X t. ( t. x 1 (t) x 2 (t) T T. t 2 (u)du. t 1 (u)du T. 0 Xd s ds X
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- Corey Powell
- 5 years ago
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1 r X X sds, X E Q e X sds X r X Xd s ds, X d, X e, ˆX f E d e Xd s ds X d, X e, ˆX f ν ν h ν H x 1 x x 1 x x 1 udu x udu
2 r X Q dr b κr d + σ dw, κ b σ W r e κ r +e κ eκs b s ds+e κ eκs σ s dw s θ e κ r +e κ eκs b s ds X e κ eκs σ s dw s θ X e κ eκs σsds r θ + X, dx κx d + σ dw, X, E Q X, E Q X e κ e κs σsds P, P, X } P, P, ; X P, P, H X + ν h + 1 νh P, } θ sds + ν H d d νh ν h h e κ H hsds ν e κ eκs σsds ν h h v e κ s νsds ν H H ν H sνsds. α σ e κ X h α sdw s u > u X u e κu α s dw s + e κu α s dw s hu X + hu P, ; X x E Q e r udu F, X x e η η u θ udu H x E Q e α s dw s. hu u α sdw s du hu u α sdw s du η Hu H u α s dw s u u Huα u dw u α s dw s H α s dw s Huα u dw u + Huα u dw u.
3 η E Q η H P, ; X x e θ u du H x + 1 αsds H Hsαsds + H sαsds θudu H x E Q e η e H 1 θudu H x } V arη αsds + H sαsds H Hsα sds }. P, θ udu + 1 P, θ udu + 1 P, P, P, ; X x H α sds + H sαsds H α sds + H sα sds H x + 1 H 1 H α sds + α sds + H x + 1 H H P, ; X x P, P, H x 1 H H H He κ e κ e κ κ P, ; X x P, P, P, P, P, P, H H H e κ H He κ αsds e κ Hsαsds h H H sαsds + H H } Hsα sds H } Hsα sds. H sαsds H } Hsαsds α sds H H α sds + H H H + H } x + e κ α sds e κ Hsαsds x + He κ αsds + e κ αsds e κ H H x + He κ αsds + 1 } H ν e κ Hsαsds. h α sds Hs e κs νsds ν h, αudu P, ; X x P, P, H x + ν h + 1 } νh, + } Hsαsds } Hsαsds. } Hsαsds } Hsαsds hs αudu ds
4 X sds, X E Q e X sds X fx V E Q e rsds fx e θsds E Q e Xsds fx e θsds E Q E Q e X sds X fx. X sds, X E Q e Xsds X Z : X sds X EX EZ v X EX, v Z EZ, ρ XZ EX Z v X v Z, c XZ EX Z. vx EX e κ eκs σsds ν X Z Z s dx s + c XZ κ X s ds κ X s Z s ds + c XZ sds + νsds. c XZ e κ e κs νsds ν h. X s ds +. d d E Z EZ X c XZ ν h vz EZ νh sds ν H vx EX ν vz EZ ν H c XZ EX Z ν h ρ XZ EX Z vx v Z νh νν H Σ Z, X EZ Σ EX Z ν EX Z EX H ν h ν h ν Σ ρ XZ X, Z 1 vz ρ XZ v X v Z ρ XZ 1 v X v Z vx gx, z; Σ 1 π e 1 x,zσ 1 x,z 1 πv X v Z 1 x 1 ρ XZ 1 ρ XZ vx ρ xz XZ v X v Z + } z vz
5 X x cxz Z N vx x, v Z c XZ vx N x νh ν, νh νh. ν E Q e Z ν h X X ν + νh νh } ν fx V E Q e rsds fx e θsds E Q e Xsds fx e θsds E Q E Q e X sds X fx } θ s ds + ν H νh E Q ν h } fx X ν ν P, νh } E Q ν h } fx X ν ν r X Q D Q Q D : dq F dq F P, /P, e. rudu d D d P, d r u du d P, P, H X + ν h + 1 νh r d d H σ dw. D Q dd /D Q dd D H σ dw L H uσ udw u D EL : L 1 L } W : W W, L W + H uσ u du Q Q X X e κ s X s + s e κ u σ u dw u s e κ u H uσ udu, X.
6 ν κν + σ h + κh 1 e κ u H uσ udu H ν H ν H ν + H ν + H ν + ν h. X h sx s + s h uh u νu + κνu du νud u h uh u + κ νu κh uh u h uh u h uh uνudu νu κh uh u + h uh u du νuh udu du + κ h uh uνudu } h uσ u dwu H ν + ν h h s H sνs + ν h s. F s X > s h sx s H ν + ν h + h s H sνs + ν h s s e κ u σudu ν h s; κνs Q r θ + X X h sx s + } s h uσ udwu H ν + ν h h s H sνs + ν h s E X X s h sx s H ν + ν h + h s H sνs + ν h s V ar X X s ν h s; κνs. W Q X X s s < F ; s, e s e s < e F ;, s, e 1 τ P, s P, e P, e τ s, e f s P, 1+τK e τ τ, 1 P, s τ P, V, σ e B, 1 < f P, e τf f ; s, e K + f σ B H e s h s f ; κν f h s ; κν
7 K, F, v, w F wφwd 1 K, F, v KwΦwd K, F, v d 1 K, F, v F /K + v / v d K, F, v F /K v /. v V P, e τ K + 1 τ, F ; s, e + 1 τ, H e f H s f ν f, 1. S S s, e : P, s P, e, < s < e e Q e S d S d H s H e X + ν h 1 ν H s H e d H s H e dx d + H e s h s σ dw e ds S H e s h s σ dw, S P, s P, e F ; s, e 1 τ S s, e 1, τ s, e f f < s < e s, e V τ P, e E e K, F, v, w F f ; s, e K + P, e E e Sf 1 + τk +. K, F, v, w F wφwd 1 K, F, v KwΦwd K, F, v d 1 K, F, v F /K + v / v d K, F, v F /K v /, v V P, e 1 + τk, P, s P,, σ e B, 1 P, e τf f ; s, e K + < f f
8 σ B H e s h s f ; κν f h s ; κν. P, 1+τK e τ τ, 1 P, s τ P, V e, σ B, 1 < f P, e τf f ; s, e K + f <, n i 1, i i i 1 i K i τ i i 1, i i V mk i P, i τ i K i, F ; i 1, i, σ mk i i, 1 σi mk i V model P, i τ i K i + 1, F ; i 1, i + 1, H i i H i 1 i ν i, 1. τ i τ i κ ν i i 1,, n K i, F ; i 1, i, σi mk i, 1 K i + 1, F ; i 1, i + 1, H i i H i 1 i ν i, 1 τ i τ i H i i H i 1 i h i 1 i H i i 1 σ σ σ i 1 1 i 1 < i } + σ n 1 1 n }, σ i 1 i 1,, n σ i 1 eκ i ν i e κ i 1 ν i 1 i i 1 e κu du ν i e κ i i 1 ν i 1 i i 1 e κu du ν i hκ; i i 1 ν i 1 Hκ; i i 1 σ σ n 1 ν ν h ν H s, e 1,, n a i, b i i 1,, n w i i < e τ s, e n w if i ; a i, b i τ s, e n w if i ; a i, b i K + τ s, e K n w if i ; a i, b i + τ s, e s, e F i ; a i, b i a i, b i F i ; a i, b i 1 P i, a i P i, b i. τ i P i, b i τ i a i, b i i a i < e b i e b i > e
9 s, e n V P, τ s, e w i F i ; a i, b i 1 i } + w i τ i Ai e BiX 1 1 <i } A i P,ai P,b i 1 ν i H a i i H b i i λ i α i + 1 λ i β } i B i λ i h i λ i Ha i i Hb i i α i H i ν i + ν h i + h i H ν + ν h β i ν i h i ; κν l i s i i n V P, τ s, e w i li F ; a i, b i + s i 1i } + w i l i τ i A i e B ix 1 s iτ i 1 <i } l i V P, τ s, e w i l i τ i A i 1 s iτ i l i A i P,ai P,b i 1 ν ih a i i H b i i λ i α i + 1 λ i ν i } λ i Ha i i Hb i i α i H i ν i + ν h i V P, τ s, e w i E F i ; a i, b i P, τ s, e w i τ i E P i, a i 1. P i, b i P i, a i P i, b i P, a i P, b i Ha i i Hb i i X i 1 ν i H a i i H b i i } i E F i ; a i, b i F i ; a i, b i < i i, X i Q i X i h i X + h i uσ u dwu H i ν i + ν h i h i H ν + ν h }. F X i h i X H i ν i + ν h i +h i H ν + ν h ν i h i ; κν e λx i E λh i X + λh i ν i + ν h i λh i H ν + ν h + 1 } λ ν i h i ; κν.
10 P E i, a i P, a i P i, b i P, b i 1 ν i H a i i H b i i } λ i h i X λ i α i + 1 } λi β i λ i Ha i i Hb i i α i H i ν i + ν h i + h i H ν + ν h β i ν i h i ; κν n V P, τ s, e w i F i ; a i, b i 1 i } + w i τ i Ai e B ix 1 1 <i } A i P,a i P,b i 1 ν i H a i i H b i i λ i α i + 1 λ i β } i B i λ i h i λ i Ha i i Hb i i α i H i ν i + ν h i + h i H ν + ν h β i ν i h i ; κν V P, τ s, e w i τ i +λ i H i ν i + ν h i + 1 λ i ν i P, ai P, b i 1 ν i H a i i H b i i } 1 l i s i i V P, τ s, e n V P, τ s, e w i li F ; a i, b i + s i 1i } + w i E l i F i ; a i, b i + s i w i l i τ i A i e B ix 1 s iτ i 1 <i} l i w V P, τ s, e E we ξ K + P, τ s, e K, eˆµ+ 1 ˆσ, ˆσ, w
11 ˆµ ˆσ i i i > ˆσ V M ˆµ M 1 ˆσ M n ii w i eµ i+ 1 Σ ii V n i,ji w i w j eµ i+µ j + 1 Σ ii+σ jj +Σ ij K n w i ii τ i + K + i 1 w if i ; a i, b i w i w i P,a i τ i P,b i 1 ν i H a i i H b i i Ha i i Hb i i ν h i } µ i Ha i i Hb i i h i X H i ν i + ν h i + h i H ν + ν h Σ ij Ha i i Hb i i Ha j j Hb j j h i + j i j ν i j h i j ; κν ˆσ V M ˆµ M 1 ˆσ M n V n V P, τ s, e E we ξ K + P, τ s, e K, eˆµ+ 1 ˆσ, ˆσ, w w i eµ i+ 1 Σ ii i,j1 w i w j eµi+µj+ 1 ii + jj + ij K n w i τ i + K w i w i P,a i τ i P,b i 1 ν i H a i i H b i i Ha i i Hb i i ν h i } µ i Ha i i Hb i i H i ν i + ν h i Σ ij Ha i i Hb i i Ha j j Hb j j h i + j i j ν i j w τ s, e w w i F i ; a i, b i wk < e X i Q F X i Q i X i h i X + h i uσ u dwu H i ν i + ν h i h i H ν + ν h }, Cov X i X j E i h i uσ u dw u j h j vσ v dw v h i + j i j ν i j h i j ; κν. i i i > w i F i ; a i, b i K w i F i ; a i, b i K + ii w i P i, a i τ ii i P i, b i n i 1 ii w i τ i + K + i j e κ i+ j w i F i ; a i, b i i 1 w i F i ; a i, b i e κu σ udu
12 K n w i ii τ i + K + i 1 w i w i τ i P, a i P, b i Z i Ha i i Hb i i X i. w if i ; a i, b i i i 1 ν i H a i i H b i i Ha i i Hb i i ν h i w i F i ; a i, b i K F Z i ii w ie Zi K. µ i Ha i i Hb i i h i X H i ν i + ν h i + h i H ν + ν h Σ ij Ha i i Hb i i Ha j j Hb j j h i + j i j ν i j h i j ; κν. ξ Nˆµ, ˆσ w i F i ; a i, b i K ii w ie Zi K e ξ K ˆσ V M ˆµ M 1 ˆσ M n ii w i eµ i+ 1 Σ ii V n i,ji w i w j eµi+µj+ 1 Σii+Σjj+Σij }, V P, τ s, e E we ξ K + P, τ s, e K, eˆµ+ 1 ˆσ, ˆσ, w. M f P f M d P d Y M d, P d, Y M f, Y P f. Q d P d, Y M f M d M d Q d Y r d θ d + X d, dx d κ d X d d + σ d dw d, X d dy Y µ e d + σ e dw e,, Y Pf M d Q d σ e W e dw e dw d ρ de d ρ de
13 P d, E d } F r d sds E d Q d dy Y r d r f d + σ e dw e M d Y, Pd Y, M f, P f. Q f M d P, d, Pf Y M f Y M f M f Q f r f θ f + X f, dx f κ f X f d + σ f dw f, X f W f Q f dw f dw d ρ df d dw f dw e ρ ef d ρ df ρ ef r f Q d Q f M d, P d, Y M f, Y P f Y M f Q f Q f Y M f r f Q d Q f dq d F dq f 1 } rs d rs f ds F Y σsdw e s e + 1 } σ e s ds σsdw e s e σsds e 1 } σ e s ds E σsdw e s e σudu e, E W e σe sds Q f W f W f, σ e sdw e s σ e udu W f + ρ ef σsds e : Ŵ f Q d r f Q d r f ˆθ f + ˆX f, d ˆX f κ f ˆXf d + σ f dŵ f, ˆXf ˆθ f θ f ρ ef e κf eκf s σ f s σ e sds W d W f i W d i W d i 1 W f i W f i 1 Q d Q f W d, W f ρ df ρ ef
14 Q d W d Ŵ f W e Q d r d θ d + X d, dx d κ d X d d + σ d dw d, X d r f ˆθ f + ˆX f, d ˆX f κ f ˆXf d + σ f dŵ f, ˆXf, ˆθ f θ f ρ ef e κf s σ eκf s f σsds e Y Y e θe +Xe, θ e θd s ˆθ s f ds 1 σe s ds, X e Xd s ˆX s f ds + σe sdws e dw d dw e ρ de d, dw d dŵ f ρ df d, dw e dŵ f ρ ef d ξ ξ fg 1 X d, g X f Y f g 1 g E d Q d V E d e rd d fg 1 X d, g X f Y e θd d E d e Xd d f g 1 X d, g ˆX f + ˆθ f θf Y e θe +Xe P d, e νh d E d e Xd d f g 1 X d, g ˆX f ρ ef e κf P f, H P d, e ν f +νh d +ρ ef e κf eκf s σ f s σe s ds 1 σe d A E d e Xd d f g 1 X d, g ˆX f B C Y e Xe, A P d, e νh d B ρ ef e κf s σ eκf s f σsds e C P f, P d, e νh f +νh d +ρ ef e κf eκf s σ f s σe s ds 1 σe d e κf s σ f s σ e sdsy e Xe Z, X d, ˆX f, Xe Z Xd d V A e 1 σ E d e µxd,xe, ˆX f, f g 1 X d, g ˆX f B C Y e Xe vd c de c df µx d, x e, x f, c zd, c ze, c zf c ed v e c ef c fd c fe vf v σ vz d c de c df c zd, c ze, c zf c ed v e c ef c fd c fe vf 1 x d x e x f 1 c dz c ez c fz Z vz c zd c ze c zf Σ E X d X e Z, X d, X e, X f c dz vd c de c df c ez c X f ed v e c ef c fz c fd c fe vf
15 Xd s ds, X d, X e, ˆX f E d e Xd s ds X d, X e, ˆX f Z Xd s ds X d e κd e κds σ d sdw d s, X e X d s ˆX f s ds + σ e sdw e s, ˆXf e κf e κf s σ f s dŵ f s, W d Ŵ f W e Q d dw d dŵ f ρ df d, dw d dw e ρ de d, dŵ f dw e ρ fe d. Z, X d, ˆX f, X e vz E d Z νd H vd Ed X d ν d ve c ze vz + νf H + σe s ds ρ ξ ξ s ef σ e κf sσ e eκf s f dsdξ vf Ed ˆX f ν f c zd E d X d Z νd h c ze ρ s s de e κd eκdu σuσ d ududs e s ξ ρ df e e κd κd +κ f ξ ξ +κ f u σ uσ d udu f dξds eκd s ξ ρ df e e κf κd +κ f ξ ξ +κ f u σ uσ d udu f dξds + vz eκd s c zf ρ df e e κf κd +κ f s s +κ f u σ uσ d udu f ds eκd c de νd h ρ s df e e κd κd +κ f s s +κ f u σ uσ d udu f ds + ρ eκd de e κd eκds σsσ d sds e c df E d X d X f ρ df e κd +κ f +κ f s σ sσ d f eκd s ds s c ef ρ df e e κf κd +κ f s s +κ f u σ d uσudu f ds ν h eκd f + ρ ef e κf s σ sσ e f eκf s ds vz c zd c ze c zf Σ c dz vd c de c df c ez c ed v e c ef. c fz c fd c fe vf X d, X e, X f, Z gx z, x d, x e, x f ; Σ 1 4π e 1 x 1 z,x d,x e,x f Σ E d e Z X d x d, X e x e, ˆX f x f. x z,x d,x e,x f. X d, X e, ˆX f Z Nµ, σ vd v de v df µx d, x e, x f, v zd, v ze, v zf ved v e v ef v fd v fe vf 1 x d x e x f v σ vz d v de v df v zd, v ze, v zf ved v e v ef v fd v fe vf v dz v ez v fz 1
16 E d e Z X d x d, X e x e, ˆX f x f µx d, x e, x f, + 1 } σ. X d v d Ed X d e κd eκds σ d s ds ν d ˆX f v f Ed ˆX f e κf eκf s σ f s ds ν f X d Z c zd E d X d Z E d X d s ds + E d vdsds κ d c zd sds. Z s κ d X d s ds + σ d sdw d s d d c zd κ d c zd + ν d c zd ν h d Z v z E d Z Ed Z s X d s ds ν H d X d X d ˆX f c df E d X d ˆX f e κd +κ f E d ρ df e κd +κ f X e e κd +κ f s σ d s σ f s ds. c de E d X d X e E d Xs d Xs d ˆX s f ds + c de c de e κd ν h d e κd ν h d ρ df e κds σ d sdw d s d d c de v d c df κ d c de + ρ de σ d σ e. e κds v ds c df s + ρ de σ d sσ e sds e κf s σ f s dŵ f s Xs e κ d Xs d ds + ρ de σsσ d sds e ρ df e κf s e κd +κ f s σuσ d ududs f + ρ de e κd e κds σsσ d sds e e κd +κ f s e κd +κ f u σuσ d udu f ds + ρ de e κd e κd s Z ˆX f e κds σ d sσ e sds. dc zf E d dz ˆXf E d ˆX f X d d + Z κ f ˆXf d c df d κ f c zf d. c zf e κf ρ df e κf s c df sds e κf e κf s e κd +κ f s ρ df e κd s e κd +κ f u σuσ d ududs f e κd +κ f u σuσ d udu f ds.
17 X e ˆX f c ef e κf e κf ρ df dc ef E d dx e ˆX f E d ˆX f X d ˆX f d + X e κ f ˆXf d + σ e dw e σ f dŵ f c df d v f d κ f c ef d + ρ ef σ e σ f d. e κf s c df s v f s + ρ ef σ e sσ f s ds ρ df e κd s e κf s e κd +κ f s Z X e c ze e κd +κ f u σ d uσ f udu e κf s e κf u σu f du ds + ρ ef e κf e κd +κ f u σuσ d udu f νf h + ρ ef e κf e κf s σ e sσ f s ds. dc ze E d dz X e E d X d X e + Z X d ˆX f d c de + c zd c zf d. c de s + c zd s c zf s ds ρ df ν h d s + ρ de e κd s e κf s ξ ν H d + ρ de ρ df e κdu σ d uσ e udu ρ df ξ e κd +κ f ξ e κd s e κdu σ d uσ e ududs e κd +κ f u σ d uσ f udu e κd s ξ + e κf s ξ e κd +κ f ξ e κd s ξ ξ dξ ds ξ e κd +κ f ξ e κd +κ f u σ d uσ f udu e κf s σ e sσ f s ds e κd +κ f u σ d uσ f udu dξds dξ X e v e d d v e E d X e dx e + E d dx e E d X e X d ˆX f d + E d σ e d c de c ef + σ e d. c de s c ef s + σ e s ds ν H d + ν H f + ρ df ρ ef ξ e κf ξ s ξ e κf ξ c ze v z + ν H f + σ e s ds ρ df e κd +κ f s e κf s σ e sσ f s dsdξ ξ σ e s ds ρ ef e κd ξ s e κd +κ f s e κd +κ f u σuσ d udu f dsdξ + ρ de ξ e κf ξ e κf s σ e sσ f s dsdξ e κd +κ f u σuσ d udu f dsdξ ξ e κd ξ e κds σ d s σ e sdsdξ
18 < 1 < < < N C f i N Ci dn Cf i N Cd i N C f i Ci d < 1 } N N C f i P f, i ; X f Ci d P d, i ; X d,. Y V E e N + N rd s ds Y C f i Xf P f, i ; X f Ci d X d P d, i ; X d Ae 1 σ E d e µxd,xe, ˆX f, f A B C g 1 x N Cd i xp d, i ; x g x N Cf i xp f, i ; x fx, y y x, } g 1 X d, g ˆX f B CY e Xe µx d, x e, x f, + V AE d e Xd s ds CY e Xe ν h; κ H; κ h; κ e κ, H; κ N C f i ˆX f BP f, i ; ˆX N f B Ci d X d P d, i ; X d hs; κds κ 1 e κ κ κ < 1 < < n 1 < n : σ σ 1 σ n 1 σ i 1 i 1, i i 1,,, n ν; κ e κ eκs σsds ν ; κ ν 1 ; κ ν n 1 ; κ ν ; κ ν; κ i 1,, n ν i ; κ e κ i i 1 i e κs σsds + e κs σsds i 1 e κ i i 1 ν i 1 ; κ + e κi σ i 1 i i 1 e κs ds h i i 1 ; κν i 1 ; κ + σ i 1H i i 1 ; κ.
19 ν ; κ ν 1 ; κ ν n 1 ; κ i 1, i ν; κ h i 1 ; κν i 1 ; κ + H i 1 ; κσ i 1 ν h ν h ; κ e κ s νs; κds. ν h ; κ ν h ; κ ν h ; κ i 1, i i 1,, n i 1 ν h ; κ e κ s νs; κds e κ e κs νs; κds + e κs νs; κds i 1 i 1, i e κ i 1 ν h i 1 ; κ + e κ i 1 e κs νs; κds ν h ; κ h i 1 ; κν h i 1 ; κ + ν i 1 ; κh i 1 ; κh i 1 ; κ + 1 σ i 1H i 1 ; κ ν H ν H ν H ; κ, κ H s, κ νs; κds, ν; κ e κ eκs σ sds H, κ e κ s ds κ κ ν H ; κ, κ ν H ; κ, 1 e κ s κ νs; κds 1 κ ν h ; κ, ν h ; κ, κ. sνs; κds νsds sνsds. ν : νsds ν1 : sνsds ν ν 1 i 1, i ν ν i 1 + νsds i 1 ν i 1 + i 1 ν i 1 + σi 1 s i 1ds κ ν i 1 + i 1 e κs i 1 ν i 1 + σi 1 1 e κs i 1 κ ds κ ν i 1 + ν i 1 i 1 + σ i 1 i 1 κ ν i 1 + ν i 1 σ i 1 κ 1 e κ i 1 κ + σ i 1 κ i 1 κ
20 ν 1 ν 1 i 1 + sνsds i 1 ν 1 i 1 + i 1 sν i 1 + σi 1 ss i 1ds κ ν 1 i 1 + i 1 s e κs i 1 ν i 1 + σi 1 1 e κs i 1 κ ds κ ν 1 i 1 + νi 1 i σ 3 i 1 i 1 3 i 1 i 1 ν 1 i 1 + σ i 1 4κ i 1 + ν i 1 κ σ i 1 κ i 1 e κ i e κ i 1 κ κ κ ν ν 1 ν H ν H ν ν 1 r r X 1 + X + θ, r r, X 1 X dx 1 κ 1 X 1 d + σ 1 dw 1, X 1 dx κ X d + σ dw, X W 1, W ρ dw 1 dw ρd, r κ 1 κ σ 1 σ ρ 1, 1 θ x 1 x x 1 e κ 1 s x 1 s + s e κ 1 u σ 1 udw 1 u, x 1 x e κ s x s + s e κ u σ udw u, x F s x 1, x µs, Ex 1, x F s µ 1 s,, µ s, h 1 sx 1 s, h sx s Σs, s e κ1 u σ 1udu s e κ1+κ u σ 1 uσ uρdu ν11 s, ν 1 s, ν 1 s, ν s, s e κ1+κ u σ 1 uσ uρdu s e κ u σudu
21 x 1 x Q Q Q ζ E Q dq P, /P,. dq rudu d ζ d P, d rudu d φudu e H i x i + 1 V, rd φd + h i x i d H i dx i + 1 V, rd x i + h i x i + 1 V, + H i κ i x i d d ζ dζ /ζ 1 dζ /ζ dζ d ζ ζ dζ d ζ + 1 dζ ζ ζ x i + h i x i + H i σ i dw i H i σi d + ρh 1 H σ 1 σ d V,. H i κ i x i d 1 + h i + H i κ i x i d H i σ i dw i, H i σ i dw i H i σ i dw i L H i uσ i udw i u ζ EL : L 1 L } W 1 W 1 W 1, L W 1 + W W W, L W + ρ H 1 uσ 1 udu + ρ H 1 uσ 1 udu + H uσ udu H uσ udu Q Q x 1 e κ 1 s x 1 s M 1 s, + s e κ 1 u σ 1 udw 1 u, x 1 x e κ s x s M s, + s e κ u σ udw u, x W1, W Q ρ M1 s, s e κ1 u σ 1 uh 1 uσ 1 u + ρh uσ udu M s, s e κ u σ uρh 1 uσ 1 u + H uσ udu
22 F s x 1, x µ s, E s x 1, x h 1 sx 1 s M 1 s,, h sx M s, Q x 1 udu x udu xudu x x κ udxu udxu + x u κxudu + σudw u + x uxudu + uσudw u + x κ κ uxudu κ κ κ κx x uxudu u e κu x + u ue κu du κ ude κu κ x + xh κ e κs σsdw s ve κv dv u e κu s σsdw s du u u u ve κv dv κ e κu σu ve κv dv dw u e κu σu u σu u e κs σsdw sd u u e κu s σsdw sdu ve κv dv ve κv dv e κu σudw u, ue κu κ ue κu u u u u e κv u dv e κv dv κ e κv dv dw u dw u
23 κ xudu κ uxudu + x + xh uσudw u + x σu u + uσudw u + x xh + σu e κv u dv dw u xh + σuh udw u u u e κv u dv dw u x 1 udu x 1 H 1 + x udu x H + σ 1 uh 1 udw 1 u σ uh udw u F µ, inegral E x 1 udu, x udu x 1 udu, x udu F H 1 x 1, H x σ Σ, inegral 1uH1 udu σ 1 uσ uh 1 H ρdu σ 1 uσ uh 1 H ρdu σuh udu V, P, E Q e rudu F e φudu E Q e x1u+xudu F φudu H 1 x 1 H x + 1 V, } σ1uh 1 udu + σuh udu + ρ σ 1 uσ uh 1 uh udu. Ω, F, P Q Q F P F D Q P F D F, P
24 D F, P F, Q M F, P M M D 1. M, D F, Q N P M, Ñ M, N M, N. D L D L 1 } L, L EL ; L L D + D 1 s dd s. Q EL P M P M M D 1 M, D M M, L Q P E L 1 Q E L Q Z 1,, Z n n µ Σ n w ie Z i w 1 w n X Nµ, Σ ϕ Ee i X e i µ 1 Σ, Ee X e µ+ 1 Σ n M : E w i e Z i w i e µ i+ 1 Σ ii. n V : E w i e Z i w i w j Ee Z i+z j i,j1 w i w j e µ i+µ j + 1 Σ ii+σ jj +Σ ij, i,j1 Z i + Z j Nµ i + µ j, Σ ii + Σ jj + Σ ij ξ Nˆµ, ˆσ Ee ξ E n w ie Zi Ee ξ E n w ie i Z eˆµ+ 1 ˆσ M e ˆµ+ˆσ V ˆσ V M ˆµ M 1 ˆσ
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