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Large deviations for optimal filtering with fractional Brownian motion
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Stochastic Processes and their Applications 123 (2013) 2340–2352 www.elsevier.com/locate/spa
Large deviations for optimal filtering with fractional Brownian motion Vasileios Maroulas a,∗ , Jie Xiong a,b a Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA b Department of Mathematics, FST, University of Macau, PO Box 3001, Macau
Received 17 January 2012; received in revised form 14 November 2012; accepted 21 February 2013 Available online 4 March 2013
Abstract We establish large deviation estimates for the optimal filter where the observation process is corrupted by a fractional Brownian motion. The observation process is transformed to an equivalent model which is driven by a standard Brownian motion. The large deviations in turn are established by proving qualitative properties of perturbations of the equivalent observation process. c 2013 Elsevier B.V. All rights reserved. ⃝ Keywords: Nonlinear filtering; Fractional Brownian motion; Large deviations
1. Introduction Motivated by the applications of the filtering theory in a plethora of branches of sciences including signal process in engineering and portfolio optimization in mathematical finance, nonlinear stochastic filtering has been studied by many authors since the pioneering work of Kushner [21,22], Kallianpur and Striebel [15], Zakai [38] and Fujisaki et al. [12]. We refer the reader to the books of Kallianpur [14], Liptser and Shiryaev [24,25], Bain and Crisan [2], and Xiong [36] for an extensive introduction to this subject. The filtering problem consists of two stochastic processes: the signal process we want to estimate and the observation process we can use to obtain information about the unobserved signal. In general, the signal is modeled by a Markov process, which is described by a certain ∗ Corresponding author. Tel.: +1 8659744302.
E-mail addresses: [email protected], [email protected] (V. Maroulas). c 2013 Elsevier B.V. All rights reserved. 0304-4149/$ - see front matter ⃝ http://dx.doi.org/10.1016/j.spa.2013.02.012
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stochastic equation or by its generator. The observation process is a function of the signal process plus a random noise which is usually modeled by white noise. Since the white noise can only be defined mathematically as a generalized function of the time variable, it is more convenient to use the integrated form of the observation process. The integrated noise is then modeled by the Brownian motion. On the other hand, it is well-recognized that memory exists in the noise for many applied problems. In this case, it is natural to model the integrated noise by a fractional Brownian motion (fBm) with a Hurst parameter H ∈ (0, 1) which is related to the memory strength of the noise (see Definition 3.3 below). Since we focus on a long-memory behavior in our framework, the Hurst parameter will be considered, H ∈ (1/2, 1). The observation model is described by the following equation: t Yt = h(X s )ds + BtH , 0
where h is called the observation function, X t is the signal process and BtH is the H -fBm. The optimal filter, which is defined as the conditional probability distribution of X t given the σ -field FtY generated by Ys , s ≤ t, is studied by some authors. Here, we list only a few of them. Kleptsyna et al. [17] consider the case that the signal process is driven by a fBm while the observation noise is still the usual BM. Kleptsyna et al. [18], Kleptsyna et al. [20], Kleptsyna and Le Breton [19] and Le Breton [23] studied the linear filtering problem with fBm as observation noise. Nonlinear filtering problem with fBm observation noise has been studied by Coutin and Decreusefond [9], Gawarecki and Mandrekar [13], Amirdjanova [1], and Xiong and Zhao [37]. In this paper, we √ study the limiting behavior of the optimal filter when the observation function h is replaced by εh with ε → 0. This setting concerns with applications where the signal to noise ratio is very small. It is clear that the optimal filter, denoted by πtε , converges to the (unconditional) probability distribution of X t . The problem we are interested in is to describe the convergence rate. To this end, we establish a large deviation principle (LDP) for the optimal filter and give a representation of the rate function. When BtH is the classical Brownian motion (H = 1/2), this problem is studied by Xiong [35]. The approach we take in this paper is different from that of [35], and it is based on weak convergence arguments. A very powerful technique in deriving LDP has been developed by Budhiraja et al. in [6], and in Maroulas [26] based on variational representations obtained by Bou´e–Dupuis [4], and Budhiraja–Dupuis [5]. This strategy is then used by Budhiraja et al. [7,8], by Maroulas [27], by Ren and Zhang [33,32], Sritharan and Sundar [34] under various setups. We apply this method to the LD problem for optimal filtering with H -fBm noise. The same methods can be applied to the classical Brownian motion case to obtain a result which is stronger than those of [35]. Finally, we would like to mention that the method in this paper might also cover the case of general Gaussian observation noise. However, the key of this paper (and that of [6]) is the variational formula for Brownian functionals. Hence, to extend the LDP result of this paper to general Gaussian noise case, a variational formula for Gaussian functional need to be established. This task is not trivial and we will look into this in a future project. Our paper is organized as follows. Section 2 reviews basic definitions of large deviations and describes the methodology we follow in order to establish such estimates. Section 3 exposes preliminary results with respect to a fractional Brownian motion and a standard Brownian motion. Section 4 is the main section which demonstrates large deviation estimates for the optimal filtering. Such results depend on several estimates of the corresponding unnormalized filter which are proved in Section 5.
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2. Large deviations This section reviews elements of large deviations, and explains how one can verify large deviation estimates via weak convergence arguments. Let {X ϵ } be an E-valued family of random variables defined on a probability space (Ω , F, P), where E is a Polish space. The theory of large deviations focuses on probabilities of events whose probability converges to zero exponentially fast. The decay rate is expressed via the rate function, denote it herein by I . In our work we will prove a tantamount argument of the large deviations principle, the Laplace principle, and we will study the Uniform Laplace Principle. The reader should refer to [11] for a proof of the aforementioned equivalence. Let E0 and E be Polish spaces. For each ϵ > 0 and y ∈ E0 let X ϵ,y be E valued random variables given on the probability space (Ω , F, P). Definition 2.1. A family of rate functions I y on E, parameterized by y ∈ E0 , is said to have . compact level sets on compacts if for all compact subsets K of E0 and each M < ∞, Λ M,K = ∪ y∈K {x ∈ E : I y (x) ≤ M} is a compact subset of E. We list below the definition of Uniform Laplace principle. Definition 2.2. Let I y be a family of rate functions on E parameterized by y in E0 and assume that this family has compact level sets on compacts. The family {X ϵ,y } is said to satisfy the Laplace principle on E with rate function I y , uniformly on compacts, if for all compact subsets K of E0 and all bounded continuous functions h mapping E into R, 1 ϵ,y lim sup ϵ log E y exp − h(X ) + inf h(x) + I y (x) = 0. ϵ→0 y∈K ϵ x∈E Next, a set of sufficient conditions for a uniform large deviation principle is discussed. . Let (Ω , F, P, {Ft }) is a filtered probability space and denote S = C([0, t], R). Consider the (S, B(S))- valued random variable. Let E be a Polish space, and for each ϵ > 0, consider the family of measurable maps, G ϵ : E0 × S → E, which are defined as follows. ϵ,x . ϵ √ X = G x, ϵβ . (2.1) Let denote the spaces, T . P2 = f : f predictable and
T
f (s)ds < ∞ a.s. 2
0
. S N = φ ∈ L 2 ([0, T ]) :
T
φ 2 (s)ds ≤ N
0
P2N
. = {u ∈ P2T : u(ω) ∈ S N , P − a.s.}.
The following condition [6,26] is the critical one which one needs to verify in order to prove uniform large deviation estimates driven by a Brownian motion. The corresponding rate function is given in (2.2). Condition 2.1. There exists a measurable map G 0 : E0 × S → E such that the following hold.
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1. For M ∈ N let ( f n , f ) ∈ S M be such that (xn , f n ) → (x, f ). Then · · f (s)ds . f n (s)ds → G 0 x, G 0 xn , 0
0
be such that, as ϵ → 0, ψϵ converges in distribution to ψ and 2. For M ∈ N let ψϵ , ψ ∈ {x ϵ } ⊂ E0 , x ϵ → x, as ϵ → 0. Then · · √ G ϵ x ϵ , ϵW (·) + ψϵ (s)ds ⇒ G 0 x, ψ(s)ds . P2M
0
0
· For φ ∈ E, define Sφ = f ∈ S M : φ = G 0 x, 0 f (s)ds . Let I x : E → [0, ∞] be defined by T 1 2 I x (φ) = inf f s ds . f ∈Sφ 2 0
(2.2)
This is the main theorem of [6,26] which verifies that under Condition 2.1, one may establish uniform large deviations with rate function I x defined in (2.2). Theorem 2.1. Let G 0 : E0 × S → E be a measurable map satisfying Condition 2.1. Suppose that for all φ ∈ E, x → I x (φ) is a lower semi-continuous map from E0 to [0, ∞]. Then for every x ∈ E0 , I x : E → [0, ∞], defined by (2.2), is a rate function on E and the family {I x , x ∈ E0 } of rate functions has compact level sets on compacts. Furthermore, the family {X ϵ,x } satisfies the Laplace principle on E with rate function I x , uniformly for x in compact subsets of E0 . 3. Preliminary results We start this section with the definition of the H -fBm. As it was mentioned in the Section 1, due to our interest in a long-memory environment, we restrict ourselves to the case of H ∈ 1 2 , 1 . The reader should refer for example to [28] for definition and techniques for a general fractional Brownian motion with Hurst parameter, H ∈ (0, 1). Definition 3.3. For a given H ∈ (1/2, 1), a stochastic process B H = (BtH , t ∈ [0, T ]) is a fractional Brownian motion with Hurst parameter H if: (i) B0H = 0; (ii) B H is a zero-mean Gaussian process with continuous sample paths and stationary increments; (iii) The covariance function is given by R H (s, t) =
VH (|s|2H + |t|2H − |t − s|2H ) 2
where VH = V ar (B1H ) =
−Γ (2 − 2H ) cos(π H ) . π H (2H − 1)
Consider the signal process, X t , governed by the following SDE: d X t = b(X t )dt + σ (X t )d Wt ,
(3.3)
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where Wt is a finite dimensional Brownian motion, and the coefficients b, σ satisfy regularity properties (Lipschitz, and linear growth) such that (3.3) attains a unique solution [31, Section 5.2]. Consider the observation process, Ytε , described in the dynamics below, t √ ε h(X s )ds + BtH , (3.4) Yt = ε 0
where h is bounded and BtH , H > 1/2 is a fBm independent of Wt . We now follow the way of thinking in [37]. Using Molchan–Golosov representation [29], and [3, Theorem 4.5], one may explicitly express the fBm, BtH , as an integral on the interval [0, t] with respect to a standard Brownian motion in the following equation, t H γ H (s, t)d Bs , (3.5) Bt = 0
t H −1/2 s 1/2−H where γ H (s, t) = Γ (H (q − s) H −3/2 dq. Now the Brownian motion, Bt , in turn −1/2) s q will be expressed with respect to the fBm below. s t H H −1/2 k H (s, q)d Bq , (3.6) s d Bt = 0
0
1 1/2−H q 1/2−H . Define now the processes where k H (s, q) = Γ (3/2−H ) (s − q) s t Z tε = s H −1/2 d k H (s, q)dYqε 0
Stε
√ = ε
t
s
H −1/2
0
s
d
k H (s, q)h(X q )dq .
0
(3.8)
0 ε
t
(3.7)
0
According to (3.7), Z tε is FtY -measurable, and it can be shown that Ytε ε ε γ H (s, t)d Z sε , [30]. Thus, the filtrations FtY = FtZ and the observation model becomes Z tε = Stε + Bt .
=
(3.9)
The assumptions below are considered in the entire manuscript. These assumptions are critical for guaranteeing existence and uniqueness [37, Theorems 4.2 and 4.3] of the Kushner–FKK and Zakai equations as defined in (4.13) and (4.14). Assumption 3.1. The coefficients on the signal process σ : R → R and b : R → R belong to Cb1 (R). Assumption 3.2. The coefficient on the observation process h : R → R is a measurable function which belongs to Cb2 (R). Let take into account the following trajectory space, . T = {(t, x t ) : t ≥ 0, x ∈ C([0, T ], R)}, where x t (s) = x(t ∧ s). According to [37, Theorem 2.2] the small perturbation, Z tε , of the observation model will be equivalent to
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Z tε =
√
ε
t
G(s, X s )ds + Bt ,
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(3.10)
0
where X s is the path of X up to time s, and G is a measurable map from T to R. The reader should note that the coefficient G listed below encapsulates the entire trajectory of X up to some time T such that it reflects the long-memory behavior which was expressed via a fractional Brownian motion in (3.4). Namely, according to [37, Theorem 2.2], G is defined as, h(X 0 )B 23 − H, 32 − H (2 − 2H ) 1 s 2 −H G(s, X s ) = Γ (3/2 − H ) 1 su 1 1 2 − 2H −H −H 2 2 + s (u(1 − u)) Lh(X r )dr du Γ (3/2 − H ) 0 0 1 3 1 1 + s 2 −H (u(1 − u)) 2 −H Lh(X su )udu Γ (3/2 − H ) 0 s 1 1 1 2 − 2H + s 2 −H (u(1 − u)) 2 −H du(σ h ′ )(X r ) r Γ (3/2 − H ) 0 s b 1 (X r )d X r − (X r )dr × σ σ s 3 1 r 12 −H − (σ h ′ )(X r ) s −1 r 2 −H 1 − Γ (3/2 − H ) s 0 1 b × (X r )d X r − (X r )dr , (3.11) σ σ where L denotes the differential operator Lh(x) = 21 σ 2 (x)h ′′ (x) + b(x)h ′ (x). As it is mentioned in [37], the signal process needs to be enriched. Let (t, X t ) be a T-valued Markov process with generator, A0 F(t, X t ) = ∂0 φ(t, X t1 ∧t , . . . , X tn ∧t ) +
n
1t
i=1
+
n 1 1t
(3.12)
where F ∈ D(A0 ), and D(A0 ) is the space such that F(t, y) = φ(t, y(t1 ∧t), . . . , y(tn ∧t)), F ∈ Cb (T) for some n ≥ 1, 0 ≤ t1 < · · · < tn ≤ T, φ ∈ Cb2 (R+ ×Rn ). The reader should note that in general Cbk (T) denotes the space of continuous bounded and smooth functions (up to derivative k ≥ 1). Here we consider bounded derivatives because it will be necessary for establishing large deviations. However, this condition does not contradict with the well-posedness of the optimal filtering framework proved in [37]. 4. Uniform large deviation principle for the optimal filtering This section is the main of our manuscript and it establishes the uniform large deviation estimates for the optimal filtering. We will proceed by first showing large deviations for the unnormalized filter (Theorem 4.3) and then employing the contraction principle one may
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demonstrate the analogous result for the optimal filtering (Theorem 4.4). However, for reasons of presentation, the proof of Theorem 4.3 is given in the next section. ε Let πtε F = E(F(t, X t )|FtY ), ∀F ∈ Cb (T) be the optimal filter. According to [37, Theorem 3.1(ii)] the process πtε satisfies the Kushner–FKK formula, i.e. t t √ (πsε (G F) − πsε Gπsε F)dξs , πtε F = π0 F + πsε (A0 F)ds + ε 0
0
∀F ∈ D(A0 ), (4.13) √ t where ξt = Z tε − ε 0 πs Gds is the innovation process and it is a Brownian motion. Letting ε → 0 we observe that πtε F → E(F(t, X t )). In this paper, we examine the rate of the aforementioned convergence by establishing the large deviation principle for the family of optimal filters {πtε }. It can be shown analogously with [37, Theorem 3.1(iii)] that the process µεt , defined by t √ ε t µεt F = πtε F exp ε πsε Gd Z sε − |πs G|2 ds , 2 0 0 for any F ∈ Cb (T), satisfies the Zakai equation t t √ ε ε µt F = µ0 F + µs (A0 F)ds + ε µεs (G F)d Z sε , 0
∀F ∈ D(A0 ).
(4.14)
0
We will first need to establish the LDP for the family {µεt } and then by applying the contraction principle to show the LDP for the family of optimal filters {πtε }. √ t Consider the innovation process ξt = Z tε − ε 0 πs Gds, and denote the collection of finite ˆ as M(T). For each ε > 0, let G ε : M(T)×C([0, T ], R) → Borel measures on the Polish space R C([0, T ], M(T)) be a measurable map. Next, define √ . µε,µ0 = G ε µ0 , ε Z ε , (4.15) where µ0 ∈ M(T) is the initial condition of the unnormalized filter. Then the following theorem holds [37]. Theorem 4.2. The unnormalized filter µε,µ0 defined in (4.15) is the unique solution of the Zakai equation (4.14) with initial condition µ0 . We now consider, for some λ ∈ L 2 ([0, T ], R), the controlled analogue of (4.14) for all F ∈ D(A0 ). t t t √ ε,λ ε,λ ε µε,λ F = µ F + µ (A F)ds + ε µ (G F)d Z + µε,λ 0 0 t s (Gλs F)ds. (4.16) s s s 0
0
0
Let µλ be the solution of the zero noise controlled version of (4.16), i.e. t µλt F = µ0 F + µλs (A0 F + Gλs F) ds,
(4.17)
0
Let G 0 : M(T) × C([0, · T ], R) → C([0, T ], M(T)) be a measurable function such that = µλ , if u = 0 λs ds, λ ∈ L 2 ([0, T ]), or 0 otherwise.
G 0 (µ0 , u)
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For v ∈ C([0, T ], M(T)) and µ0 ∈ M(T) define T 1 . 2 I1 ≡ I1,µ0 (v) = inf λs ds . {λ∈L 2 ([0,T ]):v=G 0 (µ0 , 0· λs ds)} 2 0
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(4.18)
The following theorem establishes uniform large deviations for the unnormalized filter whose proof is delegated in the next section. Theorem 4.3. Let µε,µ0 be as in (4.14). Then I1,µ0 , defined in (4.18), is a rate function on C([0, T ], M(T)), and the family of rate functions {I1,µ0 , µ0 ∈ M(T)} has compact level sets on compacts. Furthermore, {µε,µ0 } satisfies the large deviation principle on C([0, T ], M(T)) with the rate function I1,µ0 , uniformly for µ0 in compact sets of M(T). Let P(T) be the collection of all Borel probability measures on T. Take G˜ 0 : P(T) × → C([0, T ], P(T)) a measurable function, and suppose that π λ = G˜ 0 (π0 , λ), where λ π is the solution of the following controlled equation. t λ πt F = π0 F + πsλ A0 F + (G − πsλ G)λs F ds. (4.19) L 2 ([0, T ])
0
For v˜ ∈ C([0, T ], P(T)) and π0 ∈ P(T) define . I2 ≡ I2,π0 (v) ˜ =
T 1 2 λs ds . inf · ˜ G 0 (π0 , 0 λs ds )} 2 0 {λ∈L 2 ([0,T ]):v=
(4.20)
The theorem below is the main theorem of this manuscript and establishes a uniform large deviation principle for the optimal filtering defined in (4.13). Theorem 4.4. The family of solutions of the Kushner–FKK equation, {π ε,π0 } satisfies the large deviation principle on C([0, T ], P(T)) with the rate function I2,π0 , defined in (4.20), uniformly for π0 in compact sets of P(T). Proof. The reader should remark that for simplicity in this proof we suppress the initial condition µ0 or π0 in the notation of the corresponding rate functions, I1 , I2 as defined in (4.18) and (4.20), respectively. Now, similarly to [35, Lemma 5.1 and Theorem 5.2], if there exists a t0 ∈ [0, T ] such that µt0 = 0 then I1 (µ) = ∞. It is then easy to prove that µε satisfies the LDP on C([0, T ], M(T) \ {0}) with the rate function being equal to the restriction of I on C([0, T ], M(T) \ {0}). Let define the continuous map J : C([0, T ], M(T) \ {0}) → C([0, T ], P(T)), such that (J µ)t = µµtt F1 . Then, by the contraction principle [10], {π ε = J (µε )} satisfies the LDP with rate function I2′ (π ) = inf{I1 (µ) : J (µ) = π }. If I2′ (π ) < ∞ then according to the definition of I2′ (π ) for all δ > 0 there exists µ such that J (µ) = π and I1 (µ) < I2′ (π ) + δ. Let λ ∈ L 2 ([0, T ], R) such that γ (λ) = µ, where γ is the solution of T . (4.17), and 12 0 λ2s ds < I1 (µ) + δ. Thus by setting γ˜ = J ◦ γ , we have that γ˜ (λ) = π , where T γ˜ is the solution of (4.19), and 12 0 λ2s ds < I2′ (π ) + 2δ. Therefore, I2 (π ) ≤ I2′ (π ).
(4.21)
Now, suppose I2 (π ) < ∞. Then for all δ > 0, there exists λ∈ L 2 ([0, T ]) such that γ˜ (λ) = π , T t and 12 0 λ2 ds < I2 (π ) + δ. Let at = π0 exp 0 πs hλs ds and µt = at πt , where h is the
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coefficient of the observation process (3.4). Then µ = γ (λ) and I2′ (µ) = π and therefore, T I2′ (π) ≤ I1 (µ) ≤ 12 0 λ2 ds < I2 (π ) + δ which together with (4.21) yield the result. 5. Proof of Theorem 4.3 The proof of Theorem 4.3 will follow the framework described in Section 2. In other words, one may demonstrate the Condition 2.1 in order to show uniform large deviation estimates for the unnormalized filter. We first show existence and uniqueness of the controlled version of Zakai’s equation given in (4.16). Theorem 5.5. Let G ε : M(T) × C([0, T ], R) → C([0, T ], M(T)) be a measurable map as defined in (4.15). Suppose λ ∈ P2N for some N ∈ N and for ε > 0 and µ0 ∈ M(T), denote t √ ε ε,λ . ε µµ0 = G µ0 , ε Z + λs ds . 0
Then
µε,λ µ0
is the unique solution of (4.16).
Proof. Let consider the dynamics of the signal process expressed in (3.3). For any F ∈ D(A0 ), we have that t F t Nt ≡ F(t, X ) − A0 F(s, X s )ds 0
is a martingale [37]. √ t Let further consider Mtε,λ = exp ε 0 G(s, X s )d Z sε − ds . Then by Itˆo’s formula we have that d Mtε,λ = Mtε,λ G(t, X t )
ε t 2 0
G 2 (s, X s )ds +
√ εd Z tε + λt dt .
t 0
λs G(s, X s )
(5.22)
Applying the integration by parts technique, e.g. see [31, Theorem 4.1.5], on the Itˆo integrals of the Eqs. (5.22) and taking into consideration that the fractional Brownian motion, B H , and the Brownian motion W are independent, one may conclude that d Mtε,λ F(t, X t ) = Mtε,λ d F(t, X t ) + F(t, X t )d Mtε,λ = Mtε,λ A0 F(t, X t )dt + λt Mtε,λ G(t, X t )F(t, X t )dt √ + εMtε,λ G(t, X t )F(t, X t )d Z tε + Mtε,λ d NtF .
(5.23)
Now, from Eq. (5.23) we have that, Mtε,λ F(t, X t ) = F(0, X 0 ) + √ +
ε
0
t
0
t
Msε,λ A0 F(s, X s )ds +
t
Msε,λ G(s, X s )F(s, X s )d Z sε +
λs Msε,λ G(s, X s )F(s, X s )ds
0
0
t
Msε,λ d NsF .
(5.24)
Taking conditional expectations on both sides in (5.24), and applying Lemma 5.4 of [36] we get, t ε ε E(Mtε,λ F(t, X t )|FtZ ) = µ0 F(t, X t ) + E(Msε,λ A0 F(s, X s )|FsZ )ds 0
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t
+ 0
√ + ε
2349
ε
λs E(Msε,λ G(s, X s )F(s, X s )|FsZ )ds 0
t
ε
E(Msε,λ G(s, X s )F(s, X s )|FsZ )d Z sε ,
(5.25) ε
ε,λ t Z and since from the Kallianpur–Striebel formula, we have that ⟨µε,λ t , F⟩ = E(Mt F(t, X )|Ft ), ε ε,λ Eq. (5.25) results that E(Mt F(t, X t )|FtZ ) is a solution of the controlled Zakai’s equation. Now, same arguments of Theorem 4.2 of [37] yield the uniqueness of the solution.
The next theorem establishes the existence and the uniqueness of the solution of the noise free controlled version of the (4.14). Theorem 5.6. Fix µ0 ∈ M(T) and λ ∈ L 2 ([0, T ]). Then there is a unique solution vtλ ∈ C([0, T ], M(T)) of (4.17). t Proof. Let consider Mt0,λ = exp 0 λs G(s, X s )ds . Then by Feynman–Kac formula vtλ = E(Mt0,λ F(t, X t )) is the unique solution of (4.17).
The proposition below yields the first part of Condition 2.1. Proposition 5.1. For M ∈ N, consider (λn , λ) ∈ S M be such that (µn0 , λn ) → (µ0 , λ). Then for n n any F ∈ D(A0 ), µλt F → µλt F, where µλt F, µλt F are defined in (4.17). t Proof. Consider Mt0,λ = exp 0 λs G(s, X s ) . Following the strategy of the proof of Theorem 5.5 and applying Itˆo’s formula, one can show that, ηt defined by µ0 (d x)Ex Mt0,λ F(t, X t ), ηt F = R
is a solution to (4.17). However, according to Theorem 5.6, the solution is unique and thus µλt = ηt . Therefore, applying dominated convergence theorem one may easily verify the convergence below: t λn n t s n µt F = µ0 (d x)Ex F(t, X ) exp G(s, X )λs ds R 0 t t s → µ0 (d x)Ex F(t, X ) exp G(s, X )λs ds =
R µλt F.
0
We next present the definition of C-tightness [16] which is an instrumental concept in the proof of the next proposition. Definition 5.4. A sequence of probability measures {µn } on D([0, T ], R) is C-tight if it is tight and cluster points are supported on C([0, T ], R). Finally, the proposition below is the core for the establishment of the large deviation principle and it settles the second part of the Condition 2.1.
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Proposition 5.2. Let M < ∞, and suppose that µε0 → µ0 , and λε → λ in distribution, as ε,λε ,µε0
ε → 0 with {λε } ∈ P2M . Then µt unique solution (4.17).
0,λ,µ0
converges in distribution to µt
, where µ0,λ µ0 is the
Proof. Consider λ ∈ P2M and p > 1. Employing Eq. (4.16), there exists an appropriate constant K 1 such that we have 2 p f (t) ≡ E sup µε,λ s 1 s≤t
≤ K 1 (µ0 1)
2p
+ K1E 0
t
|λs |µε,λ s 1ds
2 p
t
+ K1E 0
ε,λ 2 µs 1 ds
p
.
(5.26)
Note that the second term of the upper bound in (5.26) can be further estimated by t p t ε,λ 2 K1 M pE µs 1 ds ≤ K2 f (s)ds. 0
0
Therefore, the estimate of f (t) will be given by t f (s) ≤ K 3 + K 3 f (s)ds. 0
Applying Gronwall’s inequality yields that f (t) is bounded, and hence, 2 p E sup µε,λ ≤ K4, s 1
(5.27)
s≤T
where K 4 is a constant depending on M, T and µ0 1. ε,λε ,µε0 Next, we establish the tightness of µt . It is well-known that we only need to prove the ε,λε ,µε 0 tightness of µt F in C([0, T ], R) for every test function F in Dom(A0 ). By (5.27), we have ε,λε ,µε 2 p 0 1 E sup µt ≤ K4. (5.28) t≤T
For the time steps t1 < t2 , we have t 2 p 2 ε,λε ,µε0 ε E µs (Gλs F)ds ≤ E t1
t2
t1
(λεs )2 ds
p
t2
ε,λε ,µε0
µs
(G F)2 ds
p
t1
≤ M p K 4 |t2 − t1 | p . ε,λε ,µε · 0 By Kolmogorov’s criteria, we see that the family of processes 0 µs (Gλεs F)ds is C-tight. ε,λε ,µε · 0 Similarly, we can prove that the family of processes 0 µs (A0 F)ds is C-tight. Note that ε ε the martingale parts of µε,λ ,µ0 (F) has quadratic variation processes given by t ε,λε ,µε0 Q εt ≡ ε µs (G F)2 ds 0
which is also C-tight. It follows from Theorem 6.1.1 in Kallianpur and Xiong [16] that ε ε µε,λ ,µ0 (F) is tight. It is then standard to show that any limit point will be a solution to (4.17). Furthermore, the uniqueness of the solution to (4.17) implies the desired weak convergence.
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Proof of Theorem 4.3. Condition 2.1 follows from Propositions 5.1 and 5.2 directly. The Laplace principle, which implies the large deviation principle, then follows from Theorem 2.1. Acknowledgments The authors would like to warmly thank two anonymous reviewers for their excellent comments. The first author’s research was supported partially by University of Tennessee and NIMBioS funds. The second author’s research was supported partially by NSF DMS-0906907. References [1] A. Amirdjanova, Nonlinear filtering with fractional Brownian motion, Appl. Math. Optim. 46 (2–3) (2002) 81–88. Special issue dedicated to the memory of Jacques–Louis Lions. [2] A. Bain, D. Crisan, Fundamentals of Stochastic Filtering, in: Stochastic Modelling and Applied Probability, vol. 60, Springer, New York, 2009. [3] R.J. Barton, H.V. Poor, Signal detection in fractional Gaussian noise, IEEE Trans. Inform. Theory 34 (5) (1988) 943–959. [4] M. Bou´e, P. Dupuis, A variational representation for certain functionals of Brownian motion, Ann. Probab. 26 (1998) 1641–1659. [5] A. Budhiraja, P. Dupuis, A variational representation for positive functional of infinite dimensional Brownian motions, Probab. Math. Statist. 20 (2000) 39–61. [6] A. Budhiraja, P. Dupuis, V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab. 36 (4) (2008) 1390–1420. [7] A. Budhiraja, P. Dupuis, V. Maroulas, Large deviations for stochastic flows of diffeomorphisms, Bernoulli 36 (1) (2010) 234–257. [8] A. Budhiraja, P. Dupuis, V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincare 47 (3) (2011) 725–747. [9] L. Coutin, L. Decreusefond, Abstract nonlinear filtering theory in the presence of fractional Brownian motion, Ann. Appl. Probab. 9 (4) (1999) 1058–1090. [10] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, second ed., Springer, New York, 1998. [11] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley-Interscience, New York, 1997. [12] M. Fujisaki, G. Kallianpur, H. Kunita, Stochastic differential equations for the non-linear filtering problem, Osaka J. Math. 9 (1972) 19–40. [13] L. Gawarecki, V. Mandrekar, On the Zakai equation of filtering with Gaussian noise, in: Stochastics in Finite and Infinite Dimensions, in: Trends Math, Birkh¨auser Boston, Boston, MA, 2001, pp. 145–151. [14] G. Kallianpur, Stochastic Filtering Theory, Springer-Verlag, 1980. [15] G. Kallianpur, C. Striebel, Estimation of stochastic systems: arbitrary system process with additive noise observation errors, Ann. Math. Statist. 39 (1968) 785–801. [16] G. Kallianpur, J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, in: IMS Lecture Notes Monograph Series, vol. 26, 1995. [17] M.L. Kleptsyna, P.E. Kloeden, V.V. Anh, Nonlinear filtering with fractional Brownian motion, Stoch. Anal. Appl. 16 (5) (1998) 907–914. [18] M.L. Kleptsyna, P.E. Kloeden, V.V. Anh, Linear filtering with fractional Brownian motion in the signal and observation processes, J. Appl. Math. Stoch. Anal. 12 (1) (1999) 85–90. [19] M.L. Kleptsyna, A. Le Breton, Extension of the Kalman–Bucy filter to elementary linear systems with fractional Brownian noises, Stat. Inference Stoch. Process. 5 (3) (2002) 249–271. [20] M.L. Kleptsyna, A. Le Breton, M.C. Roubaud, General approach to filtering with fractional Brownian noises — application to linear systems, Stoch. Stoch. Rep. 71 (1–2) (2000) 119–140. [21] H.J. Kushner, On the dynamic equations of conditional probability density functions with applications to optimal stochastic control theory, J. Math. Anal. Appl. 8 (1964) 332–344. [22] H.J. Kushner, Dynamic equations for nonlinear filtering, J. Differential Equations 3 (1967) 179–190.
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[23] A. Le Breton, Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion, Statist. Probab. Lett. 38 (1998) 263–274. [24] R.S. Liptser, A.N. Shiryaev, Statistics of random processes, II, applications, in: Applications of Mathematics (New York), vol. 6, second, revised and expanded ed., in: Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2001. Translated from the 1974 Russian original by A. B. Aries. [25] R.S. Liptser, A.N. Shiryaev, Statistics of random processes, I, general theory, in: Applications of Mathematics (New York), vol. 5, second, revised and expanded ed., in: Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2001. Translated from the 1974 Russian original by A. B. Aries. [26] V. Maroulas, Small noise large deviations for infinite dimensional stochastic dynamical systems. Ph.D. Thesis, The University of North Carolina at Chapel Hill, 2008. [27] V. Maroulas, Large deviations for inifinite dimensional stochastic systems with jumps, Mathematika 57 (1) (2011) 175–192. [28] Y. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Math. 1929 (2008). [29] G.M. Molchan, J.I. Golosov, Gaussian stationary processes with asymptotic power spectrum, Sov. Math. Dokl. 10 (1969) 134–137. [30] I.E. Norros, E. Valkeila, J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999) 571–587. [31] B. Øksendal, Stochastic Differential Equations, sixth ed., Springer, 2007. [32] J. Ren, X. Zhang, Freidlin–Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs, Bull. Sci. Math. 129 (8) (2005) 643–655. [33] J. Ren, X. Zhang, Freidlin–Wentzell’s large deviations for stochastic evolution equations, J. Funct. Anal. 254 (12) (2008) 3148–3172. [34] S.S. Sritharan, P. Sundar, Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (11) (2006) 1636–1659. [35] J. Xiong, Large deviation principle for optimal filtering, Appl. Math. Optim. 47 (2003) 151–165. [36] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, 2008. [37] J. Xiong, X. Zhao, Nonlinear filtering with fractional Brownian motion noise, Stoch. Anal. Appl. 23 (2005) 55–67. [38] M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 11 (1969) 230–243.
Stochastic Processes and their Applications 123 (2013) 2340–2352 www.elsevier.com/locate/spa
Large deviations for optimal filtering with fractional Brownian motion Vasileios Maroulas a,∗ , Jie Xiong a,b a Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA b Department of Mathematics, FST, University of Macau, PO Box 3001, Macau
Received 17 January 2012; received in revised form 14 November 2012; accepted 21 February 2013 Available online 4 March 2013
Abstract We establish large deviation estimates for the optimal filter where the observation process is corrupted by a fractional Brownian motion. The observation process is transformed to an equivalent model which is driven by a standard Brownian motion. The large deviations in turn are established by proving qualitative properties of perturbations of the equivalent observation process. c 2013 Elsevier B.V. All rights reserved. ⃝ Keywords: Nonlinear filtering; Fractional Brownian motion; Large deviations
1. Introduction Motivated by the applications of the filtering theory in a plethora of branches of sciences including signal process in engineering and portfolio optimization in mathematical finance, nonlinear stochastic filtering has been studied by many authors since the pioneering work of Kushner [21,22], Kallianpur and Striebel [15], Zakai [38] and Fujisaki et al. [12]. We refer the reader to the books of Kallianpur [14], Liptser and Shiryaev [24,25], Bain and Crisan [2], and Xiong [36] for an extensive introduction to this subject. The filtering problem consists of two stochastic processes: the signal process we want to estimate and the observation process we can use to obtain information about the unobserved signal. In general, the signal is modeled by a Markov process, which is described by a certain ∗ Corresponding author. Tel.: +1 8659744302.
E-mail addresses: [email protected], [email protected] (V. Maroulas). c 2013 Elsevier B.V. All rights reserved. 0304-4149/$ - see front matter ⃝ http://dx.doi.org/10.1016/j.spa.2013.02.012
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stochastic equation or by its generator. The observation process is a function of the signal process plus a random noise which is usually modeled by white noise. Since the white noise can only be defined mathematically as a generalized function of the time variable, it is more convenient to use the integrated form of the observation process. The integrated noise is then modeled by the Brownian motion. On the other hand, it is well-recognized that memory exists in the noise for many applied problems. In this case, it is natural to model the integrated noise by a fractional Brownian motion (fBm) with a Hurst parameter H ∈ (0, 1) which is related to the memory strength of the noise (see Definition 3.3 below). Since we focus on a long-memory behavior in our framework, the Hurst parameter will be considered, H ∈ (1/2, 1). The observation model is described by the following equation: t Yt = h(X s )ds + BtH , 0
where h is called the observation function, X t is the signal process and BtH is the H -fBm. The optimal filter, which is defined as the conditional probability distribution of X t given the σ -field FtY generated by Ys , s ≤ t, is studied by some authors. Here, we list only a few of them. Kleptsyna et al. [17] consider the case that the signal process is driven by a fBm while the observation noise is still the usual BM. Kleptsyna et al. [18], Kleptsyna et al. [20], Kleptsyna and Le Breton [19] and Le Breton [23] studied the linear filtering problem with fBm as observation noise. Nonlinear filtering problem with fBm observation noise has been studied by Coutin and Decreusefond [9], Gawarecki and Mandrekar [13], Amirdjanova [1], and Xiong and Zhao [37]. In this paper, we √ study the limiting behavior of the optimal filter when the observation function h is replaced by εh with ε → 0. This setting concerns with applications where the signal to noise ratio is very small. It is clear that the optimal filter, denoted by πtε , converges to the (unconditional) probability distribution of X t . The problem we are interested in is to describe the convergence rate. To this end, we establish a large deviation principle (LDP) for the optimal filter and give a representation of the rate function. When BtH is the classical Brownian motion (H = 1/2), this problem is studied by Xiong [35]. The approach we take in this paper is different from that of [35], and it is based on weak convergence arguments. A very powerful technique in deriving LDP has been developed by Budhiraja et al. in [6], and in Maroulas [26] based on variational representations obtained by Bou´e–Dupuis [4], and Budhiraja–Dupuis [5]. This strategy is then used by Budhiraja et al. [7,8], by Maroulas [27], by Ren and Zhang [33,32], Sritharan and Sundar [34] under various setups. We apply this method to the LD problem for optimal filtering with H -fBm noise. The same methods can be applied to the classical Brownian motion case to obtain a result which is stronger than those of [35]. Finally, we would like to mention that the method in this paper might also cover the case of general Gaussian observation noise. However, the key of this paper (and that of [6]) is the variational formula for Brownian functionals. Hence, to extend the LDP result of this paper to general Gaussian noise case, a variational formula for Gaussian functional need to be established. This task is not trivial and we will look into this in a future project. Our paper is organized as follows. Section 2 reviews basic definitions of large deviations and describes the methodology we follow in order to establish such estimates. Section 3 exposes preliminary results with respect to a fractional Brownian motion and a standard Brownian motion. Section 4 is the main section which demonstrates large deviation estimates for the optimal filtering. Such results depend on several estimates of the corresponding unnormalized filter which are proved in Section 5.
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2. Large deviations This section reviews elements of large deviations, and explains how one can verify large deviation estimates via weak convergence arguments. Let {X ϵ } be an E-valued family of random variables defined on a probability space (Ω , F, P), where E is a Polish space. The theory of large deviations focuses on probabilities of events whose probability converges to zero exponentially fast. The decay rate is expressed via the rate function, denote it herein by I . In our work we will prove a tantamount argument of the large deviations principle, the Laplace principle, and we will study the Uniform Laplace Principle. The reader should refer to [11] for a proof of the aforementioned equivalence. Let E0 and E be Polish spaces. For each ϵ > 0 and y ∈ E0 let X ϵ,y be E valued random variables given on the probability space (Ω , F, P). Definition 2.1. A family of rate functions I y on E, parameterized by y ∈ E0 , is said to have . compact level sets on compacts if for all compact subsets K of E0 and each M < ∞, Λ M,K = ∪ y∈K {x ∈ E : I y (x) ≤ M} is a compact subset of E. We list below the definition of Uniform Laplace principle. Definition 2.2. Let I y be a family of rate functions on E parameterized by y in E0 and assume that this family has compact level sets on compacts. The family {X ϵ,y } is said to satisfy the Laplace principle on E with rate function I y , uniformly on compacts, if for all compact subsets K of E0 and all bounded continuous functions h mapping E into R, 1 ϵ,y lim sup ϵ log E y exp − h(X ) + inf h(x) + I y (x) = 0. ϵ→0 y∈K ϵ x∈E Next, a set of sufficient conditions for a uniform large deviation principle is discussed. . Let (Ω , F, P, {Ft }) is a filtered probability space and denote S = C([0, t], R). Consider the (S, B(S))- valued random variable. Let E be a Polish space, and for each ϵ > 0, consider the family of measurable maps, G ϵ : E0 × S → E, which are defined as follows. ϵ,x . ϵ √ X = G x, ϵβ . (2.1) Let denote the spaces, T . P2 = f : f predictable and
T
f (s)ds < ∞ a.s. 2
0
. S N = φ ∈ L 2 ([0, T ]) :
T
φ 2 (s)ds ≤ N
0
P2N
. = {u ∈ P2T : u(ω) ∈ S N , P − a.s.}.
The following condition [6,26] is the critical one which one needs to verify in order to prove uniform large deviation estimates driven by a Brownian motion. The corresponding rate function is given in (2.2). Condition 2.1. There exists a measurable map G 0 : E0 × S → E such that the following hold.
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1. For M ∈ N let ( f n , f ) ∈ S M be such that (xn , f n ) → (x, f ). Then · · f (s)ds . f n (s)ds → G 0 x, G 0 xn , 0
0
be such that, as ϵ → 0, ψϵ converges in distribution to ψ and 2. For M ∈ N let ψϵ , ψ ∈ {x ϵ } ⊂ E0 , x ϵ → x, as ϵ → 0. Then · · √ G ϵ x ϵ , ϵW (·) + ψϵ (s)ds ⇒ G 0 x, ψ(s)ds . P2M
0
0
· For φ ∈ E, define Sφ = f ∈ S M : φ = G 0 x, 0 f (s)ds . Let I x : E → [0, ∞] be defined by T 1 2 I x (φ) = inf f s ds . f ∈Sφ 2 0
(2.2)
This is the main theorem of [6,26] which verifies that under Condition 2.1, one may establish uniform large deviations with rate function I x defined in (2.2). Theorem 2.1. Let G 0 : E0 × S → E be a measurable map satisfying Condition 2.1. Suppose that for all φ ∈ E, x → I x (φ) is a lower semi-continuous map from E0 to [0, ∞]. Then for every x ∈ E0 , I x : E → [0, ∞], defined by (2.2), is a rate function on E and the family {I x , x ∈ E0 } of rate functions has compact level sets on compacts. Furthermore, the family {X ϵ,x } satisfies the Laplace principle on E with rate function I x , uniformly for x in compact subsets of E0 . 3. Preliminary results We start this section with the definition of the H -fBm. As it was mentioned in the Section 1, due to our interest in a long-memory environment, we restrict ourselves to the case of H ∈ 1 2 , 1 . The reader should refer for example to [28] for definition and techniques for a general fractional Brownian motion with Hurst parameter, H ∈ (0, 1). Definition 3.3. For a given H ∈ (1/2, 1), a stochastic process B H = (BtH , t ∈ [0, T ]) is a fractional Brownian motion with Hurst parameter H if: (i) B0H = 0; (ii) B H is a zero-mean Gaussian process with continuous sample paths and stationary increments; (iii) The covariance function is given by R H (s, t) =
VH (|s|2H + |t|2H − |t − s|2H ) 2
where VH = V ar (B1H ) =
−Γ (2 − 2H ) cos(π H ) . π H (2H − 1)
Consider the signal process, X t , governed by the following SDE: d X t = b(X t )dt + σ (X t )d Wt ,
(3.3)
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where Wt is a finite dimensional Brownian motion, and the coefficients b, σ satisfy regularity properties (Lipschitz, and linear growth) such that (3.3) attains a unique solution [31, Section 5.2]. Consider the observation process, Ytε , described in the dynamics below, t √ ε h(X s )ds + BtH , (3.4) Yt = ε 0
where h is bounded and BtH , H > 1/2 is a fBm independent of Wt . We now follow the way of thinking in [37]. Using Molchan–Golosov representation [29], and [3, Theorem 4.5], one may explicitly express the fBm, BtH , as an integral on the interval [0, t] with respect to a standard Brownian motion in the following equation, t H γ H (s, t)d Bs , (3.5) Bt = 0
t H −1/2 s 1/2−H where γ H (s, t) = Γ (H (q − s) H −3/2 dq. Now the Brownian motion, Bt , in turn −1/2) s q will be expressed with respect to the fBm below. s t H H −1/2 k H (s, q)d Bq , (3.6) s d Bt = 0
0
1 1/2−H q 1/2−H . Define now the processes where k H (s, q) = Γ (3/2−H ) (s − q) s t Z tε = s H −1/2 d k H (s, q)dYqε 0
Stε
√ = ε
t
s
H −1/2
0
s
d
k H (s, q)h(X q )dq .
0
(3.8)
0 ε
t
(3.7)
0
According to (3.7), Z tε is FtY -measurable, and it can be shown that Ytε ε ε γ H (s, t)d Z sε , [30]. Thus, the filtrations FtY = FtZ and the observation model becomes Z tε = Stε + Bt .
=
(3.9)
The assumptions below are considered in the entire manuscript. These assumptions are critical for guaranteeing existence and uniqueness [37, Theorems 4.2 and 4.3] of the Kushner–FKK and Zakai equations as defined in (4.13) and (4.14). Assumption 3.1. The coefficients on the signal process σ : R → R and b : R → R belong to Cb1 (R). Assumption 3.2. The coefficient on the observation process h : R → R is a measurable function which belongs to Cb2 (R). Let take into account the following trajectory space, . T = {(t, x t ) : t ≥ 0, x ∈ C([0, T ], R)}, where x t (s) = x(t ∧ s). According to [37, Theorem 2.2] the small perturbation, Z tε , of the observation model will be equivalent to
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Z tε =
√
ε
t
G(s, X s )ds + Bt ,
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(3.10)
0
where X s is the path of X up to time s, and G is a measurable map from T to R. The reader should note that the coefficient G listed below encapsulates the entire trajectory of X up to some time T such that it reflects the long-memory behavior which was expressed via a fractional Brownian motion in (3.4). Namely, according to [37, Theorem 2.2], G is defined as, h(X 0 )B 23 − H, 32 − H (2 − 2H ) 1 s 2 −H G(s, X s ) = Γ (3/2 − H ) 1 su 1 1 2 − 2H −H −H 2 2 + s (u(1 − u)) Lh(X r )dr du Γ (3/2 − H ) 0 0 1 3 1 1 + s 2 −H (u(1 − u)) 2 −H Lh(X su )udu Γ (3/2 − H ) 0 s 1 1 1 2 − 2H + s 2 −H (u(1 − u)) 2 −H du(σ h ′ )(X r ) r Γ (3/2 − H ) 0 s b 1 (X r )d X r − (X r )dr × σ σ s 3 1 r 12 −H − (σ h ′ )(X r ) s −1 r 2 −H 1 − Γ (3/2 − H ) s 0 1 b × (X r )d X r − (X r )dr , (3.11) σ σ where L denotes the differential operator Lh(x) = 21 σ 2 (x)h ′′ (x) + b(x)h ′ (x). As it is mentioned in [37], the signal process needs to be enriched. Let (t, X t ) be a T-valued Markov process with generator, A0 F(t, X t ) = ∂0 φ(t, X t1 ∧t , . . . , X tn ∧t ) +
n
1t
i=1
+
n 1 1t
(3.12)
where F ∈ D(A0 ), and D(A0 ) is the space such that F(t, y) = φ(t, y(t1 ∧t), . . . , y(tn ∧t)), F ∈ Cb (T) for some n ≥ 1, 0 ≤ t1 < · · · < tn ≤ T, φ ∈ Cb2 (R+ ×Rn ). The reader should note that in general Cbk (T) denotes the space of continuous bounded and smooth functions (up to derivative k ≥ 1). Here we consider bounded derivatives because it will be necessary for establishing large deviations. However, this condition does not contradict with the well-posedness of the optimal filtering framework proved in [37]. 4. Uniform large deviation principle for the optimal filtering This section is the main of our manuscript and it establishes the uniform large deviation estimates for the optimal filtering. We will proceed by first showing large deviations for the unnormalized filter (Theorem 4.3) and then employing the contraction principle one may
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demonstrate the analogous result for the optimal filtering (Theorem 4.4). However, for reasons of presentation, the proof of Theorem 4.3 is given in the next section. ε Let πtε F = E(F(t, X t )|FtY ), ∀F ∈ Cb (T) be the optimal filter. According to [37, Theorem 3.1(ii)] the process πtε satisfies the Kushner–FKK formula, i.e. t t √ (πsε (G F) − πsε Gπsε F)dξs , πtε F = π0 F + πsε (A0 F)ds + ε 0
0
∀F ∈ D(A0 ), (4.13) √ t where ξt = Z tε − ε 0 πs Gds is the innovation process and it is a Brownian motion. Letting ε → 0 we observe that πtε F → E(F(t, X t )). In this paper, we examine the rate of the aforementioned convergence by establishing the large deviation principle for the family of optimal filters {πtε }. It can be shown analogously with [37, Theorem 3.1(iii)] that the process µεt , defined by t √ ε t µεt F = πtε F exp ε πsε Gd Z sε − |πs G|2 ds , 2 0 0 for any F ∈ Cb (T), satisfies the Zakai equation t t √ ε ε µt F = µ0 F + µs (A0 F)ds + ε µεs (G F)d Z sε , 0
∀F ∈ D(A0 ).
(4.14)
0
We will first need to establish the LDP for the family {µεt } and then by applying the contraction principle to show the LDP for the family of optimal filters {πtε }. √ t Consider the innovation process ξt = Z tε − ε 0 πs Gds, and denote the collection of finite ˆ as M(T). For each ε > 0, let G ε : M(T)×C([0, T ], R) → Borel measures on the Polish space R C([0, T ], M(T)) be a measurable map. Next, define √ . µε,µ0 = G ε µ0 , ε Z ε , (4.15) where µ0 ∈ M(T) is the initial condition of the unnormalized filter. Then the following theorem holds [37]. Theorem 4.2. The unnormalized filter µε,µ0 defined in (4.15) is the unique solution of the Zakai equation (4.14) with initial condition µ0 . We now consider, for some λ ∈ L 2 ([0, T ], R), the controlled analogue of (4.14) for all F ∈ D(A0 ). t t t √ ε,λ ε,λ ε µε,λ F = µ F + µ (A F)ds + ε µ (G F)d Z + µε,λ 0 0 t s (Gλs F)ds. (4.16) s s s 0
0
0
Let µλ be the solution of the zero noise controlled version of (4.16), i.e. t µλt F = µ0 F + µλs (A0 F + Gλs F) ds,
(4.17)
0
Let G 0 : M(T) × C([0, · T ], R) → C([0, T ], M(T)) be a measurable function such that = µλ , if u = 0 λs ds, λ ∈ L 2 ([0, T ]), or 0 otherwise.
G 0 (µ0 , u)
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For v ∈ C([0, T ], M(T)) and µ0 ∈ M(T) define T 1 . 2 I1 ≡ I1,µ0 (v) = inf λs ds . {λ∈L 2 ([0,T ]):v=G 0 (µ0 , 0· λs ds)} 2 0
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(4.18)
The following theorem establishes uniform large deviations for the unnormalized filter whose proof is delegated in the next section. Theorem 4.3. Let µε,µ0 be as in (4.14). Then I1,µ0 , defined in (4.18), is a rate function on C([0, T ], M(T)), and the family of rate functions {I1,µ0 , µ0 ∈ M(T)} has compact level sets on compacts. Furthermore, {µε,µ0 } satisfies the large deviation principle on C([0, T ], M(T)) with the rate function I1,µ0 , uniformly for µ0 in compact sets of M(T). Let P(T) be the collection of all Borel probability measures on T. Take G˜ 0 : P(T) × → C([0, T ], P(T)) a measurable function, and suppose that π λ = G˜ 0 (π0 , λ), where λ π is the solution of the following controlled equation. t λ πt F = π0 F + πsλ A0 F + (G − πsλ G)λs F ds. (4.19) L 2 ([0, T ])
0
For v˜ ∈ C([0, T ], P(T)) and π0 ∈ P(T) define . I2 ≡ I2,π0 (v) ˜ =
T 1 2 λs ds . inf · ˜ G 0 (π0 , 0 λs ds )} 2 0 {λ∈L 2 ([0,T ]):v=
(4.20)
The theorem below is the main theorem of this manuscript and establishes a uniform large deviation principle for the optimal filtering defined in (4.13). Theorem 4.4. The family of solutions of the Kushner–FKK equation, {π ε,π0 } satisfies the large deviation principle on C([0, T ], P(T)) with the rate function I2,π0 , defined in (4.20), uniformly for π0 in compact sets of P(T). Proof. The reader should remark that for simplicity in this proof we suppress the initial condition µ0 or π0 in the notation of the corresponding rate functions, I1 , I2 as defined in (4.18) and (4.20), respectively. Now, similarly to [35, Lemma 5.1 and Theorem 5.2], if there exists a t0 ∈ [0, T ] such that µt0 = 0 then I1 (µ) = ∞. It is then easy to prove that µε satisfies the LDP on C([0, T ], M(T) \ {0}) with the rate function being equal to the restriction of I on C([0, T ], M(T) \ {0}). Let define the continuous map J : C([0, T ], M(T) \ {0}) → C([0, T ], P(T)), such that (J µ)t = µµtt F1 . Then, by the contraction principle [10], {π ε = J (µε )} satisfies the LDP with rate function I2′ (π ) = inf{I1 (µ) : J (µ) = π }. If I2′ (π ) < ∞ then according to the definition of I2′ (π ) for all δ > 0 there exists µ such that J (µ) = π and I1 (µ) < I2′ (π ) + δ. Let λ ∈ L 2 ([0, T ], R) such that γ (λ) = µ, where γ is the solution of T . (4.17), and 12 0 λ2s ds < I1 (µ) + δ. Thus by setting γ˜ = J ◦ γ , we have that γ˜ (λ) = π , where T γ˜ is the solution of (4.19), and 12 0 λ2s ds < I2′ (π ) + 2δ. Therefore, I2 (π ) ≤ I2′ (π ).
(4.21)
Now, suppose I2 (π ) < ∞. Then for all δ > 0, there exists λ∈ L 2 ([0, T ]) such that γ˜ (λ) = π , T t and 12 0 λ2 ds < I2 (π ) + δ. Let at = π0 exp 0 πs hλs ds and µt = at πt , where h is the
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coefficient of the observation process (3.4). Then µ = γ (λ) and I2′ (µ) = π and therefore, T I2′ (π) ≤ I1 (µ) ≤ 12 0 λ2 ds < I2 (π ) + δ which together with (4.21) yield the result. 5. Proof of Theorem 4.3 The proof of Theorem 4.3 will follow the framework described in Section 2. In other words, one may demonstrate the Condition 2.1 in order to show uniform large deviation estimates for the unnormalized filter. We first show existence and uniqueness of the controlled version of Zakai’s equation given in (4.16). Theorem 5.5. Let G ε : M(T) × C([0, T ], R) → C([0, T ], M(T)) be a measurable map as defined in (4.15). Suppose λ ∈ P2N for some N ∈ N and for ε > 0 and µ0 ∈ M(T), denote t √ ε ε,λ . ε µµ0 = G µ0 , ε Z + λs ds . 0
Then
µε,λ µ0
is the unique solution of (4.16).
Proof. Let consider the dynamics of the signal process expressed in (3.3). For any F ∈ D(A0 ), we have that t F t Nt ≡ F(t, X ) − A0 F(s, X s )ds 0
is a martingale [37]. √ t Let further consider Mtε,λ = exp ε 0 G(s, X s )d Z sε − ds . Then by Itˆo’s formula we have that d Mtε,λ = Mtε,λ G(t, X t )
ε t 2 0
G 2 (s, X s )ds +
√ εd Z tε + λt dt .
t 0
λs G(s, X s )
(5.22)
Applying the integration by parts technique, e.g. see [31, Theorem 4.1.5], on the Itˆo integrals of the Eqs. (5.22) and taking into consideration that the fractional Brownian motion, B H , and the Brownian motion W are independent, one may conclude that d Mtε,λ F(t, X t ) = Mtε,λ d F(t, X t ) + F(t, X t )d Mtε,λ = Mtε,λ A0 F(t, X t )dt + λt Mtε,λ G(t, X t )F(t, X t )dt √ + εMtε,λ G(t, X t )F(t, X t )d Z tε + Mtε,λ d NtF .
(5.23)
Now, from Eq. (5.23) we have that, Mtε,λ F(t, X t ) = F(0, X 0 ) + √ +
ε
0
t
0
t
Msε,λ A0 F(s, X s )ds +
t
Msε,λ G(s, X s )F(s, X s )d Z sε +
λs Msε,λ G(s, X s )F(s, X s )ds
0
0
t
Msε,λ d NsF .
(5.24)
Taking conditional expectations on both sides in (5.24), and applying Lemma 5.4 of [36] we get, t ε ε E(Mtε,λ F(t, X t )|FtZ ) = µ0 F(t, X t ) + E(Msε,λ A0 F(s, X s )|FsZ )ds 0
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t
+ 0
√ + ε
2349
ε
λs E(Msε,λ G(s, X s )F(s, X s )|FsZ )ds 0
t
ε
E(Msε,λ G(s, X s )F(s, X s )|FsZ )d Z sε ,
(5.25) ε
ε,λ t Z and since from the Kallianpur–Striebel formula, we have that ⟨µε,λ t , F⟩ = E(Mt F(t, X )|Ft ), ε ε,λ Eq. (5.25) results that E(Mt F(t, X t )|FtZ ) is a solution of the controlled Zakai’s equation. Now, same arguments of Theorem 4.2 of [37] yield the uniqueness of the solution.
The next theorem establishes the existence and the uniqueness of the solution of the noise free controlled version of the (4.14). Theorem 5.6. Fix µ0 ∈ M(T) and λ ∈ L 2 ([0, T ]). Then there is a unique solution vtλ ∈ C([0, T ], M(T)) of (4.17). t Proof. Let consider Mt0,λ = exp 0 λs G(s, X s )ds . Then by Feynman–Kac formula vtλ = E(Mt0,λ F(t, X t )) is the unique solution of (4.17).
The proposition below yields the first part of Condition 2.1. Proposition 5.1. For M ∈ N, consider (λn , λ) ∈ S M be such that (µn0 , λn ) → (µ0 , λ). Then for n n any F ∈ D(A0 ), µλt F → µλt F, where µλt F, µλt F are defined in (4.17). t Proof. Consider Mt0,λ = exp 0 λs G(s, X s ) . Following the strategy of the proof of Theorem 5.5 and applying Itˆo’s formula, one can show that, ηt defined by µ0 (d x)Ex Mt0,λ F(t, X t ), ηt F = R
is a solution to (4.17). However, according to Theorem 5.6, the solution is unique and thus µλt = ηt . Therefore, applying dominated convergence theorem one may easily verify the convergence below: t λn n t s n µt F = µ0 (d x)Ex F(t, X ) exp G(s, X )λs ds R 0 t t s → µ0 (d x)Ex F(t, X ) exp G(s, X )λs ds =
R µλt F.
0
We next present the definition of C-tightness [16] which is an instrumental concept in the proof of the next proposition. Definition 5.4. A sequence of probability measures {µn } on D([0, T ], R) is C-tight if it is tight and cluster points are supported on C([0, T ], R). Finally, the proposition below is the core for the establishment of the large deviation principle and it settles the second part of the Condition 2.1.
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Proposition 5.2. Let M < ∞, and suppose that µε0 → µ0 , and λε → λ in distribution, as ε,λε ,µε0
ε → 0 with {λε } ∈ P2M . Then µt unique solution (4.17).
0,λ,µ0
converges in distribution to µt
, where µ0,λ µ0 is the
Proof. Consider λ ∈ P2M and p > 1. Employing Eq. (4.16), there exists an appropriate constant K 1 such that we have 2 p f (t) ≡ E sup µε,λ s 1 s≤t
≤ K 1 (µ0 1)
2p
+ K1E 0
t
|λs |µε,λ s 1ds
2 p
t
+ K1E 0
ε,λ 2 µs 1 ds
p
.
(5.26)
Note that the second term of the upper bound in (5.26) can be further estimated by t p t ε,λ 2 K1 M pE µs 1 ds ≤ K2 f (s)ds. 0
0
Therefore, the estimate of f (t) will be given by t f (s) ≤ K 3 + K 3 f (s)ds. 0
Applying Gronwall’s inequality yields that f (t) is bounded, and hence, 2 p E sup µε,λ ≤ K4, s 1
(5.27)
s≤T
where K 4 is a constant depending on M, T and µ0 1. ε,λε ,µε0 Next, we establish the tightness of µt . It is well-known that we only need to prove the ε,λε ,µε 0 tightness of µt F in C([0, T ], R) for every test function F in Dom(A0 ). By (5.27), we have ε,λε ,µε 2 p 0 1 E sup µt ≤ K4. (5.28) t≤T
For the time steps t1 < t2 , we have t 2 p 2 ε,λε ,µε0 ε E µs (Gλs F)ds ≤ E t1
t2
t1
(λεs )2 ds
p
t2
ε,λε ,µε0
µs
(G F)2 ds
p
t1
≤ M p K 4 |t2 − t1 | p . ε,λε ,µε · 0 By Kolmogorov’s criteria, we see that the family of processes 0 µs (Gλεs F)ds is C-tight. ε,λε ,µε · 0 Similarly, we can prove that the family of processes 0 µs (A0 F)ds is C-tight. Note that ε ε the martingale parts of µε,λ ,µ0 (F) has quadratic variation processes given by t ε,λε ,µε0 Q εt ≡ ε µs (G F)2 ds 0
which is also C-tight. It follows from Theorem 6.1.1 in Kallianpur and Xiong [16] that ε ε µε,λ ,µ0 (F) is tight. It is then standard to show that any limit point will be a solution to (4.17). Furthermore, the uniqueness of the solution to (4.17) implies the desired weak convergence.
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Proof of Theorem 4.3. Condition 2.1 follows from Propositions 5.1 and 5.2 directly. The Laplace principle, which implies the large deviation principle, then follows from Theorem 2.1. Acknowledgments The authors would like to warmly thank two anonymous reviewers for their excellent comments. The first author’s research was supported partially by University of Tennessee and NIMBioS funds. The second author’s research was supported partially by NSF DMS-0906907. References [1] A. Amirdjanova, Nonlinear filtering with fractional Brownian motion, Appl. Math. Optim. 46 (2–3) (2002) 81–88. Special issue dedicated to the memory of Jacques–Louis Lions. [2] A. Bain, D. Crisan, Fundamentals of Stochastic Filtering, in: Stochastic Modelling and Applied Probability, vol. 60, Springer, New York, 2009. [3] R.J. Barton, H.V. Poor, Signal detection in fractional Gaussian noise, IEEE Trans. Inform. Theory 34 (5) (1988) 943–959. [4] M. Bou´e, P. Dupuis, A variational representation for certain functionals of Brownian motion, Ann. Probab. 26 (1998) 1641–1659. [5] A. Budhiraja, P. Dupuis, A variational representation for positive functional of infinite dimensional Brownian motions, Probab. Math. Statist. 20 (2000) 39–61. [6] A. Budhiraja, P. Dupuis, V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab. 36 (4) (2008) 1390–1420. [7] A. Budhiraja, P. Dupuis, V. Maroulas, Large deviations for stochastic flows of diffeomorphisms, Bernoulli 36 (1) (2010) 234–257. [8] A. Budhiraja, P. Dupuis, V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincare 47 (3) (2011) 725–747. [9] L. Coutin, L. Decreusefond, Abstract nonlinear filtering theory in the presence of fractional Brownian motion, Ann. Appl. Probab. 9 (4) (1999) 1058–1090. [10] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, second ed., Springer, New York, 1998. [11] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley-Interscience, New York, 1997. [12] M. Fujisaki, G. Kallianpur, H. Kunita, Stochastic differential equations for the non-linear filtering problem, Osaka J. Math. 9 (1972) 19–40. [13] L. Gawarecki, V. Mandrekar, On the Zakai equation of filtering with Gaussian noise, in: Stochastics in Finite and Infinite Dimensions, in: Trends Math, Birkh¨auser Boston, Boston, MA, 2001, pp. 145–151. [14] G. Kallianpur, Stochastic Filtering Theory, Springer-Verlag, 1980. [15] G. Kallianpur, C. Striebel, Estimation of stochastic systems: arbitrary system process with additive noise observation errors, Ann. Math. Statist. 39 (1968) 785–801. [16] G. Kallianpur, J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, in: IMS Lecture Notes Monograph Series, vol. 26, 1995. [17] M.L. Kleptsyna, P.E. Kloeden, V.V. Anh, Nonlinear filtering with fractional Brownian motion, Stoch. Anal. Appl. 16 (5) (1998) 907–914. [18] M.L. Kleptsyna, P.E. Kloeden, V.V. Anh, Linear filtering with fractional Brownian motion in the signal and observation processes, J. Appl. Math. Stoch. Anal. 12 (1) (1999) 85–90. [19] M.L. Kleptsyna, A. Le Breton, Extension of the Kalman–Bucy filter to elementary linear systems with fractional Brownian noises, Stat. Inference Stoch. Process. 5 (3) (2002) 249–271. [20] M.L. Kleptsyna, A. Le Breton, M.C. Roubaud, General approach to filtering with fractional Brownian noises — application to linear systems, Stoch. Stoch. Rep. 71 (1–2) (2000) 119–140. [21] H.J. Kushner, On the dynamic equations of conditional probability density functions with applications to optimal stochastic control theory, J. Math. Anal. Appl. 8 (1964) 332–344. [22] H.J. Kushner, Dynamic equations for nonlinear filtering, J. Differential Equations 3 (1967) 179–190.
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