STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVI, Number 1, March 1 A NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT PROCESS OBSERVATION. Introduction In this note we study a state estimation of a Markovian semimartingale from a doubly stochastic point process observation. All stochastic processes below are supposed to be defined on a filtered probability space (Ω, F, (F t ) t, P ) where (F t ) is a filtration satisfying usual conditions. Consider a state estimation problem where the signal process is a real-valued continuous semimartingale X that is also a Markov process given by X t = X + H s ds + B t, t R +, (.1) where H t is a continuous process and B t is a standard Brownian motion, and the observation is a doubly stochastic point process N t driven by X t : N t is a point process of intensity λ t = λ(x t ) where λ is a nonnegative boolean function. Denote by Zt u the process exp(iux t ). We want to investigate the best state estimation π t (Zt u ) = E[Zt u Ft N ] (.) where Ft N is the natural filtration of the process N t i.e. Ft N = σ(n s, s t). In the sequel the notation π t (... ) stands for the conditional expectation given Ft N. 1. A stochastic differential equation for the best state estimation of Z u t Theorem 1. π t (Z u t ) satisfies the following equation: + π t (Z u t ) = E[Z u ] + iu π s (Z u s H s )ds u π s (Z u s )+ λ 1 s π s [(Z u s π s (Z u ))(λ s π s (λ s ))](dn s π s (λ s )ds) (1.1) 7
Proof. Applying the Ito formula to zt u = exp(iux t ) we have ) Zt u = Z u + (iuh s u ds + iu Zs u db s. Z u t is in fact a semimartingale, and the filtering equation from point process observation [] applied to Z u t : Z t (Z u ) = E[Z u ] + π s [Z u s )] (iuh s u ds+ Now + πs 1 (λ)[π s (Z u λ s ) π s (Zs u )π s (λ s )][dn s π s (λ s )ds]. π s {[Zs u π s (Zs u )][λ s π s (λ s )]} = = π s [Zs u λ s Zs u π s (λ s ) π s (Zs u )λ s + π s (Zs u )π s (λ s )] = = π s (Zs u λ s ) π s [Zs u π s (λ s )] π s [π s (Zs u )λ s ] + π s (Zs u )π s (λ s ). (1.) It follows from π s [Zs u π s (λ s )] = E[Zs u E(λ s Fs N ) Fs N ] = = E(λ s Fs N )E(Zs u Fs N ) = π s (λ s )π s (Z u ), and also from π s [π s (Zs u )λ s ] = π s (Zs u )π s (λ s ) that it remains only the first and the second terms in the left hand side of (1.) and we have: π s (Zs u λ s ) π s (Zs u )π s (λ s ) = π s [(Zs u π s (Zs u ))(λ s π s (λ s ))] and the equation (1.1) is thus completely proved. Remark. In the multidimensional case, the signal process is a vector process given by X t = X + H s ds + B t where X, H, B are multidimensional process. By Zt u we denote now the process exp(i u, X t ), where u = (u 1,..., u n ) R n, X t = (Xt 1,..., Xt n ) and, stands for the 8
A NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT PROCESS OBSERVATION scalar product in R n. And the best state estimation for Zt u based on an observation process that is a doubly stochastic point of intensity λ t = λ(x t ) is π t (Zt u ) E[Zt u Ft N ] = E[exp i u, X t Ft N ]. (1.3) The stochastic differential equation for π t (Zt u ) is the same as (1.1) with Zt u = exp u, X t. In the next Section, we will establish a connection between the characteristic function of X t and the filter of Zt u and so we will see that the laws of the signal X t can be completely determined by π t (Zt u ).. Characteristic function of X t Put 1 ψ t (u) = lim t t E[exp(iu X t) 1 X t ] (.1) is the limit in the right hand side exists, where E[ X t ] is the conditional expectation given X t. Denote by ϕ t (u) the characteristic function of X t : We note that ϕ t (u) = E[exp(iuX t )] = E[Z u t ]. ϕ t+ t (u) = E[exp(iuX t+ t )] = E[exp iu(x t + X t )] = = E[exp(iuX t exp iu X t ] = = E[exp(iuX t E(exp iu X t X t ))] ϕ t+ t (u) ϕ t (u) = E{(exp(iuX t )E[exp iu X t 1 X t ]} It follows that We have now: = lim t E { (exp iux t ) 1 } t E[exp iu X t 1 X t ]. = E[Z u t ψ t (u)] (.) ϕ (u) = E[Z u ] 9
Next, we denote by F N t ε the σ-algebra generated by all N s, s t ε for all small ε >. In noticing that by definition (.1) ψ t (u) is conditioning to the random variable χ t so it is independent of F N t ε and we have: E[Z u t ψ(u)] = E[E(Z u t ψ(u) F N t ε)] = E[ψ t (u)e(z u t F N t ε)]. Because of the left continuity of (F N t ) we have by letting ε then we have the following E[Z u t ψ t (u)] = E[ψ t (u)e(z u t F N t )] = E[ψ t (u)π t (Z u t )] Proposition 1. The law of the signal X t can be determined in term of filtering by the following equation: = E[ψ t (u)π t (Z u t )] ϕ (u) = E[Z u ] (.3) process H t. We will see in next Section that X t can be recognized by filtering and the 3. An expression of the function ψ t (u) or The equation (1.1) can be rewritten as: dx t = H t dt + db t (3.1) X t = H t t + B t (3.) where X t = X t+ t X t, B t = B t+ t B t, B t is a Brownian motion and since EB t B s = min(t, s), we have covariance t 3 E[exp iu X t 1 X t ] = exp[iuh t (X t ) t]e[exp iu B t X t ] 1 It follows from the fact that B t is normally distributed with mean and [ E[iu B t X t ] = exp 1 ] u t.
A NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT PROCESS OBSERVATION or Hence, 1 ψ t (u) = lim t t E[exp iu X t 1 X t ] = ) = exp (iuh t u ) ψ t (u) = exp (iuh t u. (3.3) A substitution of this expression of ψ t into (.3) yields Proposition. {[ )] } = E exp (iuh t u π t (Zt u ) (3.4) ϕ (u) = E[Z ] 4. A Bayes formula for the best state estimation of Z u t We know that by a change of reference probability P Q such that P t Q t for all restriction P t and Q t of P and Q respectively to (Ω, F t ), we have [1] E P [U t G t ] = E Q[U t L t G t ] E Q [L t G t ] where U t is a real-valued bounded process adapted to F t, G t is any sub σ-field of F t : G t F t and L t = dp t. dq t Now, for a doubly stochastic point process Y t of intensity λ t = λ(x t ) we have L t = { } λ(x s ) N s exp (1 λ(x s ))ds. we have References s t We note that under Q the process N t is a Poisson process of intensity 1. And π t (Z u t ) = E Q[Z u t L t F N t ] E Q [L t F N t ] = E Q[L t exp iux t Ft N ] E Q [L t Ft N. ] [1] Alan F. Karr, Point Processes and Their Statistical Inference, Second Edition, Marcel Dekker, Inc, 1991. [] Tran Hung Thao, Note on Filtering from Point Processes, Acta Mathematica, Volume 16, No.1, 1991, 39-47. Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam Institute of Mathematics, NCST, P.O. Box 631 Bo Ho, Hanoi, Vietnam 31