MAXIMUM LIKELIHOOD ESTIMATION OF HIDDEN MARKOV PROCESSES. BY HALINA FRYDMAN AND PETER LAKNER New York University

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1 he Annal of Applied Probability 23, Vol. 13, No. 4, Intitute of Mathematical Statitic, 23 MAXIMUM LIKELIHOOD ESIMAION OF HIDDEN MARKOV PROCESSES BY HALINA FRYDMAN AND PEER LAKNER New York Univerity We conider the proce dy t = u t dt + dw t, where u i a proce not necearily adapted to F Y (the filtration generated by the proce Y) and W i a Brownian motion. We obtain a general repreentation for the likelihood ratio of the law of the Y proce relative to Brownian meaure. hi repreentation involve only one baic filter (expectation of u conditional on oberved proce Y). hi generalize the reult of Kailath and Zakai [Ann. Math. Statit. 42 (1971) where it i aumed that the proce u i adapted to F Y. In particular, we conider the model in which u i a functional of Y and of a random element X which i independent of the Brownian motion W. For example, X could be a diffuion or a Markov chain. hi reult can be applied to the etimation of an unknown multidimenional parameter θ appearing in the dynamic of the proce u baed on continuou obervation of Y on the time interval [,. For a pecific hidden diffuion financial model in which u i an unoberved mean-reverting diffuion, we give an explicit form for the likelihood function of θ. For thi model we alo develop a computationally explicit E M algorithm for the etimation of θ. In contrat to the likelihood ratio, the algorithm involve evaluation of a number of filtered integral in addition to the baic filter. 1. Introduction. Let (,F,P), {F t,t } be a filtered probability pace and W ={W t,t } a tandard Brownian motion. We aume that filtration {F t,t } i complete and right continuou. We conider proce Y with the decompoition (1) dy t = u t (θ) dt + dw t, where u i a meaurable, adapted proce, and θ i an unknown, vector-valued parameter. We aume that Y i oberved continuouly on the interval [,, but that neither u ={u t,t } nor W are oberved. he purpoe of thi work i to develop method for the maximum likelihood etimation of θ. Our interet in thi work wa motivated by the following model from finance. Suppoe that we oberve continuouly a ingle ecurity with the price proce {S t ; t } which follow the equation (2) ds t = µ t S t dt + σs t dw t, Received February 21; revied July 22. AMS 2 ubject claification. 62M5, 6J6, 6J25. Key word and phrae. Hidden diffuion financial model, likelihood ratio, maximum likelihood etimation, E M algorithm, filtered integral. 1296

2 HIDDEN MARKOV PROCESSES 1297 where σ>iaknown contant, but the drift coefficient µ ={µ t,t } i an unoberved mean-reverting proce with the dynamic (3) dµ t = α(υ µ t )dt + βdwt 1. Here α>,β >andυ are unknown parameter, and W 1 i a tandard Brownian motion independent of W. he initial value of the ecurity S i an obervable contant wherea the initial value of the drift, µ, i an unknown contant. he model in (2) and (3) ha been recently found to be very flexible in modeling of ecurity price. It verion have been ued extenively in the area of dynamic aet management [ee, e.g., Bielecki and Plika (1999), Lakner (1998), Kim and Omberg (1996) and Haugh and Lo (21). We now tranform the model defined by (2) and (3) to the form in (1) by introducing the following change of variable. he reaon for the tranformation i that we want the law of {Y t,t } to be equivalent to that of a Brownian motion. Let X t = 1 β µ t 1 β µ, Y t = 1 ( ) σ log St + σ S 2 t. hen the hidden proce i (4) dx t = α(δ X t )dt + dwt 1, where δ = υ/β, and the oberved proce i (5) dy t = (λx t + ν)dt + dw t, where λ = β/σ and ν = µ /σ. Let θ = (α,δ,λ,ν). We note that X = and Y =, and that (5) i in the form (1) with u t (θ) = λx t + ν. We ummarize our reult. Our main reult i a new formula for the likelihood ratio of the law of the proce Y, pecified in (2), with repect to the Brownian meaure. Naturally, the likelihood ratio i a function of θ. hi reult generalize a reult in Kailath and Zakai (1971) that give a formula for the likelihood ratio in the cae when u i adapted to F Y ={Ft Y,t }, the filtration generated by the oberved proce Y. Here we extend thi reult to the cae when u i not adapted to F Y. We note that in our financial model example (5) u i not adapted to F Y. We etablih the new formula under a tronger (Propoition 1) and a weaker (heorem 2) et of ufficient condition. hee condition allow u to be a functional of Y, and of a random element X which i independent of a Brownian motion W. For example, X could be a diffuion or a Markov chain. he likelihood ratio involve only one baic filter, that i, the expectation of u conditional on the oberved data. he new formula make it poible to directly maximize the likelihood function with repect to the parameter. It may alo be ueful in developing the likelihood ratio tet of variou hypothee about the

3 1298 H. FRYDMAN AND P. LAKNER parameter of hidden Markov procee. For the financial model in (4) and (5) we give an explicit form of the likelihood function. We alo conider the Expectation Maximization (E M) algorithm approach to the maximization of the loglikelihood function for the model in (4) and (5). Dembo and Zeitouni (1986) extended the E M algorithm to the parameter etimation of the hidden continuou time random procee and in particular to the parameter etimation of hidden diffuion. he financial model in (4) and (5) generalize an example in Dembo and Zeitouni (1986). For thi model we develop a computationally explicit E M algorithm for the maximum likelihood etimation of θ. he expectation tep of thi algorithm require evaluation of filtered integral, that i, evaluation of the expectation of tochatic integral conditional on oberved data. A remarked by Elliott, Aggoun and Moore [(1997), page 23, Dembo and Zeitouni (1986) expre the filtered integral in the form of nonadapted tochatic integral which are not well defined. In our development we expre a required filtered integral in term of adapted tochatic integral (Propoition 4). In contrat to the likelihood function, the E M algorithm involve a number of filter in addition to the baic filter. he aymptotic propertie of the maximum likelihood etimator of θ when u i adapted to F Y have been tudied by Feygin (1976) and Kutoyant (1984). he aymptotic inference for θ when u i not adapted to F Y ha been conidered by Kutoyant (1984) and Kallianpur and Selukar (1991) in a pecial cae of our model obtained by etting δ = ν =. (hi pecial model i known a the Kalman filter.) he extenion of thee reult to a more general linear filtering model, in particular, to our model in which hidden proce ha a mean-reverting drift, i an intereting topic for the future invetigation. Wherea Kutoyant (1984) and Kallianpur and Selukar (1991) were concerned with the aymptotic propertie of the etimator in a imilar model to our, the objective of thi paper i to derive an explicit form of the likelihood function and a computationally explicit E M algorithm for the maximum likelihood etimation of the model parameter. hi paper i organized a follow. he new formula for the likelihood ratio i etablihed in Section 2. Section 3 preent the extended E M algorithm and ome of it propertie. In Section 4 and 5 we apply the E M algorithm to the etimation of θ in model in (4) and (5). he Appendix contain proof of two propoition. 2. he likelihood ratio. In thi ection we generalize a reult of Kailath and Zakai (1971), which alo can be found in Kallianpur [(198), heorem and We conider the proce in (1), but to implify notation, we uppre the dependence of u on θ, o that (6) dy t = u t dt + dw t. Our tanding aumption i (7) u 2 t dt <.

4 HIDDEN MARKOV PROCESSES 1299 Let P y be the meaure induced by Y on B(C[,) and P w be the Brownian meaure on the ame cla of Borel et. We know from Kailath and Zakai (1971) that (7) implie P y P w. he ame author give a formula for the likelihood ratio dp y dp w for the cae when u i adapted to F Y. Here we are going to extend thi reult to the cae when u i not adapted to F Y. Let {υ t,t } be the F Y -optional projection of u, thu E [ u t Ft Y (8) = υt a.. whenever E u t <. he following propoition i cloely related to the reult tated already in Lakner (1998). For the ake of the expoition we retate it here in the form mot uitable for our purpoe and include it proof in the Appendix. (9) PROPOSIION 1. Additionally to aumption (7), we aume that E u t < for all t [,, (1) v 2 d < a.. and [ { } (11) E exp u dw 1 2 u 2 d = 1. hen P y i equivalent to P w and { dp y = exp v dy 1 (12) v }. 2 dp w 2 d REMARK. he formula on the right-hand ide i an F Y -meaurable random variable, thu it i alo a path-functional of {Y t, t }. hu thi expreion repreent a random variable and a functional on C[, imultaneouly, and in (12) it appear in it capacity a a functional on the pace of continuou function. In the ret of thi ection we are going to trengthen Propoition 1 by eliminating condition (11). hi will be a generalization again of a reult by Kailath and Zakai (1971), where u i aumed to be F Y -adapted. Our objective here i to have a reult which applie to a ituation when u depend on Y and poibly on another proce which i independent of W. hi other proce may be a Brownian motion, or poibly a Markov chain. In order to achieve thi generality we aume that X i a complete eparable metric pace and X : X i an F B(X)-meaurable

5 13 H. FRYDMAN AND P. LAKNER random element, independent of the Brownian motion W.WedenotebyP X the meaure induced by X on B(X). We aume that (13) u = (Y,,X), [,, where Y repreent the path of Y up to time. Before pelling out the exact condition on we note that X may become a Brownian motion if X = C[,, or a Markov chain if X = D[, (the pace of right-continuou function with left limit) with the Skorohod topology. Other example are alo available. he condition for are the following. Firt, we give a notation that for every f C([,) and [, we denote by f C[, the retriction of f to [,. We aume the following: For every [, fixed, (,, ) i a mapping from C[, X to R uch that: (i) he C[, [, X Rmapping given by (f,,x) (f,,x) i meaurable with repect to the appropriate Borel-clae. (ii) For every [, and f C[, the random variable (f,,x) i F -meaurable. (iii) he following functional Lipchitz condition [Protter (199) hold: for every f,g C([,) and P X -almot every x X there exit an increaing (finite) proce K t (x) uch that (f t,t,x) (g t,t,x) K t (x) up f(u) g(u) u t for all t [,. Our baic etup now i Y with a decompoition (6), u of the form (13), where X i a random element independent of W and a functional atifying (i) (iii). HEOREM 2. (14) and alo (15) hen (16) Suppoe that (7) hold, that i, in our cae 2 (Y,,X)d< 2 (W,,X)d<. [ { } E exp u dw 1 2 u 2 d = 1. REMARK. It i intereting to oberve that if doe not depend on Y,thati, u i independent of W then (16) follow from (7) only.

6 REMARK. HIDDEN MARKOV PROCESSES 131 Condition (14) and (15) are obviouly atified if the mapping (f,,x) i continuou for every (f, x) C[, X. PROOF OF HEOREM 2. he main idea of the proof i to derive our theorem from the reult of Kailath and Zakai by conditioning on X = x. Let Q(x; ), x X, be a regular conditional probability meaure on F given X. he exitence of uch regular verion i well known in thi ituation. It i hown in Propoition A.1 in the Appendix that the independence of the Brownian motion W and X implie that {(W t, Ft W ), t } remain a Brownian motion under Q(x; ) for P X -almot every x X. We alo how in the Appendix that if A i a zero probability event under P then for P X -almot every x X we have Q(x; A) =. [he P X -almot every i important here; obviouly Q(x; ) i not abolutely continuou with repect to P. Accordingly, for P X -almot every x X 2 (Y,,x)d<, Q(x; )-a.. and 2 (W,,x)d<, Q(x; )-a.. Since X i aumed to be a complete meaurable metric pace, we have Q ( x;{ω : X(ω) = x} ) = 1 for P X -almot every x X [Karatza and Shreve (1988), heorem What follow i that under Q(x; ) our equation (6) and (13) become dy t = (Y t,t,x)dt + dw t. It follow from heorem V.3.7 in Protter (199) and our condition (iii) that the above equation ha a unique (trong) olution. hu the proce Y mut be adapted to the Q(x; ) augmentation of F W. However, now we are back in the ituation tudied by Kailath and Zakai (1971). For every x X let P Y (x; ) be the meaure induced by Y on B(C[,) under the conditional probability meaure Q(x; ). Kailath and Zakai, heorem 2 and 3 imply that for P X -almot every x X we have P Y (x; ) P w and { dp w dp Y (x; ) = exp (Y t,t,x)dy t (Y }. t 2,t,x)dt hi implie that the integral of the right-hand ide with repect to the meaure P Y (x; ) i 1, which can be expreed a [ { E exp u t dy t } u 2 X dt = x = 1

7 132 H. FRYDMAN AND P. LAKNER for P X -almot every x X. Formula (16) now follow. We ummarize the reult of Propoition 1 and heorem 2 in the following corollary. COROLLARY 3. Let X : X be a meaurable mapping taking value in the complete eparable metric pace X. Aume that X i independent of the Brownian motion W, Y ha the form (6) and u i a pecified in (13) where atifie condition (i) (iii). Additionally we aume E[ (Y,,X) <, v 2 d < 2 (Y,,X)d < a.., 2 (W,,X)d < a.., a.., where v i the optional projection of u to F Y. hen P y P w and { dp y = exp v dy 1 v }. 2 dp w 2 d Furthermore, [ { } E exp u dw 1 2 u 2 d [ { = E exp v dw 1 2 v 2 d } = 1. REMARK. For the ecurity price model in (4) and (5) the required optional projection i of the form υ t = λe(x t Ft Y ) + ν. We compute E(X t Ft Y ) explicitly in Section E M algorithm. We briefly decribe the extended E M algorithm preented in Dembo and Zeitouni (1986). In the next ection we apply thi algorithm to the etimation of θ in the model defined in (4) and (5). Let F X be the filtration generated by {X t, t }, F Y the filtration generated by {Y t, t }, and F X,Y the filtration generated by {X t, t } and {Y t, t }. We identify a C[, C[, and F a the Borel et of. LetX be the firt coordinate mapping proce and Y be the econd coordinate mapping proce, that i, for (ω 1,ω 2 ) C[, C[, we have X t (ω 1,ω 2 ) = ω 1 (t) and Y t (ω 1,ω 2 ) = ω 2 (t). Let P θ be a family of probability meaure uch that P θ P φ

8 HIDDEN MARKOV PROCESSES 133 for all θ,φ, where i a parameter pace. Alo let E θ denote the expectation under the meaure P θ. We denote by Pθ Y the retriction of P θ to F Y. hen obviouly (17) dp Y θ dp Y φ = E φ ( dpθ dp φ F Y ). he maximum likelihood etimator θ of θ i defined a ( dp Y ) θ = arg max θ for ome contant φ. θ dp Y φ It hould be clear that thi definition doe not depend on the choice of φ. he maximization of dpθ Y /dp φ Y i equivalent to the maximization of the log-likelihood function (18) Now by (17) and (18), L φ (θ) log dpy θ dpφ Y. L φ (θ ) L φ (θ) = log dpy θ dpθ Y and from Jenen inequality, = log E θ ( dpθ dp θ F Y ), ( L φ (θ ) L φ (θ) E θ log dp θ F Y (19) dp ), θ with equality if and only if θ = θ. he E M algorithm i baed on (19) and proceed a follow. In the firt tep we elect an initial value θ for θ. Inthe (n + 1)t iteration, n, the E M algorithm i applied to produce θ n+1. Each iteration conit of two tep: the expectation, or E, tep and the maximization, or M, tep. In the E-tep of the (n + 1)t iteration the following quantity i computed ( Q(θ, θ n ) E θn log dp θ F Y dp ). θn In the M-tep Q(θ, θ n ) i maximized with repect to θ to obtain θ n+1. Iteration are repeated until the equence {θ n } converge. It i clear from (19) that (2) L φ (θ n+1 ) L φ (θ n ), with equality if and only if θ n+1 = θ n. hu at each iteration of the E M algorithm the likelihood function increae. In the M-tep the maximizer of θ i obtained a the olution of (21) θ Q(θ, θ n) = ( θ E θ n log dp ) θ F Y dp =, θn

9 134 H. FRYDMAN AND P. LAKNER or, if we can interchange the differentiation and expectation operation, the maximizer of θ i the olution of ( E θn θ log dp ) θ F Y dp =. θn A corollary of (2) i that the maximum likelihood etimator i a fixed point of the E M algorithm. It i tated in Dembo and Zeitouni (1986) that under ome additional technical condition all limit point of the equence (θ n ) n are tationary point of the log-likelihood function. In particular, if the log-likelihood function i concave in the parameter then it follow that (θ n ) n converge to the unique maximum likelihood etimator. For a more detailed dicuion of the convergence propertie of the E M algorithm and additional reference we refer the reader to Dembo and Zeitouni (1986). 4. Etimation of the drift of the ecurity price proce with the E M algorithm. We develop the E M algorithm for the etimation of θ = (α,δ,λ,ν) in the ecurity price model in (4) and (5). By Giranov theorem, if alo X proce i completely oberved, the log-likelihood function of θ i given by { dp θ = exp α(δ X )dx 1 } α 2 (δ X ) 2 d dp W 2 { exp (λx + ν)dy 1 } (λx + ν) 2 d, 2 where W = (W 1,W) i a two-dimenional Brownian motion and P W aociated Brownian meaure on ß(C[, C[,). Hence log dp { θ = α(δ X )dx 1 } α 2 (δ X ) 2 d dp θn 2 { + (λx + ν)dy 1 } (λx + ν) 2 d 2 + f(θ n ), i the where f(θ n ) i a term which doe not depend on θ. Since, for our problem, we can interchange the operation of differentiation and integration, the equation in (21) take the form [ E θn α log dp θ F Y dp θn (22) = E θn [ (δ X )dx α (δ X ) 2 d F Y =,

10 (23) (24) (25) Let [ E θn δ log dp θ F Y dp θn = E θn [αx [ E θn λ log dp θ F Y dp θn = E θn [ HIDDEN MARKOV PROCESSES 135 X dy [ E θn υ log dp θ F Y dp θn = E θn [Y α 2 (δ X )d F Y =, (λx + ν)x d F Y (λx + ν)d F Y =. m(t,, θ n ) = E θn (X t F Y ), n(t,, θ n ) = E θn (Xt 2 F Y ). hen, recalling that X dx = 1 2 X2 2, (22) become (26) δm(,,θ n ) 1 2 n(,,θ n) + 2 αδ2 + 2αδ Equation (23) take the form (27) m(,, θ n )d α m(,,θ n ) αδ + α =, n(,, θ n )d=. m(,, θ n )d=. Multiplying (27) by δ and ubtracting the reult from (26) give 1 2 n(,,θ n) + (28) 2 + αδ m(,, θ n )d α n(,, θ n )d=. From equation (27) and (28) we obtain the following updated value of α and δ: (29) α n+1 = 1 2 n(,,θ n) + 2 /2 + m(,,θ n ) m(,, θ n )d n(,, θ n )d [, m(,, θ n )d 2 (3) δ n+1 = m(,,θ n) m(,, θ n )d +. α n+1

11 136 H. FRYDMAN AND P. LAKNER Similarly we write (25) a (31) Y λ m(,, θ n )d ν = and (24) a [ F Y (32) E θn X dy λ n(,, θ n )d ν m(,, θ n )d =. We compute the firt term of (32) in the following propoition. Let m (θ n ) = E θn (X F Y ), and to implify notation we will write (33) PROPOSIION 4. [ E θ X dy F Y = m(t, ) = m(t,, θ), n(t, ) = n(t,, θ), m = m (θ). + ν m dy m(, ) d + λ m (ν + λm )d m(, )m d PROOF. Let (34) W t = Y t νt λ m d. It i known that the proce ( W t, Ft Y ), t, i a Brownian motion [ee, e.g., Lemma 11.3 in Lipter and Shiryayev (1978). By heorem 12.1 of Lipter and Shiryayev (1978), m t atifie the following equation: (35) dm t = α(δ m t )dt λγ t (ν + λm t )dt + λγ t dy t, m =, where γ t = V(X t Ft Y ) i a determinitic function atifying (36) From (34) (37) and thu (38) dγ t dt = λ 2 γ 2 t 2αγ t + 1, γ =. d W t = dy t (ν + λm t )dt, dm t = α(δ m t )dt + λγ t d W t.

12 We ee from (38) that m t i F W t [ E θ (39) F Y X dy = E θ [ = E θ [ HIDDEN MARKOV PROCESSES 137 F W X dy X d W + -adapted. hen Y t i F W t -adapted by (34). Hence, X (ν + λm )d F W [ = E θ X d W F W + ν m(, ) d + λ m(, )m d. Now, ince the aumption of heorem 5.14 in Lipter and Shiryayev [(1977), page 185 hold in our cae, it follow by thi theorem that [ E θ X d W F W (4) = m d W. hu by (4), (39) and (37), [ F Y E θ X dy = = + λ m d W + ν m dy m(, )m d. m(, ) d + λ m (ν + λm )d+ ν m(, )m d m(, ) d By ubtituting (33) into (32) and uing (31), we obtain the updated value for λ and ν: λ n+1 = m (θ n )dy Y m (θ n )d ( [ (41) m 2 (θ n ) + n(,, θ n ) m(,, θ n )m (θ n ) d ) 1 m(,, θ n )d m (θ n )d and ν n+1 = Y λ n+1 m(,, θ n )d (42). o implement the E M algorithm uing equation (29), (3), (41) and (42) we need the expreion for m,m(,)and n(, ). hee expreion are obtained in the next ection.

13 138 H. FRYDMAN AND P. LAKNER 5. Expreion for filter. We obtain m t a the olution of (35). Letting ) (43) t = exp ( αt λ 2 γ d, the olution i (44) m t = t [λ γ 1 (dy νd)+ αδ 1 d. he function γ t i obtained a the olution of (36) and i given by (45) γ t = b exp(2at) d λ 2 exp(2at) + d α λ 2, where b = α 2 + λ 2,d = (b a)/(b + a). For evaluation of m t it i ueful to note the following expreion. Subtituting (45) into (43), after ome algebra, give t = 2b/[(b + α)exp(bt) + (b α) exp( bt), and alo γ 1 = inh(b)/b. Application of Lemma 12.3 and heorem 12.3 in Lipter and Shiryayev [(1978), page 36 and 37 to our model how that and (46) m(, t) = (ϕ t ) 1 m t αδ (ϕ u ) 1 du λ (ϕ u ) 1 γ (u, )(dy u νdu), t 1 γ(,t)= [1 + λ 2 γ (ϕ du u )2 γ, t, repectively, where ϕ t,t, i the olution of dϕ t (47) = ( α λ 2 γ(t,) ) ϕ t dt,ϕ = 1, and for t>,γ(t,)= V(X t Ft Y,X ) atifie dγ(t,) (48) = 2αγ(t,) + 1 λ 2 γ(t,) 2, γ(,)=. dt It can be eaily verified that the olution to (48) i (49) γ(t,)= b exp[2b(t ) d λ 2 exp[2b(t )+d α λ 2, t. Uing (49), the olution to (47) i ϕ [ b t = exp exp[2b(u ) d exp[2b(u )+d du exp[b(t ) = (1 + d) exp[2b(t )+d.

14 HIDDEN MARKOV PROCESSES 139 For evaluation of m(, t) it i ueful to note that (ϕ t ) 1 γ(t,)= inh(b(t ))/b, and to evaluate γ(,t)we compute the integral (ϕ u exp[2b(t ) 1 )2 du= (b + α)exp[2b(t )+b α. Clearly n(, t) = γ(,t)+ m(, t) 2, t. PROOF OF PROPOSIION 1. { Z t = exp APPENDIX We define the proce u dw 1 2 u 2 d }, t [,, and the probability meaure P by dp dp = Z, which i a bona fide probability meaure by condition (11). We are going to repeatedly ue the following form of Baye theorem : for every F t -meaurable random variable V we have (5) E [V F Y t = 1 E[Z t Ft Y E[Z tv F Y t, t [,, where E i the expectation operator correponding to the probability meaure P. According to Giranov theorem Y i a Brownian motion under P. Alo, one can ee eaily that [ dp y 1 F Y = E dp w Z = ( E[Z F Y ) 1. he equivalence of P y and P w already follow, ince Z > (under P and P ) thu alo [ 1 F Y E Z >, hence the ame expreion, a a path-functional of {Y t, t } i P y -almot everywhere poitive. We till need to how (12). We tart with the identity 1 Z t = Z u dy. heorem 5.14 in Lipter and Shiryayev (1977) implie that (51) E [ 1 Z t F Y t = 1 + E [ 1 Z u F Y dy,

15 131 H. FRYDMAN AND P. LAKNER a long a the following two condition hold: [ 1 (52) E u <, Z (53) ( E [ 1 Z u F Y ) 2 d < However, (52) follow from condition (9). In order to how (53) we ue the rule (5) and compute ( [ 1 ) 2 F Y ( E u Z d = E[Z F Y ) 2 v 2 d = a.. ( E [ 1 Z F Y ) 2 v 2 d. Now we oberve that by (51), (, ω) E [ Z 1 F Y i a P -local martingale (and alo a martingale but thi i not important at thi point), adapted to the filtration generated by the P -Brownian motion Y, thu it mut be continuou [Karatza and Shreve (1988), Problem hi fact and condition (1) imply (53). Now uing (51) and (5), we get E [ 1 Z t F Y t = 1 + which implie (12). 1 t [ 1 F Y E[Z F Y v dy = 1 + E Z v dy, PROPOSIION A.1. Suppoe that {(W t, F t ), t } i a Brownian motion, X i a complete eparable metric pace, X : X i a meaurable mapping, and {Q(x; ), x X} i a family of regular conditional probability meaure for F given X. Let P X be the meaure induced by X on B(X). hen: (a) for every A F uch that P(A)= we have Q(x; A) = for P X -almot every x X, (b) for P X -almot every x X, the proce {(W t, Ft W ), t } i a Brownian motion under the probability meaure Q(x; ). PROOF. Statement (a) i an obviou conequence of the definition of regular conditional probability meaure. Indeed, for every A F we have Q(x; A) = P(A X = x), P X -a.e., x X and thu Q(X; A) = P(A X), P -a.. hi implie that if A i a P -zero event then E[Q(X; A)=P(A)= and(a)now follow. Next we are going to prove (b). It follow from (a) that the proce W

16 HIDDEN MARKOV PROCESSES 1311 ha Q(x; )-almot urely continuou path for P X -almot every x X. We need to how that the finite dimenional ditribution of W under Q(x; ) coincide with thoe under the probability meaure P. Let u fix a poitive integer n and time point t 1 t n. We denote by Q the et of rational number. For every q = (q 1,...,q n ) Q n there exit a et B q B(X), P X (B q ) = 1 uch that for all x B q, Q ( x; W(t 1 ) q 1,...,W(t n ) q n ) = P ( W(t 1 ) q 1,...,W(t n ) q n X = x ) = P ( W(t 1 ) q 1,...,W(t n ) q n ), where the lat tep follow from independence of X and W under P. Now we define B = B q, q Q n and note that P X (B) = 1inceQ n i countable. It follow that the equation above hold for every x B which complete the proof. Acknowledgment. Halina Frydman completed part of thi work while viiting the Department of Biotatitic at the Univerity of Copenhagen. She thank the department for the hopitality. We alo thank the anonymou referee for the comment which ubtantially improved the paper. REFERENCES BIELECKI,. R. and PLISKA, S. R. (1999). Rik enitive dynamic aet management. J. Appl. Math. Optim DEMBO, A. and ZEIOUNI, O. (1986). Parameter etimation of partially oberved continuou time tochatic procee. Stochatic Proce. Appl ELLIO,R. J., AGGOUN, L. P. andmoore, J. B. (1997). Hidden Markov Model. Springer, Berlin. FEYGIN, P. D. (1976). Maximum likelihood etimation for continuou-time tochatic procee. Adv. in Appl. Probab HAUGH,M.B.andLO, A. W. (21). Aet allocation and derivative. Quant. Finance KAILAH,. and ZAKAI, M. (1971). Abolute continuity and Radon Nikodym derivative for certain meaure relative to Wiener meaure. Ann. Math. Statit KALLIANPUR, G. (198). Stochatic Filtering heory. Springer, New York. KALLIANPUR, G. and SELUKAR, R. S. (1991). Parameter etimation in linear filtering. J. Multivariate Anal KARAZAS, I. andshreve, S. E. (1988). Brownian Motion and Stochatic Calculu. Springer, New York. KIM,. and OMBERG, E. (1996). Dynamic nonmyopic portfolio behavior. Rev. Financial Studie KUOYANS, Y. A. (1984). Parameter Etimation for Stochatic Procee. Helderman, Berlin. LAKNER, P. (1998). Optimal trading trategy for an invetor: he cae of partial information. Stochatic Proce. Appl

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