lemma we conclude that
(!0 ) (!
0
(!0 )
) = lim
n (!0 )
= lim
n (!0 )
n n
= lim inf ¿
(!0 ) q
n
lim inf n q
= =
q
(!0 ) (!
(!0 ) (!
0
0
n (!0 )
ˆ !) (!0 ; !) ˆ d P( ˆ
n (!0 )
ˆ !) (!0 ; !) ˆ d P( ˆ
ˆ !) ; !) ˆ d P( ˆ
):
Thus, ¿ a.s. Q. By Fubini and the de nition of it follows that EQ [ ] = 1. Also, ( t ) is a mean one-continuous martingale and hence EQ [ ] = 1: These observations give us Q( = ) = 1
for all stopping times :
(2.3)
Since and are predictable processes, (2.3) implies that Q(t = t for all t) = 1
(2.4)
see (MQetivier, 1982, Theorem 14:2). The last part follows from the fact that t is a continuous martingale and Q(T ¿ 0) = 1. We are now ready to prove our main result. Theorem 2.2. Assume that the signal X is continuous in probability. Then a.s. the paths of (Ft ) are continuous.
78
A.G. Bhatt, R.L. Karandikar / Statistics & Probability Letters 54 (2001) 75 – 78
Proof. Let N1 = {!0 : t (!0 ) is not continuous in t}. By the previous lemma, Q(N1 ) = 0. Fix tn → t. Then for f bounded continuous on E, f(Xtn ) → f(Xt ) in Pˆ probability: Also, for !0 ∈ N1 , qtn (!0 ; !) ˆ → qt (!0 ; !) ˆ and
ˆ !) qtn (!0 ; !) ˆ d P( ˆ →
∀!ˆ
ˆ !): qt (!0 ; !) ˆ d P( ˆ
ˆ Thus, for !0 ∈ N1 ; {qtn (!0 ; ·): n¿1} is P-uniformly integrable. Since f is bounded, it now follows that ˆ {f(Xtn )qtn (!0 ; ·): n¿1} is P–uniformly integrable: Thus, Ftn (!0 ); f → Ft (!0 ); f for all !0 ∈ N1 : Since this holds for all bounded continuous f, the result follows. The above theorem now gives the path continuity of the lter mentioned in Section 1. Theorem 2.3. Consider the 7ltering model (1:1). Assume that the function h satis7es (1:3). Further; assume that the signal process is continuous in probability. Then the paths of the optimal 7lter t de7ned by (1:8) are continuous a.s. P. Proof. Note that t (!0 ) = Ft (!0 ); 1 and hence t (!) =
Ft (Y (!)) : t (Y (!))
Since the law P ◦ Y −1 of Y is mutually absolutely continuous with respect to the Wiener measure Q, the Q-a.s. path continuity of Ft , t (proved above) implies that Ft (Y (!)) and t (Y (!)) are both P-a.s. continuous. Now (2.1) implies the required result. References Bhatt, A.G., Kallianpur, G., Karandikar, R.L., 1999. Robustness of the optimal lter. Stoch. Processes Appl. 81, 247–254. Kallianpur, G., 1980. Stochastic Filtering Theory. Springer, Berlin. Kallianpur, G., Karandikar, R.L., 1988. White Noise Theory of Prediction, Filtering and Smoothing. Gordon and Breach, New York. MQetivier, M., 1982. Semimartingales. Walter de Gruyter, Berlin. Yor, M., 1977. Sur les thQeories du ltrage et de la prQediction. SQeminaire de ProbabilitQes XI LNM, Vol. 581. Springer, Berlin, pp. 257–297.