Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise

SPA: 3463

Model 1

pp. 1–29 (col. fig: NIL)

Available online at www.sciencedirect.com

ScienceDirect Stochastic Processes and their Applications xxx (xxxx) xxx www.elsevier.com/locate/spa

Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise Vasileios Maroulasa ,∗, Xiaoyang Pana , Jie Xiongb b

a Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

Received 2 January 2018; received in revised form 1 February 2019; accepted 21 February 2019 Available online xxxx

Abstract In this paper, we focus on the asymptotic behavior of the optimal filter where both signal and observation processes are driven by Lévy noises. Indeed, we study large deviations for the case where the signal-to-noise ratio is small by considering weak convergence arguments. To that end, we first prove the uniqueness of the solution of the controlled Zakai and Kushner–Stratonovich equations. For this, we employ a method which transforms the associated equations into SDEs in an appropriate Hilbert space. Next, taking into account the controlled analogue of Zakai and Kushner–Stratonovich equations, respectively, the large deviation principle follows by employing the existence, uniqueness and tightness of the solutions. c 2019 Elsevier B.V. All rights reserved. ⃝ Keywords: Nonlinear filtering; Lévy noise; Uniqueness; Large deviations

1. Introduction Stochastic filtering deals with the estimation problem under partial information. Given two stochastic processes, the signal process and observation process, the filtering problem aims to estimate a functional of the signal based on the partially observed data. This paper focuses on a general model in which signal and observation processes are driven by L´evy noises. Several models in finance engineering, biology etc., e.g. see the partial list [1,8,9,12,15,17,22, 23,25,26,31,32], have considered such stochastic dynamics driven by a pertinent L´evy noise for describing partially received information with discontinuities or jumps in a time interval. By the ∗ Corresponding author.

E-mail address: [email protected] (V. Maroulas). https://doi.org/10.1016/j.spa.2019.02.009 c 2019 Elsevier B.V. All rights reserved. 0304-4149/⃝

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

1

2 3 4 5 6 7 8

SPA: 3463

2

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

42

same token, there exists an array of studies which encounter the analysis of optimal filter in a L´evy environment, for instance, [6,7,13,20,28,29]. We investigate a small-noise large deviation principle (LDP) of the optimal filter, which is basically a solution of the Kushner–Stratonovich stochastic partial differential equation (SPDE). Such analysis is associated with the rare events of a small signal-to-noise ratio. The early work [14] derived large deviations for the conditional density for diffusion systems in which both the signal and the observation noises were small. The study [27] established a quenched large deviation principle with small-noise observations. In a similar setup as herein, [33] took aim in a model where the signal was a diffusion process and the observation process was driven by a Brownian motion. A fractional Brownian motion model was studied in [24]. This work investigates the large deviations for the optimal filter where both signal and observation processes are driven by L´evy noises. The strategy we apply is to prove the Laplace principle, which is equivalent to the large deviation principle, by using a weak convergence argument, proposed in [3,4]. This weak convergence method is an approach that has been increasingly used, e.g., the large deviations for a variety of SPDEs [2,3,5,10,21,30,35,36], based on variational representations of the functionals of driving Brownian motions and Poisson random measures. The novelty of such a method is that, it does not require the exponential continuity or exponential tightness, and in contrast only basic qualitative properties of existence, uniqueness of controlled analogues of the stochastic dynamical systems of interest are needed to be shown. To that end, we first prove the uniqueness of the controlled unnormalized filtering equation, i.e. the controlled Zakai equation, and subsequently this of the controlled filtering equation, the controlled Kushner–Stratonovich equation. Some studies have been devoted to verifying the uniqueness of the filtering equation. In [7], the uniqueness was shown by the Filtered Martingale Problem approach which was proposed in [18], however, that model has a limitation that the signal and observation are driven by the same Poisson random measure having common jump times. A more general setting was suggested in [29] and the uniqueness was also proved by the approach of Filtered Martingale Problem, however, it was shown under the assumption that the correlated Poisson random measure is independent of the signal. Moreover, in these studies the uniqueness of the Filtered Martingale Problem was assumed when the Filtered Martingale Problem method was used, which requires the regularity conditions of the coefficients of the equations. In our work, based on a method using the Brownian motion semigroup, we bypass these restrictions and establish the uniqueness for the general model with a mild assumption on the coefficients of the Poissonian noise. The paper is organized as follows. Section 2 discusses preliminaries on the introduction to the controlled Poisson random measure and a general criterion of large deviations. We prove the uniqueness of the solutions to the Zakai and Kushner–Stratonovich equations in Section 3 which are given in Theorems 3.3 and 3.4 respectively. Section 4 focuses on the establishment of the large deviation principle for the optimal filter. We start with deriving the controlled version and zero-noise version of Zakai equation, then the existence and uniqueness of these two versions of Zakai equation are verified and finally the results are concluded by demonstrating the sufficient conditions presented in Propositions 4.2 and 4.3.

43

2. Preliminaries

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

44 45 46 47

Our problem concerns with a small-noise large deviation principle for the optimal filter where the respective SPDEs are driven by a pertinent Brownian motion and a Poisson random measure. This section lists definitions and notations used later on. Most of these notations can be also found in [4] but they are presented here for the sake of completeness. Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

3

2.1. Definitions and conventional notations

1

Let U be a locally compact Polish space and denote by M(U) the space of all measures ν on (U, B(U)), satisfying ν(A) < ∞ for every compact subset A of U, and B(U) is the Borel σ -field on U. Endow M(U) with the weakest topology ∫ such that for every continuous function f on U with compact support, the function ν → U f (u)ν(du), ν ∈ M(U) is continuous. This topology can be metrized such that M(U) is a Polish space, e.g. see [4]. For a fixed T ∈ (0, ∞), denote by M = M(UT ) the space of measures on UT = [0, T ] × U and let νT = λT ⊗ ν, λT is the Lebesgue measure on [0, T ]. We recall that a general Poisson random measure (PRM) n on UT with intensity measure νT is an M-valued random variable such that for each A ∈ B(UT ) with νT (A) < ∞, n(A) is Poisson distributed with mean νT (A) and for disjoint A1 , . . . , Ak ∈ B(UT ), n(A1 ), . . . , n(Ak ) are mutually independent random variables. For any m ≥ 1, let Wm = C([0, T ], Rm ) be the space of all continuous functions from [0, T ] to Rm , and D([0, T ], E) denote the space of right continuous functions with left limits from [0, T ] to a Polish space E. Take V = Wm × Wn × M and let P be the probability measure on (V, B(V)) such that (i) N : V → M is a Poisson random measure with intensity measure θνT , and νT (A) < ∞ for all A ∈ B(UT ); (ii) W : V → Wm is a Rm -valued Brownian motion and B : V → Wn is a Rn -valued Brownian motion; and (iii) {Wt }t∈[0,T ] , {Bt }t∈[0,T ] and {N ([0, t] × A), t ∈ [0, T ]} are Gt -martingales for every A ∈ B(U), where the filtration Gt := σ {N ([0, t] × A) − θ tν(A), Ws , Bs : 0 ≤ s ≤ t, A ∈ B(U)}. To adopt the strategy of weak convergence arguments in order to prove the large deviations, we introduce a properly controlled Poisson random measure. Define YT = [0, T ] × Y, where ¯ = M(YT ). Suppose N¯ is a Poisson random measure with Y = U × [0, ∞) and then denote M ¯ ¯ points on V = Wm × Wn × M with intensity measure ν¯ T = λT ⊗ ν ⊗ λ∞ where λ∞ is the Lebesgue measure on [0, ∞). Similarly abusing notations, B and W , are Brownian motions ¯ Next define (P, ¯ {G¯t }) on (V, ¯ B(V)) ¯ analogous to (P, {Gt }) by replacing (N , θ νT ) with on V. ¯ ¯ ( N , ν¯ T ). Consider the P-completion of the filtration {Gt } and denote it by {F¯ t }. We denote by ¯ with the filtration {F¯ t : 0 ≤ t ≤ T } on (V, ¯ B(V)). ¯ P¯ the predictable σ -field on [0, T ] × V ¯ → [0, ∞). For Let A¯ be the class of all (P¯ ⊗ B(U)) \ B[0, ∞) measurable maps φ : UT × M the variable φ ∈ A¯ which basically controls the intensity at time s on location u, define the counting process N φ on UT by ∫ N φ ((0, t] × A) = 1[0,φ(s,u)] (r ) N¯ (ds, du, dr ) (1)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

31

(0,t]×A×(0,∞)

where t ∈ [0, T ], A ∈ B(U). If φ(s, u) = θ for all (s, u) ∈ UT , then we write N φ = N θ , where ¯ with respect to P¯ as N has on M with respect to P. For N θ has the same distribution on M ¯ any φ ∈ A, the quantity ∫ L T (φ) = l(φ(t, u)) ν(du)ds (2)

32 33 34

35

UT

is well-defined as a [0, ∞]-valued random variable where l(r ) := r log r − r + 1, r ∈ [0, ∞). 2 2 m m We ∑m denote 2 by L ([0, T ], R ) the Hilbert space from [0, T ] to R satisfying |ψ(s)| = i=1 ψi (s) < ∞. Define P2 = { } ∫ T m 2 ¯ ¯ ψ = (ψi )i=1 : ψi is P \ B(R) measurable and |ψ(s)| ds < ∞, P − a.s. 0 Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

36 37 38

39

SPA: 3463

4 1

2

3

4

5

6 7 8 9 10

11 12 13 14

15

16 17 18

19

20 21 22

23

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

¯ For ψ ∈ P2 define and set U = P2 × A. ∫ T 1 ψ(s)2 ds L˜ T (ψ) := 2 0 and for u = (ψ, φ) ∈ U, set

(3)

L¯ T (u) = L T (φ) + L˜ T (ψ).

(4)

2.2. A general criterion of large deviations The theory of small-noise large deviations concerns with the asymptotic behavior of solutions of SPDEs, say {X ϵ }, ϵ > 0 defined on a probability space (Ω , F, P), which converge exponentially fast as ϵ → 0. The decay rate is expressed via a rate function. An equivalent argument of the large deviations principle is the Laplace principle. A reader may refer to [11, Theorem 1.2.1 and Theorem 1.2.3]. Definition 2.1. A function I : E → [0, ∞] is called a rate function on E , if for each M < ∞ the level set {x ∈ E : I (x) ≤ M} is a compact subset of E. The family {X ϵ } is said to satisfy the Laplace principle on E with rate function I , if for all bounded continuous functions h mapping E into R, ⏐ ⏐ { [ ]} ⏐ ⏐ 1 ϵ ⏐ lim ⏐ϵ log Ex0 exp − h(X ) + inf {h(x) + I (x)}⏐⏐ = 0. ϵ→0 x∈E ϵ Next, a set of sufficient conditions for a uniform large deviation principle for functionals of a Brownian motion and Poisson random measure is presented. Consider the family of measurable maps, for any ϵ > 0, G ϵ : Wm × M → E defined as follows: √ −1 · {X ϵ = G ϵ ( ϵW, ϵ N ϵ )}. Define the space, for some M ∈ N, S˜ M := {ψ ∈ L 2 ([0, T ], Rm ) : L˜ T (ψ) ≤ M} and S M := {φ : UT → [0, ∞) : L T (φ) ≤ M}, where L T and L˜ T are defined in Eqs. (2) and φ (3), respectively. For a function φ, define a measure νT ∈ M, such that ∫ φ νT (A) = φ(s, u)ν(du)ds, A ∈ B(UT ). A

24 25 26 27 28 29 30

31 32

33

34

Throughout we encapsulate the topology on S M obtained through this identification which makes S M a compact space. Let S¯ M = S˜ M × S M with the usual product topology and M ¯M = {u = (ψ, φ) ∈ U : u(ω) ∈ S¯ M , P¯ a.e. ω}, S = ∪∞ M=1 S . Define the space of controls U ¯ where U = P2 × A. The following condition in [4] plays a key role in proving large deviation estimates for the filtering equation driven by a Brownian motion and an independent Poisson random measure. The corresponding rate function is given in Eq. (5). Condition 2.1. holds.

There exists a measurable map G 0 : Wm × M → E such that the following

1. For M ∈ N, let ( f n , gn ), ( f, g) ∈ S¯ M be such that ( f n , gn ) → ( f, g) as n → ∞, then (∫ · ) (∫ · ) gn g 0 0 f n (s)ds, νT → G f (s)ds, νT . G 0

0

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx d

5

d

2. For M ∈ N, let ξ ϵ = (ψ ϵ , φ ϵ ), ξ = (ψ, φ) ∈ U M be such that ξ ϵ → ξ , where → denotes convergence in distribution, as ϵ → 0. Then ∫ · ∫ · √ d φ ϵ ϵ −1 φ ϵ 0 ϵ ψ (s)ds, ϵ N ) → G ( ψ(s)ds, νT ). G ( ϵW + 0

For ζ ∈ E, define Sζ = {u ∈ S : ζ = G 0 (

1 2

3

0

∫· 0

φ

ψ(s)ds, νT )}. Let I : E → [0, ∞] be defined

by

4 5

} I (ζ ) = inf L¯ T (u) . {

u∈Sζ

(5)

By convention, I (ζ ) = ∞ if Sζ = ∅. Under Condition 2.1, we have the following theorem shown in [4].

6

7 8

√ −1 · Theorem 2.1. For ϵ > 0, let X ϵ = G ϵ ( ϵW, ϵ N ϵ ), and suppose that Condition 2.1 holds. Then I : E → [0, ∞], defined by Eq. (5), is a rate function on E and the family {X ϵ } satisfies the large deviation principle on E with rate function I .

11

3. Existence and uniqueness of filtering equations

12

3.1. Existence

13

9 10

This section first presents the filtering model and filtering equations for the system driven by L´evy noise. Consider the following signal-observation system (X t , Yt ) on Rd × Rm : ∫ f 1 (X t− , u) N˜ p (dt, du) (6a) d X t = b1 (X t )dt + σ1 (X t )d Bt + U1 ∫ + g1 (X t− , u)N p (dt, du), U\U1 ∫ dYt = b2 (t, X t )dt + σ2 (t)d Wt + f 2 (t, u) N˜ λ (dt, du) (6b) U2 ∫ g2 (t, u)Nλ (dt, du), + U\U2

where Bt , Wt are n-dimensional and m-dimensional Brownian motions respectively defined on the filtered probability space (V, B(V), {Ft }t∈[0,T ] , P). N p is a Poisson random measure such that ∫ EN2p ([0, t] × A) = tν1 (A), for any A ∈ B(U) with ν1 (A) < ∞, and ν1 (U \ U1 ) < ∞, U1 ∥u∥U ν1 (du) < ∞ where ∥ · ∥U denotes the norm on the measurable space (U, B(U)) and U1 ⊂ U. Denote the compensated measure N˜ p ([0, t] × du) = N p ([0, t], du) − tν1 (du). Let Nλ (dt × du) be an integer-valued random measure and its predictable compensator is given by ˜ λ(t, X t− , u)dtν2 (du), ∫ t ∫ where the function λ(t, x, u) ∈ [l, 1), 0 < l < 1, and Nλ ([0, t] × A) = Nλ ([0, t] × A) − 0 A λ(s, X s− , u)ν2 (du)ds such that for each A ∈ B(U), ν2 (A) < ∞, and ∫ ν2 (U \ U2 ) < ∞, U2 ∥u∥2U ν2 (du) < ∞ with U2 ⊂ U. Moreover, Bt , Wt , N p , Nλ are mutually independent. We assume that the mappings b1 : Rd → Rd , b2 : [0, T ] × Rd → Rm , σ1 : Rd → Rd×n , σ2 : [0, T ] → Rm×m , f 1 : Rd × U1 → Rd , f 2 : [0, T ] × U2 → Rm , g1 : Rd × (U \ U1 ) → Rd , and g2 : [0, T ] × (U \ U2 ) → Rm are all Borel measurable, and satisfy the following conditions: Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

14 15 16 17 18 19 20 21 22 23 24 25 26

SPA: 3463

6

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

For each x1 , x2 ∈ Rd , there exists a constant K > 0 such that ∫ |b1 (x1 ) − b1 (x2 )|2 + |σ1 (x1 ) − σ1 (x2 )|2 + | f 1 (x1 , u) − f 1 (x2 , u)|2 ν1 (du)

Assumption 3.1.

U1

≤ K |x1 − x2 |2 , ∫

|g1 (x1 , u) − g1 (x2 , u)|ν1 (du) ≤ K |x1 − x2 |,

U\U1 1

where | · | denotes the Hilbert–Schmidt norm for a matrix and the length for a vector. Assumption 3.2. σ2 (t) is invertible for t ∈ [0, T ], and for each x ∈ Rd , ∫ ∫ 2 | f 2 (t, u)|2 ν2 (du) ≤ K . | f 1 (x, u)|2 ν1 (du) + |b2 (t, x)|2 + |σ2−1 (t)| + |σ1 (x)|2 + U2 U1 ∫ |g2 (t, u)|ν2 (du) ≤ K . U\U2

7

Note that Yt = f 2 (t, pλ (t)) + g2 (t, pλ (t))1 Dλ (t) is observable, where Dλ and pλ are the jumping times and locations of the random measure Nλ . As g2 describes the large jumps while f 2 the small ones, we assume they have disjoint ranges and, for each t, f 2 (t, ·) and g2 (t, ·) are invertible functions. Namely, we assume that Nλ is observable. Let Y˜t be given by ˜ N d Y˜t = b2 (t, X t )dt + σ2 (t)d Wt . Then, FtY = FtY ∨ Ft λ . Based on the discussion above, we make the following assumption throughout this article.

8

Assumption 3.3. For each t, f 2 (t, ·) and g2 (t, ·) are invertible functions with disjoint ranges.

2 3 4 5 6

Set πt (F) := E(F(X t )|FtY ), for any F ∈ Cb2 (Rd ), where Cb2 (Rd ) denotes the set of all bounded functions F : Rd → Rd that have continuous second order derivative. Taking into account the Radon–Nikodym derivative there is an equivalent probability measure P˜ such that, d P˜ = Λ−1 T dP, where ( ∫ T ∫ 1 T −1 2 −1 ∗ |σ2 (s)b2 (s, X s )| ds (7) Λ−1 = exp − σ (s)b (s, X ) d W − 2 s s T 2 2 0 0 ) ∫ T∫ ∫ T∫ − log λ(s, X s− , u)Nλ (ds, du) − (1 − λ(s, X s , u))ν2 (du)ds . 0 9 10 11

0

U

U

where ∗ denotes the transpose operator. In turn, the Kallianpur–Striebel formula [34] gives Y ˜ ˜ , where µt (F) := E(F(X πt (F) = µµtt(F) t )Λt |Ft ) and E denotes expectation under the measure (1) ˜P. Now we are ready to derive Zakai and Kushner–Stratonovich equations. Proposition 3.1 (Zakai Equation). Assume Assumptions 3.1–3.3. For F ∈ D(L), the Zakai equation of Eq. (6) is given by ∫ t m ∫ t ∑ µt (F) = µ0 (F) + µs (LF)ds + µs (F(σ2−1 (s)b2 (s, ·))i )d W˜ si 0

∫ t∫

0

µs− (F(λ(s, ·, u) − 1)) N˜ (ds, du),

+ 0

i=1

(8)

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

where W˜ t = Wt + f ∈

Cb2 (Rd ),

σ2−1 (s)b2 (s, X s )ds and N˜ (dt, du) = Nλ (dt, du) − dtν2 d(u), and for any

0

the infinitesimal generator, L, is given by

L f (x) =

d ∑ ∂ f (x)

∂ xi

i=1 ∫

b1i (x) +

d n 1 ∑ ∑ ∂ 2 f (x) ik jk σ1 (x)σ1 (x) 2 i, j=1 k=1 ∂ xi ∂ x j

U\U1

∫ [ f (x + f 1 (x, u)) − f (x) −

+ U1

where

1

2

[ f (x + g1 (x, u)) − f (x)]ν1 (du)

+

σ1ik

7

t

∫

d ∑ ∂ f (x) i=1

∂ xi

(9) f 1i (x, u)]ν1 (du),

is the (i, k)th entry of the diffusion coefficient σ1 .

4

Proof. By Itˆo’s formula, we have dΛt = Λt σ2−1 (t)b2 (t, X t )d W˜ t +

3

5

∫

Λt− (λ(t, X t− , u) − 1) N˜ (dt, du),

6

U

and

7

d F(X t ) = LF(X ∫ t )dt + ∇ F(X t )σ1 (X t )d Bt + [F(x + g1 (x, u)) − F(x)] N˜ p (dt, du) U\U1 ∫ + [F(x + f 1 (x, u)) − F(x)] N˜ p (dt, du).

8

9

10

U1

By Itˆo’s formula again, we have d(Λt F(X t )) =

Λt F(X t )σ2−1 (t)b2 (t, ∫

11

X t )d W˜ t

12

Λt− F(X t− )(λ(t, X t− , u) − 1) N˜ (dt, du)

+

13

U

+Λ ∫t LF(X t )dt + ∇ F(X t )σ1 (X t )d Bt

14

Λt− [F(x + g1 (x, u)) − F(x)] N˜ p (dt, du)

+

15

U\U1

∫

Λt− [F(x + f 1 (x, u)) − F(x)] N˜ p (dt, du).

+

16

U1

Taking conditional expectations on both sides, we arrive at (8).

■

Using the Kallianpur–Striebel formula, we obtain the following Kushner–Stratonovich equation. Proposition 3.2. Assume Assumptions 3.1–3.3. For F ∈ D(L), the solution of the following equation exists ∫ t πt (F) = π0 (F) + πs (LF)ds 0 ∫ m t ∑ + (πs (F(σ2−1 (s)b2 (s, ·))i ) − πs (F)πs (σ2−1 (s)b2 (s, ·))i )d Wˆ si 0 i=1 ∫ t∫ πs− (Fλ(s, ·, u)) − πs− (F)πs− (λ(s, ·, u)) ˆ + N (ds, du), (10) πs− (λ(s, ·, u)) 0 U Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

17 18 19

SPA: 3463

8

1

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

where Wˆ t = W˜ t −

∫ 0

t

πs (σ2−1 (s)b2 (s, ·))ds is the innovation process and Nˆ (dt, du) =

2

Nλ (dt, du) − πt− (λ(t, ·, u))ν2 (du)dt.

3

3.2. Uniqueness

4 5 6 7 8 9 10 11 12 13

14 15

16 17 18 19 20 21 22 23 24 25 26 27

28

In this section, we prove the uniqueness for the solutions to the Zakai and Kushner– Stratonovich equations for the signal-observation model (6). Although the uniqueness was investigated in [29], it was assumed that the Poisson noise in the observation is independent of the signal, i.e., λ(t, x, u) = λ(t, u). This reduces the complexity of the Zakai equation; that is, the Zakai equation is independent of the Poisson noise. This makes the problem more tractable since the Poissonian part in Eq. (10) vanishes, see [29, Section 4]. Furthermore, therein, the uniqueness of the Filtered Martingale Problem is assumed, and regularity conditions on the coefficients of the signal and observation processes are required (see [29, Remark 4.1] and [18]). Next, we show the uniqueness of Zakai and Kushner–Stratonovich equations by bypassing the above restrictive assumptions and instead imposing the following mild assumption. ⏐ ( ⏐ ( )⏐ )⏐ 1 1 and ⏐det Jg1 + I ⏐ > for a constant C > 0, where Assumption 3.4. ⏐det J f1 + I ⏐ > C C J f1 and Jg1 are the Jacobian matrices of f 1 and g1 with respect to x, respectively. The uniqueness for the solution to Zakai equation is proved by transforming it to an SDE in a pertinent Hilbert space and by making use of estimates based on Hilbert-space techniques, which was studied in [19,34]. Recall that the optimal filter E(F(X t )|FtY ) is the solution to the filtering Eq. (10) characterized by the conditional probability πt . Denote P(Rd ) the collection of all Borel probability measures on Rd such that πt ∈ P(Rd ). Denote by ⟨ν, F⟩ the integral of a function F with respect to a measure ν, e.g., for any F ∈ Cb (Rd ), E(F(X t )|FtY ) = ⟨πt , F⟩. Let M F (Rd ) be the collection of all finite Borel measures on Rd such that the unnormalized filter µt is an M F (Rd )-valued process. Let H0 = L 2 (Rd ) be the Hilbert space consisting of square-integrable functions on Rd ∫with the usual L 2 -norm and the inner product given by ∫ 2 2 ∥φ∥0 = Rd |φ(x)| d x and ⟨φ, ψ⟩0 = Rd φ(x)ψ(x)d x. We introduce an operator to transform a measure-valued process to an H0 -valued process. Denote by MG (Rd ) the space of finite signed measures on Rd . For any ν ∈ MG (Rd ) and δ > 0, let ∫ (Tδ ν)(x) = G δ (x − y)ν(dy), (11) Rd

33

( ) 2 d where G δ is the heat kernel given by G δ (x) = (2π δ)− 2 exp − |x| . We use the same notation 2δ as in Eq. (11) for the Brownian motion semigroup on H , i.e., for t ≥ 0, define operator 0 ∫ Tt : H0 → H0 by Tt φ(x) = Rd G t (x − y)φ(y)dy, for any φ ∈ H0 . Then Lemma 3.1 obtains bounds for the partial derivative of Tδ , and Lemma 3.2 is directly applied to Theorem 3.1. The reader should refer to [34] for the proofs of these two lemmas.

34

Lemma 3.1.

29 30 31 32

35 36 37 38 39

(i) The family of operators {Tt : t ≥ 0} forms a contraction semigroup on H0 , i.e. for any t, s ≥ 0 and φ ∈ H0 , we have Tt+s = Tt Ts and ∥Tt φ∥0 ≤ ∥φ∥0 . (ii) If ν ∈ MG (Rd ) and δ > 0, then Tδ ν ∈ H0 . (iii) If ν ∈ MG (Rd ) and δ > 0, then ∥T2δ |ν|∥0 ≤ ∥Tδ |ν|∥0 , where |ν| is the total variation measure of ν. Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

9

For any δ > 0, ν ∈ MG (Rd ) and φ ∈ H0 , we have

Lemma 3.2.

1

(i) ⟨Tδ ν, φ⟩0 = ⟨ν, Tδ φ⟩.

(12)

(ii) If, in addition, ∂i φ ∈ H0 , where ∂i φ =

∂φ ∂ xi

, then

∂i Tδ φ = Tδ ∂i φ.

3

(13)

The next theorem presents an expression for the expectation of the transformation applying to the solution to Zakai equation. The proof is delegated to Appendix A. Theorem 3.1. Let µt ∈ M F (Rd ) be a solution to Zakai equation (8) and let Z tδ = Tδ µt . Considering the probability measure P˜ defined by (7), the following holds. ˜ tδ ∥20 = A1 − 2A2 + A3 + 2A4 + A5 + A6 + A7 , E∥Z

2

(14)

4

5 6

7 8

9

where A1 = ∥Z 0δ ∥20 ,

A2 =

d ∫ ∑ i=1

A3 =

d ∑ n ∫ ∑ 0

i, j=1 k=1 ∫ t∫

d ∑

∫

˜ sδ , ∂i2j Tδ (σ1ik σ k j µs )⟩ ds, E⟨Z 1 0 (15)

˜ sδ , Z˜ sδ, f1 ⟩ − E∥Z ˜ sδ ∥20 [E⟨Z 0

A5 = −

˜ sδ , ∂i Tδ (b1i µs )⟩ ds, E⟨Z 0

U\U1

∫ t∫ 0

0

˜ sδ , Z˜ sδ,g1 ⟩ − E∥Z ˜ sδ ∥20 ]ν1 (du)ds, [E⟨Z 0

A4 = 0

t

t

U1

˜ sδ , ∂i Tδ ( f 1i (·, u)µs )⟩ ]ν1 (du)ds, E⟨Z 0

i=1 t

−1 2 ˜ E∥⟨T δ (σ2 (s)b2 (s, ·)µs )⟩∥0 ds, ∫0 t ∫ ˜ δ ((λ(s, ·, u) − 1)µs ) ∥20 ν2 (du)ds; A7 = E∥T

A6 =

0

U

and δ, f ⟨Z tδ , Z˜ t 1 ⟩0 =

∑

⟨Z tδ , η⟩0 ⟨Z tδ , η(· + f 1 (·, u))⟩0 ,

(16)

⟨Z tδ , η⟩0 ⟨Z tδ , η(· + g1 (·, u))⟩0 ,

(17)

η δ,g ⟨Z tδ , Z˜ t 1 ⟩0 =

∑ η

here the set of functions {η} is a complete orthonormal system (CONS) of H0 . In order to get estimates of the terms as defined in Eq. (15), we proceed with the following lemmas. Lemma 3.3. Suppose Assumption 3.4 holds. Then there exists a constant C0 > 0 such that ⏐ ⏐ ⏐ δ ˜ δ, f1 ⏐ (18) ⏐⟨Z t , Z t ⟩0 ⏐ ≤ C0 ∥Z tδ ∥20 , Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

10

11 12

13

14

SPA: 3463

10 1

2

3 4 5

6

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

and ⏐ ⏐ ⏐ δ ˜ δ,g1 ⏐ ⏐⟨Z t , Z t ⟩0 ⏐ ≤ C0 ∥Z tδ ∥20 .

(19)

Proof. We only need to show inequality (18). Let y = x + f 1 (x, u) and assume x = h(y, u). Denote of h with respect to y by |∂ y h(y, u)|. According to Assumption 3.4, ⏐ ( the Jacobian )⏐ ⏐det I + J f ⏐ > 1 , then 1 C ⏐ ⏐ ⏐ ( )⏐ ⏐∂ y h(y, u)⏐ = ⏐det I + J f ⏐−1 ≤ C. (20) 1 Note that ⟨Z tδ , η(·

7

8

∫

Z tδ (h(y, u))η(y)|∂ y h(y, u)|dy Rd =⟨Z tδ (h(·, u))|∂ y h(·, u)|, η⟩0 .

+ f 1 (·, u))⟩0 =

Furthermore, summing over {η} in a CONS of H0 , we have ∑ ⟨Z tδ (h(·, u))|∂ y h(·, u)|, η⟩0 ⟨Z tδ , η⟩0 = ⟨Z tδ (h(·, u))|∂ y h(·, u)|, Z tδ ⟩0 . η

Hence, by Eq. (20) we get ∫ |Tδ µt (h(y, u))|2 |∂ y h(y, u)|2 dy ∥Z tδ (h(y, u))|∂ y h(y, u)|∥20 = d R ∫ 2 |Tδ µt (x)| |∂ y h(y, u)|d x ≤ C∥Z tδ ∥20 , =

(21)

Rd

9

10 11

12 13 14

15

16

17

18

and the bound of Eq. (18) then follows from the Cauchy–Schwarz inequality.

■

Lemma 3.4, verified in [19, Lemma 3.2], is useful to estimate the transformation Tδ and the derivatives of Tδ in Lemma 3.5 and Theorem 3.2. Lemma 3.4. Let (H, H, η) be a measure space and H = Ł2 (η). Let φi : Rd → H, i = 1, 2 such that there exists a constant K > 0, for any x ∈ Rd , ∥φi (x)∥H ≤ K . Let ζ ∈ MG (Rd ). Then there exists a constant K 1 ≡ K 1 (φi ) such that ∥∥Tδ (φ1 ζ )∥0 ∥H ≤ K 1 ∥Tδ (|ζ |)∥0 .

(22)

If, additionally, (we are interested in H = U with η = ν1 ) ∥φi (x) − φi (y)∥H ≤ K |x − y|, then |⟨Tδ (φ2 ζ ), ∂i Tδ (φ1 ζ )⟩ H0 ⊗H | ≤ K 1 ∥|Tδ (|ζ |)∥20 ,

(23)

where ⊗ denotes convolution.

20

The next lemma gives a bound of the derivatives of the transformation on σ1 . The proof is shown in Appendix B.

21

Lemma 3.5.

19

22

There exists constant K 1 such that for any ζ ∈ MG (Rd ), we have  d 2 d n ∑  ∑ ∑  2 ∗ ik  ⟨Tδ ζ, ∂i j Tδ (σ1 σ1 )i j ζ ⟩0 + ∂i Tδ (σ1 ζ ) ≤ K 1 ∥Tδ (|ζ |)∥20 .   

i, j=1 23

i=1

i=1

(24)

0

Now applying Lemmas 3.3–3.5 yields the following theorem. Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

11

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

Theorem 3.2. then

If µ is a measure-valued solution of the Zakai equation (8) and Z δ = Tδ µ,

1 2

˜ tδ ∥20 ≤ ∥Z 0δ ∥20 + K 1 E∥Z

∫

t

˜ δ (|µs |)∥20 ds, E∥T

(25)

3

0

where K 1 is a suitable constant.

4

˜ tδ ∥2 = A1 − 2A2 + A3 + 2A4 + A5 + A6 + A7 and Proof. Consider Eq. (14) such that E∥Z 0 A1 , . . . , A7 are defined in Eq. (15). Then by inequality (23), A2 is bounded by a constant times ∥Tδ (|µs |)∥20 . The bound for A3 follows from inequality (24). A4 follows from Lemma 3.3 and inequality (22), and A5 is bounded by Lemma 3.3, inequalities (22) and (23). The bounds for A6 and A7 follow from inequality (22). ■ Corollary 3.1. If µ is a measure-valued solution to Eq. (8) and µ0 ∈ H0 , then µt ∈ H0 a.s. ˜ t ∥2 < ∞, for all t ≥ 0. and E∥µ 0 ∫

δ→0

j

δ→0

δ→0 Rd Rd

δ→0

∑ ∑ Let µ˜ t = j ⟨µt , φ j ⟩φ j . Then µ˜ t ∈ H0 and ⟨µ˜ t , F⟩0 = j ⟨µt , φ j ⟩⟨F, φ j ⟩0 = ⟨µt , F⟩. Hence, ˜ t ∥2 < ∞. µt ∈ H0 and E∥µ ■ 0 Theorem 3.3.

Suppose that µ0 ∈ H0 . Then the solution of Zakai equation (8) is unique.

Proof. Let µ1t and µ2t be two measure-valued solution with the same initial value µ0 . By ˜ δ D t ∥2 ≤ Corollary 3.1, µ1t and µ2t ∈ H0 a.s. Let Dt = µ∫1t − µ2t . Then Dt ∈ H∫0 and E∥T 0 ∫ t t t 2 2 2 ˜ ˜ ˜ K1 E∥Tδ (|Ds |)∥0 ds. Note that by Lemma 3.1, E∥Tδ (|Ds |)∥0 ds ≤ E∥Ds ∥0 ds < ∞. 0 0 0∫ t ˜ t ∥2 ≤ K 1 ˜ s ∥20 ds, and Then letting δ → 0, by dominated convergence we have E∥D E∥D 0 Gronwall’s inequality yields Dt = 0.

■

The solution of Kushner–Stratonovich equation (10) is unique.

Proof. Let πt1 and πt2 be two solutions to the Kushner–Stratonovich equation (10). Note that for i = 1, 2 and F ∈ Cb (R) we have πti (F)µit (1) = µit (F), where µit are the corresponding solutions to Zakai equation. From Theorem 3.3, we have µ1t = µ2t , a.s. for all t ≥ 0. Hence µ2 (F) µ1 (F) for all t ≥ 0, πt1 (F) = µt1 (1) = µt2 (1) = πt2 (F), a.s. ■ t

8 9

10 11

12

13

14

15

16 17

18

19 20

21

22

0

Consequently, the uniqueness of Kushner–Stratonovich equation then follows from Theorem 3.3. Theorem 3.4.

7

0

⟨µt , F⟩. Let {φ j } be a CONS of H0 such that φ j ∈ Cb (R). Then by Fatou’s lemma ⎛ ⎞ ⎛ ⎞ ∑ ∑ 2 ˜ tδ ∥20 ≤ ∥Z 0δ ∥20 e K 1 t . E˜ ⎝ ⟨µt , φ j ⟩2 ⎠ = E˜ ⎝ lim ⟨Z tδ , φ j ⟩0 ⎠ ≤ lim inf E∥Z j

6

t

˜ sδ ∥20 ds. By Gronwall’s inequality, E∥Z ∫ ∫ δ δ 2 K1t δ 2 ˜ G δ (x − y)F(x)d xµt (dy) = we get E∥Z t ∥0 ≤ ∥Z 0 ∥0 e . Note that lim ⟨Z t , F⟩0 = lim ˜ tδ ∥2 ≤ ∥Z δ ∥2 + K 1 Proof. Eq. (25) yields that E∥Z 0 0 0

5

t

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

23

24 25

26

27 28 29 30

SPA: 3463

12

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

1

4. Large deviation principle

2

4.1. LDP result for the optimal filter We study the limiting behavior of the optimal filter with a small signal-to-noise ratio, i.e., consider the signal given in Eq. (6a) and the observation process below, for ϵ ↓ 0, ∫ t ∫ t √ b2 (s, X s )ds + σ2 (s)d Ws (26) Ytϵ = ϵ 0 ∫ t∫ ∫ t ∫0 g2 (s, u)Nλϵ (ds, du), f 2 (s, u) N˜ λϵ (ds, du) + + 0

0

U2

U\U2

where N (dt, du) is a Poisson random measure with intensity λϵ (t, x, u)ν2 (du)dt and ϵ λϵ (t, x, u) = ϵλ(t, x, u) + 1 − ϵ. For any test function F ∈ Cb2 (Rd ), set πtϵ (F) = E(F(X t )|FtY ) and define similarly to Eq. (7) an equivalent probability measure which makes the signal and observation processes independent, i.e., { ∫ t √ ϵ Λt = exp ϵ (σ2−1 (s)b2 (s, X s ))∗ d Ws 0 ∫ ∫ t∫ ϵ t −1 2 + |σ2 (s)b2 (s, X s )| ds + (1 − λ(s, X s , u))ν2 (du)ds 2 0 U } ∫ t ∫0 −1 log(ϵλ(s, X s− , u) + 1 − ϵ)Nλϵϵ (ds, du) , + λϵ

0

3 4 5 6

U

−1 Nλϵϵ (dt, du) ˜ϵ

is a Poisson random measure with intensity ϵ −1 λϵ (t, X t , u) ν2 (du)dt. where ϵ Consider P such that d P˜ ϵ /dP = (ΛϵT )−1 and µϵt (F) = E˜ ϵ (Λϵt F(X t )|FtY ), where E˜ ϵ denotes the expectation under the measure P˜ ϵ . First, we establish the existence of the small-noise optimal filter, given in Proposition 4.1. Its proof is delegated to Appendix C. Proposition 4.1. Let the signal be defined as in Eq. (6a) and the observation process, Ytϵ , be as in Eq. (26). Then we have the following small-noise Zakai equation, for any F ∈ D(L) ∫ t m ∫ t √ ∑ ϵ ϵ µt (F) = µ0 (F) + µs (LF)ds + ϵ µϵs (F(σ2−1 (s)b2 (s, ·))i )d W˜ sϵ,i 0

∫ t∫ + 0

i=1

0

µϵs− (F(λ(s, ·, µ) − 1))(ϵ Nλϵϵ (ds, du) − ν2 (du)ds). −1

(27)

U

The corresponding small-noise Kushner–Stratonovich equation is given by ∫ t πtϵ (F) = π0 (F) + πsϵ (LF)ds 0 m √ ∑ + ϵ i=1

t

∫ 0

(πsϵ (F(σ2−1 (s)b2 (s, ·))i ) − πsϵ (F)πsϵ (σ2−1 (s)b2 (s, ·))i )d Wˆ sϵ,i

ϵ ϵ ϵ πs− (Fλ(s, ·, u)) − πs− (F)πs− (λ(s, ·, u)) ϵ −1 ϵ Nˆ (ds, du), ϵ πs− (ϵλ(s, ·, u) + 1 − ϵ) 0 U ∫ t √ ϵ ϵ ˆ ˜ where Wt = Wt − ϵ πsϵ (σ2−1 (s)b2 (s, ·))ds is the innovation process and

∫ t∫

+

7

(28)

0

8

ˆ ϵ −1

ϵN

ϵ (dt, du) = ϵ Nλϵϵ (dt, du) − πt− (ϵλ(s, ·, u) + 1 − ϵ)ν2 (du)dt. −1

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

13

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

What it follows verifies Condition 2.1 such that we show the LDP for the signal described in Eq. (6a) and observation process as in Eq. (26). To proceed with the demonstration of the LDP, the assumption on the boundedness of the infinitesimal generator L is necessary. However, this condition does not contradict with the well-posedness of the optimal filtering framework, as seen in [29] and the proof of Proposition 4.1. Assumption 4.1. The test function F has continuous and bounded derivatives up to order 2. The images of G ϵ , G 0 considered in Condition 2.1 are solutions of versions of Zakai equation with or without noise respectively. Recall that M F (Rd ) denotes the collection of finite Borel measure on Rd , and µϵt is an M F (Rd )-valued process. For each ϵ > 0, let G ϵ : C([0, T ], Rm ) × M(UT ) → D([0, T ], M F (Rd )) be a measurable map, such that √ −1 (29) µϵ := G ϵ ( ϵ W˜ ϵ , ϵ Nλϵϵ ). Adopting the arguments of Theorem 3.3, the following holds.

1 2 3 4 5

6

7 8 9 10

11

12

Theorem 4.1. Under Assumptions 3.1–3.4, the unnormalized filtered µϵ defined in Eq. (29) is the unique solution of the Zakai equation (27).

13 14

Let ξ = (ψ, φ) ∈ U M . The controlled version of Eq. (27) for all F ∈ D(L), is given by ∫ t m ∫ t √ ∑ ϵ,ξ −1 ϵ,ξ i ˜ ϵ,i µs (LF)ds + ϵ µt (F) = µ0 (F) + µϵ,ξ s (F(σ2 (s)b2 (s, ·)) )d Ws 0

∫

i=1

0

t

−1 µϵ,ξ s (F(σ2 (s)b2 (s, ·)))ψ(s)ds ∫0 t ∫ ϵ −1 φ ϵ,ξ µs− (F(λ(s, ·, µ) − 1))(ϵ Nλϵ (ds, du) − ν2 (du)ds). +

+

0

(30)

U

0,ξ

Let µt

be the solution of the noise-free controlled version of Eq. (30), i.e. ∫ t ∫ t 0,ξ −1 µ0,ξ (LF)ds + µt (F) = µ0 (F) + µ0,ξ s (F(σ2 (s)b2 (s, ·)))ψ(s)ds s 0 0 ∫ t∫ + µ0,ξ s (F(λ(s, ·, µ) − 1))(φ(s, u) − 1)ν2 (du)ds. 0

(31)

U

∫ g For g : B(UT ) → [0, ∞), define ν2 (A) = A g(s, u)ν2 (du)ds for any A ∈ B(UT ) where ν2 is the intensity measure of Nλϵ in the observation process. Let G 0 : M F (Rd ) × C([0, T ], Rm ) × 0,ξ M(UT ) →∫ C([0, T ], M F (Rd )) be a measurable map such that G 0 (µ0 , w, m) = µt if · φ (w, m) = ( 0 ψ(s)ds, ν2 ) ∈ C([0, T ], Rm ) × M(UT ) for ξ = (ψ, φ), otherwise G 0 = 0. For µ ∈ C([0, T ], M F (Rd )) define I1 ≡ I1 (µ) :=

inf

∫ φ {ξ =(ψ,φ)∈Sµ :µ=G 0 ( 0· ψ(s)ds,ν2 )}

L¯ T (ξ ),

(32)

where L¯ T is defined in Eq. (4) by replacing ν with ν2 . The following theorem is the main one that establishes the uniform large deviations for the unnormalized filter. Its proof is delegated to the next section. Theorem 4.2. Let µϵ be as in Eq. (29). Then I1 , defined in (32), is a rate function and {µϵ } satisfies the large deviation principle on D([0, T ], M F (Rd )) with the rate function I1 . Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

15 16 17 18 19

20

21 22 23

24 25

SPA: 3463

14

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

Recall that P(Rd ) denotes the collection of the optimal filter πt . We define G˜ 0 : L 2 ([0, T ], R ) × [0, ∞) → C([0, T ], P(Rd )) a measurable function and suppose that π 0,ξ = G˜ 0 (ψ, φ), where π 0,ξ is the solution of the following controlled equation. { } ∫ t ( ) 0,ξ πt (F) = π0 (F) + πs0,ξ LF + σ2−1 (s)b2 (s, ·) − πs0,ξ (σ2−1 (s)b2 (s, ·)) ψ(s)F ds 0 ∫ t∫ [ 0,ξ ] + πs (Fλ(s, ·, u)) − πs0,ξ (F)πs0,ξ (λ(s, ·, u)) (φ(s, u) − 1)ν2 (du)dt. (33) m

0 1

U

For π ∈ C([0, T ], P(Rd )) define I2 ≡ I2 (π) :=

2

inf

∫ φ {ξ =(ψ,φ)∈Sπ :π=G 0 ( 0· ψ(s)ds,ν2 )}

L¯ T (ξ ).

(34)

3

For ξ = (ψ, φ) ∈ U M , let (∫ t ξ Mt = exp πs (σ2−1 (s)b2 (s, ·)ψ(s))ds 0 ) ∫ t∫ + πs (λ(s, ·, µ) − 1)(φ(s, u) − 1)ν2 (du)ds .

Lemma 4.1.

0 4

5

6

Then

0,ξ µt (F)

=

U

0,ξ ξ πt (F)Mt

satisfies the noise-free Zakai equation (31).

Proof. We first note that ( ) ∫ ξ ξ d Mt = Mt πt (σ2−1 (t)b2 (t, ·)ψ(t))dt + πt (λ(t, ·, µ) − 1)(φ(t, u) − 1)ν2 (du)dt . U

Differentiating the product,

0,ξ ξ πt (F)Mt ,

we have

0,ξ ξ πt (F)Mt πt (σ2−1 (t)b2 (t, ·)ψ(t))dt 0,ξ

ξ

+ πt (F)Mt + −

(35)

∫

πt (λ(t, ·, µ) − 1)(φ(t, u) − 1)ν2 (du)dt U ξ ξ 0,ξ πt (LF)Mt dt + πt (Fσ2−1 (t)b2 (t, ·)ψ(t))Mt dt 0,ξ ξ πt (F)Mt πt (σ2−1 (t)b2 (t, ·)ψ(t))dt

∫

] ξ [ Mt πs0,ξ (Fλ(s, ·, u)) − πs0,ξ (F)πs0,ξ (λ(s, ·, u)) (φ(s, u) − 1)ν2 (du)dt.

+ U 7 8

9 10

11 12 13

14 15

Regroup the right-hand side of Eq. (35) and then the integral form coincides with Eq. (31). ■ The following theorem establishes a uniform large deviation principle for the optimal filtering defined in Eq. (28). Theorem 4.3. Suppose πtϵ is the optimal filter described by the Kushner–Stratonovich equation (28). Then {π ϵ } satisfies the large deviation principle on D([0, T ], P(Rd )) with the rate function I2 , defined in Eq. (34). Proof. Define a map G : D([0, T ], M F (Rd ) \ {0}) → D([0, T ], P(Rd )) such that (Gµ)t = µt (F) . Then, by the contraction principle, {π ϵ = G(µϵ )} satisfies the large deviation principle µt (1) Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

15

with rate function I2′ (π) = inf{I1 (µ) : G(µ) = π }. Suppose I2′ (π ) < ∞, then for all δ > 0 there exists µ such that G(µ) = π and I1 (µ) < I2′ (π) + δ. Choose a control ξ ∈ Sπ such that · γ (ξ ) = µ, where γ is the solution of Eq. (31) and L¯ T (ξ ) < I1 (µ) + δ. Taking γ˜ = G ◦ γ we have γ˜ (ξ ) = ξ , where γ˜ is the solution of Eq. (33), and L¯ T (ξ ) < I2′ (π ) + 2δ. Thus, by definition of I2 we have I2 (π ) ≤ I2′ (π ).

(36)

Now if I2 (π ) < ∞. Then for all δ > 0, there exists ξ ∈ Sπ such that γ˜ (ξ ) = π and ξ L¯ T (ξ ) < I2 (π ) + δ. By Lemma 4.1, we have µt = Mt πt . Then µ = γ (ξ ) is the solution of Eq. (31) and G(µ) = π . Hence I2′ (π ) ≤ I1 (µ) ≤ L¯ T (ξ ) < I2 (π ) + δ. Eqs. (36) and (37) give I2 = I2′ and then the result follows.

(37) ■

4.2. Proof of Theorem 4.2 In the subsection we verify Condition 2.1 in order to show the large deviation estimates for the unnormalized filter. The following theorem establishes the existence and uniqueness of the controlled version of Zakai equation given in Eq. (30). √ −1 Theorem 4.4. Suppose G ϵ is given by µϵ = G ϵ ( ϵ W˜ ϵ , ϵ Nλϵϵ ), and ξ = (ψ, φ) ∈ U M for ∫ √ · ϵ −1 φ some M > 0. For ϵ > 0, define µϵ,ξ = G ϵ ( ϵ W˜ ϵ + 0 ψ(s)ds, ϵ Nλϵ ). Then µϵ,ξ is the unique solution of Eq. (30). Proof. Take a control ξ = (ψ, φ) ∈ U M , and consider { ∫ t∫ ∫ t 1 ϵ,ξ ϵ ˜ log φ(s, u) N¯ λϵ (ds, du, dr ) ψ(s)d Ws + Mt : = exp − √ ϵ 0 0 U×[0,ϵ −1 ] } ∫ t ∫ t∫ 1 2 − |ψ(s)| ds + (1 − φ(s, u))¯ν2 (ds, du, dr ) , 2ϵ 0 U×[0,ϵ −1 ] 0 where N¯ λϵ and the corresponding intensity ν¯ 2 satisfy the definitions of N¯ and ν¯ T in Section 2.1 respectively. By Itˆo’s formula, ( 1 1 ϵ,ξ ϵ,ξ d Mt = Mt − √ ψ(t)d W˜ tϵ − |ψ(t)|2 dt 2ϵ ϵ ) ∫ + (1 − φ(s, u))¯ν2 (ds, du, dr ) U×[0,ϵ −1 ] ∫ 1 ϵ,ξ ϵ,ξ + Mt |ψ(t)|2 dt + Mt− (φ(t, u) − 1) N¯ λϵ (ds, du, dr ). −1 2ϵ U×[0,ϵ ] Thus, ϵ,ξ

Mt

∫ t 1 =1− √ M ϵ,ξ ψ(s)d W˜ sϵ ϵ 0 s ∫ t∫ ( ) ϵ,ξ + Ms− (φ(s, u) − 1) N¯ λϵ (ds, du, dr ) − ν¯ 2 (ds, du, dr ) . 0

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

1 2 3 4 5

6

7 8 9

10

11

12

13 14 15

16 17 18

SPA: 3463

16 1

2

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx dPϵ,ξ d P˜ ϵ

ϵ,ξ

= MT , then Pϵ,ξ is an equivalent probability ∫ t ϵ,ξ 1 ϵ ϵ ˜ ˜ √ measure with respect to P . By Girsanov’s theorem, Wt := Wt + ϵ ψ(s)ds is a

Define a new probability measure Pϵ,ξ by

0

3 4 5

6 7

8 9

Pϵ,ξ -Brownian motion and N ϵ,ξ (dt, du) := Nλϵ (dt, du) − φ(t, u)ν2 (du)dt is a Pϵ,ξ -Poisson martingale measure. Hence, the desired result follows from replacing (W˜ , Nλϵ ) in Eq. (27) with (W ϵ,ξ , N ϵ,ξ ) and Theorem 4.1. ■ We now verify the existence and uniqueness of the solution of the zero-noise controlled Zakai equation (31). Theorem 4.5. For a pair of well-defined control ξ = (ψ, φ) ∈ U M , M > 0, there is a unique 0,ξ solution µt ∈ C([0, T ], M F (Rd )) of Eq. (31). Proof. We start with showing the existence of the solution. Let (∫ t 0,ξ Λt = exp σ2−1 (s)b2 (s, X s )ψ(s)ds ) ∫ t∫ 0 (λ(s, X s , µ) − 1)(φ(s, u) − 1)ν2 (du)ds . + 0

10 11

12

U 0,ξ

0,ξ

Next, we define µt = E(Λt F(X t )) and we show that it is a solution of Eq. (31). First of all, ) ( ∫ 0,ξ 0,ξ −1 dΛt = Λt σ2 (t)b2 (t, X t )ψ(t)dt + (λ(t, X t , µ) − 1)(φ(t, u) − 1)ν2 (du)dt . U

Note that for any F ∈ D(L) we have MtF ≡ F(X t ) −

∫

t

LF(X s )ds is a martingale. Then by 0

Itˆo’s formula, we have

) 0,ξ 0,ξ ( 0,ξ dΛt F(X t ) = Λt d MtF + LF(X t )dt + F(X t )Λt σ2−1 (t)b2 (t, X t )ψ(t)dt ∫ 0,ξ + F(X t )Λt (λ(t, X t , µ) − 1)(φ(t, u) − 1)ν2 (du)dt. U

Hence, ∫ t 0,ξ −1 Λt F(X t ) = F(X 0 ) + Λ0,ξ s (LF(X s ) + σ2 (s)b2 (s, X s )ψ(s))ds. 0 ∫ t∫ ∫ t 0,ξ F + Λs F(X s )(λ(s, X s , µ) − 1)(φ(s, u) − 1)ν2 (du)ds + Λ0,ξ s d Ms . 0 13 14

(38)

0

U

0,ξ

0,ξ

Taking expectation on both sides of Eq. (38) yields Eq. (31). Thus µt (F) = E(Λt F(X t )) is a solution of Eq. (31). Next, we verify the uniqueness by a similar strategy to Theorem 3.3. Recall that for any ∫ µ ∈ M F (Rd ), ⟨µ, F⟩ = R F(x)µ(d x). Then Eq. (31) is written as ∫ t ∫ t −1 0,ϵ ⟨µ0,ϵ ⟨µ0,ϵ , F⟩ = ⟨µ , F⟩ + ⟨µ , LF⟩ds + 0 t s s , F(σ2 (s)b2 (s, ·)ψ(s))⟩ds 0 0 ∫ t∫ + ⟨µ0,ϵ s , F(λ(s, ·, u) − 1)(φ(s, u) − 1)⟩ν2 (du)ds. 0

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

17

Let Z tδ = Tδ µ0,ϵ be defined as in Eq. (11). Replacing F by Tδ F and noting that ⟨Tδ ν, F⟩ = t ⟨ν, Tδ F⟩, we have that ⟨Z tδ , F⟩0 equals to ∫ t ∫ t −1 0,ϵ ⟨µ0 , Tδ F⟩ + ⟨µs , LTδ F⟩ds + ⟨µ0,ϵ s , Tδ F(σ2 (s)b2 (s, ·)ψ(s))⟩ds 0 0 ∫ t∫ 0,ϵ + ⟨µs , Tδ F(λ(s, ·, u) − 1)(φ(s, u) − 1)⟩ν2 (du)ds. (39) 0

1 2

3

4

U

Employing Lemma 3.2, Eq. (39) is given by ∫ ∫ t 1 t 2 kj ⟨∂i j Tδ (σ1ik σ1 µ0,ϵ ⟨∂i Tδ (b1i µ0,ϵ ), F⟩ ds + ⟨Z tδ , F⟩0 = ⟨Z 0δ , F⟩ + s ), F⟩0 ds s 0 2 0 0 ∫ t + ⟨Tδ (σ2−1 (s)b2 (s, ·)ψ(s)µ0,ϵ s ), F⟩0 ds 0 ∫ t∫ [ ] 0,ϵ ⟨Tδ µ0,ϵ + s , F(· + g1 (·, u))⟩0 − ⟨Tδ µs , F⟩0 ν1 (du)ds 0 U\U ∫ t∫ 1 [ 0,ϵ + ⟨Tδ µ0,ϵ s , F(· + f 1 (·, u))⟩0 − ⟨Tδ µs , F⟩0 0 U1 ] −⟨∂i Tδ ( f 1i (·, u)µ0,ϵ s ), F⟩0 ν1 (du)ds ∫ t∫ + ⟨Tδ (λ(s, ·, u) − 1)((φ(s, u) − 1)µ0,ϵ s ), F⟩0 ν2 (du)ds. 0

U 2

Furthermore, summing ⟨Z tδ , η⟩0 = ⟨Z 0δ , η⟩0 + 2

∫ 0

t

2⟨Z sδ , η⟩0 d⟨Z sδ , η⟩0 over η in a CONS of

L 2 -space, there exists a constant K 1 > 0 such that ∫ t 2 ∥Z tδ ∥20 ≤ ∥Z 0δ ∥20 + K 1 ∥Tδ µ0,ϵ s ∥0 ds.

5

6

(40)

7

0 0,ϵ Now let µ0,ϵ same initial value µ0 . Then by Corollary 3.1 1,t and µ2,t be two solutions with the ∫ t 0,ϵ 2 0,ϵ 2 and Eq. (40), we get ∥Tδ (µ0,ϵ ∥Tδ (µ0,ϵ 1,t − µ2,t )∥0 ≤ K 1 1,s − µ2,s )∥0 ds. Taking δ → 0 gives 0 ∫ t 0,ϵ 2 0,ϵ 2 0,ϵ 0,ϵ ∥µ0,ϵ − µ ∥ ≤ K ∥µ0,ϵ 1 1,t 2,t 0 1,s − µ2,s ∥0 ds, and by Gronwall’s inequality, we have µ1,t − µ2,t

= 0.

9

10

0

■

11

The following proposition demonstrates the first statement in Condition 2.1. Proposition 4.2. For M < ∞, suppose that ξn = (ψn , φn ), ξ = (ψ, φ) ∈ U M such that ξn → ξ , as n → ∞. Then ∫ · ∫ · φ φ G 0 ( ψn (s)ds, ν2 n ) → G 0 ( ψ(s)ds, ν2 ), as n → ∞. 0

0

Proof. Consider 0,ξ Λt

8

(∫

t

= exp σ2−1 (s)b2 (s, X s )ψ(s)ds 0 ) ∫ t∫ + (λ(s, X s , µ) − 1)(φ(s, u) − 1)ν2 (du)ds . 0

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

12

13 14

15

SPA: 3463

18

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

∫ 0,ξ The proof of the existence in Theorem 4.5 shows that ρt F = Rd µ0 (d x)Ex Λt F(X t ) is a solution of zero-noise version of Zakai equation (31). Furthermore, the solution is unique 0,ξ according to Theorem 4.5 and hence we get µt = ρt . Notice that, by the boundedness of −1 M σ2 , b2 and (ψn , φn ) ∈ U , there exists a constant K > 0 such that ∫ T |σ2−1 (s)b2 (s, X s )ψn (s)|ds 0

(∫ ≤ 0 1

2

T

|σ2−1 (s)b2 (s, X s )| 2 ds

)1/2 (∫

T

|ψn (s)|2 ds

)1/2

≤ (K T )1/2 M 1/2 .

0

In addition, by the inequality, for any φ ∈ [0, ∞), |φ − 1| ≤ 2 + φ log φ − φ + 1, we have ∫ T∫ 0

U

∫ ≤2 0

(41)

|(λ(s, X s , µ) − 1)(φn (s, u) − 1)|ν2 (du)ds T ∫ (2 + φn (s, u) log φn (s, u) − φn (s, u) + 1) ν2 (du)ds U

≤ 4T ν2 (U) + 2M. The dominated convergence theorem yields the convergence below, as n → ∞, (∫ t ∫ 0,ξ σ2−1 (s)b2 (s, X s )ψn (s)ds µt n (F) = µ0 (d x)Ex F(X t ) exp 0 R ) ∫ t∫ (λ(s, X s , µ) − 1)(φn (s, u) − 1)ν2 (du)ds + 0 U (∫ t ∫ µ0 (d x)Ex F(X t ) exp σ2−1 (s)b2 (s, X s )ψ(s)ds → 0 R ) ∫ t∫ + (λ(s, X s , µ) − 1)(φ(s, u) − 1)ν2 (du)ds 0

U

0,ξ

= µt (F). 3

4

■

Then the proof is completed.

The next proposition verifies the second part of Condition 2.1. d

5 6

7

Proposition 4.3. For M < ∞, let ξ ϵ = (ψ ϵ , φ ϵ ), ξ = (ψ, φ) ∈ U M be such that ξ ϵ → ξ as ϵ → 0. Then ∫ · ∫ · √ d ϵ −1 φ ϵ φ ψ ϵ (s)ds, ϵ Nλϵ ) → G 0 ( ψ(s)ds, ν2 ), as ϵ → 0. G ϵ ( ϵ W˜ ϵ + 0

8

9

0

Proof. First, we prove that the family ∫ · √ ϵ ϵ −1 φ ϵ {µϵ,ξ = G ϵ ( ϵ W˜ ϵ + ψ ϵ (s)ds, ϵ Nλϵ ), ϵ ∈ (0, ϵ0 )} 0

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

19

is tight in D([0, T ], M F (Rd )) for some ϵ0 > 0. Note that ϵ,ξ ϵ

µt

(1) = µ0 (1) +

m √ ∑ ϵ i=1

t

∫

t

∫

ϵ,ξ ϵ ,µϵ0

µs 0

1

((σ2−1 (s)b2 (s, ·))i )d W˜ sϵ,i

2

ϵ

µϵ,ξ (σ2−1 (s)b2 (s, ·))ψ ϵ (s)ds s 0 ∫ t∫ ϵ (λ(s, ·, µ) − 1) (φ ϵ (s, u) − 1) ν2 (du)ds µϵ,ξ + s 0 U ∫ t∫ ϵ −1 φ ϵ ϵ,ξ ϵ +ϵ µs− (λ(s, ·, µ) − 1)(Nλϵ (ds, du) − ϵ −1 φ ϵ (s, u)ν2 (du)ds).

+

0

3

4

5

U ϵ,ξ ϵ

By Itˆo’s formula, we get µt

(1)2 = A1t + A2t + A3t + A4t + A5t + A6t + A7t where

∫ t ϵ ϵ ϵ,ξ ϵ µϵ,ξ A1t = µ0 (1)2 , A2t = 2 (1)µϵ,ξ (σ2−1 (s)b2 (s, ·))ψ ϵ (s)ds, s s 0 ∫ t∫ ϵ ϵ 3 µϵ,ξ (1)µϵ,ξ (λ(s, ·, µ) − 1) (φ ϵ (s, u) − 1) ν2 (du)ds, At = 2 s s 0 U ∫ t⏐ ⏐2 ⏐ ϵ,ξ ϵ −1 ⏐ A4t = ϵ ⏐µs (σ2 (s)b2 (s, ·))⏐ ds, ∫ t0∫ ϵ 5 ϵ 2 µϵ,ξ At = (λ(s, ·, µ) − 1)2 φ ϵ (s, u)ν2 (du)ds, s 0

U

m √ ∑ A6t = 2 ϵ i=1

∫ t∫ [

∫ 0

t

ϵ

ϵ

µϵ,ξ (1)µϵ,ξ ((σ2−1 (s)b2 (s, ·))i )d W˜ sϵ,i , s s ϵ,ξ ϵ

ϵ,ξ ϵ

2ϵµs− (1)µs− (λ(s, ·, µ) − 1) , 0 U ] ( −1 ϵ ) ϵ φ ϵ,ξ ϵ +ϵ 2 µs− ((λ(s, ·, µ) − 1))2 Nλϵ (ds, du) − ϵ −1 φ ϵ (s, u)ν2 (du)(ds) .

A7t =

Considering Assumption 3.2 and Cauchy–Schwarz inequality, A3t is bounded by ∫

t

ϵ

µϵ,ξ (1)2 |ψ ϵ (s)| ds s 0 (∫ t )1/2 (∫ t )1/2 ϵ 2 2 ϵ ϵ,ξ ϵ 2 |ψ ≤ 2K µϵ,ξ (1) (s)| ds µ (1) ds s s ∫ t 0 ∫ t 0 ϵ ϵ ≤K µϵ,ξ (1)2 |ψ ϵ (s)|2 ds + K µϵ,ξ (1)2 ds, s s

2K

0

(42)

0

and A3t + A4t + A5t is bounded by ∫ t∫

ϵ

µϵ,ξ (1)2 |φ ϵ (s, u) − 1| ν2 (du)ds s ∫ t ∫ t∫ ϵ ϵ 2 2 + Kϵ µϵ,ξ (1) ds + 4ϵ µϵ,ξ (1)2 |φ ϵ (s, u)|ν2 (du)ds. s s

4

0

(43)

U

0

0

U 6

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

20 1

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

Set ∫ t∫

A7,1 t

) ( −1 ϵ ϵ φ ϵ,ξ ϵ ϵ,ξ ϵ 2ϵµs− (1)µs− (λ(s, ·, µ) − 1) Nλϵ (ds, du) − ϵ −1 φ ϵ (s, u)ν2 (du)(ds)

= 0

U

(44)

2

3

4

and ∫ t∫

A7,2 t

( −1 ϵ ) ϵ φ ϵ,ξ ϵ ϵ 2 µs− ((λ(s, ·, µ) − 1))2 Nλϵ (ds, du) − ϵ −1 φ ϵ (s, u)ν2 (du)(ds) . (45)

= 0

U

Combining Eqs. (42) and (43) implies that ϵ,ξ ϵ

µt

ϵ,ξ ϵ

7,2 (1)2 ≤ µ0 (1)2 + sup |A6t | + sup |A7,1 t | + sup |At | 0≤t≤T 0≤t≤T 0≤t≤T ( ) ∫ t ∫ ( ϵ ) 2 ϵ,ξ ϵ 2 ϵ 2 ϵ |φ (s, u) − 1| + K ϵ + 4ϵ φ (s, u) ν2 (du) ds. + µs (1) K |ψ (s)| + K + 0

U

Furthermore, inequality (41) gives ∫ T∫ ∫ T∫ |φ ϵ (s, u)| ν2 (du)ds ≤ (|φ ϵ (s, u) − 1| + 1) ν2 (du)(ds) 0 U 0 U ∫ T∫ ϵ ϵ ≤ (3 + φ (s, u) log φ (s, u) − φ ϵ (s, u) + 1) ν2 (du)(ds) 0

(46)

U

≤ 3T ν2 (U) + M. 5

6

Then by Gronwall’s inequality and Eq. (46), we have ( ) ϵ,ξ ϵ ϵ,ξ ϵ 7,2 µt (1)2 ≤ C0 µ0 (1)2 + sup |A6t | + sup |A7,1 | + sup |A | , t t 0≤t≤T

7

8

0≤t≤T

(47)

0≤t≤T

( ) where C0 = exp K M + K T + K ϵT + 2T ν2 (U) + M + 4ϵ 2 (3T ν2 (U) + M) . Next, we have √ ) ( ) ( ∫ T √ √ √ ϵ ϵ,ξ ϵ 2 (1) . (48) E˜ ϵ sup |A6t | ≤ 8K ϵ E˜ ϵ µs (1)4 ds ≤ 8K T ϵ E˜ ϵ sup µϵ,ξ s 0≤t≤T

0≤t≤T

0

For Eq. (44), we have ( ) ˜Eϵ sup |At7,1 |

(49)

0≤t≤T

≤ 8ϵ E˜ ϵ

(∫

T

∫

⎛0

ϵ,ξ ϵ ϵ,ξ ϵ µs− (1)2 µs− (λ(s, ·, µ)

≤ 16ϵ E˜ ϵ ⎝ sup µt

(∫

1 ˜ϵ E C

0 ϵ,ξ ϵ

sup µt

T

∫

(1) ·

0≤t≤T

≤

ϵ −1 φ ϵ Nλϵ (ds, du)

) 21

U ϵ,ξ ϵ

(

− 1)

2

(1)2

ϵ,ξ ϵ

µs− (1)2 N

ϵ −1 φ ϵ λϵ

⎞ ) 21 (ds, du) ⎠

U

)

0≤t≤T

(∫ T ∫ ) ϵ,ξ ϵ ϵ 2 ϵ −1 φ ϵ ˜ + 64ϵ C E µs− (1) Nλϵ (ds, du) 0 U ( ) 1 ˜ϵ ϵ,ξ ϵ 2 = E sup µt (1) C 0≤t≤T 2

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

21

T ∫ ϵ + 64ϵC µϵ,ξ (1)2 φ ϵ (s, u)ν2 (du)ds s 0 U ) ) ( ( 1 ϵ,ξ ϵ 2 ϵ ˜ sup µt (1) , + 64ϵC(3T ν2 (U) + M) E ≤ C 0≤t≤T

(∫

)

where C is any positive number. Eq. (44) is bounded by ) ( E˜ ϵ sup |At7,2 |

(50)

0≤t≤T

≤ E˜ ϵ

T

(∫

∫

0

(∫

U T

ϵ ∫

2

ϵ,ξ ϵ µs− ((λ(s, ·, µ)

2

− 1))

)

ϵ −1 φ ϵ Nλϵ (ds, du)

ϵ + ϵ E˜ ϵ µϵ,ξ ((λ(s, ·, µ) − 1))2 φ ϵ (s, u)ν2 (du)(ds) s 0 U (∫ T ∫ ) ϵ ϵ,ξ ϵ 2 ϵ ˜ ≤ 8ϵ E µs (1) φ (s, u)ν2 (du)(ds) 0 U ( ) (∫ T ∫ ) ϵ,ξ ϵ ϵ 2 ϵ ˜ ≤ 8ϵ E sup µt (1) φ (s, u)ν2 (du)(ds) 0≤t≤T 0 U ( ) ϵ,ξ ϵ ϵ 2 ˜ ≤ ϵ (24T ν2 (U) + 8M) E sup µt (1) ,

)

0≤t≤T

where inequality (46) was used. By Eqs. (48), (49) and (50), Eq. (47) turns out to be ( ) ϵ,ξ ϵ ϵ,ξ ϵ E˜ ϵ sup µt (1)2 ≤ C0 µ0 (1)2 0≤t≤T ( ) √ √ 1 + C0 8K T ϵ + ϵ (24T ν2 (U) + 8M) + + 64ϵC(3T ν2 (U) + M) . C Recall that the constant C can be arbitrarily large, we hence select C and ϵ0 small enough such that ) ( √ √ 1 1 C0 8K T ϵ + ϵ (24T ν2 (U) + 8M) + + 64ϵC(3T ν2 (U) + M) < . C 2 Therefore, there is a constant K 1 such that ) ( ϵ,ξ ϵ sup E˜ ϵ sup µt (1)2 ≤ K 1 . 0<ϵ≤ϵ0

1 2

3

4

(51)

5

0≤t≤T ϵ

We now establish the tightness of {µϵ,ξ }. It is well-known, e.g. see [16] that we only need to ϵ,ξ ϵ prove the tightness of {µt (F)} in D([0, T ], R) for every test function F in D(L). By Eq. (51) and the boundness of F, we have ( ) ϵ,ξ ϵ ϵ 2 ˜ sup E sup µt (F) ≤ K 2 . (52) 0<ϵ≤ϵ0

0≤t≤T

Denote ∫ t ϵ ϵ µϵ,ξ (LF)ds + µϵ,ξ (F(σ2−1 (s)b2 (s, ·)))ψ ϵ (s)ds s s 0 0 ∫ t∫ ϵ + µϵ,ξ (F(λ(s, ·, µ) − 1)) (φ ϵ (s, u) − 1) ν2 (du)ds. s

Aϵt =

t

∫

0

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

6 7 8

9

SPA: 3463

22 1

2

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

Let Mtϵ = Mtϵ,1 + Mtϵ,2 , where m ∫ t √ ∑ ϵ µϵ,ξ Mtϵ,1 = ϵ (F(σ2−1 (s)b2 (s, ·))i )d W˜ sϵ,i s i=1

3

4

and Mtϵ,2

∫ t∫

6

7

( ) ϵ −1 ,φ ϵ ϵ,ξ ϵ (ds, du) − φ ϵ (s, u)ν2 (du)ds . µs− (F(λ(s, ·, µ) − 1)) ϵ Nλϵ

= 0

5

0

U

To verify the tightness of Aϵt in D([0, T ], R), it suffices to show (see Lemma 6.1.2 of [16]) that for all δ > 0, there exists τ = τδ > 0 such that ( ) ϵ ϵ ϵ sup P˜ sup |At − At | > δ < δ. 0
0≤ϵ≤ϵ0

1

2

Then for arbitrary τ > 0 and a fixed δ > 0, ( ) ϵ ϵ ϵ sup |At1 − At2 | > δ sup P˜

(53)

0
0≤ϵ≤ϵ0

( ˜ϵ

≤ sup P

t2

sup

⏐∫ ⏐ ⏐ ⏐

t2

sup

⏐∫ ⏐ ⏐ ⏐ ⏐∫ ⏐ ⏐ ⏐

t2

0
0≤ϵ≤ϵ0

( + sup P˜ ϵ 0≤ϵ≤ϵ0

0
( + sup P˜ ϵ 0≤ϵ≤ϵ0

sup

0
t1

⏐ ⏐

ϵ µϵ,ξ (LF)ds ⏐⏐ s

ϵ,ξ ϵ

µs

t1

t1

∫

δ > 3

)

⏐ ⏐

(F(σ2−1 (s)b2 (s, ·)))ψ ϵ (s)ds ⏐⏐

δ > 3

)

ϵ

µϵ,ξ (F(λ(s, ·, µ) − 1)) s

U

δ × (φ (s, u) − 1) ν2 (du)ds| > 3

)

ϵ

:= D1 + D2 + D3 . The term D1 in Eq. (53) is bounded by )2 ( ∫ t2 9 ˜ϵ ϵ,ξ ϵ µs (LF)ds sup 2 E sup 0≤ϵ≤ϵ0 δ 0
By Eq. (52) and Assumption 4.1, we can find τ1 > 0 such that for all τ ≤ τ1 , Eq. (54) is bounded by δ/3. For D2 in Eq. (53), we note that ⏐∫ t ⏐2 ⏐ 2 ϵ,ξ ϵ ⏐ E˜ ϵ ⏐⏐ µs (F(σ2−1 (s)b2 (s, ·)))ψ ϵ (s)ds ⏐⏐ t1 (∫ t2 ⏐ ) ⏐2 ∫ t 2 ⏐ ϵ,ξ ϵ ⏐ ≤ E˜ ϵ |ψ ϵ (s)|2 ds ≤ M K 2 |t2 − t1 |. ⏐µs (F(σ2−1 (s)b2 (s, ·)))⏐ ds t1

10 11

(54)

t1

Thus, we can find τ2 > 0 such that for all τ ≤ τ2 the second term D2 in Eq. (53) is bounded by δ/3. Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

23

Now let us consider D3 in Eq. (53). Using the fundamental inequality, for a, b ∈ (0, ∞) and any c ∈ [1, ∞), ab ≤ eac + 1c (b log b − log b + 1) = eac + 1c l(b), twice (once with b = φ ϵ and once with b = 1), for any C1 ∈ (1, ∞) we have ⏐ ⏐∫ t ∫ ⏐ ⏐ 2 ϵ,ξ ϵ ϵ ⏐ ⏐ µ (55) (F(λ(s, ·, µ) − 1)) (s, u) − 1) ν (du)ds (φ 2 s ⏐ ⏐ t1 U ∫ t2 ∫ ϵ,ξ ϵ (φ ϵ (s, u) + 1)ν2 (du)ds ≤ 2 sup µt (F) 0≤t≤T t1 U ( ) M ϵ,ξ ϵ ≤ 2 sup µt (F) 2τ ν2 (U)eC1 + . C1 0≤t≤T Given any ϵ > 0, we can choose C1 such that CM ≤ ϵ, and choose τ > 0 such that 1 2τ ν2 (U)eC1 ≤ ϵ. Hence combining Eqs. (52) and (55) we proved that ⏐ ⏐∫ t ∫ ⏐ ⏐ 2 ϵ,ξ ϵ ϵ ⏐ lim sup ⏐ µs (F(λ(s, ·, µ) − 1)) (φ (s, u) − 1) ν2 (du)ds ⏐⏐ = 0. τ →0 0
t1

1 2

3

U

Thus we find τ3 > 0 such that for τ ≤ τ3 , D3 in Eq. (53) is bounded by δ/3. Consequently, the tightness of {Aϵ }ϵ≤ϵ0 follows from Eq. (53) by taking τ = min{τ1 , τ2 , τ3 }. ∫ T ϵ,ξ ϵ Next, considering M ϵ we have E˜ ϵ ⟨M ϵ,1 ⟩T ≤ ϵ K E˜ ϵ 0 µt (F)2 ds and (∫ T ∫ ) ϵ,ξ ϵ µt (F(λ(s, ·, µ) − 1))2 φ ϵ (s, u)ν2 (du)ds E˜ ϵ ⟨M ϵ,2 ⟩T = ϵ E˜ ϵ 0 U ( ) ∫ T∫ ϵ,ξ ϵ ≤ ϵ4E˜ ϵ sup µt (F)2 φ ϵ (s, u)ν2 (du)ds . 0≤t≤T

0

5

U

By Eqs. (46) and (52), we have E˜ ϵ sup0≤t≤T ⟨M ϵ ⟩t converges to 0 as ϵ → 0. Then according ϵ,ξ ϵ to Theorem 6.1.1 in [16], for any F ∈ D(L), the sequence of semimartingales µt (F) = ϵ ϵ,ξ µ0 (F) + Aϵt + Mtϵ is tight ∫ · in D([0, Tφ ], R). ϵ Next we show that G 0 ( 0 ψ(s)ds, ν2 ) is the weak limit of µϵ,ξ . Note that ( ) ∫ T ⏐ ⏐ ϵ,ξ ϵ ˜Eϵ sup ⏐ Mtϵ,1 ⏐2 ≤ ϵ K E˜ ϵ µt (F)2 ds 0≤t≤T

4

6 7 8 9

10

0

and

11

E˜ ϵ

(

⏐ ⏐2 sup ⏐ Mtϵ,2 ⏐

)

≤ ϵ K E˜ ϵ

0≤t≤T

(

ϵ,ξ ϵ

sup µt 0≤t≤T

(F)2

∫ 0

T

)

∫

φ ϵ (s, u)ν2 (du)ds . U

Therefore, Mtϵ,1 and Mtϵ,2 converge to 0 in ϵdistribution as ϵ → 0. Let (µt , ψ, φ, 0, 0) be ϵ,ξ any limit point of the tight sequence (µt , ψ ϵ , φ ϵ , Mtϵ,1 , Mtϵ,2 ) for ϵ ∈ (0, ϵ0 ). Without loss of generality, we assume that the convergence is almost sure by using the Skorokhod representation theorem. Note that for any test function F in D(L), we have ∫ t ϵ ϵ,ξ ϵ ϵ,ξ ϵ µt (F) = µ0 (F) + µϵ,ξ (LF)ds + Mtϵ,1 s 0 ∫ t ϵ,ξ ϵ + µs (F(σ2−1 (s)b2 (s, ·)))ψ ϵ (s)ds 0 ∫ t∫ ϵ ϵ,2 +Mt + µϵ,ξ (F(λ(s, ·, µ) − 1)) (φ ϵ (s, u) − 1) ν2 (du)ds. s 0,ξ

0

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

12

SPA: 3463

24

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

2

Taking ϵ → 0, along the lines of Theorem 4.5 and Proposition 4.2 we see that µ0,ξ must solve Eq. (31). Then the uniqueness of the solution to Eq. (31) completes the proof. ■

3

Acknowledgments

1

4 5 6 7 8 9

10

The authors would like to thank an anonymous reviewer for their comments which substantially improved the manuscript. VM’s research has been partially supported by Army Research Office, W911NF-17-1-0313 and the National Science Foundation, MCB-1715794 and DMS-1821241. JX’s research is supported partially by Southern University of Science and Technology Start up fund Y01286120 and National Science Foundation of China grants 61873325 and 11831010. Appendix A. Proof of Theorem 3.1

11

Proof. Considering Eq. (12) and replacing F by Tδ F in the Zakai equation (8), we have that ⟨Z sδ , F⟩0 equals to ∫ t m ∫ t ∑ ⟨µs , LTδ F⟩ds + ⟨µs , (Tδ F)(σ2−1 (s)b2 (s, ·))i ⟩d W˜ si ⟨µ0 , Tδ F⟩ + 0

i=1

∫ t∫

⟨µs− , (Tδ F)(λ(s, ·, u) − 1)⟩ N˜ (ds, du).

+ 0

0

(A.1)

U

Note that for any ν ∈ M F (Rd ), ⟨ν, LTδ F⟩ = ⟨ν,

d ∑

∂i (Tδ F)b1i +

i=1

n d 1 ∑∑ 2 jk ∂ (Tδ F)σ1ik σ1 ⟩ 2 i, j=1 k=1 i j

∫ [Tδ F(x + g1 (x, u)) − Tδ F(x)]ν1 (du)⟩

+⟨ν,

(A.2)

U\U1

∫ [Tδ F(x + f 1 (x, u)) − Tδ F(x) −

+⟨ν, U1

d ∑

∂i (Tδ F) f 1i (x, u)]ν1 (du)⟩.

i=1

Furthermore, by Lemma 3.2, we have ⟨ν,

d ∑ i=1

∂i (Tδ F)b1i

d n 1 ∑∑ 2 jk + ∂ (Tδ F)σ1ik σ1 ⟩ 2 i, j=1 k=1 i j

(A.3)

d d n ∑ 1 ∑ ∑ ik jk i = ⟨b1 ν, Tδ ∂i F⟩ + ⟨σ1 σ1 ν, Tδ ∂i2j F⟩ 2 i=1 i, j=1 k=1

=

d ∑

⟨Tδ (b1i ν), ∂i F⟩0

i=1

=−

d ∑

⟨∂i Tδ (b1i ν),

i=1

d n 1 ∑∑ jk + ⟨Tδ (σ1ik σ1 ν), ∂i2j F⟩0 2 i, j=1 k=1

d n 1 ∑∑ 2 jk F⟩0 + ⟨∂ Tδ (σ1ik σ1 ν), F⟩0 . 2 i, j=1 k=1 i j

In addition, = ⟨Tδ (σ2−1 (s)b2 (s, ·)i ν), F⟩0 , where (Tδ F)(σ2−1 (s) i b2 (s, ·)) is the ith entry of vector (Tδ F)(σ2−1 (s)b2 (s, ·)), as well as ⟨ν, (Tδ F)(λ(s, ·, u) − 1)⟩ ⟨ν, (Tδ F)(σ2−1 (s)b2 (s, ·))i ⟩

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

25

= ⟨Tδ ((λ(s, ·, u) − 1)v), F⟩0 . Suppose that F ≥ 0, then the case for general F follows from the linearity. By Fubini’s theorem, ∫ ∫ ∫ ⟨ν, Tδ F(x + g1 (x, u))ν1 (du)⟩ = Tδ F(x + g1 (x, u))ν(d x)ν1 (du) d U\U1 ∫U\U1 R = ⟨ν, Tδ F(x + g1 (x, u))⟩ν1 (du). (A.4) U\U1

Consequently, taking into consideration Eqs. (A.2), (A.3) and (A.4) into Eq. (A.1), one can get ⟨Z sδ , F⟩0 equals to ⟨Z 0δ ,

F⟩0 −

d ∫ ∑

+

d n ∫ 1 ∑∑ t 2 jk ⟨∂ Tδ (σ1ik σ1 µs ), F⟩0 ds F⟩0 ds + 2 i, j=1 k=1 0 i j

[ ] ⟨Tδ µs , F(· + g1 (·, u))⟩0 − ⟨Tδ µs , F⟩0 ν1 (du)ds

+ 0

⟨∂i Tδ (b1i µs ),

0

i=1

∫ t∫

t

U\U1

m ∫ ∑

t

⟨Tδ (σ2−1 (s)b2 (s, ·))i µs , F⟩0 d W˜ si

0

i=1

∫ t∫

⟨Tδ ((λ(s, ·, u) − 1)µs− ) , F⟩0 N˜ (ds, du)

+ 0

U

∫ t∫ [ + 0

⟨Tδ µs , F(· + f 1 (·, u))⟩0 − ⟨Tδ µs , F⟩0 −

U1

d ∑

⟨∂i Tδ ( f 1i (·, u)µs ),

] F⟩0

i=1

× ν1 (du)ds. 2

Then by Itˆo’s formula, we have ⟨Z tδ , F⟩0 equals to 2

⟨Z 0δ , F⟩0 + 2 ∫

∫ 0

t

{ ∑ d d n 1 ∑∑ 2 jk ⟨∂i j Tδ (σ1ik σ1 µs ), F⟩0 ⟨Z sδ , F⟩0 − ⟨∂i Tδ (b1i µs ), F⟩0 + 2 i=1 i, j=1 k=1

] [ δ ⟨Z s , F(· + g1 (·, u))⟩0 − ⟨Z sδ , F⟩0 ν1 (du)

+ U\U1

} [ δ ] δ i ⟨Z s , F(· + f 1 (·, u))⟩0 − ⟨Z s , F⟩0 − ⟨∂i Tδ ( f 1 (·, u)µs ), F⟩0 ν1 (du) ds

∫ + U1

+2

m ∫ ∑

∫

i=1 t⏐

0

t

⟨Z sδ , F⟩0 ⟨Tδ ((σ2−1 (s)b2 (s, ·))i µs ), F⟩0 d W˜ si

⏐ ⏐⟨Tδ (σ −1 (s)b2 (s, ·)µs ), F⟩ ⏐2 ds 2 0 ∫ 0t ∫ [ ] ( δ )2 2 δ + ⟨Z s− , F⟩0 + ⟨Tδ ((λ(s, ·, u) − 1)µs− ) , F⟩0 − ⟨Z s− , F⟩0 N˜ (ds, du) ∫0 t ∫U [ ( δ )2 2 + ⟨Z s , F⟩0 + ⟨Tδ ((λ(s, ·, u) − 1)µs ) , F⟩0 − ⟨Z sδ , F⟩0 0 U ] −2⟨Tδ ((λ(s, ·, u) − 1)µs ) , F⟩0 ⟨Z sδ , F⟩0 ν2 (du)ds. +

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

26 1

2

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

Summing over η in a CONS of H0 we have ∑ ∑ 2 ⟨Z 0δ , η⟩0 = ∥Z 0δ ∥20 , ⟨Z tδ , η⟩0 ⟨∂i Tδ (b1i µt ), η⟩0 = ⟨Z t , ∂i Tδ (b1i µt )⟩0 . η

η

Similarly, we eventually get that ∥Z tδ ∥20 equals to d ∫ t d ∑ n ∫ t ∑ ∑ jk δ 2 δ i ∥Z 0 ∥0 − 2 ⟨Z s , ∂i Tδ (b1 µs )⟩0 ds + ⟨Z sδ , ∂i2j Tδ (σ1ik σ1 µs )⟩0 ds 0

i=1

∫ t∫

[⟨Z sδ , Z˜ sδ,g1 ⟩0 − ∥Z sδ ∥20 ]ν1 (du)ds

+2 0

[⟨Z sδ , Z˜ sδ, f1 ⟩0 − ∥Z sδ ∥20 −

+ +2

U1

m ∫ ∑

+ ∫0 t ∫U +

d ∑

⟨Z sδ , ∂i Tδ ( f 1i (·, u)µs )⟩0 ]ν1 (du)ds

i=1 t

0

∫ i=1 t∫

0

(A.5)

U\U1

∫ t∫ 0

0

i, j=1 k=1

∫

⟨Z sδ , Tδ ((σ2−1 (s)b2 (s, ·))i µs )⟩0 d W˜ si

t

+ 0

∥Tδ (σ2−1 (s)b2 (s, ·)µs )∥20 ds

δ [2⟨Z s− , Tδ ((λ(s, ·, u) − 1)µs )⟩0 + ∥Tδ ((λ(s, ·, u) − 1)µs− ) ∥20 ] N˜ (ds, du)

∥Tδ ((λ(s, ·, u) − 1)µs ) ∥20 ν2 (du)ds,

U δ,g

δ, f

4

provided Z t 1 , Z t 1 are defined as Eqs. (16) and (17) respectively. Eventually, taking expectation on Eq. (A.5) gives Eq. (14). ■

5

Appendix B. Proof of Lemma 3.5

3

6 7

8

∑ Proof. Note that i,d j=1 ⟨Tδ ζ, ∂i2j Tδ (σ1 σ1∗ )i j ζ ⟩0 equals to ∫ ∫ n d ∫ ∑ ∑ jk 2 σ1ik (z)σ1 (z). ζ (dy)G δ (x − y) ζ (dz)∂xi x j G δ (x − z) dx Rd

Rd

d i, j=1 R

(B.1)

k=1

∫ Employing the semigroup property of G δ in Lemma 3.1, we have Rd(G δ (x − y)G)δ (x − z)d x = x x 1 G 2δ (y − z). Noticing that ∂i G δ (x) = − xδi G δ (x), we have ∂i2j G δ (x) = δi 2 j − i=δ j G δ (x). Due to the fact that ∂x2i x j G δ (x − z) = ∂z2i z j G δ (x − z), Eq. (B.1) is written as d ∫ ∑ d i, j=1 R

=

ζ (dy)

d ∫ ∑ d i, j=1 R

∫ Rd

ζ (dy)

ζ (dz)

n ∑ k=1

∫ Rd

ζ (dz)

kj

σ1ik (z)σ1 (z)∂z2i z j

(

∫ Rd

G δ (x − y)G δ (x − z)d x

(z i − yi )(z j − y j ) 1i= j − 4δ 2 2δ

) G 2δ (z − y)

n ∑

(B.2) jk

σ1ik (z)σ1 (z).

k=1

By symmetry of y and z in Eq. (B.2), Eq. (B.1) is further given by ( ) ∫ d ∫ ∑ (z i − yi )(z j − y j ) 1i= j ζ (dy) ζ (dz) − d 4δ 2 2δ Rd i, j=1 R n

× G 2δ (z − y)

) 1 ∑ ( ik jk jk σ1 (z)σ1 (z) + σ1ik (y)σ1 (y) . 2 k=1

(B.3)

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

27

Similarly, the second term on the left hand side of Eq. (24) can be expressed as ) ( ∫ d ∫ ∑ (z i − yi )(z j − y j ) 1i= j − − ζ (dy) ζ (dz) d 4δ 2 2δ Rd i, j=1 R × G 2δ (z − y)

n ) 1 ∑ ( ik jk jk σ1 (y)σ1 (z) + σ1ik (z)σ1 (y) . 2 k=1

(B.4)

Now adding Eqs. (B.3) and (B.4), the left-hand side of (24) is given by ( ) ∫ d ∫ ∑ (z i − yi )(z j − y j ) 1i= j − ζ (dy) ζ (dz) d 4δ 2 2δ Rd i, j=1 R n ) ) ( jk 1 ∑ ( ik jk σ1 (y) − σ1ik (z) σ1 (y) − σ1 (z) . 2 k=1 ) ( 2 2d/2 G 2δ (x), the Lipschitz continuity of σ1 and Using the identity that G δ (x) = exp − |x| 4δ Lemma 3.1, we have the quantity above estimated by ( ) ( ) ∫ d ∫ ∑ 1 |z − y|2 |z − y|2 + ζ (dy) ζ (dz) exp − d 4δ 2 2δ 4δ Rd i, j=1 R

× G 2δ (z − y)

1 × 2d/2 G 4δ (z − y) K 2 |z − y|2 2 ∫ d ∫ ∑ ≤ 4K 2 |ζ |(dy) |ζ |(dz)2d/2 G 4δ (z − y) d i, j=1 R

Rd

= d 2 22+d/2 K 2 ∥T2δ (|ζ |)∥20 ≤ d 2 22+d/2 K 2 ∥Tδ (|ζ |)∥20 . The lemma then follows with K 1 = d 2 22+d/2 K 2 .

■

Appendix C. Proof of Proposition 4.1

1

2

3

Proof. By Itˆo’s formula, { √ −1 ϵ ϵ dΛt = Λt ϵ(σ2 (t)b2 (t, X t ))∗ d Wt } ∫ ϵ −1 2 + |σ2 (t)b2 (t, X t )| dt + (1 − λ(t, X t , u))ν2 (du)dt 2 U ∫ ϵ ϵ −1 −1 2 + Λt |σ2 (t)b2 (t, X t )| dt + Λϵt (λ(t, X t− , u) − 1)ϵ Nλϵϵ (dt, du). 2 U Then ∫ t √ Λϵt = 1 + ϵ Λϵs (σ2−1 (s)b2 (s, X s ))∗ d W˜ sϵ 0 ∫ t∫ −1 + Λs− (λ(s, X s− , u) − 1)(ϵ Nλϵϵ (ds, du) − ν2 (du)ds). 0

U

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

28

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

√ ∫t By Girsanov’s theorem, W˜ tϵ = Wt + ϵ 0 σ2−1 (s)b2 (s, X s )ds, is a P˜ ϵ -Brownian motion, −1 −1 and N˜ ϵ = Nλϵϵ (dt, du) − ϵ −1 ν2 (du)dt, is a P˜ ϵ -Poisson martingale measure. The Zakai equation (27) is obtained by the same argument of Proposition 3.1. We now show the derivation of the Kushner–Stratonovich equation (28). Note that m ∫ t √ ∑ µϵt (1) = µ0 (1) + ϵ µϵs ((σ −1 (s)b2 (s, ·))i )d W˜ sϵ,i i=1

∫ t∫ + 0

0

2

µϵs− (λ(s, ·, µ) − 1)(ϵ Nλϵϵ (ds, du) − ν2 (du)ds). −1

U

By Itˆo’s formula, we have that d µϵ1(1) equals to t √ ∑ m ϵ ϵ − ϵ 2 µϵt ((σ2−1 (t)b2 (t, ·))i )d W˜ tϵ,i + ϵ 3 µϵt (σ2−1 (t)b2 (t, ·))dt µt (1) i=1 µt (1) )( ∫ ( ) 1 1 −1 + − Nλϵϵ (dt, du) − ϵ −1 ν2 (du)dt ϵ ϵ ϵ µt (1) U µt (1) + ϵµt (λ(t, ·, u) − 1) ) ∫ ( 1 1 1 1 ϵ ν2 (du)dt. + − + ϵµ (λ(t, ·, u) − 1) t ϵ U µϵt (1) + ϵµϵt (λ(t, ·, u) − 1) µϵt (1) µϵt (1)2 Hence, the quadratic variation, d[µϵ (F), µϵ (1)−1 ]t , is given by m ϵ ∑ ϵ −1 µ ((σ (t)b2 (t, ·))i )µϵt (F(σ2−1 (t)b2 (t, ·))i )dt µϵt (1)2 i=1 t 2 ) ∫ ( 1 1 −1 + − ϵµϵs− (F(λ(s, ·, µ) − 1))Nλϵϵ (ds, du). ϵ ϵ ϵ µt (1) U µt (1) + ϵµt (λ(t, ·, u) − 1)

−

1 2

µϵ (F)

Eventually, one can easily get Eq. (28) by the following the product of Itˆo’s formula d µtϵ (1) = t 1 dµϵt (F) + µϵt (F)d µϵ1(1) + d[µϵ (F), µϵ (1)−1 ]t . ■ µϵ (1) t

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

t

References [1] F. Bao, V. Maroulas, Adaptive meshfree backward SDE filter, SIAM J. Sci. Comput. 39 (6) (2017) A2664–A2683. [2] J. Bao, C. Yuan, Large deviations for neutral functional SDEs with jumps, Stochastics 87 (1) (2015) 48–70. [3] A. Budhiraja, J. Chen, P. Dupuis, Large deviations for stochastic partial differential equations driven by a Poisson random measure, Stochastic Process. Appl. 123 (2) (2013) 523–560. [4] A. Budhiraja, P. Dupuis, V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat. 47 (3) (2011) 725–747. [5] Y. Cai, J. Huang, V. Maroulas, Large deviations of mean-field stochastic differential equations with jumps, Statist. Probab. Lett. 96 (2015) 1–9. [6] C. Ceci, K. Colaneri, Nonlinear filtering for jump diffusion observations, Adv. Appl. Probab. 44 (3) (2012) 678–701. [7] C. Ceci, K. Colaneri, The Zakai equation of nonlinear filtering for jump-diffusion observations: existence and uniqueness, Appl. Math. & Optim. 69 (1) (2014) 47–82. [8] D.D. Creal, Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models, Comput. Statist. Data Anal. 52 (6) (2008) 2863–2876. [9] H. Dong, Z. Wang, D.W. Ho, H. Gao, Robust H∞ filtering for markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case, IEEE Trans. Signal Process. 59 (7) (2011) 3048–3057. [10] J. Duan, A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl. 119 (6) (2009) 2052–2081. Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

SPA: 3463

V. Maroulas, X. Pan and J. Xiong / Stochastic Processes and their Applications xxx (xxxx) xxx

29

[11] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Vol. 902, John Wiley & Sons, 2011. [12] E. Evangelou, V. Maroulas, Sequential empirical Bayes method for filtering dynamic spatiotemporal processes, Spatial Statistics 21 (A) (2017) 114–129. [13] B. Grigelionis, R. Mikulevicius, Nonlinear filtering equations for stochastic processes with jumps, in: The Oxford Handbook of Nonlinear Filtering, 2011, pp. 95–128. [14] O. Hijab, Asymptotic Bayesian estimation of a first order equation with small diffusion, Ann. Probab. (1984) 890–902. [15] M.S. Johannes, N.G. Polson, J.R. Stroud, Optimal filtering of jump diffusions: extracting latent states from asset prices, Rev. Financ. Stud. 22 (7) (2009) 2759–2799. [16] G. Kallianpur, J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, in: Lecture Notes-Monograph Series, vol. 26, 1995, iii–342. [17] K. Kang, V. Maroulas, I. Schizas, F. Bao, Improved distributed particle filters for tracking in a wireless sensor network, Comput. Statist. Data Anal. 117 (2018) 90–108. [18] T.G. Kurtz, D.L. Ocone, Unique characterization of conditional distributions in nonlinear filtering, Ann. Probab. 16 (1) (1988) 80–107. [19] T.G. Kurtz, J. Xiong, Particle representations for a class of nonlinear SPDEs, Stochastic Process. Appl. 83 (1) (1999) 103–126. [20] V. Mandrekar, T. Meyer-Brandis, F. Proske, A Bayes formula for nonlinear filtering with Gaussian and Cox noise, J. Probab. Stat. (2011). [21] V. Maroulas, Large deviations for infinite-dimensional stochastic systems with jumps, Mathematika 57 (1) (2011) 175–192. [22] V. Maroulas, A. Nebenführ, Tracking rapid intracellular movements: a Bayesian random set approach, Ann. Appl. Stat. 9 (2) (2015) 926–949. [23] V. Maroulas, X. Pan, J. Xiong, Statistical inference for the intensity in a partially observed jump diffusion, J. Math. Anal. Appl. 472 (1) (2019) 1–10. [24] V. Maroulas, J. Xiong, Large deviations for optimal filtering with fractional Brownian motion, Stochastic Process. Appl. 123 (6) (2013) 2340–2352. [25] T. Meyer-Brandis, F. Proske, Explicit solution of a non-linear filtering problem for Lévy processes with application to finance, Appl. Math. Optim. 50 (2) (2004) 119–134. [26] X. Pan, S.M. Djouadi, Estimation and identification for wireless sensor network undergoing uncertain jumps, in: 2017 Proceedings of the Conference on Control and its Applications, SIAM, 2017, pp. 46–53. [27] É. Pardoux, O. Zeitouni, Quenched large deviations for one dimensional nonlinear filtering, SIAM J. Control Optim. 43 (4) (2004) 1272–1297. [28] S. Popa, S. Sritharan, Nonlinear filtering of Itô–Lévy stochastic differential equations with continuous observations, Commun. Stoch. Anal. 3 (3) (2009) 313–330. [29] H. Qiao, J. Duan, Nonlinear filtering of stochastic dynamical systems with Lévy noises, Adv. Appl. Probab. 47 (03) (2015) 902–918. [30] M. Röckner, T. Zhang, X. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim. 61 (2) (2010) 267–285. [31] I. Sgouralis, A. Nebenführ, V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci. 10 (2) (2017) 871–899. [32] P. Shi, M. Mahmoud, S.K. Nguang, A. Ismail, Robust filtering for jumping systems with mode-dependent delays, Signal Process. 86 (1) (2006) 140–152. [33] J. Xiong, Large deviation principle for optimal filtering, Appl. Math. Optim. 47 (2) (2003) 151–165. [34] J. Xiong, An Introduction to Stochastic Filtering Theory, Vol. 18, Oxford University Press, 2008. [35] J. Xiong, J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, arXiv preprint arXiv:1605.06618. [36] X. Yang, J. Zhai, T. Zhang, Large deviations for SPDEs of jump type, Stoch. Dyn. 15 (04) (2015) 1550026.

Please cite this article as: V. Maroulas, X. Pan and J. Xiong, Large deviations for the optimal filter of nonlinear dynamical systems driven by L´evy noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.02.009.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49