Propagation of initial errors on the parameters for linear and Gaussian state space models
aa r X i v : . [ s t a t . O T ] M a r Propagation of initial errors on the parametersfor linear and Gaussian state space models
Salima El Kolei
Abstract
For linear and Gaussian state space models parametrized by θ ∈ Θ ⊂ R r , r ≥ θ is perfectly known. Inmost real applications, this assumption is not realistic since θ is unknownand has to be estimated. In this paper, we analysis the Kalman filter fora biased estimator θ of θ . We show the propagation of this bias on theestimation of the hidden state. We give an expression of this propagationfor linear and Gaussian state space models and we extend this result foralmost linear models estimated by the Extended Kalman filter. An illus-tration is given for the autoregressive process with measurement noiseswidely studied in econometrics to model economic and financial data. Keywords:
Kalman filter, Extended Kalman filter, State space mod-els, Autoregressive process
Let (Ω , F , P θ ) be a probability space parametrized by θ ∈ Θ ⊂ R r , r ≥ { x t , t ∈ N } defined on (Ω , F , P θ ) with value in X and { y t , t ∈ N ∗ } defined on(Ω , F , P θ ) with value in Y . The process { x t , t ∈ N } (respectively { y t , t ∈ N ∗ } )is called the unobserved signal process (resp. the observation process).The Kalman filter (KF) and the Extended Kalman filter (EKF) commonlyused in some engineering applications have been successfully employed in variousareas. These filters may be easily understood by reading the first publication ofKalman in 1960 [Kal60] or the Bayesian interpretation of Harrison and Stevensin 1971 [HS71]. Affiliation: S. El Kolei, Laboratoire de Math´ematiques J.A. Dieudonn´eUMR n 7351 CNRS UNSAUniversit´e de Nice - Sophia Antipolis06108 Nice Cedex 02 FranceTel.: [email protected] .1 The (Extended) Kalman filter: motivation Let y , · · · , y T be the data (which may be either a scalar or a vector) at time1 , · · · , T . We assume that y t depends on the unobservable variable x t . The aimof the (Extended) Kalman filter is to make inference about the hidden state x t (which may also be a scalar or a vector) conditionally to the data y , · · · , y t . Therelationship between the observed variable y t and the hidden state x t is linearand described by a function h depending on the unknown vector of parameters θ . This relation is specified by the following observation equation: y t = h ( θ , x t ) + σ εθ ε t , t ≥ ε t is the vector of noises assumed to be normally distributed with meanzero and unit variance, denoted as: ε t ∼ N (0 , I n y × n y ) where n y is the dimensionof the observation space Y .The hidden state x t is assumed to be varying with time and its dynamic featureis given by the following state equation: x t = b ( θ , x t − ) + σ ηθ η t where b is a known function and η t is the state error assumed to be normallydistributed with mean zero and unit variance, i.e η t ∼ N (0 , I n x × n x ) where n x is the dimension of the state space X .In addition to the usual Kalman filter assumptions (see [Kal60]), we also assumethat the noises ε t and η t are independent.Hence, this paper is concerned with the following discrete time state space modelwith additives noises: (cid:26) y t = h ( θ , x t ) + σ εθ ε t , t ≥ x t = b ( θ , x t − ) + σ ηθ η t (1)Under the usual Kalman assumptions, the model (1) can be rewritten asfollows: (cid:26) y t = d t ( θ ) + C θ x t + σ εθ ε t t ≥ ,x t = u t ( θ ) + A θ x t − + σ ηθ η t , (2)If the vector of parameters θ is perfectly known, the optimal filtering p θ ( x t | y t )is Gaussian and the Kalman filter gives exactly the two first conditional mo-ments: ˆ x t = E θ [ x t | y t ] and P t = E θ [( x t − ˆ x t )( x t − ˆ x t ) ′ | y t ] where ′ stands forthe transpose. In particular, the Kalman filter estimator is the BLUE (Best Lin-ear and Unbiased Estimator) among linear estimators. Nevertheless, in mostapplications the linearity assumption of the functions h and b is not alwayssatisifed. A linearization by a one order Taylor series expansion can be per-formed and the Extended Kalman filter consists in applying the Kalman filteron this linearized model.For the EKF, the matrix C θ is the differential of the function h with respect to(w.r.t.) x computed at the point ( θ , ˆ x − t ) where ˆ x − t corresponds to the condi-tional expectation E θ [ x t | y t − ]. Additionally, the matrix A θ is the differentialof the function b w.r.t. x computed at the point ( θ , ˆ x t − ). Furthermore, thefunctions u t ( θ ) and d t ( θ ) are defined as:2 u t ( θ ) = b ( θ , ˆ x t − ) − A θ ˆ x t − d t ( θ ) = h ( θ , ˆ x − t ) − C θ ˆ x − t In this paper, we assume that the vector of parameters θ is not perfectlyknown such that the inference of the hidden state x t conditionnally to y t ismade with errors of specification. This typical case is frequent in practice sincein general the vector of parameters is unknown and need to be estimated by anordinary method. The resulting estimator can be biased and consequently thisbias is propagated on the estimation of the hidden state. More precisely, if wedenote by ˆ θ a biased estimator of θ such that E θ [ˆ θ ] = θ = θ + ǫ where ǫ isa fixed and unknown error corresponding to the bias, we want to evaluate thepropagation of the error a posteriori and of the residues a posteriori given by: e t = x t − E θ [ x t | y t ] (3) ξ t = y t − E θ [ y t | y t ] . (4)Many papers concerned the propagation of the initial error on the state( x − ˆ x ) through the filter, and, to the best of our knowledge, there don’t ex-ist in the literature, an analysis of the propagation of the initial errors on thevector of parameters. In this paper, we derive an expression of these propaga-tions for the Kalman and the Extended Kalman filters. Our main result showsthat a correlation between the error a posteriori e t and the unobserved state x t appeared at each time t of the filter. The Kalman filter is now a biased estima-tor and a new Lyapunov dynamic equation for the variance matrix P t is induced.Applications of this result include epidemiology, meteorology, neuroscience,ecology (see [IBAK11]) and finance (see [JPS09]). For example, our result canbe applied to the five ecological state space models described in [PHH10]. Al-though the scope of our method is general, we have chosen to focus on theso-called autoregressive process AR(1) with measurement noise which has beenwidely studied and on which our main result can be easily applied and under-stood. A full illustration of this result is given for a more complex model asthe Heston model which is very used in finance for pricing options and hedgingportfolios (see [ElK12]).The paper is organized as follows. Section 2 presents the model assumptionsand states all of the theoretical results. The application is given and discussedin Section 3. Some concluding remarks are provided in the last section. Theproofs are gathered in Appendix 4. 3 Main result
In this section, we introduce some preliminary main notations and provide theassumptions of model (2).
Subsequently, we denote by E t the pair of the unobservable states vectors givenby (cid:18) e t x t (cid:19) and by E t the pair of the observations vectors (cid:18) ξ t y t (cid:19) where e t and ξ t are defined in (3) respectively. Their variances matrix are denoted by Σ xt andΣ yt respectively.Regarding the partial derivatives, for any function h , [ ∂h/∂θ ] is the vectorof the partial derivatives of h w.r.t θ .Finally, R θ denotes σ εθ σ ′ εθ and Q θ denotes σ ηθ σ ′ ηθ and are the covariancesmatrix of ε t and of η t respectively. We consider the state space models (2), the following assumption ensures somesmoothness for the functions h and b . (A1) The functions b and h are differentiable with respect to θ and x . Before running into the main theorem of this paper, let us explain some existingresults. It is well known that if the vector of parameters is exactly known, theerror a posteriori e t is given by the following formula: e t = ( I n x × n x − K t C θ ) A θ e t − − K t ( σ εθ ε t + C θ σ η θ η t ) + σ ηθ η t where K t is called the Kalman matrix that minimizes the variance matrix P t .Under some assumptions on the model (2), a CLT is obtained for e t as t tends toinfinity (see [dNCdL94]). The following Theorem gives the propagation of theerror a posteriori e t and of ξ t for the Kalman filter and the Extended Kalmanfilter when θ is not exactly known. In this respect, we further assume thatassumption (A1) holds true and the usual Kalman assumptions are satisfied. Theorem 2.1.
Consider the model (2). If ǫ << , then: e t = ( I n x × n x − K t C θ ) A θ e t − − K t ( σ εθ ε t + C θ σ ηθ η t ) + σ ηθ η t + E ǫx ( θ, t ) + F ǫx ( θ, t ) x t − + W ǫx ( θ, t ) + o ( ǫ ) (5) with: ǫx ( θ, t ) = − ǫ (cid:18) ( I n x × n x − K t C θ ) ∂u t ∂θ ( θ ) − K t ∂d t ∂θ ( θ ) − K t ∂C θ ∂θ u t ( θ ) (cid:19) (6) F ǫx ( θ, t ) = − ǫ (cid:18) ( I n x × n x − K t C θ ) ∂A θ ∂θ − K t ∂C θ ∂θ A θ (cid:19) (7) W ǫx ( θ, t ) = − ǫ (cid:18) ∂σ ηθ ∂θ η t − K t C θ ∂σ ηθ ∂θ η t − K t σ ηθ ∂C θ ∂θ η t − K t ∂σ εθ ∂θ ε t (cid:19) (8) Additionally, the propagation of ξ t is equal to: ξ t = C θ e t + σ εθ ε t + E ǫy ( θ, t ) + F ǫy ( θ, t ) x t + W ǫy ( θ, t ) + o ( ǫ ) (9) with: E ǫy ( θ, t ) = − ǫ ∂d t ∂θ ( θ ) , F ǫy ( θ, t ) = − ǫ ∂C θ ∂θ , W ǫy ( θ, t ) = − ǫ ∂σ εθ ∂θ ε t (10) Moreover, when the linearity assumption of the funtions b and h is not sat-isfied, the formulas above remain true with the notations of the EKF defined inSection 1.Proof. See Appendix (A).We note that the terms depending on ǫ : E ǫx ( θ, t ), F ǫx ( θ, t ) and W ǫx ( θ, t ) (resp. E ǫy ( θ, t ), F ǫy ( θ, t ) and W ǫy ( θ, t )) are the corrective terms arising from the bias ofthe parameters estimates.Besides, we can see in Eq.(5) that at time t , the propagation of the state error e t depends on e t − but also on the true state variable x t − . Therefore, the vari-ance of the error e t depends on the variance of e t − but also on the covariancebetween e t − and x t − .Theorem 2.1 gives an expression of the error a posteriori e t and of the residues ξ t which can be rewritten as follows: e t = x t − E θ [ x t | y t ]= ( x t − E θ [ x t | y t ]) | {z } (1) error of estimation + ( E θ [ x t | y t ] − E θ [ x t | y t ]) | {z } (2) correctives terms arising from the bias of parameters. Additionally, ξ t = x t − E θ [ ξ t | y t ]= ( ξ t − E θ [ ξ t | y t ]) | {z } (1) true residues + ( E θ [ ξ t | y t ] − E θ [ ξ t | y t ]) | {z } (2) correctives terms arising from the bias of parameters. The expression of the correctives terms (2) are given in Corollary 1.5 orollary 1.
Let e t given in Eq.(5), the mean of the error a posteriori is givenby: E θ [ e t | y t ] = E θ [ x t | y t ] − E θ [ x t | y t ]= (cid:0) ( I n x × n x − K t C θ ) A θ + F ǫx ( θ, t ) (cid:1) E θ [ e t − | y t − ]+ E ǫx ( θ, t ) + F ǫx ( θ, t ) E θ [ x t − | y t − ] + o ( ǫ ) (11) where E ǫx ( θ, t ) and F ǫx ( θ, t ) are given in Eq.(6) and Eq.(7).Besides, let ξ t given in Eq.(9), the mean of ξ t is given by: E θ [ ξ t | y t ] = E θ [ ξ t | y t ] − E θ [ ξ t | y t ]= (cid:0) C θ + F ǫy ( θ, t ) (cid:1) E θ [ e t | y t ] + E ǫy ( θ, t ) + F ǫy ( θ, t ) E θ [ x t | y t ] + o ( ǫ )(12) where E ǫy ( θ, t ) and F ǫy ( θ, t ) are given in Eq.(10). Corollary 1 which is just a consequence of Theorem (2.1) gives a computablerecursive expression of the expected error E θ [ e t | y t ]. Given E θ [ e ] one can de-duce all the values of this expectation for all t = 1 , · · · , T . Let us consider the linear AR(1) model with measurement noise given by: (cid:26) y t = x t + σ ε ε t , t = 1 , · · · , Tx t +1 = φ x t + σ η η t +1 . (13)Since this model is linear and Gaussian we can apply Eq.(11) in Corollary 1to recover the expectation of e t when the state x t is estimated with a biasedvector of parameters. For this straighforward example, θ is equal to φ . Forthe simulation, we take φ = 0 . σ η = 0 . σ ε = 0 . We run a Kalman filter by assuming that the parameter estimate φ is biasedand we take φ = 0 .
85, that is ǫ = 0 .
15. For this model, the functions b and h are given by: b ( θ , x ) = φ x and h ( θ , x ) = x The variable A θ is equal to φ and C θ is equal to one. The control variables u t ( θ ) and d t ( θ ) are equal to zero.Furthermore, the functions E ǫx ( θ, t ) , F ǫx ( θ, t ) are easily computable and given inthe following lemma. 6 emma 1. For the linear AR(1) model, the functions E ǫx ( θ, t ) and F ǫx ( θ, t ) areequal to: E ǫx ( θ, t ) = 0 , and F ǫx ( θ, t ) = − ǫ (1 − K t ) Therefore, by using Eq.(11) of Corollary 1, the expectation E θ [ e t | y t ] is givenby E θ [ e t | y t ] = (cid:0) (1 − K t )( φ − ǫ ) (cid:1) E θ [ e t − | y t − ] − ǫ (1 − K t ) E θ [ x t − | y t − ] + o ( ǫ )(14)Figure 1: Red: True Error E θ [ e t | y t ] × × E θ [ e t | y t ] for an easy model. The term F ǫx ( θ, t ) corresponds to the bias of x t induced by the bias of the parameter estimate. Furthermore, we can see thatthe error between the true expectation E θ [ e t | y t ] and the approximation (14)corresponds to o ( ǫ ). A full application is given in [ElK12].The following Theorem regards the expression of the variances matrix Σ xt and Σ yt of E t and E t respectively. Theorem 3.1.
The variance matrix Σ xt is given by: Σ xt = (cid:18) V xt S ′ xt S xt P xt (cid:19) where: V xt = ( I − K t C θ ) A θ V xt − A ′ θ ( I − K t C θ ) ′ + F ǫx ( θ ) S t − ( I − K t C θ ) A ′ θ +( I − K t C θ ) A θ S ′ xt − F ′ ǫx ( θ ) + F ǫx ( θ ) P xt − F ′ ǫx ( θ ) + V θ [ ˜ W ǫx ( θ )] S xt = A θ S xt − A ′ θ ( I − K t C θ ) ′ + A θ P xt − F ′ ǫx ( θ ) + Cov θ (cid:16) ˜ W ǫx ( θ ) , σ ηθ η t (cid:17) P xt = A θ P xt − A ′ θ + Q θ ith: ˜ W ǫx ( θ ) = W ǫx ( θ ) + σ ηθ η t − K t σ εθ ε t − K t C θ σ ηθ η t (15) where E ǫx ( θ ) , F ǫx ( θ ) and W ǫx ( θ ) are given in Eq.(6), Eq.(7) and Eq.(8) inTheorem 2.1.If ǫ << , then V θ [ ˜ W ǫx ( θ )] = Q θ + K t ( C θ Q θ C ′ θ + R θ ) K ′ t and Cov θ (cid:16) ˜ W ǫx ( θ ) , σ ηθ η t (cid:17) is given by: Cov θ (cid:16) ˜ W ǫx ( θ ) , σ ηθ η t (cid:17) = − ǫ (cid:18) ∂σ ηθ ∂θ Q θ σ ′ ηθ − K t (cid:18) C θ ∂σ ηθ ∂θ + σ ηθ (cid:19) Q θ σ ′ ηθ (cid:19) Additionally, the variance matrix Σ yt is given by Σ yt = (cid:18) V yt S ′ yt S yt P yt (cid:19) where: V yt = C θ V xt C ′ θ + F ǫy ( θ ) S t C ′ θ + C θ S ′ t F ′ ǫy ( θ ) + ˜ F ǫy ( θ ) P xt ˜ F ǫy ( θ ) + V θ [ ˜ W ǫy ( θ )] S yt = C θ S xt C ′ θ + C θ P xt F ′ ǫy ( θ ) + Cov θ (cid:16) ˜ W ǫy ( θ ) , σ εθ ε t (cid:17) P yt = C θ P xt C ′ θ + R θ with: ˜ W ǫy ( θ ) = W ǫy ( θ ) + σ εθ ε t (16) where E ǫy ( θ ) , F ǫy ( θ ) and W ǫy ( θ ) are given in Eq.(10) Theorem 2.1.If ǫ << , then V θ [ ˜ W ǫy ( θ )] = R θ and Cov θ (cid:16) ˜ W ǫy ( θ ) , σ εθ ε t (cid:17) is given by: Cov θ (cid:16) ˜ W ǫy ( θ ) , σ εθ ε t (cid:17) = − ǫ (cid:18) ∂σ εθ ∂θ R θ σ ′ εθ (cid:19) Proof.
See Appendix (B).The quantities F ǫx ( θ, t ) and W ǫx ( θ, t ) (resp. F ǫy ( θ, t ) and W ǫy ( θ, t )) correspond tothe correctives terms arising from the bias of the parameters and in particularfrom the correlation between e t and the true state x t (see Eq.(5)). This corre-lation induces a new Lyapunov dynamic equation for the variance matrix V xt .For unbiased parameters estimates, these terms are dropped and a CLT is givenin [dNCdL94]. 8 Concluding remarks and discussion
In this paper we provide an expression of the propagation errors on the hiddenstate for an initial and fixed error on the vector of parameters.We showed that the hidden state x t appaered in the propagation equation in-ducing a correlation between e t and the true state x t and most importantly anew Lyapunov dynamic equation for the variance matrix. By using the sameassumptions than in [dNCdL94] and adding smoothness assumptions on thefunctions b and h and on their derivatives, one can again obtain a CLT for e t .Nevertheless, it is not the subject of this paper.Another remark concerns the case where ǫ is not fixed and is supposed to be arandom variable. This particular case refers to the approach proposed in [HK01]for which the parameters are supposed time varying. A dynamical artificial evo-lution is assumed for θ such that θ t = θ t − + σ Z Z where Z is a centered andstandard gaussian random variable. To the best of our knowledge, there doesnot exist results about the convergence of this approach. This method fails inpractice when the variance σ Z is not small. Some authors use σ Z decreasingwith time. Hence, at each step of the filter, a small perturbation is added tothe parameters. This can be seen as a small bias ǫ introduced at the first stepof the filter. 9 Proof of Theorem 2.1:
The proof is essentially based on a one order Taylor expansion of the functions b and h with respect to θ . e t = x t − E θ [ x t | y t ] = x t − E θ [ x t | y t − ] − K t ( y t − ˆ y − t )= u t ( θ ) + A θ x t − + σ ηθ η t − E θ [ u t ( θ ) + A θ x t − + σ ηθ η t | y t − ] − K t ( y t − ˆ y − t )= u t ( θ ) + A θ x t − + σ ηθ η t − E θ [ u t ( θ ) + A θ x t − | y t − ] − K t ( y t − ˆ y − t )= u t ( θ ) + A θ x t − + σ ηθ η t − u t ( θ ) − A θ E θ [ x t − | y t − ] − K t ( y t − ˆ y − t )= u t ( θ ) + A θ x t − + σ ηθ η t − u t ( θ ) − A θ ˆ x t − − K t ( y t − ˆ y − t ) (17)Note that one can write: u t ( θ ) = u t ( θ ) − ǫ ∂u∂θ ( θ ) + o ( ǫ ) , A θ = A θ − ǫ ∂A θ ∂θ + o ( ǫ ) , σ ηθ = σ ηθ − ǫ ∂σ ηθ ∂θ + o ( ǫ )Pluging into (17), one gets: e t = A θ e t − − ǫ ∂u t ∂θ ( θ ) − ǫ ∂A θ ∂θ x t − + σ ηθ η t − ǫ ∂σ ηθ ∂θ η t − K t ( y t − ˆ y − t ) + o ( ǫ ) (18)Furthermore,( y t − ˆ y − t ) = d t ( θ ) + C θ x t + σ εθ ε t − E θ [ y t | y t − ]= d t ( θ ) + C θ x t + σ εθ ε t − d t ( θ ) − C θ E θ [ x t | y t − ]and d t ( θ ) = d t ( θ ) − ǫ ∂d∂θ ( θ ) + o ( ǫ ) , C ( θ ) = C ( θ ) − ǫ ∂C∂θ ( θ ) + o ( ǫ ) , σ εθ = σ εθ − ǫ ∂σ εθ ∂θ + o ( ǫ )So that:( y t − ˆ y − t ) = − ǫ ∂d t ∂θ ( θ ) + ( σ εθ − ǫ ∂σ εθ ∂θ ) ε t + ( C θ − ǫ ∂C θ ∂θ ) (cid:0) u t ( θ ) + A θ x t − + σ ηθ η t (cid:1) − C θ E θ [ x t | y t − ] + o ( ǫ )Rewrite, E θ [ x t | y t − ] = A θ E θ [ x t − | y t − ] + u t ( θ )we get: = C θ A θ x t − − C θ A θ ˆ x t − + σ εθ ε t + C θ σ ηθ η t − ǫ (cid:18) ∂d t ∂θ ( θ ) + C θ ∂A θ ∂θ x t − + ∂σ εθ ∂θ ε t + C θ ∂u t ∂θ ( θ ) + C θ ∂σ ηθ ∂θ η t + ∂C θ ∂θ u t ( θ ) + ∂C θ ∂θ A θ x t − + ∂C θ ∂θ σ ηθ η t (cid:19) + ǫ (cid:18) ∂C θ ∂θ ∂u t ∂θ ( θ ) + ∂C θ ∂θ ∂A θ ∂θ x t − + ∂C θ ∂θ ∂σ ηθ ∂θ η t (cid:19) + o ( ǫ ) (19)Define, ǫy − ( θ ) = − ǫ (cid:18) ∂d t ∂θ ( θ ) + C θ ∂u t ∂θ ( θ ) + ∂C θ ∂θ u t ( θ ) (cid:19) , F ǫy − ( θ ) = − ǫ (cid:18) C θ ∂A θ ∂θ + ∂C θ ∂θ A θ (cid:19) , W ǫy − ( θ ) = − ǫ (cid:18) ∂C θ ∂θ σ ηθ η t + C θ ∂σ ηθ ∂θ η t + ∂σ εθ ∂θ ε t (cid:19) , we obtain: ξ − t = y t − E θ [ y t | y t − ]= C θ A θ e t − + σ εθ ε t + C θ σ ηθ η t + E ǫy − ( θ ) + F ǫy − ( θ ) x t − + W ǫy − ( θ ) + o ( ǫ )By combining Eq.(18) and Eq.(19), we have: e t = x t − E θ [ x t | y t ]= ( I n x × n x − K t C θ ) A θ e t − − K t σ εθ ε t − K t C θ σ ηθ η t + σ ηθ η t + E ǫx ( θ ) + F ǫx ( θ ) x t − + W ǫx ( θ ) + o ( ǫ )where, E ǫx ( θ ) = − ǫ (cid:18) ( I n x × n x − K t C θ ) ∂u t ∂θ ( θ ) − K t ∂d t ∂θ ( θ ) − ∂C θ ∂θ u t ( θ ) (cid:19) , F ǫx ( θ ) = − ǫ (cid:18) ( I n x × n x − K t C θ ) ∂A θ ∂θ − ∂C θ ∂θ A θ (cid:19) , W ǫx ( θ ) = − ǫ (cid:18) ∂σ ηθ ∂θ η t − K t C θ ∂σ ηθ ∂θ η t − K t σ ηθ ∂C θ ∂θ η t − K t ∂σ εθ ∂θ ε t (cid:19) , One can deduce the
Propagation of the residues a posteriori: ξ t = y t − E θ [ y t | y t ]= d t ( θ ) + C θ x t + σ εθ ε t − E θ [ d t ( θ ) + C θ x t + σ εθ ε t | y t ]= d t ( θ ) − d t ( θ ) + ( C θ − ǫ ∂C θ ∂θ ) x t − C θ E θ [ x t | y t ] + ( σ εθ − ǫ ∂σ εθ ∂θ ) ε t + o ( ǫ )= C θ e t + σ εθ ε t − ǫ (cid:18) ∂d t ∂θ ( θ ) + ∂C θ ∂θ x t + ∂σ εθ ∂θ ε t (cid:19) + o ( ǫ )By defining: E ǫy ( θ ) = − ǫ ∂d t ( θ ) ∂θ F ǫy ( θ ) = − ǫ ∂C θ ∂θ W ǫy ( θ ) = − ǫ ∂σ εθ ∂θ ε t Eq.(9) follows.The proof of Corollary 1 is obtained by taking the expectations in Eq.(5) andEq.(9). Proof of Theorem 3.1:
By using the system (2) and Eq.(5)-Eq.(9) we can rewrite the model as follows: E t = (cid:18) E ǫx ( θ ) u t ( θ ) (cid:19) + (cid:18) ( I − K t C θ ) A θ F ǫx ( θ )0 A θ (cid:19) E t − + (cid:18) ˜ W ǫx ( θ ) σ ηθ η t (cid:19) (20)and, E t = (cid:18) E ǫy ( θ ) d t ( θ ) (cid:19) + (cid:18) C θ F ǫy ( θ )0 C θ (cid:19) E t + (cid:18) ˜ W ǫy ( θ ) σ εθ ε t (cid:19) (21)Hence, the variance matrix Σ xt is given by:Σ xt = (cid:18) ( I − K t C θ ) A θ F ǫx ( θ )0 A θ (cid:19) Σ xt − (cid:18) ( I − K t C θ ) A θ F ǫx ( θ )0 A θ (cid:19) ′ + V [ ˜ W ǫx ( θ )] C ov ( ˜ W ǫx ( θ ) , σ ηθ η t ) C ov ( ˜ W ǫx ( θ ) , σ ηθ η t ) σ ηθ σ ′ ηθ ! Additionally, the variance matrix Σ yt is given by:Σ yt = (cid:18) C θ F ǫy ( θ )0 C θ (cid:19) Σ xt (cid:18) C θ F ǫy ( θ )0 C θ (cid:19) ′ + V [ ˜ W ǫy ( θ )] C ov ( ˜ W ǫy ( θ ) , σ εθ η t ) C ov ( ˜ W ǫx ( θ ) , σ εθ η t ) σ εθ σ ′ εθ ! Proposition 2.1 gives that: W ǫx ( θ ) = − ǫ (cid:18) ∂σ ηθ ∂θ η t − K t C θ ∂σ ηθ ∂θ η t − K t σ ηθ ∂C θ ∂θ η t − K t ∂σ εθ ∂θ ε t (cid:19) W ǫy ( θ ) = − ǫ ∂σ εθ ∂θ ε t Hence, if ǫ <<
1, then V [ W ǫx ( θ )] = Q θ + K t (cid:16) C θ Q θ C ′ θ + R θ (cid:17) K ′ t and V [ W ǫy ( θ )] = R θ Furthermore, the covariances are given by:
Cov θ (cid:16) ˜ W ǫx ( θ ) , σ ηθ η t (cid:17) = − ǫCov θ (cid:18) ∂σ ηθ ∂θ η t , σ ηθ η t (cid:19) + ǫ (cid:18) K t C θ ∂σ ηθ ∂θ η t , σ ηθ η t (cid:19) + ǫCov θ (cid:0) K t σ ηθ η t , σ ηθ η t (cid:1) + ǫCov θ (cid:18) K t ∂σ ηθ ∂θ ε t , σ ηθ η t (cid:19) = ǫ ∂σ ηθ ∂θ Q θ σ ′ ηθ + ǫK t C θ ∂σ ηθ ∂θ Q θ σ ′ ηθ + ǫK t σ ηθ Q θ σ ′ ηθ by assumption A2 = − ǫ (cid:18) ∂σ ηθ ∂θ Q θ σ ′ ηθ − K t (cid:18) C θ ∂σ ηθ ∂θ + σ ηθ (cid:19) Q θ σ ′ ηθ (cid:19) Additionally,
Cov θ (cid:16) ˜ W ǫy ( θ ) , σ εθ ε t (cid:17) = − ǫCov θ (cid:18) ∂σ εθ ∂θ ε t , σ εθ ε t (cid:19) = − ǫ (cid:18) ∂σ εθ ∂θ R θ σ ′ εθ (cid:19) eferences [dNCdL94] B. d’Andr´ea Novel and M. Cohen de Lara. Commande lin´eaire dessyst`emes dynamiques . Mod´elisation. Analyse. Simulation. Commande.[Modeling. Analysis. Simulation. Control]. Masson, Paris, 1994. With apreface by A. Bensoussan.[ElK12] S. ElKolei.
Estimation des mod`eles `a volatilit´e stochastique par filtrage etd´econvolution . PhD thesis, 2012.[HK01] M. H¨urzeler and H. R. K¨unsch. Approximating and maximising the likeli-hood for a general state-space model. In
Sequential Monte Carlo methodsin practice , pages 159–175. Springer, New York, 2001.[HS71] P. J. Harrison and C. F. Stevens. A Bayesian approach to short-termforecasting.
Operational Res. Quart. , 22:341–362, 1971.[IBAK11] Edward L. Ionides, Anindya Bhadra, Yves Atchad´e, and Aaron King.Iterated filtering.
Ann. Statist. , 39(3):1776–1802, 2011.[JPS09] Michael S. Johannes, Nicholas G. Polson, and Jonathan R. Stroud. Op-timal Filtering of Jump Diffusions: Extracting Latent States from AssetPrices.
Review of Financial Studies , 22(7):2559–2599, July 2009.[Kal60] Rudolph Emil Kalman. A new approach to linear filtering and predic-tion problems.
Transactions of the ASME–Journal of Basic Engineering ,82(Series D):35–45, 1960.[PHH10] Gareth W. Peters, Geoffrey R. Hosack, and Keith R. Hayes. Ecologicalnon-linear state space model selection via adaptive particle Markov chainmonte carlo (adpmcmc).
Preprint:arXIv-1005.2238v1 , 2010.
Acknowledgements.