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1 Nonlinear Filtering with Perfect Discrete Time Observations Marc Joannides Department of Mathematics Imperial College London SW7 2B, United Kingdom Francois LeGland IRISA / INRIA Campus de Beaulieu 3502 Rennes Cedex, France legland@irisa.fr Abstract : We consider the problem of estimating the state of a diusion process, based on discrete time observations in singular noise. We reduce the problem to a static problem, and we show that the solution is provided by the area or co{area formula of geometric measure theory, provided the observed value is a regular value of the observation function. In order to address the case of singular values, we propose another approach, based on small{noise perturbation and asymptotics of Laplace integrals. 1 INTRODUCTION One major limiting assumption in the nonlinear ltering literature is the non degeneracy of the observation noise covariance matrix. However, there are numerous situations of practical interest, where some perfect noise{free information is available about the unknown state, and yet the problem of state estimation with degenerate observation noise has received little attention, except in the linear case. An additional motivation for studying this problem, is the existing connection with various problems of state estimation for non classical dynamical systems, including : hybrid systems, i.e. systems with state constraints, stochastic dierential{algebraic systems, systems with colored noise, see Korezlioglu and Runggaldier [8], systems with state{dependent observation noise, see Takeuchi and Akashi [11], etc. Finally, a better understanding of the nonlinear ltering problem with noise{free observations should help designing robust and ecient numerical approximation schemes in the important case where the observation noise is small. To be more specic, we consider the following state equation in R m dx t b(x t ) dt + (X t ) dw t ; (1) where fw t ; t 0g is a Wiener process of appropriate dimension, and the noise{free d{dimensional discrete time observations z k h(x tk ) : This work was partially supported by the Army Research Oce, under grant DAAH0{95{1{016. The objective of nonlinear ltering is to compute the conditional probability distribution k (dx) P[X tk 2 dx j z 1 ; ; z k ] : What makes this problem singular is that the state X tk is known exactly to belong to the level set k, where for all z 2 R d h?1 (z) fx 2 R m : h(x) zg : Therefore, the conditional probability distribution k is supported by the set k, which in general has null Lebesgue measure, and k does not have a density w.r.t. the Lebesgue measure on R m. The question naturally arises whether k has a density w.r.t. some canonical measure on the level set k. The objective of this paper is to give conditions under which this holds, and to provide an explicit expression for the density. As usual, the transition from k to k+1 consists in two steps. In the prediction step, i.e. between times t k and t k+1, we solve the Fokker{Planck k L k t ; k t k k ; where L is the adjoint of the innitesimal generator L associated with equation (1). For all t k t t k+1 it holds k t (dx) P[X t 2 dx j z 1 ; ; z k ] : and in particular at nal time t k+1 we obtain the prior probability distribution? k+1 k t k+1 of the state X tk+1. In the correction step, i.e. at time t k+1, we have to combine the prior probability distribution? k+1 and the new observation z k+1 which denes the level set k+1, so as to obtain the posterior probability distribution k+1 of the state X tk+1. We make now the following two assumptions.

2 Assumption A : For all 0 s < t, and all x 2 R m, the conditional probability distribution s;x t (dy) P[X t 2 dy j X s x] ; which is the solution of the Fokker{Planck s;x L s;x t ; s;x s x ; has a continuous density w.r.t. the Lebesgue measure on R m. Assumption B : The mapping h is continuously dierentiable from R m to R d. The purpose of Assumption A is to make sure that, even though k is supported by a set of null Lebesgue measure, the prior probability distribution? has k+1 a continuous density w.r.t. the Lebesgue measure on R m. Assumption B is used to describe the level sets. A stronger assumption (Assumption C) will be introduced later. Under Assumptions A and B, the original problem reduces as follows : Let X be a r.v. in R m, with absolutely continuous probability distribution, and continuous density p w.r.t. the Lebesgue measure on R m. Let the mapping h be continuously dierentiable from R m to R d. Compute the conditional probability distribution z (dx) P[X 2 dx j h(x) z] ; for all z 2 R d. Our objective is to provide an explicit expression for the conditional probability distribution z. In particular, assuming that a canonical measure z exists on the level set, we want to give conditions under which z is absolutely continuous w.r.t. z, and to provide an explicit expression for the density. 2 DIRECT APPROACH FOR REGULAR VALUES We begin this section with some denitions, and properties of the level sets associated with the continuously dierentiable mapping h from R m to R d. For all z 2 R d, the level set is dened by h?1 (z) fx 2 R m : h(x) zg : Notice that for all x 2 R m, h 0 (x) is a linear mapping from R m to R d, i.e. a d m matrix. By denition, x 2 R m is a regular point i the matrix h 0 (x) has full rank min(m; d). We distinguish two cases : In the case m d, x 2 R m is a regular point i h 0 (x) has rank m, i.e. J h (x) q det? [h 0 (x)] h 0 (x) > 0 : In the case m d, x 2 R m is a regular point i h 0 (x) has rank d, i.e. J h (x) q det? h 0 (x) [h 0 (x)] > 0 : Remark 2.1 If the mapping h is only Lipschitz continuous, it follows from the theorem of Rademacher, see Federer [6] or Evans and Gariepy [5, Chapter 3], that h is dierentiable a.e., and the Jacobian J h (x) is dened as above for a.e. x 2 R m. By denition, z 2 R d is a regular value i all x 2 are regular points. We distinguish again the same two cases : In the case m d, if z 2 R d is a regular value, then is a collection of isolated points, i.e. a 0{ dimensional submanifold of R m, and the canonical measure z on is the counting measure. In the case m d, if z 2 R d is a regular value, then is a (m?d){dimensional submanifold of R m, and the canonical measure z on is the Lebesgue measure on, see Berger and Gostiaux [1, Chapter 6]. Under additional smoothness assumption on the mapping h, it follows from the theorem of Sard that the set of regular values has full Lebesgue measure on R d, see Berger and Gostiaux [1, Chapter ] or Sternberg [10]. We are therefore focusing our attention below on regular values z 2 R d. We rst recall the area and co{area formula, see Federer [6] or Evans and Gariepy [5, Chapter 3]. Theorem 2.2 (Area formula) Let the mapping h be Lipschitz continuous from R m to R d, and assume m d. Then for every g 2 L 1 (R m ) R m g(x) J h (x) dx R d X x2 g(x) H m (dz) ; where H m denotes the m{dimensional Hausdor measure on R d. Theorem 2.3 (Co{area formula) Let the mapping h be Lipschitz continuous from R m to R d, and assume m d. Then for every g 2 L 1 (R m ), the restriction of g to the level set is H m?d {integrable for almost all z 2 R d, and g(x) J h (x) dx R m R d g(x) H m?d (dx) dz ; where H m?d denotes the (m? d){dimensional Hausdor measure on R m.

3 Remark 2. For a regular value z 2 R d, the restriction to of the (m? d){dimensional Hausdor measure agrees with the canonical measure z on. Remark 2.5 If the mapping h is only locally Lipschitz continuous from R m to R d, the formulas still hold for every g 2 L 1 (R m ) vanishing outside a compact subset of R m. Remark 2.6 If the mapping h is proper (i.e. h?1 (K) R m is compact, whenever K R d is compact), then all the level sets are compact, and in addition the set of regular values is open in R d. Indeed, assume the contrary, i.e. assume that there exists a sequence fz n ; n 1g in R d, converging to the regular value z. By denition, there exists a sequence fx n ; n 1g in R m such that h(x n ) z n and J h (x n ) 0 for all n 1. The converging sequence fz n ; n 1g stays in a compact K R d, hence the sequence fx n ; n 1g stays in h?1 (K) R m, which is compact since h is proper. Any limit point x of the sequence fx n ; n 1g satis- es h(x ) z and J h (x ) 0, which contradicts the assumption that z is a regular value. This motivates the introduction of the following stronger assumption. Assumption C : The mapping h is continuously dierentiable and proper from R m to R d. Under Assumption C, we deduce from the area and co{area formula, the following characterization of the conditional probability distribution z for a regular value z 2 R d. Theorem 2.7 Let Assumption C hold, and assume m d. Let z 2 R d be a regular value, with nite level set fx i ; i 2 Ig. The following summability condition holds X p(x i ) J h (x i ) < 1 : i2i If in addition the density p does not identically vanish on, then with density z X i2i q i x i ; p(x i ) J q i h (x i ) X : p(x j ) J j2i h (x j ) Theorem 2.8 Let Assumption C hold, and assume m d. Let z 2 R d be a regular value, and let z denote the canonical measure on the compact level set. The following integrability condition holds J h (x) z(dx) < 1 : If in addition the density p does not identically vanish on, then z (dx) q(x) z (dx) ; with density q(x) J h (x) : p(y) J h (y) z(dy) Proof. Under Assumption C, there exists a compact neighbourhood V z of the regular value z 2 R d, such that all values in V z are regular. For any test function dened in R m, and any Borel set B V z in R d, dene g(x) (x) 1 (h(x) 2 B) J h (x) : This mapping is Lebesgue{integrable and vanishes outside h?1 (V z ), which is a compact subset of R m. Therefore, it follows from Theorem 2.3 and Remark 2.5 that E[(X) 1 (h(x) 2 B) ] R m g(x) J h (x) dx R d 0 B 0 B 0 By taking (x) 1, we get P[h(X) 2 B] g(x) H m?d (dx) dz 0 (x) J h (x) Hm?d (dx) dz 0 (x) J h (x) z 0(dx) dz0 : B 0 J h (x) z 0(dx) dz0 : The result follows from the denition of E[(X) j h(x) z], see e.g. Breiman [2]. 2 3 ASYMPTOTICS OF LAPLACE INTEGRALS To obtain an expression of the conditional probability distribution z for any (regular or singular) value z 2 R d, and motivated by the design of robust and ef- cient numerical approximation schemes in nonlinear ltering with small observation noise, we introduce the following perturbation procedure : For " > 0, dene " + V " h(x) + V " ; where V " is a d{dimensional Gaussian r.v. with covariance matrix " I d, independent of the r.v. X. Using the Bayes formula, it is easy to dene a regular conditional probability distribution for the r.v. X given " : For any test function dened in R m E[(X) j " ] R m (x) exp? 1 2" 2 j"? h(x)j 2 dx exp? 1 R m 2" 2 j"? h(x)j 2 dx Two questions arise at this point : :

4 Does the left{hand side have a limit as " # 0, and does this limit provide a version of the conditional probability distribution of the r.v. X given, i.e. given h(x)? How can we compute the limit of the right{hand side as " # 0? The answer to the rst question is provided by the following result. Proposition 3.1 For any test function dened in R m E[ E[(X) j " ] j ]?! E[(X) j ] ; in L 1, as " # 0. Proof. Given a d{dimensional Gaussian white noise sequence fv n ; n 0g with unit covariance matrix, we dene e n + 1X kn k V k + e Vn ; with k "k? " k+1, and f" n ; n 0g is a sequence of positive numbers decreasing to zero. It follows from Takeuchi and Akashi [11, Lemma A.1] that E[(X) j e n ]?! E[(X) j ] ; a.s. and in L 1, as n! 0. As a result E[ E[(X) j e n ] j ]?! E[(X) j ] ; in L 1, as n! 0. Notice that (X; e n ) and (X; "n ) are identically distributed, hence E[ E[(X) j e n ] j ] E[ E[(X) j "n ] j ] a.s., and and therefore E[ E[(X) j "n ] j ]?! E[(X) j ] ; in L 1, as n! 0. Since the convergence holds for an arbitrary sequence f" n ; n 0g of positive numbers decreasing to zero, the result is proved. 2 Corollary 3.2 For any test function dened in R m R d " z? E[(X) j h(x) z] (dz)?! 0 ; as " # 0, where " z E[ E[(X) j " ] j h(x) z] ; and where denotes the probability distribution of the r.v. h(x). In the remaining of this section, we are going to answer the second question, i.e. to study the asymptotic behaviour of " z as " # 0, for a given z 2 Rd. For every x 2 R m, v 2 R d, we dene? " z(x; v) exp? 1 " 2 H" z(x; v) ; with and we notice that " z H " z (x; v) 1 2 jh(x)? (z + " v)j2 ; R d (x)? " R m z (x; v) dx R m? " z (x; v) dx e? 1 2 jvj2 dv (2) d2 : The rst step is to show that the behaviour of " z is determined only by the points x 2 R m in a neighbourhood of the level set. For this purpose, we introduce the following identiability condition. Assumption D : For all z 2 R d, r > 0 where M r z g z (r) inf x62m r z jh(x)? zj > 0 ; fx 2 R m : d(x; ) < rg : Remark 3.3 Under Assumption C, this identiability condition is always satised. Indeed, assume the contrary, i.e. assume that g z (r) 0 for some z 2 R d, r > 0. By denition, there exists a sequence fx n ; n 1g in R m nm r z such that h(x n) converges to z as n! 1. The converging sequence fh(x n ) ; n 1g stays in a compact K R d, hence the sequence fx n ; n 1g stays in h?1 (K) R m, which is compact since h is proper. Any limit point x of the sequence fx n ; n 1g satises h(x ) z and d(x ; ) r > 0, which is a contradiction. We dene D z and we notice that " z with D z In addition fv 2 R d : j" vj < 1 2 g z(r)g ; (x)? " z(x; v) dx R m? " z(x; v) dx R m e? 1 2 jvj2 dv (2) d2 + R z ; jr z j kk P[ jv " j 1 2 g z(r) ] : (2) R m (x)? " z(x; v) dx + R m nm r z and for any x 62 M r z, v 2 D z hence M r z (x)? " z(x; v) dx (x)? " z(x; v) dx ; jh(x)? (z + " v)j jh(x)? zj? j" vj 1 2 g z(r) ;? " z(x; v) exp? 1 8" 2 g2 z(r) ;

5 and j (x)? " z(x; v) dxj R m nmz r kk exp? 1 8" 2 g2 z(r) ; (3) for any v 2 D z. As a consequence, we concentrate below on studying the asymptotic behaviour of z (v) "?d (x)? (2) M " z(x; v) dx ; d2 z r as " # 0, for a given v 2 R d. For the sake of brevity, we will limit ourselves to recover the expression given in the Theorem 2.8 above for a regular value z 2 R d, in the case m d, following the method used in Ellis and Rosen [3], [] and Hwang [7]. Proposition 3. Let Assumption C hold, and assume m d. Let z 2 R d be a regular value, and let z denote the canonical measure on the compact (m? d){ dimensional submanifold. Then, for any xed v 2 R d, and any r > 0 small enough as " # 0. z (v)?! (x) J h (x) z(dx) ; Proof. We prove the result under the additional assumption that the mapping h has a continuous second order derivative. For any x 2, let N x denote the normal space to the submanifold at point x. This is a d{dimensional ane subspace of R m, isomorphic to R d. Dene also N r x f 2 N x : jj < rg ; N r f(x; ) 2 Mz R m : 2 N r x g : Since is compact, we know that for r > 0 small enough, the canonical mapping (x; ) 2 N r 7! x + 2 M r z ; is a dieomorphism, see Berger and Gostiaux [1, Chapter 2]. Moreover, the Jacobian of this dieomorphism is a continuous mapping G z (x; ), with G z (x; 0) 1, for all x 2. The change of variable formula gives z (v) "?d (2) d2 N r x (x + )? " z(x + ; v) G z (x; ) z;x (d) z (dx) ; where z;x is the canonical measure on the d{dimensional linear space N x. To evaluate the inner integral z (x; v) 1 (x + " ) (2) N d2 x r"? " z(x + " ; v) G z (x; " ) z;x (d) ; we use a rst order Taylor expansion for the mapping 7! H " z (x+" ; v), where x 2 and v 2 R d are xed, i.e. H " z(x + " ; v) H " z(x; v) + " [H " z] 0 (x; v) + " (1? ) [H " z ]00 (x + " ; v) d : Notice that if x 2, i.e. if x satises h(x) z, then for every v 2 R d H " z (x; v) 1 2 "2 jvj 2 ; [H " z] 0 (x; v)?" v h 0 (x) ; [H " z] 00 (x; v)?" v h 00 (x) + [h 0 (x)] h 0 (x) : It follows from these observations that 1 " 2 H" z(x + " ; v)?! 1 2 jv? h0 (x) j 2 ; as " # 0. Therefore, by the Lebesgue dominated convergence theorem z (x; v)?! (x) I z (x; v) ; as " # 0, where the integral I z (x; v) is dened as 1 (2) d2 N x exp? 1 2 jv? h0 (x) j 2 z;x (d) : To evaluate this integral we select an orthonormal basis fu 1 ; ; u d g of N x, and we dene the linear mapping U : 2 R d 7! dx k1 This mapping is orthogonal, i.e. k u k 2 N x ' R d : U U IR d and U U I R(U) ; and the change of variable formula gives where I z (x; v) 1 p det (x) ; (x) [h 0 (x) U] h 0 (x) U : Since h 0 (x) U is a square d d matrix, we obtain det (x) det? h 0 (x) U [h 0 (x) U] : Notice that the rows of h 0 (x), i.e. the columns of [h 0 (x)], are d linearly independent m{dimensional vectors in N x R(U). It follows that, for any ; 2 R d h 0 (x) U U [h 0 (x)] h 0 (x) [h 0 (x)] ; since [h 0 (x)] 2 R(U). Therefore p det (x) Jh (x) ; which nishes the proof. 2 Collecting this result with the estimates (2) and (3), we obtain the following limit, i.e. we recover the expression given in the Theorem 2.8 above.

6 Theorem 3.5 Let Assumption C hold, and assume m d. Let z 2 R d be a regular value, and let z denote the canonical measure on the compact (m? d){dimensional submanifold. Then (x) " z?! J h (x) z(dx) ; J h (x) z(dx) as " # 0. APPLICATION TO HYBRID SYSTEMS With the approach presented in this paper, it is possible to address the following estimation problems, which for the sake of simplicity are stated in the static case, i.e. for nite{dimensional r.v.'s rather than for random sequences. Let X be a r.v. in R m, with absolutely continuous probability distribution, and continuous density p w.r.t. the Lebesgue measure on R m. Assume that we learn that X satises the constraint 0 h(x) ; where x 7! h(x) is a proper continuously dierential mapping from R m to R d. If 0 is a regular value of the mapping x 7! h(x), then the conditional probability distribution P [X 2 dx j h(x) 0] can be computed explicitly, and is supported by M fx 2 R m : h(x) 0g. Assume that we learn that X satises instead the constraint 0 g(x; Y ) ; where Y is some observed deterministic variable in R p, e.g. input, parameter, etc., and where x 7! g(x; y) is a proper and continuously dierential mapping from R m to R d, for any y 2 R p. If we observe Y y, and if 0 is a regular value of the mapping x 7! g y (x) g(x; y), then the conditional probability distribution P [X 2 dx j g y (X) 0] can be computed explicitly, and is supported by M y fx 2 R m : g y (x) 0g. 5 EXTENSIONS In the case of a singular value z 2 R d, some partial results could be obtained in the case m d, using ideas contained in Ellis and Rosen [3]. Only the most singular points will contribute to the conditional probability distribution z. The case of observations in continuous time should also be considered. A possible approach would consist to sample the observations and take the limit as the sampling rate goes to innity, but this has not been addressed yet. Let us nally mention another related problem, arising in parameter estimation for a diusion process with small noise. In the case where the standard identi- ability condition does not hold, i.e. where the corresponding Kullback{Leibler information does not have a unique minimum, the consistency of the Bayesian estimate could be obtained as an application of the asymptotics of Laplace integrals. Notice that the case of a nite number of minima of the contrast function has already been considered in Kutoyants [9]. ACKNOWLEDGEMENT The authors gratefully acknowledge Matthew James for bringing the papers [] and [3] to their attention. 6 REFERENCES [1] M. BERGER and B. GOSTIAUX. Dierential Geometry : Manifolds, Curves and Surfaces. Volume 115 of Graduate Texts in Mathematics, Springer{Verlag, New York, [2] L. BREIMAN. Probability. Addison{Wesley, Reading, MA, [3] R.S. ELLIS and J.S. ROSEN. Asymptotic analysis of Gaussian integrals. I : Isolated minimum points. Transactions of the American Mathematical Society, 273(2):7{81, October [] R.S. ELLIS and J.S. ROSEN. Asymptotic analysis of Gaussian integrals, II : Manifold of minimum points. Communications in Mathematical Physics, 82:153{181, [5] L.C. EVANS and R.F. GARIEPY. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL, [6] H. FEDERER. Geometric Measure Theory. Volume 153 of Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin, [7] HWANG Chii{Ruey. Laplace's method revisited : weak convergence of probability measures. The Annals of Probability, 8(6):1177{1182, [8] H. KORELIOGLU and W.J. RUNGGALDIER. Filtering for nonlinear systems driven by nonwhite noises : an approximation scheme. Stochastics and Stochastics Reports, :65{102, [9] Yu.A. KUTOYANTS. Nonconsistent estimation by diusion type observations. Statistics & Probability Letters, 20(1):1{7, May 199. [10] Sh. STERNBERG. Lectures on Dierential Geometry. Prentice Hall, Englewood Clis, NJ, 196. [11] Y. TAKEUCHI and H. AKASHI. Least{squares state estimation of systems with state{dependent observation noise. Automatica, 21(3):303{313, 1985.

Abstract. Jacobi curves are far going generalizations of the spaces of \Jacobi

$Abstract. Jacobi curves are far going generalizations of the spaces of \Jacobi$ Principal Invariants of Jacobi Curves Andrei Agrachev 1 and Igor Zelenko 2 1 S.I.S.S.A., Via Beirut 2-4, 34013 Trieste, Italy and Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia; email: