Estimation for Parameters in Partially Observed Linear Stochastic System

Transcription

1 Esimaion for Parameers in Parially Observed Linear Sochasic Sysem Chao Wei Absrac This paper is concerned wih he esimaion problem for parially observed linear sochasic sysem. The sae esimaion equaion is provided by applying general filer heory, hen he sae esimaor is obained. The likelihood funcion is given and esimaors of wo parameers are derived. The srong consisency and asympoic normaliy of parameer esimaors are proved by using ergodic heorem, Borel-Canelli lemma, cenral limi heorem for sochasic inegrals and he srong law of large numbers for Brownian moion. A numerical simulaion example is presened o demonsrae he effeciveness of he esimaors. Index Terms linear sochasic sysem, general filer heory, sae and parameer esimaion, srong consisency, asympoic normaliy. I. INTRODUCTION Sochasic differenial equaions have been widely used in many applicaion areas such as biology, chemisry and medical science( [4], [5]). Recenly, sochasic differenial equaions have been applied o describe he dynamics of a financial asse, asse porfolio and erm srucure of ineres raes, such as he popular Black-Scholes opion pricing model ( [6]), Vasicek and Cox-Ingersoll-Ross pricing formulas for he zero coupon bond ( [8], [9], [27]), Chan-Karloyi- Longsaff-Sanders model ( []), Consaninides model ( []), Ai-Sahalia model( []). Some parameers in pricing formulas describe he relaed asses dynamic, however, hese parameers are always unknown. In he pas few decades, some auhors sudied he parameer esimaion problem for economic models. For example, Yu ( [33]) used Gaussian approach o sudy he parameer esimaion for coninuousime shor-erm ineres raes model, Overback( [23]), Rossi( [25]) and Wei( [3]) invesigaed he parameer esimaion problem for Cox-Ingersoll-Ross model by applying he maximum likelihood mehod, leas- square mehod and Gaussian mehod respecively. Moreover, some popular mehods have been used o esimae he parameers in general nonlinear sochasic differenial equaion. For insance, Bayes esimaion( [7], [9]), maximum likelihood esimaion( [3], [3], [32]) and leas-square esimaion( [2], [2]). Some oher mehods such as minimum conras esimaion( [6]), generalized mehod of momens( [3]), M-esimaion( [26]) and hreshold mehod( [7]) have been discussed as well. A variey of sochasic sysems are defined by sochasic differenial equaions( [24]), and someimes he saes of a sochasic sysem can no be observed compleely. Many auhors sudied he sae esimaion problem for sochasic sysems by using Kalman filering or exend Kalman filering( [5], [2], [28], [29]). Furhermore, he parameers and This work was suppored in par by he key research projecs of universiies under Gran 8A6. Chao Wei is wih he School of Mahemaics and Saisics, Anyang Normal Universiy, Anyang 455, China.( chaowei86@aliyun.com.. saes of a sochasic sysem are always unknown a he same ime. Therefore, he parameer and sae esimaion needed o be solved simulaneously. In recen years, he parameer esimaion problem for parially observed linear sochasic sysems has been invesigaed. For example, Deck( [2]) used Kalman filering and Bayes mehod o sudy he linear homogenous sochasic sysems. Kan( [4]) discussed he linear nonhomogenous sochasic sysems based on he mehods used in ( [2]). Mbalawaa [22] applied Kalman filering and maximum likelihood esimaion o invesigae he parameer and sae esimaion for linear sochasic sysems. Alhough he parameer esimaion for parially observed linear sochasic sysems has been sudied by some auhors ( [2], [4], [22]), he asympoic propery of he parameer esimaor has no been discussed in ( [22]), and only one parameer has been considered in ( [2], [4]). In his paper, he parameer esimaion problem for parially observed coninuous-ime linear sochasic sysem wih wo parameers is invesigaed and he srong consisency and asympoic normaliy of wo parameer esimaors are analyzed. Firsly, he sae esimaor is obained by using he general filer heory. Secondly, he likelihood funcion is given based on he Girsanov heorem, he parameer esimaor and he error of esimaion are provided. Then, he srong consisency and asympoic normaliy of wo parameer esimaors are proved by applying ergodic heorem, Borel-Canelli lemma, cenral limi heorem for sochasic inegrals and he srong law of large numbers for Brownian moion. Finally, he simulaion is made o verify he effeciveness of he esimaors. This paper is organized as follows. In Secion 2, we derive he sae esimaion equaion and obain he sae esimaor. In Secion 3, he likelihood funcion is given, he parameer esimaors and he error of esimaion are obained, and he srong consisency and asympoic normaliy of wo parameer esimaors are discussed. In Secion 4, a numerical simulaion example is provided. The conclusion is given in Secion 5. II. PROBLEM FORMULATION AND PRELIMINARIES In his paper, he esimaion problem for parially observed linear sochasic sysem wih wo parameers is invesigaed. The sochasic sysem is described as follows: { dx (α βx )d + dw, X () dy (α β X )d + dv, Y, where α > and β > are wo unknown parameers, α > and β > are consans, (W, ) and (V, ) are independen Wiener processes, {Y } is observable while {X } is unobservable. The likelihood funcion can no be given direcly due o he unobservabiliy of {X }. Therefore, he esimaion problem of {X } should be solved firsly.

2 According o (), one has X (α βx s )ds + W. (2) h X, H α βx, x W, (3) i can be checked ha h H s ds + x, (4) where x is maringale. m E[X Y s, s ]. (5) Then, from he general filer heory [8], we have where m m + W (α βm s )ds β (E[X 2 s Y u, u s] m 2 s)dw s, dy s (α β m s )ds. (6) γ E[X 2 Y s, s ] m 2. (7) Subsiuing (7) ino (6), we obain dm (α βm )d β γ dw, m. (8) According o ()and Iô lemma, one has X 2 (2X s (α βx s ) + )ds + 2 h X 2, H( ) 2X (αβx )+, x 2 Therefore, h X s dw s. (9) X s dw s. () H ( s)ds + x, () where x ( ) is maringale. n E[X 2 Y s, s ]. (2) Then, i can be checked ha n n + (2αm s 2βn s + )ds β (E[Xs 3 Y u, u s] n s m s )dw s. From (8), we have m 2 m 2 + (2m s (αβm s )+β 2 γ 2 s )ds 2β m s γ s dw s. (3) Hence, γ n m 2 γ + Since we obain (2βγ s + σ 2 β 2 γ 2 s )ds β (E[Xs 3 Y u, u s] n s m s 2m s γ s )dw s. Namely, γ γ + E[X 3 s Y u, u s] 3n s m s 2m 3 s n s m s + 2m s γ s, I is easy o check ha (2βγ s + β 2 γ 2 s )ds. (4) dγ β 2 γ 2 d 2βγ d +. (5) γ γ β + β 2 + β 2 β 2. (6) We assume ha sysem (8) has reached he seady sae, i.e., dm (α βm )d β γdw, m. (7) Therefore, on has m α β ( eβ ) β γe β e βs dw s. (8) From (6, we have dy (α β m )d + dw. (9) Remark : Sysem (8) reaches he seady sae means ha he Riccai equaion saisfies dγ d. Hence, we obain γ γ. The deails can be found in [5]. III. MAIN RESULT AND PROOFS In he following heorem, he maximum likelihood esimaors are obained and he srong consisency of he maximum likelihood esimaors are proved by applying ergodic heorem, maximal inequaliy for maringale, Borel-Canelli lemma and he srong law of large numbers for Brownian moion. Theorem : The maximum likelihood esimaors α and β have he following expressions: α (Y Y ) m2 sds m sdy s m sds m2 sds ( m β (Y Y ) m sds m sdy s m2 sds ( m. Moreover, when T, α and β are srong consisen, i.e. α α, β β. Proof:

3 According o (9), he likelihood funcion has he following expression: l (α, β ) (α β m s )dy s 2 Solving he equaion se l (α, β ) α l (α, β ), β we obain he maximum likelihood esimaors (α β m s ) 2 ds. α (Y Y ) m2 sds m sdy s m sds m2 sds ( m β (Y Y ) m sds m sdy s m2 sds ( m. From (9), one has m s dy s α m s ds β m 2 sds + (2) (2) (22) m s dw s, (23) Y Y α β m s ds + W. (24) Subsiuing (23) and (24) ino he expression of α, we obain α α W m2 sds m sdw s m sds m2 sds ( m W m2 sds m sdw s m sds m2 sds ( m. By using ergodic heorem, i can be checked ha m s ds α β, (25) m 2 sds α2 β 2 + β2 γ 2 2β. (26) According o he srong law of large numbers for Brownian moion, we have W. (27) From he maximal inequaliy for maringale and Borel- Canelli lemma, one has From above resuls, we obain From (24) and l(α,β) α m s dw s. (28) α α. (29) β β, i is easy o check ha ( α α ) W m sds ( α α ) W m. sds Since α α, W, m sds α β, one has β β. (3) Therefore, he maximum likelihood esimaors α and β are srong consisen. The proof is complee. In he following heorem, he asympoic normaliy of he error of esimaion is proved by using he cenral limi heorem for sochasic inegrals and he ergodic heorem. Theorem 2: When, and and Proof: Since ( α α ) d N(, 2α + β2 γ 2 ββ 2 ). ( β β ) d N(, E[ α2 E m s dw s ] 2 E[m s ] 2 ds m s dw s, 2β β 2 ). β 2 ( eβ ) 2 + β2 γ 2 [ e 2β ]. 2β According o he cenral limi heorem for sochasic inegrals, i follows ha m s dw s d N(, α 2 β 2 + β2 γ 2 2β ). As W d N(, ), ogeher wih (25) and (26), one has ( α α ) d N(, 2α + β2 γ 2 ββ 2 ). By applying he same mehod, i can be checked ha ( β β ) d 2β N(, The proof is complee. β 2 ). Remark 2: When he parameers α and β are unknown, we have obained he esimaors and proved he consisency and asympoic normaliy of he esimaors. However, when α and β are unknown, he likelihood funcion and he proof will be differen. We will give he specific seps below. According o (7), he likelihood funcion has he following expression: l (α βm s ) β 2 dm s 2 (α βm s ) 2 ds. β 2 Since β is in he expression of γ, i is difficul o obain he explici formula of he esimaor of β. Thus, we discuss he parameer esimaion problem of α and β separaely. When α is unknown, i is easy o check ha α m + β m s ds.

4 Since m and m sds α β, i follows ha α α. When β is unknown, i is difficul o obain he explici formula of he esimaor. We assume ha β is he rue parameer. According o (7), i can be checked ha 2 (α βm s ) l (β) β 2 dm s (α βm s ) 2 2 β 2 ds (α βm s ) β 2 (α β m s )ds β γ dw s ) (α βm s ) 2 ds β 2 (α βm s )(α β m s ) β 2 ds (α βm s )γ β γ 2 dw s (α βm s ) 2 2 β 2 ds α 2 2β 2 αβ β 2 (α βm s )γ β γ 2 dw s. I is obviously ha wih (25) and (26), i follows ha m s ds + 2ββ β 2 2β 2 (αβm s)γ β γ 2 4β 2 β2 m 2 sds dw s, ogeher l (β) (2α2 β + ββ 2 βγ 2 )(2β β). Since (2α2 β+β 2 ββγ 2 )(2β β) reaches he maximum value 4β 2β2 when β β, we have ha β β. IV. SIMULATION In sysem (), le α 8, β 7, sep size.5. For every given rue value of he parameers-α, β, he size of he sample is represened as Size n and given in he firs column of he able. In Table, he size is increasing from o. This able liss he value of α MLE, β M LE and he Absolue Errors (AE). The able illusraes ha he Absolue Errors of α and β depend on he size of given value of α and β. Bu under he hypohesis of normal disribuion, here is no obvious difference beween esimaor and rue value, esimaors- α and β are good. V. CONCLUSION The aim of his paper is o esimae wo parameers for parially observed linear sochasic sysem. The sae esimaor has been obained by using general filer heory. The likelihood funcion has been given and he parameer esimaors and he error of esimaion have been derived. The srong consisency and asympoic normaliy of wo parameer esimaors are proved by applying ergodic heorem, Borel-Canelli lemma and he srong law of large numbers for Brownian moion. Furher research opics will include he parameer esimaion for parially observed nonlinear sochasic sysems. TABLE I MLE SIMULATION RESULTS OF α AND β.5, α 8, β 7 True Aver AE (α, β ) Size n α β α β (.5,) (2,) (2.5,.5) (3,2) REFERENCES [] Y Ai-Sahalia, Tesing coninuous-ime models of he spo ineres rae, Review of Financial Sudies, vol. 9, no. 2, pp , 996. [2] J Bai, Leas squares esimaion of a shif in linear processes, The Review of Financial Sudies, 994, vol. 9, no., pp , 994. [3] M Barczy, G Pap, Asympoic behavior of maximum likelihood esimaor for ime inhomogeneous diffusion processes, Journal of Saisical Planning and Inference, vol. 4, no. 6, pp , 2. [4] B Øksendal, Sochasic Differenial Equaions: an Inroducion wih Applicaions, Springer-Verlag, Berlin, 23. [5] J P N Bishwal, Parameer esimaion in sochasic differenial equaions, Springer-Verlag, London, 28. [6] F Black, M Scholes, The pricing of opions and corporae liabiliies, Journal of Poliical Economy, vol. 8, no. 3, pp , 973. [7] J W Cai, P Chen, X Mei, Range-Based Threshold Spo Volailiy Esimaion for Jump Diffusion Models, IAENG Inernaional Journal of Applied Mahemaics, vol. 47, no., pp , 27. [8] J Cox, J Ingersoll, S Ross, An ineremporal general equilibrium model of asse prices, Economerica, vol. 53, no. 2, pp , 985. [9] J Cox, J Ingersoll, S Ross, A heory of he erm srucure of ineres raes, Economerica, vol. 53, no. 2, pp , 985. [] K C Chan, An empirical comparision of alernaive models of he shor-erm ineres rae, The Journal of Finance, vol. 47, no. 3, pp , 992. [] G M Consaninides, A heory of he nominal erm srucure of ineres raes, Review of Financial Sudies, vol. 5, no. 4, pp , 992. [2] T Deck, T G Theing, Robus parameer esimaion for sochasic differenial equaions, Aca Applicandae Mahemaicae, vol. 84, no. 3, pp , 24. [3] L P Hansen, Large sample properies of generalized mehod of momens esimaors, Economerica, vol. 5, no. 4, pp , 982. [4] X Kan, H S Shu, Y Che, Asympoic parameer esimaion for a class of linear sochasic sysems using Kalman-Bucy filering, Mahemaical Problems in Engineering, Aricle ID: 34275, DOI:.55/22/34275, 22. [5] R Kannan, Orienaion esimaion based on LKF using differenial sae equaion, IEEE Sensors Journal, vol. 5, no., pp , 25. [6] M Kessler, Esimaion of an ergodic diffusion from discree observaions, Journal of Saisics, vol. 24, no. 2, pp , 997. [7] Y A Kuoyans, Saisical Inference for Ergodie Diffusion Processes, Springer-Verlag, London, 24.

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