A practical threshold estimation for jump processes

Transcription

1 A practcal threshold estmato for jump processes Yasutaka Shmzu (Osaka Uversty, Japa) WORKSHOP o Face ad Related Mathematcal ad Statstcal Kyoto, JAPAN, 3 6 Sept., 2008.

2 Itroducto O (Ω, F,P; {F t } t 0 ), cosder a 1-dm semmartgale X: ε>0, dx t = a t dt + b t dw t + c(x t,z) µ(dt, dz)+ c(x t,z) µ(dt, dz), X 0 = x where: 0< z ε a, b: càdlàg, adapted processes; c: R R \{0} R; W : a Weer process; µ(dt, dz): a Posso radom measure o R + R wth the compesator ν(dz)dt; ν({0}) =0, (1 z 2 ) ν(dz) < ; R µ W ad µ := µ ν dt. z >ε Fte actvty: R ν(dz) <, Ifte actvty: R ν(dz) =. 1

3 Observatos: X := {X t } =0,1,...,, t = h, T = h. Statstcally, we are terested the ferece for e.g. Itegrated volatlty / quartcty: T 0 b p t dt (p =2, 4); I the parametrc model: a t (ξ),b t (σ),c(x, z; ϑ) ad ν ϑ (dz); Fuctoals of ν: ν(h) := H(z)ν(dz) (semparametrc model); R\{0}...etc. Asymptotc Iferece: h 0, T T or T ( ). 2

4 Threshold Estmato Method Proposed by: Mac (2004); S.&Yoshda (2006), depedetly. Notato: X := X t X t 1 r = Ch ρ ad ɛ = C h ρ for some 0 <ρ<ρ < 1/2, C, C > 0; Estmato of cotuous part: Use { X ; =0, 1,...,, X r }. Estmato of jump part: Use { X ; =0, 1,...,, X >r }. { t P b t u dw u + t } (( ) q h 1 t 1 z ɛ c d µ >r O + r ( ɛ r ) p ) 0. X1 { X >r } has formato of large jumps. Fte actvty case: Each jump sze s approxmated by X1 { X >r }. 3

5 Examples of Threshold Estmators Example 1. [Mac (2006), Jacod (2008)] Ifte actvty case: T T. T ( X)2 1 { X r } p b 2 t dt; 0 1 T ( X) 4 1 { 3h X r } p b 4 t dt. 0 Example 2 [S. (2008)] Fte actvty case: T. 1 H ( c 1 (X t h, 1 X) ) 1 { X >r } p ν(h) for measurable H uder some regulartes, where c 1 (x, y) s the verse of the map z y = c(x, z) for each x. 4

6 X: Markova: a t (ω; ξ) =a(x t (ω); ξ), b t (ω; σ) =b(x t (ω); σ), ad ergodc case. Example 3. [S. (2006), S.&Yoshda (2006)] Ifte actvty case (MLE-type): T. ) T (ˆξ ξ 0, (ˆσ σ 0 ) d Normal, where (ˆξ, ˆσ ) := arg f ξ, σ { ( X h a () ξ,ϑ 0 (X t 1 ))2 h b 2 (X t 1 ; σ) where a () ξ,ϑ (x) =a(x; ξ)+ z >ɛ c(x, z; ϑ)ν ϑ (dz). Fte actvty case: ɛ 0. + log b 2 (X t 1 ; σ) } 1 { X r }, 5

7 X: Ergodc case. Example 4 [S.& Yoshda (2006)] Fte actvty case (MLE-type): T. T (ˆϑ ϑ 0 ) d Normal, where { ˆϑ := arg sup ϑ log dν ϑ (c 1 (X t 1, X; ϑ))j,(ϑ)1 { X >r } T dν s the Lévy desty ad J, (ϑ) = ϑ c 1 (X t, 1 X; ϑ). Example 5 [S. (2006)] Ifte actvty case: (Momet type): T. T (ˆϑ ϑ 0 ) d Normal (T ), where e.g. R ν ϑ (dz) }, for p 2. ˆϑ := arg f ϑ 1 T ( X)p 1 { X >r } 1 R c p (X t,z; ϑ)ν 1 ϑ(dz) 6

8 Problem practce Questo: How should we choose r (or C, ρ) from the data X? Asymptotcally, ay C>0 ad ρ (0, 1/2) are OK. Practcally, we eed to choose r for fxed, sutably. Threshold estmators are sestve to chage of r. I ths talk: We propose a systematc way to choose the optmal r from X. 7

9 Itegrated volatlty: Bas estmates Itegrated volatlty: v := T Threshold estmator: ˆv (r ):= 0 b 2 t dt. ( X)2 1 { X r }. Cosder the asymptotc property of the followg bas; B (r )=E [ˆv (r ) v ]. Uder some regulartes, ˆv (r )sasymptotcally ubased: lm B (r )=0 both cases where T T s fxed ad T. 8

10 Notato: ε : a determstc sequece s.t. ε 0. J (ε ):=#{t (t 1,t ]; c 1 (X t, X t ) >ε }; λ := ν(dz) ad ν := z 2 ν(dz). z >ε z ε Assumpto H[p, q]: For some p, q > 0, lm sup a Lp (t 1 0,t) =0, lm sup b/ h Lp (t 1 0,t) <, sup 0 where g Lp (s,t) := sup c(x t,z) κ z a.s., t (t 1,t] { [ t 1/p E g(u) du]} p. s lm E [ ] κ q+2 <, 9

11 Theorem 1 Suppose Assumpto H[p, q] for p 4 ad q 2. The ( ) B (r) [ξ (r) η (r)] = O T h 1/2 λ + T ν, sup r>0 where [ ] ξ (r) :=E ( X) 2 1 { X r, J (ε )=1} = O ( T λ r 2), [ ] ( [ ( h ) p/2 ]) ( η (r) :=E ( X) 2 ε ) q 1 { X >r, J (ε )=0} = O T + ν. r r I partcular, we ca take ε 0 (.e. λ <, ν 0) fte actvty case. 10

12 Corollary 1 Suppose the same Ass. as Theorem 1, ad that r ad ε satsfy [ lm T λ r 2 + h ] + ν r 2 =0 ( ) ad lm sup ε r 1 <. The ˆv (r ) s asymptotcally ubased for v : lm B (r )=0 uder the both cases where T T ad T. I the fte actvty case, Codto ( ) ca be reduced by ε 0 to [ lm T r 2 + h ] =0. r 2 11

13 Oe way to choose r : Mmsg B (r) r for fxed. Defe our target ( optmal threshold) as r opt := arg m r>0 B (r) for each Practcally, B (r) ξ (r) η (r) : put c 1 (z) :=c(x t,z). 1 [ ] ξ (r) =E ( X)2 1 { X r, J (ε )=1} E {z; c 1 (z) r} [ ] η (r) =E ( X) 2 1 { X >r, J (ε )=0} c 2 1 (z) λ 1 1 { z >ε }ν(dz) λ h e λ h, [ E z >r z 2 Φ(dz;0, Σ, ) where Φ(z;0, Σ) s Gaussa d.f. wth mea 0, varace Σ, ad t Σ, = b 2 t dt + c 2 1(z) ν(dz). z ε t 1 ] e λ h, 12

14 As a result, B (r) e λ h E [ B (r) ], B (r) = [h c 2 1(z) ν(dz) ζ (r) where ζ (r) :={z [ ε,ε ] c ; c 1 (z) r}. I practce, put ε = δr (0 <δ<1) ad z >r ] z 2 Φ(dz;0, Σ, ), r opt := arg m r>0 B (r) codtoal o X Note: B (r) stll cludes ukow quattes: We ca ot drectly fd r opt. A c 2 1(z) ν(dz) ad Σ,. Below, we propose a algorthm to fd r opt, umercally. 13

15 Notato: d, (r) := ζ (r) c 2 1(z) ν(dz). ˆd, (r; r ): Threshold estmator of d, (r) wth the flter { X (>)r }. Smlarly, so s Σ, (r ). [ B (r; r ):= h ˆd, (r; r ) Threshold estmator of B (r). z >r ] z 2 Φ(dz; Σ, (r )) ; 14

16 Plug- Algorthm: 0-stage: Choose a plot threshold r (0) ad make ˆd, (r; r (0) ) ad Σ, (r (0) ); k-stage (k 1): Fd r (k) Iterate: Set γ = lm r (k) k := arg m r>0 < max 1 X, B (r; r (k 1) ) ;, ad use γ as the threshold. We hope that γ r opt. Let us llustrate ths algorthm by a smple example. 15

17 Smulatos: fte actvty model where dx t = µx t dt + σdw t + d ( Nt U ), X 0 = x 0, N: a Posso process wth the testy λ, U s:..d. N(α, β). The ν λ,α,β (dz) = λ 2πβ exp ) (z α)2 ( 2β dz. Set (µ, σ, λ, α, β; x 0 )=(0.3, 0.7, 10, 0, 0.5; 5). 16

18 Path of Jump Dffuso Value of X tme 17

19 Threshold Estmators [S.&Yoshda (2006), S. (2008)]: ˆµ (r) = X t 1 X1 { X r} h ; X2 t 11 { X r} ( ) ˆσ (r) = ( X +ˆµ X t h 1/2 1 ) 2 1 { X r} h 1 ; { X r} ˆλ (r) = 1 1 { h e X >r}, where e satsfes e = Φ(dz;ˆα (r), ˆβ (r)); z r 1 ˆα (r) = X1 { h ˆλ (r) X >r} ; 1 ˆβ (r) = ( X ˆβ (r)) 2 1 { h ˆλ (r) X >r} ; all of whch are asymptotcally effcet f r = O(h ρ ) for some ρ (0, 1/2) ad h p 0 for some p (1, 1+ρ). 18

20 For sample szes = 500, 1000 ad 3000, Geerate X wth h = 0.7, ad choose r (0) < max X ; Fd the fal threshold γ := lm r (k) by Plug- Algorthm wth k B (r; r )=T z 2 νˆλ,ˆα,ˆβ (r ) (dz) z 2 Φ ( dz;0,h ˆσ (r 2 ) ) z r Retur ˆµ (γ ), ˆσ (γ ), ˆλ (γ ), ˆα (γ ) ad ˆβ (γ ). Iterate these steps 1000 tmes, ad calculate the mea ad the stadard error (s.e.) of each estmator. z >r 19

21 Sample() TRUEs γ ( ) s.e ˆµ s.e ˆσ s.e ˆλ s.e ˆα s.e ˆβ s.e ( ) r opt : ( = 500); ( = 1000); ( = 3000). 20

22 Summary We dealt wth Threshold (r ) selecto the flter. Crtero s the bas fucto (B (r)) of Threshold Estmator of I.V. Asymptotcally ubased both as T s fxed ad as T. Optmal threshold s r opt := arg m B (r) for fxed. r>0 We used a rough approxmato: B (r) Approx. of [ξ (r) η (r)]. Crtero stll cludes some ukow quattes Plug- Algorthm. We show a smulato oly for a fte actvty model. Our approach ca also be appled to fte actvty cases. Thak you for your atteto! 21