参 2

1

参 2

閲 3

4

5 { } i τ

6

7

参 参 8

9

10

11

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15 [ ) 0, ( ], st

τ { } i τ = t t ( t = 0) i i i 1 0 { } τ i { } t τ t i { } { } τ i i { } i t 0 t 1 t 2 t 3 t 4 t n τ 1 τ 2 τ3 τ 4 () λ t 16

( ( ) ) P N t 0 = 0 = 1 () N t t t 0 ( ) P N ( t + t) N ( t) = H t ( ) 1 t λ t H t 1 λ() t = lim P N( t + t) N( t) 1 t 0 t N( t t) N( t) ( = H ) ( + H ) = ο( t) P t t 2 lim ( ) = 0 t 0 o t t () ο ( λ t) H t τ τ λτ P [( N( t τ ) N( t) ) = k] ( λτ ) λτ e + = k k! 1 t = lim P N t+ t N t = 1 = k, t 0 t ( λ ) ( ( ) ( ) ) P( N( t + t) N( t) ) = ο( t) 2 k k 17

P ( N( t ) = 0) 1 0 = ( ) λ t 1 t = lim P N( t+ t) N( t) = 1 = g( t) t 0 t [ P N( t+ t) N( t) 2 ] = o( t) ( λ ) [ ] P N( t ) 0 = 0 = 1 gt () b λ λ( ) = a, b a t dt ( ] ab, 参 λ ab, λ e P ( N( b) N( a) ) = k = a, b ( λa, b) k! k k τ, τ, τ, 18 1 2 3

( λ t) H t ( ) λ t 1 t = lim P N t+ t N t = 1 Ht t 0 t ( λ t, Ht ) ( ) ( λ ) ( ) ( ) [ P N( t+ t) N( t) 2 ] = o( t) 0 = = 1 0 P Nt ( ) { X t } t1, t2,..., t n X t1, Xt2,..., Xtn X t 19

w w w i i 1 i w w i ( w 1, w) ( w1, w2) ( w 2, w 3 ) i i w i 1 w i wi 1 ( w, w ) ( w, w ) 5 6 3 4 w wi i i i 1 20

1. w 1 w 2 2. w 2 w 3 3. w 3 w 4 4. w 4 w 5 5. w 5 w 6 5. w 5 w 6 3. w 3 w 4 4. w 4 w 5 1. w 1 w 2 2. w 2 w 3 (0,T] 0 < u < u <... < u <... < u T K 1 2 uk k uk u k+ 1 uk ( ) u k f t H, θ t t > u k k uk k K H u 21

( u θ ) k f t H (, ) k ( ) ( ) 1 2 2 θ 1 p 1 t u p k µ Hu θ + + 3 2 k u k θ, = exp 2 π ( t u ) 2 µ H, θ t u k { Hu = uk, w k, w k 1,..., w k p+ 1} wk = uk uk 1 k k µ RR µ ( Hu θ) θ k 0 θ jwk j+ 1 p =, = + > 0 ( H, ) j= 1 1 3 1 2 RR u p 1 σ = µ θ θ k + θ + > θ = θ0, θ1,..., θ p + 1 p 1 0 ( ) ( ) ( 0 1 ) u k k p = 0 f t H, θ = f t u, θ, θ ( H, ) uk 0 µ θ = θ θ0 σ (0,T] l t [ l, T] t l ( t, t] u u, u,..., u l l t = ( 1 2 ) t n t n t t l < u 1 < u 2 <... < u t n t t l 22

w 2 w 3 t l u1 u2 u 3 0... T t l t θˆt u nt t t l log n t ( t-l t θt ) f u ut l t ( i) log ( i i 1 u, θ ) ( ) log 1 (, i t nt u θ n t t) = wt u f u u H + wt u f vh dv i= 2 t u () t u w t u t lt wt u = e α n t ( ) t l < u1 < u2 <... < u t t u 1 > t u 2 >... > t u nt 0 ( ) n t ( t u) wt u = e α α ( t u ) ( ) ( t u ) 1 α t u 2 e e... e α < < < nt 1 ( ) w t ( t u) ( wt u) = e α = 1 [ ] t l, T t l ( t l, t] t l t l ( ) 23

l t = T t l 参 ( θ0, σ ) θ σ θ0 σ θ σ 0 θ σ 0 ( θ, σ ) = (0.85,0.004) 参 0 θ θ 1 参 0 24 0

σ 0.004 0.003 0.002 (0.001,0.65) 0.75 0.85 0.95 θ 0 t θˆ θ ( H ˆ ) t ˆ µ = µ, θ RR t t t 1 3 1 2 ( ˆ ) ˆ ˆ RR =, t Ht t p 1, t σ µ θ θ + 25

26

27

wt wt 1. 3 wt 1. 3 1. 2 1. 2 1. 1 1. 1 1. 0 1. 0 0. 9 0. 9 0. 8 0. 8 0. 7 0. 7 0. 6 0. 6 0. 5 0. 5 0 1000 2000 3000 4000 time 0 1000 2000 3000 4000 time wt 1. 3 wt 1. 3 1. 2 1. 2 1. 1 1. 1 1. 0 1. 0 0. 9 0. 9 0. 8 0. 8 0. 7 0. 7 0. 6 0. 6 0. 5 0. 5 0 1000 2000 3000 4000 time 0 1000 2000 3000 4000 time wt 1. 3 1. 2 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0 1000 2000 3000 4000 time 28

wt 1. 5 1. 4 1. 3 1. 2 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 wt 1. 5 1. 4 1. 3 1. 2 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0 1000 2000 3000 4000 time 0 1000 2000 3000 4000 time wt 1. 5 1. 4 1. 3 1. 2 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 wt 1. 5 1. 4 1. 3 1. 2 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0 1000 2000 3000 4000 time 0 1000 2000 3000 4000 time wt 1. 5 1. 4 1. 3 1. 2 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0 1000 2000 3000 4000 time 29

30

n432443 n645005 n881798 0 5 10 15 0 5 10 15 0 5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt n1000038 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt n1050515 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 0 5 10 15 0 5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 784229 879663 882502 0 5 10 15 0 5 10 15 0 5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 1011902 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 5073527 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 0 5 10 15 0 5 10 15 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 0.6 0.7 0.8 0.9 1.0 1.1 1.2 wt 31

32

33

34

θ σ θ0σ θ σ θ σ θ 0 θ0 35 0 0 0

θ θ 0 θ0 θ0 θ 0 θ0 0 36

θ0 σ θ 0 σ θ 0 σ θ 0 σ θ 0 σ 37

θ0 σ θ 0 σ θ 0 σ θ 0 σ θ 0 σ 38

39

40

41

θ σ 0 θ0 σ θ 0 θ θ0 θ0 0 θ0 θ0 42

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44 堃

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n432443 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0.7 0.8 0.9 1.0 1.1 w1 0.7 0.8 0.9 1.0 1.1 w2 0.7 0.8 0.9 1.0 1.1 w3 0 20 40 60 80 0 20 40 60 80 0.7 0.8 0.9 1.0 1.1 w4 n645005 0.7 0.8 0.9 1.0 1.1 w5 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0.7 0.8 0.9 1.0 1.1 w1 0.7 0.8 0.9 1.0 1.1 w2 0.7 0.8 0.9 1.0 1.1 w3 0 20 40 60 80 0 20 40 60 80 0.7 0.8 0.9 1.0 1.1 w4 0.7 0.8 0.9 1.0 1.1 w5 46

n881798 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0.6 0.7 0.8 0.9 1.0 w1 0.6 0.7 0.8 0.9 1.0 w2 0.6 0.7 0.8 0.9 1.0 w3 0 20 40 60 80 100 0 20 40 60 80 100 0.6 0.7 0.8 0.9 1.0 w4 0.6 0.7 0.8 0.9 1.0 w5 n1000038 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0.8 0.9 1.0 1.1 1.2 w1 0.8 0.9 1.0 1.1 1.2 w2 0.8 0.9 1.0 1.1 1.2 w3 0 20 40 60 80 0 20 40 60 80 0.8 0.9 1.0 1.1 1.2 w4 0.8 0.9 1.0 1.1 1.2 w5 47

n1050515 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0.6 0.7 0.8 0.9 1.0 w1 0.6 0.7 0.8 0.9 1.0 w2 0.6 0.7 0.8 0.9 1.0 w3 0 20 40 60 80 100 0 20 40 60 80 100 0.6 0.7 0.8 0.9 1.0 w4 0.6 0.7 0.8 0.9 1.0 w5 48

784229 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0.6 0.7 0.8 0.9 1.0 w1 0.6 0.7 0.8 0.9 1.0 w2 0.6 0.7 0.8 0.9 1.0 w3 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0.6 0.7 0.8 0.9 1.0 w4 879663 0.6 0.7 0.8 0.9 1.0 w5 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0.9 1.0 1.1 1.2 1.3 w1 0.9 1.0 1.1 1.2 1.3 w2 0.9 1.0 1.1 1.2 1.3 w3 0 10 20 30 40 50 0 10 20 30 40 50 0.9 1.0 1.1 1.2 1.3 w4 0.9 1.0 1.1 1.2 1.3 w5 49

882502 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0.6 0.7 0.8 0.9 1.0 w1 0.6 0.7 0.8 0.9 1.0 w2 0.6 0.7 0.8 0.9 1.0 w3 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0.6 0.7 0.8 0.9 1.0 w4 0.6 0.7 0.8 0.9 1.0 w5 1011902 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0.6 0.7 0.8 0.9 1.0 w1 0.6 0.7 0.8 0.9 1.0 w2 0.6 0.7 0.8 0.9 1.0 w3 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0.6 0.7 0.8 0.9 1.0 w4 0.6 0.7 0.8 0.9 1.0 w5 50

5073527 0 50 100 150 0 50 100 150 0 50 100 150 0.5 0.6 0.7 0.8 0.9 w1 0.5 0.6 0.7 0.8 0.9 w2 0.5 0.6 0.7 0.8 0.9 w3 0 50 100 150 0 50 100 150 0.5 0.6 0.7 0.8 0.9 w4 0.5 0.6 0.7 0.8 0.9 w5 51