Efficient nonlinear filtering of a singularly perturbed stochastic hybrid ...

LMS J. Comput. Math. 0 (2011) 1–17 Ce 2011 Authors doi:10.1112/S146115701000029X 01 02 03 04

Efficient nonlinear filtering of a singularly perturbed stochastic hybrid system

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Jun H. Park, Boris Rozovskii and Richard B. Sowers

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Abstract Our focus in this work is to investigate an efficient state estimation scheme for a singularly perturbed stochastic hybrid system. As stochastic hybrid systems have been used recently in diverse areas, the importance of correct and efficient estimation of such systems cannot be overemphasized. The framework of nonlinear filtering provides a suitable ground for on-line estimation. With the help of intrinsic multiscale properties of a system, we obtain an efficient estimation scheme for a stochastic hybrid system.

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1. Introduction

The theory of filtering gives a recursive procedure for estimating an evolving signal or state from a noisy observation process. Since the state is usually hidden and evolves according to its own dynamics, the objective is to compute the conditional distribution of the state given noisy observations. Aside from several special cases where the distribution of the state can be described with a finite number of moments or modes (for example, the celebrated Kalman filter [11] for linear systems and Bene˘s [2] and Daum [4] filters for special nonlinear systems), filtering problems in general deal with infinite-dimensional objects such as stochastic partial differential equations (PDEs) for posterior densities and, thus, require enormous amounts of computation. In this work, our interest lies in a filtering problem for stochastic hybrid systems. As stochastic hybrid systems have been used in diverse areas to model complex random phenomena which were not captured by state models with either continuous or discrete dynamics alone, their estimation has become an active research field for the last decade (cf. [9, 13, 19, 20, 24] and references therein). The computation required to solve multidimensional nonlinear filtering problems might be quite intensive. This may hinder the practical implementation of stochastic hybrid systems in time-critical applications such as target tracking, fault detection, volatility estimation in financial markets, etc. However, if the system can be cast into a multiscale setting, significant reduction in the computational complexity may be available. Singularly perturbed dynamical systems are a natural framework for dealing with multiscale systems [12, 14, 21]. In several riveting areas including biology [3], optimal control [6], and finance [26], various phenomena have been successfully modeled by singularly perturbed stochastic hybrid systems. We here consider nonlinear filtering for continuous–discrete state processes given by a pair of fast–slow processes. More specifically, we choose a fast diffusion process with a slow switching process [24]. Both the fast and slow processes are coupled so that neither process on its own is Markovian. More specifically, we will consider a state process (Xtε , Θεt ), where ε is a small parameter, ε Xt is the Rd -valued diffusion process governed by the following stochastic differential equation

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Received 17 August 2010; revised 14 March 2011. 2000 Mathematics Subject Classification 60H15 (primary), 35R60, 60G35, 34C29, 70K65 (secondary).

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(SDE): 1 1 dXtε = − ΛΘεt (Xtε − Θεt ) dt + √ dWt , ε ε

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J. H. PARK ET AL.

(1.1)

and Θεt is a (continuous-time) conditionally Markov process taking values in a finite set S. Namely, P{Θεt+∆ = θ1 | Θεt = θ2 , Xtε = x} = qθ1 ,θ2 (x)∆ + o(∆),

06 07 08

where qθ1 ,θ2 (x) > 0 if θ1 6= θ2 , X qθ,θ (x) = − θ0 ∈S qθ,θ0 (x), θ ∈ S.

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θ 0 6=θ

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dYtε = h(Xtε , Θεt ) dt + dVt

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PR OO

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In the literature, the process Θεt is often called the parameter process or simply the ‘parameter’ and we will use this term as well. We assume that the qθ,θ0 are all bounded and measurable. We also assume that for each θ ∈ S, the eigenvalues of the matrix Λθ = (λθi,i0 )16i,i0 6d are all strictly positive, and dW in (1.1) is a d-dimensional Brownian motion† . Note that under these assumptions, the distribution of Xtε quickly relaxes to a locally ‘invariant’ distribution centered at the current value of the parameter process. Parameter process Θεt evolves on a much slower scale, and its dynamics depend on X ε . The small parameter ε measures the ratio of slow and fast scales. We observe a corrupted function of X ε ; that is, the n-dimensional observation process Y ε is given by

12

for some bounded and continuous sensor function h from Rd × S to Rn and where V is a standard n-dimensional Brownian motion. We also assume that (X0ε , Θε0 ) is independent of the other sources of randomness in our system and that there is a ρ ∈ C0 (Rd × S) such that X Z P{X0ε ∈ A, Θε0 ∈ A0 } = ρ(x, θ) dx θ∈A0

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P{Θεt ∈ A | Ytε },

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x∈A

for all A ∈ B(Rd ) and A0 ⊂ S. Let us also assume that Y0ε = 0. Based on the observation process, we want to reconstruct the law of Θε ; that is, to compute

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(1.2)

where def

Ytε = σ{Ysε : 0 6 s 6 t}.

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We want to do this efficiently; that is, to find an effective filter which works as the scaling parameter ε & 0. The standard equations of filtering (which we will develop in a moment) require us to evolve a conditional law for the full state; that is, the pair (X ε , Θε ). Since the fast X ε quickly relaxes to its local invariant measure, it should not have too much information. Our objective is to show that we can track Θε without fully resolving the conditional density of X ε . Thus, instead of solving an Rd × S-dimensional Zakai equation, we can effectively solve an approximate Zakai equation whose state space is the finite set S. The resulting equation can be used in place of the original more complex equations to provide qualitatively accurate and computationally feasible descriptions either for simulation and prediction or for real-time control. † By means of various coordinate changes, we can transform the problem so that dW can have any positivedefinite covariance matrix, which may, in fact, depend on Θεt . Note that we have included Θεt -dependence in our sensor function in (1.2).

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This type of result has been covered in the literature within the framework of homogenization theory; we refer to [16] and the references therein for more detail. Methodologically, this study is similar to [15] and [16]. In the former work, the observation becomes independent of the system in the limit, while the latter has an explicit dependence on the slow variable in the limit. In [15, 16], the fast motion was a fast angular drift. In contrast to these papers, the fast motion (1.1) has both drift and diffusion, so the speed of averaging depends on a spectral gap. In this work, we show that the observation in the limit still has crucial information for estimation even though there is no explicit dependency on the system. This property is quite useful in many practical applications such as molecular motors [22] and rare-event simulations, where the observation could be given in terms of the fast variable. To the authors’ knowledge, there is no previous work in this setting.

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2. The Zakai equation 13

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Our first step is to recall the known framework of nonlinear filtering; that is, the Zakai equation [1, 18, 25]. Let us start with the generator of the fast motion for a fixed value of the parameter. For each θ ∈ S, define the second-order partial differential operator X 1 X ∂2f ∂f def (x) (x) + (Lθ f )(x) = − λθi,j (xj − θj ) 16i,j6d ∂xi 2 ∂x2i ∞

d

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16i6d

for f ∈ C (R ) and x = (x1 , x2 , . . . , xd ). We also define the generator of the parameter process as X def (Qf )(x, θ) = f (θ0 )qθ,θ0 (x)

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PR OO

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θ 0 ∈S

for all f ∈ B(Rd × S), x ∈ Rd , and θ ∈ S. These are the generators of the fast and slow motions, and we propagate densities by their adjoints; define X ∂ 1 X ∂2f def (Lθ∗ f )(x) = λθi,j ((xj − θj )f )(x) + (x) ∂xi 2 ∂x2i 16i,j6d

∞

d

16i6d

for f ∈ C (R ) and x = (x1 , x2 . . . xd ) and

def

(Q ∗ f )(x, θ) =

X

f (θ0 )qθ0 ,θ (x)

θ 0 ∈S

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d

d

for all f ∈ B(R × S), x ∈ R , and θ ∈ S. The Zakai equation in our setting is given by 1 duε (t, x, θ) = Lθ∗ uε (t, x, θ) dt + Q ∗ uε (t, x, θ) dt + uε (t, x, θ)h(x, θ)T dYtε ε 1 ∗ ε = Lθ u (t, x, θ) dt + Q ∗ uε (t, x, θ) dt + uε (t, x, θ)h(x, θ)T dVt ε

(2.1)

+ uε (t, x, θ)h(x, θ)T h(Xtε , Θεt ) dt, uε (0, x, θ) = ρ(x, θ). Under the assumptions of this paper, one could show that for every ε > 0, R P ε θ∈A0 x∈A u (t, x, θ) dx ε ε 0 ε R P{Xt ∈ A, Θt ∈ A | Yt } = P , A ∈ B(Rd ), A0 ⊂ S ε (t, x, θ) dx u d θ∈S x∈R with probability 1. Note that in the literature on nonlinear filtering the Zakai equation is usually considered on a new probability space, where Ytε is a Brownian motion. This space changes when the

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parameter ε changes. This is inconvenient for our purposes. Therefore, we will consider (2.1) on a fixed probability space for all ε. To see the asymptotic behavior of the solution of (2.1) as ε & 0, we construct the invariant measure of the fast motion. For θ ∈ S, define Z∞ def Bθ = exp[−Λθ s] exp[−ΛTθ s] ds s=0

and   1 0 −1 exp − (x − θ) Bθ (x − θ) , µθ (x) = p 2 (2π)d det Bθ def

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J. H. PARK ET AL.

1

Then

Z Lθ∗ µθ (x)

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=0

and

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for all θ ∈ S. We want to find the effective behavior of the jumps by averaging over the invariant distribution of the fast motion. For each θ1 and θ2 in S, define Z def q¯θ1 ,θ2 = qθ1 ,θ2 (x)µθ1 (x) dx;

PR OO

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µθ (x) dx = 1 Rd

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x ∈ Rd

x∈Rd

we still have that q¯θ1 ,θ2 > 0 if θ1 6= θ2 and q¯θ,θ = − def (Q¯∗ f )(θ) =

X

P

¯θ,θ0 θ 0 ∈S q θ 0 6=θ

for all θ ∈ S. Define

f (θ0 )¯ qθ0 ,θ

θ 0 ∈S

for all f ∈ B(S). We also need to average the sensor function and the initial condition; for each θ ∈ S, define Z Z def def ¯ h(θ) = h(x, θ)µθ (x) dx and ρ¯(θ) = ρ(x, θ)µθ (x) dx. (2.2) x∈Rd

x∈Rd

ε

The effective Zakai equation for θ should be given by averaging the coefficients of (2.1) with respect to the invariant measure of the fast motion; that is, ¯ T dY ε , dv ε (t, θ) = Q¯∗ v ε (t, θ) dt + v ε (t, θ)h(θ) t (2.3) ε v (0, θ) = ρ¯(θ) (see also Il’in et al. [10]). Note that while we can find the effective behavior of the x variable in the coefficients of the Zakai equation (2.1), we cannot really average the observations since they are the inputs to the system. Thus, (2.3) is not a true Zakai equation; this is clear upon writing ¯ T {h(X ε , Θε ) dt + dVt } dv ε (t, θ) = Q¯∗ v ε (t, θ) dt + v ε (t, θ)h(θ) t t ¯ T {h(Θ ¯ ε ) dt + dVt } = Q¯∗ v ε (t, θ) dt + v ε (t, θ)h(θ) t

¯ T {h(X ε , Θε ) − h(Θ ¯ ε )} dt. + v ε (t, θ)h(θ) t t t The last term captures the deviation from a true Zakai equation. Let us now collect our thoughts and formulate our results. For each t > 0, define R P ε def θ∈A x∈Rd u (t, x, θ) dx ε R ; A ⊂ S; πt (A) = P ε θ∈S x∈Rd u (t, x, θ) dx then πtε (A) = P{Θεt ∈ A | Ytε }. Let us also define P v ε (t, θ) def π ¯tε (A) = Pθ∈A ε ; θ∈S v (t, θ)

A ⊂ S.

We note that the evolution of v is essentially an |S|-dimensional SDE, whereas that of uε is a stochastic partial differential equation (SPDE). Thus, π ¯tε is a much simpler process to compute.

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Note also that the only dependence of π ¯ ε on ε is through the observation process Y ε ; the actual dynamics are ε-independent. Our main result is that as ε & 0, π ¯ ε is a good substitute for π ε .

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Theorem 2.1. For each t > 0,

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lim E[dP(S) (πtε , π ¯tε )] = 0.

ε→0

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Here dP(S) is the Prohorov metric on P(S).

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3. Asymptotic analysis

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Several preliminary steps will help make the proof of Theorem 2.1 more natural. Firstly, we will consider a slightly more intrinsic formulation of the Zakai equation. Secondly, we will rewrite (2.3) in a way which facilitates comparison to the dynamics of the original problem. To begin, let us define

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then

Z

uε (t, x, θ) ; µθ (x)

PR OO

def

u ˜ε (t, x, θ) =

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t > 0, x ∈ Rd , θ ∈ Rd ;

Z

ε

u ˜ε (t, x, θ)f (x)µθ (x) dx,

u (t, x, θ)f (x) dx =

x∈Rd

t > 0, θ ∈ S

x∈Rd

for all f ∈ Cc∞ (Rd ) and

def πtε (A) =

R P ˜ε (t, x, θ)µθ (x) dx θ∈A x∈Rd u R P ˜ε (t, x, θ)µθ (x) dx θ∈S x∈Rd u

A ⊂ S.

In other words, let us make our reference measure the invariant measure of the fast motion. If we define def 1 L˜θ∗ f = (L ∗ (f µθ )) µθ θ for f ∈ C ∞ (Rd ) and x = (x1 , x2 , . . . , xd ) and def

(Q˜∗ f )(x, θ) =

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1 X µθ0 (x)f (θ0 )qθ0 ,θ (x) µθ (x) 0 θ ∈S

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for all f ∈ B(Rd × S), x ∈ Rd , and θ ∈ S, we then have that 1 ˜∗ ε L u ˜ (t, x, θ) dt + Q˜∗ u ˜ε (t, x, θ) dt + u ˜ε (t, x, θ)h(x)T dYtε ε θ 1 = L˜θ∗ u ˜ε (t, x, θ) dt + Q˜∗ u ˜ε (t, x, θ) dt + u ˜ε (t, x, θ)h(x)T h(Xtε , Θεt ) dt ε +u ˜ε (t, x, θ)h(x)T dVt , ρ(x, θ) u ˜ε (0, x, θ) = . µθ (x)

d˜ uε (t, x, θ) =

(3.1)

We next observe that v ε satisfies a similar SPDE. Since µθ is the invariant measure for the generator Lθ , Lθ∗ µθ ≡ 0. Defining 1 : Rd → R as 1 : Rd 7→ 1, we thus have that L˜θ∗ 1 ≡ 0. Hence, the function (t, x, θ) 7→ v ε (t, θ) = v ε (t, θ)1(x) from R+ × Rd × S satisfies 1 ¯ T dY ε , dv ε (t, θ) = (L˜θ∗ v ε )(t, x, θ) dt + Q¯∗ v ε (t, θ) dt + v ε (t, θ)h(θ) t ε ε v (0, θ) = ρ¯(θ).

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We also note, of course, that

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π ¯tε (A)

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R P ε θ∈A x∈Rd v (t, θ)µθ (x) dx R ; =P ε θ∈S x∈Rd v (t, θ)µθ (x) dx

A ⊂ S.

Our immediate goal is then the following lemma.

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Lemma 3.1. For each t > 0, we have that  X Z ε ε 2 lim |˜ u (t, x, θ) − v (t, θ)| µθ (x) dx = 0. E ε&0

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θ∈S

x∈Rd

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This will be a crucial step towards the proof of Theorem 2.1; see Section 4. The value of the linear dynamics of (1.1) is that they allow us to get explicit rates at which the fast motion achieves its stationary behavior (if the dynamics of X ε had nonlinearities, one could in general get only abstract bounds on the rate of convergence to a stationary distribution). To formalize the notation surrounding this, fix θ ∈ S. For each x ∈ Rd , define Zt θ,x def ˜ Xt = θ + exp[−Λθ t]x + exp[−Λθ (t − s)] dWs , t > 0.

PR OO

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s=0

˜ θ,x satisfies the SDE Then X

˜ tθ,x = −Λθ (X ˜ tθ,x − θ) dt + dWt , dX θ,x ˜ X0 = x;

this is a Markov process with generator Lθ . For t > 0 and f ∈ B(Rd ), define def ˜ tθ,x )]. (Ptθ f )(x) = E[f (X

This is the semigroup on B(Rd ) generated by Lθ (and of course limt&0 Ptθ f = f pointwise). We can write a kernel representation for Ptθ . For every t > 0, define Zt Zt def Bθ (t) = exp[−Λθ (t − s)] exp[−ΛTθ (t − s)] ds = exp[−Λθ s] exp[−ΛTθ s] ds. s=0

s=0

Of course, limt→∞ Bθ (t) = Bθ . Define def

pθx (t, z) = (2π)−d/2 (det Bθ (t))−1/2

× exp[− 21 (z − (θ + exp[−Λθ t]x))T Bθ−1 (t)(z − (θ + exp[−Λθ t]x))] for all t > 0 and x and z in Rd . Then

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Z

(Ptθ f )(x) =

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(3.2)

z∈Rd

pθx (t, z)f (z) dz

for all t > 0, x ∈ Rd , and f ∈ B(Rd ). For each f ∈ Cc∞ (Rd ), let us next define Z 1 def (Stθ f )(x) = f (z)pθz (t, x)µθ (z) dz µθ (x) z∈Rd for all t > 0 and x ∈ Rd , so that Z Z (Stθ f )(x)g(x)µθ (x) dx = x∈Rd

z∈Rd

f (z)µθ (z)(Ptθ g)(z) dz

(3.3)

for all f and g in Cc∞ (Rd ) and all t > 0. In fact, we should think of Stθ as the adjoint of Ptθ . For all f and g in Cc (Rd ), define Z def hf, giθ = f (x)g(x)µθ (x) dx x∈Rd

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def p and define kf kθ = hf, f iθ for all f ∈ Cc (Rd ); this is a norm on Cc (Rd ). Define L2 (µθ ) as the closure of Cc (Rd ) with respect to this norm; then h·, ·iθ can uniquely be extended to an inner product on L2 (θ). Then (3.3) can be written as

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hStθ f, giθ = hf, Ptθ giθ

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for all f and g in Cc (Rd ) and all t > 0.

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def

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Lemma 3.2. Fix f ∈ Cc∞ (Rd ). For t > 0 and x ∈ Rd , define w(t, x) = (Stθ f )(x). Then w satisfies the PDE ∂w (t, x) = L˜θ∗ w(t, x) t > 0, x ∈ Rd , ∂t w(0, ·) = f. Proof. The proof is fairly standard, but, for the sake of completeness, we will outline it. Fix g ∈ Cc∞ (Rd ) and set Z def ξt = w(t, x)g(x)µθ (x) dx = hf, Ptθ giθ

PR OO

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x∈Rd

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for all t > 0. Clearly,

Z

lim ξt =

19

t&0

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Secondly,

f (x)g(x)µθ (x) dx.

x∈Rd

ξ˙t = hf, (Ptθ (Lθ g))iθ

= hStθ f, Lθ giθ Z = w(t, x)µθ (x)(Lθ g)(x) dx d Zx∈R = (L˜θ∗ w)(t, x)g(x)µθ (x) dx. x∈Rd

Collecting things together, we get the result.

An important result which will form the basis for our averaging estimates is the following.

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Lemma 3.3. There are a K > 0 and a ν > 0 such that, for all t > 0, kPtθ f − hf, 1iθ 1kθ ≤ Ke−νt kf kθ

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for all f ∈ L2 (θ) and all θ ∈ S.

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Proof. This is Proposition 4.3 of [7].

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This gives us our central averaging estimate (the proof of which is also fairly standard).

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Proposition 3.4.

There are a K > 0 and a ν > 0 such that kStθ f kθ ≤ Ke−νt kf kθ

for all θ ∈ S, t > 0, and f ∈ Cc (Rd ) such that hf, 1iθ = 0. Proof. Fix g ∈ Cc∞ (Rd ). Then, since hf, 1iθ = 0, hStθ f, giθ = hf, Ptθ giθ − hf, Ptθ g − hg, 1iθ 1iθ ,

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so |hStθ f, giθ | ≤ Ke−νt kf kθ kgkθ .

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Take g = Stθ f , and the claim follows. To proceed, we make the decomposition

06

uε (t, x, θ) = v ε (t, θ) + Φε (t, x, θ) + Rε (t, x, θ),

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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

where dΦε (t, x, θ) =

1 ˜∗ ε L Φ (t, x, θ) dt + {(Q˜∗ v)(t, x, θ) − (Q¯∗ v)(t, θ)} dt ε θ ε ε T ¯ + v ε (t, θ){h(x, θ) − h(θ)}h(X t , Θt ) dt T ¯ + v ε (t, θ){h(x, θ) − h(θ)} dVt ,

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Φε (0, x) = ρ(x, θ) − ρ¯(θ), 1 dRε (t, x, θ) = L˜θ∗ Rε (t, x, θ) dt + Q˜∗ Rε (t, x, θ) dt ε + Rε (t, x, θ)h(x, θ)h(Xtε , Θεt )T dt + Rε (t, x, θ)h(x, θ)T dVt + Q˜∗ Φε (t, x, θ) dt + Φε (t, x, θ)h(x, θ)h(Xtε , Θεt )T dt + Φε (t, x, θ)h(x, θ)T dVt ,

Rε (0, x) = 0.

We want to show that Φε is small since it reflects an averaging correction. We will then use ε standard SPDE estimates to show that Rε , which is driven P4by Φ ε, is also small. ε ε Let us further split Φ into several parts, writing Φ = i=1 Φi , where ∂Φε1 1 (t, x, θ) = L˜θ∗ Φε1 (t, x, θ), ∂t ε Φε1 (0, x, θ) = ρ(x, θ) − ρ¯(θ),

1 ∂Φε2 (t, x, θ) = L˜θ∗ Φε2 (t, x, θ) + {(Q˜∗ v)(t, x, θ) − (Q¯∗ v)(t, θ)}, ∂t ε Φε2 (0, x, θ) = 0,

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1 ∂Φε3 ε ε T ¯ (t, x, θ) = L˜θ∗ Φε3 (t, x, θ) + v ε (t, θ){h(x, θ) − h(θ)}h(X t , Θt ) , ∂t ε

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1 T ¯ dΦε4 (t, x, θ) = L˜θ∗ Φε4 (t, x, θ) + v ε (t, θ){h(x, θ) − h(θ)} dVt , ε

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Φε4 (0, x, θ) = 0.

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Let us start to bound the various terms. Lemma 3.5. For each t > 0, we have that Z  lim max E |Φε1 (t, x, θ)|2 µθ (x) dx = 0. ε&0 θ∈S

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(3.5)

Φε3 (0, x, θ) = 0,

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(3.4)

PR OO

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x∈Rd

Proof. For convenience, define def

ρˇ(x, θ) = ρ(x, θ) − ρ¯(θ).

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By definition, hˇ ρ(·, θ), 1iθ = 0. We then have that

02

θ Φε1 (t, x, θ) = (St/ε ρˇ(·, θ))(x),

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so kΦε (t, ·)kθ ≤ Ke−νt/ε kˇ ρ(·, θ)kθ .

05 06 07

This gives the claimed result.

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Before proceeding, we need some uniform bounds on v ε .

10

Lemma 3.6. For each t > 0, we have that sup E[|v ε (s, θ)|2 ] < ∞.

12

ε∈(0,1) θ∈S 0≤s≤t

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Proof. Define

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PR OO

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F

11

def

V ε (t) =

17

X

(v ε (t, θ))2 .

θ∈S

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We have that dV ε (t) = 2

v ε (t, θ)¯ qθ0 ,θ v ε (t, θ0 ) dt + 2

θ,θ 0 ∈S

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X

Define now

23

X

ε ¯ h(θ)(v (t, θ))2 dYtε +

θ∈S

X

ε ¯ (h(θ)v (t, θ))2 dt.

θ∈S

 X  X 0 2 Q = sup q¯θ,θ0 f (θ)f (θ ) : f ∈ B(S), f (θ) = 1 ; def

24

θ,θ 0 ∈S

25 26

then Q < ∞. Thus,

27

E[V ε (t)] ≤

28

X

Zt

ρ¯(θ)2 + {2Q + 8khkB }K

E[V ε (s)] ds.

s=0

θ∈S

29 30

θ∈S

Gronwall’s inequality then implies the claim.

31 32 33 34

Lemma 3.7. For each t > 0, we have that Z  lim sup E |Φε2 (t, x, θ)|2 µθ (x) dx = 0. ε&0 θ∈S

x∈Rd

35 36

Proof. Let us start by writing (Q˜∗ v ε )(t, x, θ) − (Q¯∗ v ε )(t, θ) =

37 38 39

42 43 44 45 46

qˇθ0 ,θ (x)v ε (t, θ0 ),

θ 0 ∈S

where

40 41

X

def

qˇθ0 ,θ (x) =

µθ0 (x) qθ0 ,θ (x) − q¯θ0 ,θ µθ (x)

for all θ and θ0 in S and all x ∈ Rd . Note that hˇ qθ0 ,θ , 1iθ = 0 for all θ and θ0 in S. We then have that X Zt θ Φε2 (t, x, θ) = (S(t−s)/ε qˇθ0 ,θ )(x)v ε (s, θ0 ) ds. θ 0 ∈S

s=0

Marked Proof

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J. H. PARK ET AL.

Thus,

X Zt

02

kΦε2 (t,

03

·, θ)kθ ≤ K

04 05

≤ Kε

06 07

e−ν(t−s)/ε kˇ qθ0 ,θ kθ |v ε (s, θ0 )| ds

s=0

θ 0 ∈S

X

Z t/ε e−νs |v ε (t − sε, θ0 )| ds.

kˇ qθ,θ0 k s=0

θ 0 ∈S

The claim follows.

08 09

Let us next define the function

10

def ˇ ¯ h(x, θ) = h(x, θ) − h(θ);

11

13

ˇ j (·, θ), 1iθ = 0 for all j ∈ {1, 2, . . . , n}. from (2.2), we have that hh ε The bound on Φ3 follows from arguments similar to those of Lemma 3.7.

F

12

14

16 17

Lemma 3.8. For each t > 0, we have that Z  ε 2 lim max E |Φ3 (t, x, θ)| µθ (x) dx = 0. ε&0 θ∈S

18 19

Proof. We have that

20

Φε3 (t,

21

24 25 26 27 28 29 30

x, θ) =

x∈Rd

n Zt X

θ ˇ j )(x)hj (X ε , Θε )v ε (s, θ) ds. (S(t−s)/ε h s s

j=1 s=0

22 23

PR OO

15

x ∈ Rd , θ ∈ S;

Thus,

kΦε3 (t, ·, θ)kθ ≤

n Zt X

θ ˇ j k|hj (X ε , Θε )||v ε (s, θ)| ds kS(t−s)/ε h s s

j=1 s=0 n Zt X

ˇ j kθ |hj (X ε , Θε )||v ε (s, θ)| ds e−ν(t−s)/ε kh s s

≤K

j=1 s=0 n X

Zt

ˇ j kθ kh

≤ Kε

ε e−νs |hj (Xt−sε , Θεt−sε )||v ε (t − sε, θ)| ds.

s=0

j=1

31 32 33 34 35 36 37

This gives us the result. The bound on Φε4 follows from similar arguments once we use Ito’s isometry. Lemma 3.9. For each t > 0, we have that Z  lim sup E |Φε4 (t, x, θ)|2 µθ (x) dx = 0. ε&0 θ∈S

38 39

x∈Rd

Proof. We have that

40 41

Φε4 (t,

x, θ) =

The Ito isometry thus gives us that

44 45 46

θ ˇ j )(x)v ε (s, θ) dV j . (S(t−s)/ε h s

j=1 s=0

42 43

n Zt X

E[kΦε4 (t, ·, θ)k2θ ] ≤

n Zt X

θ ˇ j k2 E[|v ε (s, θ)|2 ] ds kS(t−s)/ε h

j=1 s=0

Marked Proof

Ref: 53180

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EFFICIENT NONLINEAR FILTERING FOR A HYBRID SYSTEM

01

≤ K2

02 03 04

≤ Kε

05

n Zt X

Page 11 of 17

ˇ j kθ E[|v ε (s, θ)|2 ] ds e−2ν(t−s)/ε kh

j=1 s=0 n X

Z t/ε

ˇ j k2 kh θ

e−2νs E[|v ε (t − sε, θ)|2 ] ds. s=0

j=1

06 07 08

The result follows. Summarizing, we have that Φε is small.

09

11 12

Lemma 3.10. For each t > 0, we have that Z  lim sup E |Φε (t, x, θ)| dx dt = 0. ε&0 θ∈S

13

x∈Rd

14

16 17 18 19 20 21 22 23 24 25 26 27 28

Proof. Collect Lemmas 3.5, 3.7, 3.8, and 3.9.

PR OO

15

3.1. Proof of Lemma 3.1

By standard SPDE methods [18], we have that Z X 2X t ε 2 E[kR (t, ·, θ)kθ ] ≤ E[hLθ Rε (s, ·, θ), Rε (s, ·, θ)iθ ] ds ε s=0 θ∈S θ∈S X Zt +2 E[hQRε (s, ·, θ), Rε (s, ·, θ)iθ ] ds θ∈S

+2

30

θ∈S 1≤j≤n

+2

+2

32

+

33

37 38

E[hQRε (s, ·, θ), Φε (s, ·, θ)iθ ] ds

s=0

X Zt

E[hRε (s, ·, θ), Φε (s, ·, θ)hj (·, θ)iθ hj (Xsε , Θεs )] ds

s=0

X Zt

1≤j≤n s=0

34

36

E[hRε (s, ·, θ), Rε (s, ·, θ)hj (·, θ)iθ hj (Xsε , Θεs )] ds

s=0

X Zt

θ∈S 1≤j≤n

31

35

s=0

X Zt

θ 0 ∈S

29

F

10

2   X

ε ε

E {R (s, ·, θ) + Φ (s, ·, θ)}hj (·, θ)

ds. θ

θ∈S

The bound on the 1/ε term is standard. For any f ∈ C ∞ (Rd ), we have that X  ∂f 2 > 2f Lθ f Lθ f 2 = 2f Lθ f + ∂xi 1≤i≤d

39 40 41 42 43 44 45 46

and, hence, since µθ is an invariant distribution, Z Z 1 hLθ f, f iθ = f (x)Lθ f (x)µθ (x) dx ≤ (Lθ f 2 )(x)µθ (x) dx = 0. 2 x∈Rd x∈Rd Let us also define  X  X X def 0 d 2 2 Q = sup qθ,θ0 (x)f (θ)g(θ ) : x ∈ R , f, g ∈ B(S), f (θ) = g (θ) = 1 ; θ,θ 0 ∈S

θ∈S

Marked Proof

Ref: 53180

θ∈S

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Page 12 of 17 01 02 03

J. H. PARK ET AL.

then Q < ∞. Hence, X E[kRε (t, ·, θ)k2θ ] θ∈S

05

≤ 2Q

06

+2

 X

11

θ∈S

13

+

 X

s=0

X Z t

1≤j≤n

θ∈S

18 19

30 31 32 33

s=0

θ∈S

X Z t

 E[kΦε (s, ·, θ)k2θ ] ds

s=0

s=0

θ∈S

ε

E[kR (s, ·,

θ)k2θ ]

X Zt

ds +

s=0

θ∈S

ε

E[kΦ (s, ·,

θ)k2θ ]

 ds .

s=0

θ∈S

From standard calculations, we have that v ε (t, θ) > 0 for all t > 0 and θ ∈ S.

28 29

X Zt

We finally want to return to our analysis of πtε . We want to use Lemma 3.1 to show that πtε and π ¯tε are close. To start, define X def V¯ ε (t) = v ε (t, θ).

26 27

E[kRε (s, ·, θ)k2θ ] ds +

4. Proof of Theorem 2.1

22

25

s=0

Apply Gronwall’s inequality and use Lemma 3.10 to bound Rε . Combining things together, the claim follows.

21

24

khj k2B

1≤j≤n

20

23

 Zt

 E[kΦε (s, ·, θ)k2θ ] ds

PR OO

+2

X Zt θ∈S

khj k2B

 X

17

E[kRε (s, ·, θ)k2θ ] ds

s=0

E[kRε (s, ·, θ)k2θ ] ds +

15 16

 X Zt θ∈S

X Z t +Q

12

14

khj k2B

1≤j≤n

09 10

E[kRε (s, ·, θ)k2θ ] ds

s=0

θ 0 ∈S

07 08

X Zt

F

04

Lemma 4.1. For all t > 0, ε ∈ (0, 1), and L > 0, sZ 2 X ε ε dP(S) (πt , π ¯t ) ≤ ε |˜ uε (t, x, θ) − v ε (t, θ)|2 µθ (x) dx V (t) x∈Rd

(4.1)

θ∈S

if V¯ ε (t) > 0.

34 35 36 37 38

Proof. For each t > 0 and ε > 0, define the random σ-finite measures πt◦,ε and π ¯t◦,ε on d d (R × S, B(R × S)) as XZ def ◦,ε πt (A) = u ˜ε (t, x, θ)µθ (x) dx, d θ∈A x∈R

39

def

π ¯t◦,ε (A) =

40

θ∈A

41 42 43 44 45 46

X

v ε (t, θ) =

XZ θ∈A

v ε (t, θ)µθ (x)dx x∈Rd

for all A ⊂ S. Then πt◦,ε (A) π ¯t◦,ε (A) − πt◦,ε (S) π ¯t◦,ε (S) ◦,ε π (A) 1 = ◦,ε t ◦,ε {¯ πt◦,ε (S) − πt◦,ε (S)} + ◦,ε {¯ π ◦,ε (A) − πt◦,ε (A)}. πt (S)¯ πt (S) π ¯t (S) t

πtε (A) − π ¯tε (A) =

Marked Proof

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EFFICIENT NONLINEAR FILTERING FOR A HYBRID SYSTEM

01

For any A0 ⊂ S,

Z X |πt◦,ε (A0 ) − π ¯t◦,ε (A0 )| =

02 03

θ∈A0

x∈Rd

{˜ uε (t, x, θ) − v ε (t, θ)}µθ (x) dx

X Z ε ε {˜ u (t, x, θ) − v (t, θ)}µθ (x) dx ≤ d θ∈S x∈R sZ X |˜ uε (t, x, θ) − v ε (t, θ)|2 µθ (x) dx. ≤

04 05 06 07 08

θ∈S

09 10

Page 13 of 17

x∈Rd

Of course, π ¯t◦,ε (S) = V¯ ε (t). The claim follows.

11

14 15

In fact, we have that

16

dV¯ ε (t) =

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

F

13

Proof of Theorem 2.1. From Lemma 3.1, it suffices to show that   1 < ∞. sup E ¯ ε (V (t))2 ε∈(0,1) X

(4.2)

 ε ¯ h(θ)v (t, θ) dYtε .

PR OO

12

θ∈S

Of course, V¯ ε (0) = 1. For each n ∈ N, define   1 def ε ¯ τn = inf t > 0 : V (t) ≤ . n def

Define τ = limn→∞ τn = inf{t > 0 : V¯ ε (t) = 0}. For t ∈ [0, τ ), define P ε ¯ def θ∈S h(θ)v (t, θ) ; a(t) = P ε θ∈S v (t, θ) since the v ε (t, θ) are non-negative, we have that

¯ ka(t)kRn ≤ sup kh(θ)k Rn .

(4.3)

θ∈S

For every n ∈ N, we have that Z t∧τn  Z t∧τn Z 1 t∧τn V¯ ε (t ∧ τn ) = exp a(s)T h(Xsε , Θεs ) ds + a(s)T dVs − ka(s)k2 ds . 2 s=0 s=0 s=0 Letting n → ∞, we get that  Z t∧τ Z Z t∧τ 1 t∧τ V¯ ε (t ∧ τ ) = exp a(s)T h(Xsε , Θεs ) ds + a(s)T dVs − ka(s)k2 ds . 2 s=0 s=0 s=0

35

38

Since the exponential term is finite thanks to (4.3), we must have that τ > t. Thus,  Z t∧τ  Z t∧τ Z t∧τ 1 T ε ε T 2 = exp −2 a(s) h(X , Θ ) ds − 2 a(s) dV + ka(s)k ds . s s s (V¯ ε (t))2 s=0 s=0 s=0

39

This implies (4.2), completing the proof.

36 37

40 41 42 43 44 45 46

5. A numerical example We consider a simple system in a continuous–discrete set-up; the fast variable X ε ∈ R2 is given as a continuous process 1 1 dXtε = − ΛΘεt (Xtε − Θεt ) dt + √ dWt , X0ε = ξ, ε ε

Marked Proof

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Page 14 of 17

J. H. PARK ET AL.

01

6

02

4

4

03

2

04

06 07

Y2

05

X2

2 0

0 –2

–2

08 09 10

–4

–4 –6

–4

–2

0

4

6

8

–6

4

15

3

16

2

17

1

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

0 Y1

2

4

6

(b) Observation, Y

PR OO

14

19

–2

F

(a) Signal, X

13

18

–4

X1

11 12

2

0

0.5

1

1.5 Time

2

2.5

3

(c) 1 : (2, 2), 2 : (–2, 2), 3 : (– 2, –2), 4 : (2, –2)

Figure 1. Signal and observation.

while the slow variable Θε is a jump process with a finite state space S = {(2, 2), (−2, 2), (−2, −2), (2, −2)}. The matrix Λθ is given for each θ ∈ S as     def 2 −2 def 1 −2 , , Λ(−2,2) = Λ(2,2) = 2 2 1   2 def 3 −2 def 4 −2 Λ(−2,−2) = , Λ(2,−2) = 2 3 2 4 and the generator of Θε is defined as   −kxk2 kxk2 0 0 2 −2kxk2 0 kxk2  def  kxk  × 10−5 , Q(x) =  2 2  kxk 0 −2kxk kxk2  0 0 kxk2 −kxk2 def p where kxk = x21 + x22 . In this example, observations are made at equally spaced discrete points as follows: Ytεk = sin Xtεk + Btk , where Btk is a standard Gaussian white noise sequence. Figure 1(a) and (b) show typical plots for the fast process, Xtε , of the above multiscale hybrid system and the observation process Ytε , respectively. Figure 1(c) shows the evolution of the slow process, Θε , in time, where the original state S is mapped into {1, 2, 3, 4}. To show the validity and efficiency of the homogenized filter, we applied the particle filter (PF) [5] and the homogenized hybrid particle filter (HHPF) [8, 17] algorithms for a comparison. Figure 2(a) and (b) show maximum a posteriori (MAP) estimates with error bars representing one standard deviation, where 400 particles are used.

Marked Proof

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Page 15 of 17

EFFICIENT NONLINEAR FILTERING FOR A HYBRID SYSTEM

01 02

5

03

4 04

3 05

2 06

1 07

0 08 09

0

0.5

1

1.5

2

3

2.5

Time

10

(a)

11

F

12 13

5

15

4

16

3

17

2 1

18

0

19 20

0

21

(b)

23

25

29

E1Θε

b {·} tk 6=Θk

30

32 33 34 35 36

1.5

2

3

2.5

We also compare the errors for PF and HHPF in Table 1. The errors are obtained from a 0–1 error estimate given by

28

31

1

Figure 2. MAP and standard deviation.

24

27

0.5

Time

22

26

PR OO

14

≈

T 1 X 1Θε 6=Θ b {·} , tk k T l=1

b HHPF are MAP estimates at a discretized point k obtained respectively from b PF and Θ where Θ k k PF and HHPF algorithms and Θεtk represents the value of Θε at k. The values in Table 1 are based on 50 Monte Carlo simulations. The mean times taken for these simulations with Intel Xeon 5540 (2.53 GHz) quad-core Nehalem processors are given in the parentheses (the unit is 103 seconds). While the errors for both algorithms are comparable, the time taken for HHPF is much less that that of PF.

37 38 39

Acknowledgements. The third author would like to thank the Departments of Mathematics and Statistics of Stanford University for their hospitality in the Spring of 2010 during

40 41 42

Table 1. Errors of PF and HHPF.

43 44 45

Algorithm/N PF HHPF

50

100

200

400

800

0.2246 (0.26) 0.2607 (0.007)

0.2061 (0.52) 0.2187 (0.02)

0.2026 (1.07) 0.2036 (0.07)

0.1972 (2.23) 0.2010 (0.23)

0.1937 (4.86) 0.1970 (0.89)

46

Marked Proof

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J. H. PARK ET AL.

a sabbatical stay. The first author acknowledges the support from OSD/AFOSR grant 9550-051-0613. The second author acknowledges the support from NSF grant DMS 0604863, ARO grant W911NF-07-1-0044, and OSD/AFOSR grant 9550-05-1-0613. The third author acknowledges the support from AFOSR grant FA9550-08-1-0206 and NSF grant DMS 0604249 during the preparation of this work.

06 07 08

11 12

Q1

13 14 15

Q2

16 17 18 19 20 21 22 23 24

Q3

25 26 27

Q4

28 29 30 31

Q5

32 33 34 35 36 37 38 39 40 41

Q6

42 43 44 45 46

F

10

References 1. A. Bain and D. Crisan, Fundamentals of stochastic filtering, Stochastic Modelling and Applied Probability 60 (Springer, New York, 2009). 2. V. E. Bene˘ s, ‘Exact finite-dimensional filters for certain diffusions with nonlinear drift’, Stochastics 5 (1981) 65–92. 3. A. Crudu, A. Debussche and O. Radulescu, ‘Hybrid stochastic simplifications for multiscale gene networks’, BMC Syst. Biol. 3 (2009) no. 89,. 4. F. Daum, ‘Exact finite-dimensional nonlinear filters’, IEEE Trans. Automat. Control 31 (1986) no. 7, 616–622. 5. A. Doucet, ‘On sequential simulation-based methods for Bayesian filtering’, Technical report (Department of Engineering, University of Cambridge, Cambridge, UK, 1998). 6. J. Filar, V. Gaitsgory and A. Haurie, ‘Control of singularly perturbed hybrid stochastic systems’, IEEE Trans. Automat. Control 46 (2001) no. 2, 179–190. 7. M. Fuhrman, ‘Hypercontractivity properties of nonsymmetric Ornstein–Uhlenbeck semigroups in Hilbert spaces’, Stoch. Anal. Appl. 16 (1998) no. 2, 241–260. 8. D. Givon, P. Stinis and J. Weare, ‘Variance reduction for particle filters of systems with time scale separation’, IEEE Trans. Signal Process. 57 (2009) no. 2, 424–435. 9. I. Hwang, H. Balakrishnan and C. Tomlin, ‘State estimation for hybrid systems: applications to aircraft tracking’, IEE Proc. – Control Theory Appl. 153 (2006) no. 5, 556–566. 10. A. M. Il’in, R. Z. Khasminskii and G. Yin, ‘Singularly perturbed switching diffusions: rapid switchings and fast diffusions’, J. Optim. Theory Appl. 102 (1999) no. 3, 555–591. 11. R. E. Kalman, ‘A new approach to linear filtering and prediction problems’, J. Basic Eng. 82 (1960) no. 1, 35–45. ´, H. Khalil and J. O’Reilly, Singular perturbation methods in control: analysis and design 12. P. Kokotovic (Society for Industrial Mathematics, 1999). 13. M. Miller, U. Grenander, J. O’Sullivan and D. Snyder, ‘Automatic target recognition organized via jump-diffusion algorithms’, IEEE Trans. Image Process. 6 (1997) no. 1, 157–174. 14. G. C. Papanicolaou, ‘Asymptotic analysis of stochastic equations’, Stud. Probab. Theory (1978) 111–179. 15. J. H. Park, N. S. Namachchivaya and R. B. Sowers, ‘A problem in stochastic averaging of nonlinear filters’, Stoch. Dyn. 8 (2008) no. 3, 543–560. 16. J. H. Park, R. B. Sowers and N. S. Namachchivaya, ‘Dimensional reduction in nonlinear filtering’, Nonlinearity 23 (2010) 305–324. 17. J. H. Park, N. S. Namachchivaya and H. C. Yeong, Particle filters in a multiscale environment: homogenized hybrid particle filter (HHPF). Under review. 18. B. L. Rozovskii, Stochastic evolution systems, Mathematics and its Applications (Soviet Series) 35 (Kluwer Academic, Dordrecht, 1990) Linear theory and applications to nonlinear filtering, translated from the Russian by A. Yarkho. 19. B. Rozovskii, R. Blazek and A. Petrov, Interactive banks of Bayesian matched filters, In SPIE Proceedings (Volume 4048): Signal and Data Processing of Small Targets (Orlando, FL, 2000) (ed. O. E. Drummond; SPIE (The International Society for Optical Engineering), Bellingham, WA, 2000). 20. D. D. Sworder and J. Boyd, Estimation problems in hybrid systems (Cambridge University Press, Cambridge, UK, 1999). 21. F. Verhulst, Methods and applications of singular perturbations: boundary layers and multiple timescale dynamics (Springer, Berlin, 2005). 22. H. Wang, ‘Mathematical theory of molecular motors and a new approach for uncovering motor mechanisms’, IEE Proc. – Nanobiotechnology 150 (2003) no. 3, 127–133. 23. E. Wong and J. B. Thomas, ‘On polynomial expansions of second-order distributions’, J. Soc. Ind. Appl. Math. 10 (1962) no. 3, 507–516. 24. G. Yin and C. Zhu, Hybrid switching diffusions: properties and applications, Stochastic Modelling and Applied Probability 63 (Springer, Berlin, 2010). 25. M. Zakai, ‘On the optimal filtering of diffusion processes’, Z. Wahrscheinlichkeitstheorie verw. Geb. 11 (1969) 230–243. 26. Q. Zhang and G. Yin, ‘Nearly-optimal asset allocation in hybrid stock investment models’, J. Optim. Theory Appl. 121 (2004) no. 2, 419–444.

PR OO

09

Marked Proof

Ref: 53180

jcm2010-029

5 October 2011

EFFICIENT NONLINEAR FILTERING FOR A HYBRID SYSTEM

01 02 03 04 05

Jun H. Park 182 George Street Providence, RI 02912 USA

Boris Rozovskii 182 George Street Providence, RI 02912 USA

jun [email protected]

boris [email protected]

Page 17 of 17

06 07 08 09

Richard B. Sowers 1409 W. Green Street Urbana, IL 61801 USA

10 11

[email protected]

F

12 13 14

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

PR OO

15

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

Marked Proof

Ref: 53180

jcm2010-029

5 October 2011

AUTHOR QUERIES

01 02 03 04 05

Q1 (page 16)

Please check that the journal title given in Ref. [3] is OK as set. Also, please give the page range.

06 07 08 09

Q2 (page 16)

Please check added organization details in Ref. [5]. Q3 (page 16)

Please confirm the publisher name in Ref. [12]. Maybe “Society for Industrial and Applied Mathematics”?

12

Q4 (page 16)

13

Please give the volume number in Ref. [14]. Q5 (page 16)

15

Please update Ref. [17], if possible.

16

Q6 (page 16) 17 18 19 20 21 22 23 24 25 26 27 28 29 30

PR OO

14

F

11

10

Reference [23] is not cited in the text. Please cite or remove from the list.

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

Marked Proof

Ref: 53180

jcm2010-029

5 October 2011