Parameter Estimation for Discretely Observed Vasicek Model Driven by Small Lévy Noises

Transcription

1 arameter Etimatio for Dicretely Oberved Vaicek Model Drive by Small Lévy Noie Chao Wei Abtract Thi paper i cocered with the parameter etimatio problem for Vaicek model drive by mall Lévy oie from dicrete obervatio The explicit formula of the leat quare etimator are obtaied the etimatio error i give By uig Cauchy-Schwarz iequality, Growall iequality, Markov iequality domiated covergece, the coitecy of the leat quare etimator are proved whe a mall diperio coefficiet ε imultaeouly The imulatio i made to verify the effectivee of the etimator Idex Term Leat quare etimator, Lévy oie, dicrete obervatio, coitecy I INTRODUCTION Stochatic differetial equatio are of great importace for tudyig rom pheomea are widely ued i the modelig of tochatic pheomea i the field of phyic, chemitry, medicie fiace [4], [6] However, part or all of the parameter i tochatic differetial equatio are alway ukow I the cae of tochatic model drive by Browia motio, the popular method are maximum likelihood etimatio Baye etimatio whe the procee ca be oberved cotiuouly [7], [9], [2], [2] Whe the proce i oberved oly at dicrete time, the explicit expreio of the likelihood fuctio ca ot be give Hece, ome approximate likelihood method have bee propoed [], [3], [5], [4], [5] The leat quare etimatio i aymptotically equivalet to the maximum likelihood etimatio ha bee ued to etimate the parameter for tochatic differetial equatio [2], [7] But, i fact, o-gauia oie ca more accurately reflect the practical rom perturbatio Lévy oie, a a kid of importat o-gauia oie, ha attracted wide attetio i the reearch practice i the field of egieerig, ecoomy ociety From a practical poit of view i parametric iferece, it i more realitic iteretig to coider aymptotic etimatio for tochatic differetial equatio with mall Lévy oie Recetly, a umber of literature have bee devoted to the parameter etimatio for the model drive by mall Lévy oie Whe the coefficiet of the Lévy jump term i cotat, drift parameter etimatio ha bee ivetigated by ome author [8], [], [] Vaicek model, which wa itroduced by Oldrich Alfo Vaicek i 977( [9]), i a mathematical model decribig the evolutio of iteret rate It i a type of oe-factor hort rate model a it decribe iteret rate movemet a drive by oly oe ource of market rik The model ca Thi work wa upported i part by the key reearch project of uiveritie uder Grat 8A6 Ayag Normal Uiverity cietific reearch fud project uder Grat AYNUK-27-B2 Chao Wei i with the School of Mathematic Statitic, Ayag Normal Uiverity, Ayag 455, Chia( chaowei86@aliyucom be ued i the valuatio of iteret rate derivative, ha alo bee adapted for credit market It i kow that parameter etimatio for Vaicek model drive by Browia motio ha bee well developed( [3], [8], [22]) However, ome feature of the fiacial procee caot be captured by the Vaicek model, for example, dicotiuou ample path heavy tailed propertie Therefore, it i atural to replace the Browia motio by the Lévy proce Recetly, the parameter etimatio problem for Vaicek model drive by mall Lévy oie have bee tudied by ome author For example, Davi( [6]) ued Malliavi calculu Mote Carlo etimatio to tudy the etimator of the Vaicek model drive by jump proce, Bao( [2]) developed the approximate bia of the ordiary leat quare etimator of the Vaicek model drive by cotiuou-time Lévy procee But, i ( [2]), oly oe parameter ha bee coidered, the explicit expreio of the etimatio error the coitecy of the etimator have ot bee dicued i both Davi( [6]) ( [2]) I thi paper, we coider the parameter etimatio for Vaicek model drive by mall Lévy oie from dicrete obervatio The explicit formula of all parameter etimator the etimatio error are derived the coitecy of the etimator are proved Firtly, the proce i dicreted baed o Euler-Maruyama cheme, the leat quare method i ued to obtai the explicit formula of the etimator the etimatio error are give a well The, the coitecy of the leat quare etimator are proved by applyig the Cauchy-Schwarz iequality, Growall iequality, Markov iequality domiated covergece whe the mall diperio coefficiet ε imultaeouly Fially, the imulatio reult i provided to verify the effectivee of the obtaied etimator Thi paper i orgaized a follow I Sectio 2, the Vaicek model drive by mall Lévy oie i itroduced, the cotrat fuctio i give the explicit formula of the leat quare etimator are obtaied I Sectio 3, the etimatio error are derived the coitecy of the etimator are proved I Sectio 4, ome imulatio reult are made The cocluio i give i Sectio 5 II ROBLEM FORMULATION AND RELIMINARIES Let (Ω, F, ) be a baic probability pace equipped with a right cotiuou icreaig family of σ-algebra ({F t } t ) Let (L t, t ) be a ({F t })-adapted Lévy oie with decompoitio L t = B t + zn(d, dz) + zñ(d, dz), z () where (B t, t ) i a tard Browia motio, N(d, dz) i a oio rom meaure idepedet of (B t, t (Advace olie publicatio: 28 May 28)

2 ) with characteritic meaure dtν(dz), Ñ(d, dz) = N(d, dz) ν(dz) i a martigale meaure We aume that ν(dz) i a Lévy meaure o R\ atifyig ( z 2 )ν(dz) < I thi paper, we tudy the parameter etimatio for Vaicek model drive by mall Lévy oie decribed by the followig tochatic differetial equatio: { drt =(a br t )dt + εdl t, t [, ] (2) R =r, where a b are ukow parameter Without lo of geerality, it i aumed that ε (, ] Coider the followig cotrat fuctio Y (a, b) = R ti R ti (a br ti ) t i 2, (3) where t i = t i t i = It i eay to obtai the etimator â = (R t i R ti )R ti R t i ( R t i ) 2 R2 t i (R t i R ti ) R2 t i ( R t i ) 2 R2 t i b = (R t i R ti ) R t i ( R t i ) 2 R2 t i 2 (R t i R ti )R ti ( R t i ) 2 R2 t i Before givig the mai reult, we itroduce ome aumptio below Let R = (R t, t ) be the olutio to the uderlyig ordiary differetial equatio uder the true value of the parameter: dr t = (a b R t )dt, R = r Aumptio : a b are poitive true valve of the parameter Aumptio 2: if t {R t } > I the ext ectio, the coitecy of the leat quare etimator are derived the imulatio i made to verify the effectivee of the etimator III MAIN RESULT AND ROOFS Firt of all, we itroduce ome lemma which are of great importace for provig the mai reult Lemma : Let N t = R [t]/, i which [t] deote the iteger part of t The equece {N t } coverge to the determiitic proce {Rt } uiformly i probability a ε roof: Oberve that (4) R t Rt = b (R R)d + εl t (5) By uig the Cauchy-Schwarz iequality, we fid that 2b 2 R t R t 2 2b 2 t (R R )d 2 + 2ε 2 L t 2 R R 2 d + 2ε 2 up L 2 t Accordig to the Growall iequality, we obtai R t Rt 2 2ε 2 e 2b2 t2 up L 2 (6) t The, it follow that up R t Rt 2εe b2 T 2 up L t (7) tt tt Therefore, for each T >, it i eay to check that up R t Rt (8) tt A [t]/ t whe, we coclude that the equece {N t } coverge to the determiitic proce {Rt } uiformly i probability a ε The proof i complete Lemma 2: A ε, R ti (L ti L ti ) RdL roof: Note that R ti (L ti L ti ) = The, it i elemetary to ee that = N dl (N R )db (N (N z (N (N (N z R )db R dl N dl (9) R )zn(d, dz) R)zÑ(d, dz) R )zn(d, dz) R)zÑ(d, dz) It ca be eaily to check that (N R)zN(d, dz), N up N R R z N(d, dz) z N(d, dz) a ε By uig the Markov iequality domiated covergece, we have (N R)dB (N z R )zñ(d, dz) Thu, combiig the previou reult, it follow that R ti (L ti L ti ) RdL () (Advace olie publicatio: 28 May 28)

3 The proof i complete I the followig theorem, the coitecy i probability of the leat quare etimator are proved by uig Cauchy- Schwarz iequality, Growall iequality, Markov iequality domiated covergece Theorem : Whe ε, the leat quare etimator â b are coitet i probability, amely â a, b b roof: By uig the Euler-Maruyama cheme, from (2), we have R ti R ti = (a b R ti ) t i + ε(l ti L ti ) () The, it i eay to ee that (R ti R ti ) = a b R ti +ε (L ti L ti ), (2) (R ti R ti )R ti (3) = a + ε R ti b (L ti L ti )R ti R 2 t i Subtitutig (2) (3) ito the expreio of â, it follow that Let â a = ε R t i R t i (L ti L ti ) ( R t i ) 2 R2 t i ε R2 t i (L t i L ti ) ( R t i ) 2 R2 t i = ε R t i R t (L i t i L ) ti ( R t )2 i R2 t i (L t i L ) ti ε R2 t i ( R t i ) 2 R2 t i R N = R M = if {R t i }, (4) t i up {R ti } (5) t i We make a aumptio that R N R M From (5), it follow that R ti R M <, Rt 2 i RM 2 <, Therefore, from Lemma Lemma 2, whe ε, we have ε R ti R ti (L ti L ti ), (6) ε Rt 2 i (L ti L ti ) (7) Uder the aumptio that R N R M, it i obviouly that ( R ti ) 2 Rt 2 i < (8) From (4) (5), it follow that ( R ti ) 2 Rt 2 i > RN 2 RM 2 (9) The, we have ( R t i ) 2 < R2 t i RN 2 < R2 M (2) Combiig the previou argumet, whe ε,, we have â a (2) R 2 N Sice b b = (â a ) R t i ε R2 t i (L t i L ti )R ti R2 t i A R2 t i R2 N >, we get that < R2 t i Together with the reult that â a ε R t i (L ti L ti ), it follow that b b, (22) a ε Therefore, â b are coitet i probability The proof i complete Theorem 2: Whe ε, ε (â a ) R d R dl L (R ) 2 d ( R d) 2, (R ) 2 d ε ( b b ) R dl L R d ( R d) 2 (R ) 2 d roof: Sice = ε (â a ) R t i R t (L i t i L ) ti ( R t i ) 2 R2 t i R2 t i (L t i L ti ) ( R t )2 i R2 t i Accordig to the Lemma 2, it i eay to check that R ti R d, (Advace olie publicatio: 28 May 28)

4 Rt 2 i (R) 2 d Together with the reult that R t i (L ti L ti ) R dl (L t i L ti ) = L, it follow that ε (â a ) R d R dl L (R ) 2 d ( R d) 2 (R ) 2 d Sice ε ( b b ) = ε (â a ) R t i R2 t i (L t i L ti )R ti R2 t i From above reult, we obtai that ε ( b b ) R dl L R d ( R d) 2 (R ) 2 d The proof i complete IV SIMULATION I thi experimet, we geerate a dicrete ample (R ti ),, compute â b from the ample We let r = 5 For every give true value of the parameter-(a, b ), the ize of the ample i repreeted a Size give i the firt colum of the table I Table, ε =, the ize i icreaig from 5 to 5 I Table 2, ε =, the ize i icreaig from 5 to 5 The table lit the value of a, b the abolute error (AE) of, mea leat quare etimator Two table illutrate that whe i large eough ε i mall eough, the obtaied etimator are very cloe to the true parameter value Therefore, the method ued i thi paper are effective the obtaied etimator are good TABLE I SIMULATION RESULTS OF a AND b True Aver AE (a, b ) Size a (,) (2,3) (4,5) b a b TABLE II SIMULATION RESULTS OF a AND b True Aver AE (a, b ) Size a (,) (2,3) (4,5) b a b V CONCLUSION I thi paper, the parameter etimatio for Vaicek model drive by mall Lévy oie ha bee tudied from dicrete obervatio The leat quare method ha bee ued to obtai the etimator The explicit formula of the etimatio error ha bee give the coitecy of the leat quare etimator ha bee proved Further reearch topic will iclude the parameter etimatio for geeral oliear tochatic differetial equatio drive by lévy oie REFERENCES [] Y Ait-Sahalia, Maximum likelihood etimatio of dicretely-ampled diffuio: a cloed form approximatio approach, Ecoometrica, vol 7, o, pp , 22 [2] Y Bao, A Ullah, Y Wag, J Yu, Bia i the etimatio of mea reverio i cotiuou-time lévy procee, Ecoomic Letter, vol 34, o, pp 6-9, 25 [3] B Bibby, M Sqree, Martigale etimatio fuctio for dicretely oberved diffuio procee, Beroulli, vol, o, pp 7-39, 995 [4] J N Bihwal, arameter etimatio i tochatic differetial equatio, Spriger-Verlag, Lodo, 28 [5] J W Cai, Che, X Mei, Rage-baed threhold pot volatility etimatio for jump diffuio model, IAENG Iteratioal Joural of Applied Mathematic, vol 47, o, pp 43-48, 27 [6] M H A Davi, M Johao, Malliavi mote carlo greek for jump diffuio, Stochatic rocee Their Applicatio, vol 6, o, pp -29, 26 [7] T Deck, Aymptotic propertie of Baye etimator for Gauia Itô procee with oiy obervatio, Joural of Multivariate Aalyi, vol 97, o 2, pp , 26 [8] Z H Li, C H Ma, Aymptotic propertie of etimator i a table Cox-Igeroll-Ro model, Stochatic rocee Their Applicatio, vol 25, o 8, pp , 25 [9] C Li, H B Hao, Likelihood Bayeia etimatio i tre tregth model from geeralized expoetial ditributio cotaiig outlier, IAENG Iteratioal Joural of Applied Mathematic, vol 46, o 2, pp 55-59, 26 [] H W Log, Leat quare etimator for dicretely oberved Ortei- Uhlebeck procee with mall Lévy oie, Statitic robability Letter, vol 79, o 9, pp , 29 [] H Log, Y Shimizu, W Su, Leat quare etimator for dicretely oberved tochatic procee drive by mall Lévy oie, Joural of Multivariate Aalyi, vol 6, o, pp , 23 [2] I S Mbalawata, S Särkkä, H Haario, arameter etimatio i tochatic differetial equatio with Markov chai mote carlo o-liear Kalma filterig, Computatioal Applied Statitic, vol 28, o 3, pp , 23 (Advace olie publicatio: 28 May 28)

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