sr s t - - PowerPoint PPT Presentation
sr s t r Pt Prss rt tt s
❘♦❜❡rt ❏✳ ❊❧❧✐♦tt ∗ ❍❛s❦❛②♥❡ ❙❝❤♦♦❧ ♦❢ ❇✉s✐♥❡ss ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧❣❛r② ❈❛❧❣❛r②✱ ❆❧❜❡rt❛✱ ❈❛♥❛❞❛
∗❚❤✐s ✐s ❛ ❥♦✐♥t ✇✐t❤ ❚❛❦ ❑✉❡♥ ❙✐✉ ❛♥❞ ❍❛✐❧✐❛♥❣ ❨❛♥❣✳ ❘♦❜❡rt ❊❧❧✐♦tt ✇♦✉❧❞
❧✐❦❡ t♦ t❤❛♥❦ ❙❙❍❘❈ ❢♦r ✐ts ❝♦♥t✐♥✉❡❞ s✉♣♣♦rt✳ ❍❛✐❧✐❛♥❣ ❨❛♥❣ ✇♦✉❧❞ ❧✐❦❡ t♦ ❛❝❦♥♦✇❧❡❞❣❡ t❤❡ ❘❡s❡❛r❝❤ ●r❛♥ts ❈♦✉♥❝✐❧ ♦❢ t❤❡ ❍♦♥❣ ❑♦♥❣ ❙♣❡❝✐❛❧ ❆❞♠✐♥✐str❛t✐✈❡ ❘❡❣✐♦♥✱ ❈❤✐♥❛ ✭Pr♦❥❡❝t ◆♦✳ ✼✹✷✻✴✵✻❍✮✳
❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r❡s❡♥t❛t✐♦♥✿
➓✶✳ ❇❛❝❦❣r♦✉♥❞ ✶✳✶✳ ❈♦♠♣♦✉♥❞ P♦✐ss♦♥ ♣r♦❝❡ss ❢♦r ❛❝t✉❛r✐❛❧ ✉s❡
❙t❛♥❞❛r❞ ❛♥❞ ❝❧❛ss✐❝❛❧ ♠♦❞❡❧ ❢♦r ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s ✐♥ r✉✐♥ t❤❡♦r②✳
③♦♥✿ ✶✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝❧❛✐♠s ♠♦❞❡❧❡❞ ❛s ❛ P♦✐ss♦♥ ♣r♦❝❡ss✳ ✷✳ ❚❤❡ ❛♠♦✉♥ts ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❝❧❛✐♠s ♠♦❞❡❧❡❞ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳
✸✳ ❈❧❛✐♠ ❢r❡q✉❡♥❝② ❛♥❞ ❝❧❛✐♠ ❛♠♦✉♥ts ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳
✶✳✷✳ ❘❡❣✐♠❡✲s✇✐t❝❤✐♥❣ ♠♦❞❡❧s ❢♦r ❛❝t✉❛r✐❛❧ ✉s❡
✐♥s✉r❛♥❝❡ ❝❧❛✐♠s✿ ❘❡✐♥❤❛r❞ ✭✶✾✽✹✮ ❛♥❞ ❆s♠✉ss❡♥ ✭✶✾✽✾✮✳
s❡♥t ❞✐✛❡r❡♥t ❡❝♦♥♦♠✐❝ ❡♥✈✐r♦♥♠❡♥ts✳
t✉r❛❧ ❝❤❛♥❣❡s ✐♥ t❤❡ ✭♠❛❝r♦✮✲❡❝♦♥♦♠✐❝ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❜✉s✐✲ ♥❡ss ❝②❝❧❡s✳
♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♠♦❞❡❧✿ ✶✳ ❍♦✇ t♦ ❡st✐♠❛t❡ t❤❡ ❤✐❞❞❡♥ r✐s❦ st❛t❡❄ ✷✳ ❍♦✇ t♦ ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧❄
✶✳✸✳ ❑❡② ♣♦✐♥ts ♦❢ ♦✉r ✇♦r❦
♦❢ ▼❛r❦♦✈✲♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♣r♦❝❡ss❡s ❜❛s❡❞ ♦♥ t❤❡ ♦❜s❡r✈❡❞ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❝❧❛✐♠s ❛♥❞ t❤❡ ❝❧❛✐♠ s✐③❡s✳
❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳
t❤❡ ❝❧❛✐♠ s✐③❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳
✐♥ ❜♦t❤ ❝❛s❡s✳
❛♥❞ s♠♦♦t❤❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠s✳
➓✷✳ ▼♦❞❡❧ ❉②♥❛♠✐❝s ❛♥❞ ❈❤❛♥❣❡ ♦❢ ▼❡❛s✉r❡s
❛ ▼❛r❦♦✈✲s✇✐t❝❤✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ♦♥❧② ❢♦r ❛❣❣r❡❣❛t❡ ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s✳
❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳
✇✐t❤ st❛t❡ s♣❛❝❡ {e1, e2, . . . , eN}✱ ❛ ✜♥✐t❡ s❡t ♦❢ ✉♥✐t ✈❡❝t♦rs ✇✐t❤ ei = (0, . . . , 1, . . . , 0) ∈ ℜN✳
Xt = X0 +
t
0 AXsds + Mt ,
✇❤❡r❡ {Mt}t∈T ✐s ❛ ℜN✲✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ (FX, P)✳
st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ✐s✿ λt := λ, Xt =
K
λ, ei I{Xt=ei}, ✇❤❡r❡ λ := (λ1, λ2, . . . , λK) ∈ ℜK ❛♥❞ λk ≥ 0✱ ❢♦r ❡❛❝❤ k = 1, 2, . . . , K✳
t✐♦♥s FX := {FX
t }t∈T ,
FX
t
:= σ{Xu|u ∈ [0, t]} , FN := {FN
t }t∈T ,
FN
t
:= σ{Nu|u ∈ [0, t]} , G := {Gt}t∈T , Gt := FX
t
∨ FN
t
▼❛❧❝♦❧♠ ✭✷✵✵✺✮✮✿ Nt =
t
0 λ, Xu du + Vt ,
✇❤❡r❡ V := {Vt}t∈T ✐s ❛ (P, σ{Gu|u ∈ [0, t]})✲♠❛rt✐♥❣❛❧❡✳
FY (·) ✐s ❣✐✈❡♥✳
Zt =
t ∞
ydFY (y)dNu
✐s ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ✉♥✐t ✐♥t❡♥s✐t② ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ X✳
Qt := Nt − t , ✐s ❛ ♠❛rt✐♥❣❛❧❡✳
Λ0,t :=
Xu, λ∆Nu exp
t
0 (1 − Xu, λ)du
1 +
t
0 Λ0,u−(Xu−, λ − 1)dQu
Λ0,t := dP dP†
t ✱ ❡st✐♠❛t❡ γt ❜② ✐ts ❧❡❛st✲sq✉❛r❡ ❡st✐♠❛t❡ E[γt|FN t ]✳
E[γt|FN
t ] = E†[Λ0,tγt|FN t ]
E†[Λ0,t|FN
t ]
= σt(γ) σt(1) , ✇❤❡r❡ E† ❞❡♥♦t❡s ❡①♣❡❝t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ P†✳
➓✸✳ ❋✐❧t❡rs ❛♥❞ ❙♠♦♦t❤❡rs
❝❡ss ♦❢ t❤❡ ✜❧t❡r❡❞ ❡st✐♠❛t❡s✳
t❤❡ s♠♦♦t❤❡❞ ❡st✐♠❛t❡s ❛♥❞ t❤❡ ♣r♦❝❡ss ♦❢ ❡①tr❛ ✐♥❢♦r♠❛t✐♦♥✱ r❡s♣❡❝t✐✈❡❧②✳
❝♦✉rs❡ t♦ st♦❝❤❛st✐❝ ✐♥t❡❣r❛t✐♦♥✳
✸✳✶✳ ❋✐❧t❡r❡❞ ❡st✐♠❛t❡s
t ] ∈ ℜK✱ t❤❡ ✉♥♥♦r♠❛❧✐③❡❞ ❝♦♥❞✐✲
t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ X ❣✐✈❡♥ t❤❡ ♦❜s❡r✈❛t✐♦♥s✳
qt = q0 +
t
0 Aqudu +
t
0 diag{λ, ek − 1}qu−dQu
pt := E[Xt|FN
t ] =
qt qt, 1 , ✇❤❡r❡ 1 := (1, 1, . . . , 1) ∈ ℜK✳
ℜK×K✱ t ∈ T ✱ ❛♥❞ Γt := diag{γ1
t , γ2 t , . . . , γK t } ,
✇❤❡r❡ γk
t := exp[(1 − λ, ek)t] λ, ekNt✱ k = 1, 2, . . . , K✳
qt := Γ−1
t
qt✳
qt s❛t✐s✜❡s t❤❡ ❢♦r✇❛r❞ ❧✐♥❡❛r ❖✳❉✳❊✳✿ d¯ qt dt = Γ−1
t
AΓt¯ qt , ¯ q0 = q0 ∈ ℜK
π(Xt) := Γt¯ qt Γt¯ qt, 1 . ❚❤❡♥✱ π(Xt) ✐s ❛ ✈❡rs✐♦♥ ♦❢ E[Xt|FN
t ]✱ ✇❤✐❝❤ ✐s ❝♦♥t✐♥✉♦✉s ✐♥
t❤❡ ♦❜s❡r✈❛t✐♦♥ ♣r♦❝❡ss N ✐♥ t❤❡ ❙❦♦r♦❦❤♦❞ t♦♣♦❧♦❣②✳
✸✳✷✳ ❙♠♦♦t❤❡❞ ❡st✐♠❛t❡s
T ]✱ ❢♦r t ∈ [0, T]✳
E[Xt|FN
T ] = E†[Λ0,TXt|FN T ]
E†[Λ0,T|FN
T ]
T ]✱ ✇❤✐❝❤ ✐s t❤❡ ✉♥♥♦r♠❛❧✐③❡❞ s♠♦♦t❤❡r
♦❢ Xt ❣✐✈❡♥ FN
T ✳
rt = E†[Λ0,tXtE†[Λt,T|FN
T ∨ FX t ]|FN T ]
E†[Λt,T|FN
T ∨ FX t ] = E†[Λt,T|FN T ∨ σ{Xt}]
νk
t := E†[Λt,T|FN T , Xt = ek] ,
❛♥❞ νt := (ν1
t , ν2 t , . . . , νK t ) ∈ ℜK✳
k=1 Xt, ek = 1✱ t❤❛t νk t ✐s FN T ✲♠❡❛s✉r❛❜❧❡✱ ❛♥❞
t❤❛t N ❤❛s ✐♥❞❡♣❡♥❞❡♥t ✐♥❝r❡♠❡♥ts ✉♥❞❡r P†✳
rt =
K
qk
t νk t ek ∈ ℜK
pt := E[Xt|FN
T ] =
rt rt, 1
rt, 1 = qt, rt
qt, νt✳
d dt rt, 1 = d dt qt, νt = 0
pt = 1 ¯ qt, ¯ νt
K
¯ qt, ek ¯ νt, ek ek , ✇❤❡r❡ ¯ qt s❛t✐s✜❡s✿ d¯ qt dt = Γ−1
t
AΓt¯ qt , ¯ q0 = q0 , ❛♥❞ ¯ νt s❛t✐s✜❡s✿ d¯ νt dt = −ΓtA∗Γ−1
t
¯ νt , ¯ νT = ΓT1
➓✹✳ ▼❛r❦♦✈✲❙✇✐t❝❤✐♥❣ ❙t♦❝❤❛st✐❝ ■♥t❡♥s✐t② ❛♥❞ ❈❧❛✐♠ ❙✐③❡s
{Zt}t∈T ✉♥❞❡r P✳
(0, ∞)✳
γ(dy, dt; ω) =
δ(YTk(ω))δ(Tk(ω)) s♦ t❤❛t ❢♦r s✉✐t❛❜❧❡ ✐♥t❡❣r❛♥❞s f : (Ω × (0, ∞) × [0, ∞)) → ℜ✱
t ∞
f(ω, y, u)γ(dy, du) =
f(ω, YTk(ω), Tk(ω))
Zt =
t ∞
yγ(dy, du)
s✐③❡ y := Zu − Zu− ✇❤❡♥ Xu− = ek✱ k = 1, 2, . . . , K✳
r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t②✿ λt := λ, Xt , ✇❤❡r❡ λ := (λ1, λ2, . . . , λK) ∈ ℜK✳
s✐③❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳
ν(dy, du|Xu−) :=
K
Xu−, ek λkfk(y)dydu
Gt := FZ
t ∨ FX t ✳
ˆ Mt := Zt −
t ∞
yν(dy, du|Xu−) , ✐s ❛ (¯ G, P)✲❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡✳
❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ✉♥✐t ✐♥t❡♥s✐t② ❛♥❞ ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❝❧❛✐♠ s✐③❡s f(y)✳
ν†(dy, du) := f(y)dydu
M†
t := Zt −
t ∞
yν†(dy, du) , ✐s ❛ ❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡ ✉♥❞❡r P†✳
f(y) ✱ k = 1, 2, . . . , K✳
Λ := {¯ Λ0,t}t∈T ❣✐✈✐♥❣ t❤❡ ❝❤❛♥❣❡ ♦❢ ♠❡❛s✉r❡✿ ¯ Λ0,t = exp
t
K
Xu−, ek
∞
0 (hk(y) − 1)f(y)dydu
+
t
K
Xu−, ek
∞
log hk(y)γ(dy, du)
Λ ✐s ❛ ( ¯ G, P†)✲❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡ ❛♥❞ ❛ss✉♠❡ ✐t ✐s ❛ ( ¯ G, P†)✲ ♠❛rt✐♥❣❛❧❡✳
¯ Λ0,t := dP dP†
Gt
ˆ M ✐s ❛ (¯ G, P)✲❧♦❝❛❧ ♠❛rt✐♥❣❛❧❡✳
t )✱
✇❤❡r❡ FZ
t = σ{Zu|u ∈ [0, t]}✳
❛♥❞ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ f ✉♥❞❡r P†✳
Hk(u, ω) := λkfk(Yu(ω)) f(Yu(ω)) = hk(Yu(ω))
diag(H − 1) := diag(H1(u, ω) − 1, . . . , HK(u, ω) − 1) , diag(λ − 1) := diag(λ1 − 1, . . . , λK − 1)
Λ0,tXt|FZ
t )✳ ❚❤❡♥✱
qt = q0 +
t
0 Aqudu +
t
0 diag(H − 1)qu−dNu
−
t
0 diag(λ − 1)qudu
γk
t := exp
t
0 log Hk(u, ω)dNu
t , γ2 t , . . . , γK t )✳
qt := Γ−1
t
qt✳ ❚❤❡♥✱ ¯ q s❛t✐s✜❡s t❤❡ ❧✐♥❡❛r ❖✳❉✳❊✳✿ ¯ qt = q0 +
t
0 Γ−1 u AΓu¯
qudu
T )✳
t
r❡♣❧❛❝❡❞ ❜② FZ
t ✳
➓✺✳ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ❜② t❤❡ ❊▼ ❆❧❣♦r✐t❤♠
♣♦✐♥t ♣r♦❝❡ss ✇✐t❤ ▼❛r❦♦✈✲s✇✐t❝❤✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ❛♥❞ ❝❧❛✐♠ s✐③❡s ✉s✐♥❣ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠✳
t❤❡ ▼❛r❦♦✈✲♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♠♦❞❡❧ ✇✐t❤ st♦❝❤❛s✲ t✐❝ ✐♥t❡♥s✐t②✳
(λ1, λ2, . . . , λK)✳
❢♦r ❡❛❝❤ r❡❣✐♠❡ k = 1, 2, . . . , K✳
✶✳ ❉✐✈✐❞❡ ❛ ❣✐✈❡♥ s❡t ♦❢ ❝❧❛✐♠s ❞❛t❛ ✐♥t♦ K ❣r♦✉♣s ✉s✐♥❣ s♦♠❡ s✐♠♣❧❡ ❝r✐t❡r✐❛✳ ✷✳ ❋♦r ❡①❛♠♣❧❡✱ ❞❡t❡r♠✐♥❡ ❛ s❡t ♦❢ t❤r❡s❤♦❧❞ ♣❛r❛♠❡t❡rs✱ s❛② R1 < R2 < · · · < RK−1✱ ❛♥❞ ❛❧❧♦❝❛t❡ t❤❡ ❝❧❛✐♠ ❞❛t❛ Yt ✐♥t♦ t❤❡ kth ❣r♦✉♣ ✐❢ Yt ∈ [Rk−1, Rk]✳ ✸✳ ❊st✐♠❛t❡ Fk(y) ✉s✐♥❣ t❤❡ ❝❧❛✐♠s ❞❛t❛ ✐♥ t❤❡ kth ❣r♦✉♣ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦✉t❧✐♥❡❞ ✐♥ ❑❧✉❣♠❛♥ ❡t ❛❧✳ ✭✷✵✵✹✮✳
s♠♦♦t❤❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠✳
♠❛t✐♦♥ s❝❤❡♠❡ ❜② ❝♦♠♣✉t✐♥❣ t✐♠❡ ❞♦♠❛✐♥ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡s❡ r♦❜✉st ❞②♥❛♠✐❝s✳
✺✳✶✳ ❘♦❜✉st ✜❧t❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠
❝❡ss❡s N ❛♥❞ Z ♣r♦✈✐❞❡ t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ ✐♥❢♦r♠❛t✐♦♥ t♦ ❡st✐♠❛t❡ aij ❛♥❞ λi✳
aij ❛♥❞ ˆ λi ❛r❡ ✭s❡❡ ❉❡♠❜♦ ❛♥❞ ❩❡✐t♦✉♥✐ ✭✶✾✽✻✮✮✿ ˆ aij = E[Nij
T |FN T ]
E[Oi
T|FN T ] = E[Nij T |FZ T ]
E[Oi
T|FZ T ] ,
❛♥❞ ˆ λi = E[Gi
T|FN T ]
E[Oi
T|FN T ] = E[Gi T|FZ T ]
E[Oi
T|FZ T ]
G✲❛❞❛♣t❡❞ ✐♥t❡❣r❛❜❧❡ ♣r♦❝❡ss γ := {γt}t∈T ✱ σ(γt) := E†[¯ Λ0,tγt|FZ
t ]
ˆ aij = σ(Nij
T )
σ(Oi
T) ,
ˆ λi = σ(Gi
T)
σ(Oi
T)
Oi
t :=
t
0 Xu, ei du ∈ ℜ ,
Nij
t
:=
t
0 Xu−, ei
❛♥❞ Gi
t :=
t
0 Xu, ei dNu ∈ ℜ
t Xt)✱
σ(Oi
tXt) ❛♥❞ σ(Gi tXt) ❛r❡✿
σ(Gi
tXt)
=
t
0 Aσ(Gi uXu)du +
t
0 diag(H − 1)σ(Gi u−Xu−)dNu
−
t
0 diag(λ − 1)σ(Gi uXu)du
+
t
0 h, ei qu−, ei dNuei
−
t
0 λ − 1, ei qu, ei duei ,
σ(Nij
t Xt)
=
t
0 Aσ(Nij u Xu)du +
t
0 qu−, ei
+
t
0 diag(H − 1)σ(Nij u−Xu−)dNu
−
t
0 diag(λ − 1)σ(Nij u Xu)du ,
σ(Oi
tXt)
=
t
0 Aσ(Oi uXu)du +
t
0 qu, ei duei
+
t
0 diag(H − 1)σ(Oi u−Xu−)dNu
−
t
0 diag(λ − 1)σ(Oi uXu)du
t Xt), 1
t Xt], 1
t ) ,
tXt), 1
t) ,
tXt), 1
t)
ˆ aij =
T XT), 1
TXT), 1
,
ˆ λi =
TXT), 1
TXT), 1
st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s t♦ ❝♦♠♣✉t❡ t❤❡ ❡st✐♠❛t♦rs✳
✭✶✾✼✽✮ t♦ ❡❧✐♠✐♥❛t❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ❛♥❞ t♦ ❞❡✈❡❧♦♣ r♦❜✉st ✜❧t❡rs ✇❤✐❝❤ ❞♦ ♥♦t ✐♥✈♦❧✈❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s✳
tXt)✱
σ(Nij
t Xt) ❛♥❞ σ(Oi tXt)✳
t , γ2 t , . . . , γK t )✱ ✇❤❡r❡ γk t := exp(Xk t ) ❛♥❞
Xk
t := (1 − λk)t +
t
0 log Hk(u, ω)dNu
σ(Gi
tXt) := Γ−1 t
σ(Gi
tXt)✳
¯ σ(Gi
tXt)
=
t
0 Γ−1 u AΓu¯
σ(Gi
uXu)du + diag
H
−
t
0 Nu h, ei
H
qu
¯ σ(Nij
t Xt)
=
t
0 Γ−1 u AΓu¯
σ(Nij
u Xu)du +
t
0 ¯
qu, ei
¯ σ(Oi
tXt)
=
t
0 Γ−1 u AΓu¯
σ(Oi
uXu)du +
t
0 ¯
qu, ei duei ,
¯ qt = q0 +
t
0 Γ−1 u AΓu¯
qudu , ✇❤❡r❡ diag
H
Hk(t, ω), . . . , 1 HK(t, ω)
✜❧t❡r✐♥❣ ❡q✉❛t✐♦♥s✳
qtm ≈ Φm[I + A∆]qtm−1 , ✇❤❡r❡ Φm := ΓtmΓ−1
tm ✳
tXt)✿
σ(Gi
tmXtm)
≈ σ(Gi
tm−1Xtm−1) + Γtm−1 h, ei
Htm
qtm, ei Ntmei −diag
Htm−1
qtm−1, ei
Htm
qtm, ei
−Γtm−1Ntm−1 h, ei
Htm−1
tm−1(A − I)qtm−1, ei
t Xt)✿
σ(Nij
tmXtm)
≈ Φm[I + ∆A]σ(Nij
tm−1) + Φm
tXt)✿
σ(Oi
tmXtm) ≈ Φm[I + ∆A]σ(Oi tm−1Xtm−1) + Φm
aij ❛♥❞ ˆ λi ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② t❤❡ t❤r❡❡ st❡♣s ♦❢ t❤❡ ✜❧t❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠✳ ✶✳ ❙t❡♣ ■✿ ❙❡❧❡❝t t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s ˆ aij(0) ❛♥❞ λi(0)✳ ✷✳ ❙t❡♣ ■■✿ ❈♦♠♣✉t❡ t❤❡ ▼▲❊s ˆ aij =
T XT), 1
TXT), 1
,
ˆ λi =
TXT), 1
TXT), 1
✺✳✷✳ ❙♠♦♦t❤❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠
✜❧t❡r✐♥❣ s❝❤❡♠❡s✳
❡st✐♠❛t❡s✱ r❛t❤❡r t❤❛♥ ✜❧t❡r❡❞ ❡st✐♠❛t❡s✱ ✐♥ s♦♠❡ ♣r❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠✳
✇❤✐❝❤ ✐♥✈♦❧✈❡ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ❡✈♦❧✈✲ ✐♥❣ ❜❛❝❦✇❛r❞s ✐♥ t✐♠❡✳
✐❝s t♦ ❞❡✈❡❧♦♣ s♠♦♦t❤✐♥❣ ❛❧❣♦r✐t❤♠s✱ ✇❤✐❝❤ ❞♦ ♥♦t ✐♥✈♦❧✈❡ st♦❝❤❛st✐❝ ✐♥t❡❣r❛❧s ❛t ❛❧❧✳
ν ❢♦r ν✱ ✇❤❡r❡ νt := (ν1
t , ν2 t , . . . , νK) ∈ ℜK ✇✐t❤
νk
t := E†[Λt,T|FN T , Xt = ek]
ν s✉❝❤ t❤❛t t❤❡ ❞✉❛❧✐t② ❤♦❧❞s✿ ¯ qt, ¯ νt =
t
qt, Γtνt
νt := Γtνt✳ ❇② ❚❤❡♦r❡♠ ✸✳✹✱ d¯ νt dt = −ΓtA∗Γ−1
t
¯ νt , ✇❤❡r❡ ¯ νT = ΓTνT = ΓT1✳
TXT), νT
σ(Gi
TXT), ¯
νT
diag
H
qT, ei ei, ¯ qT −
T
0 Nt h, ei ei, ¯
νt
H
qt
T XT), νT
σ(Nij
T XT), ¯
νT
T
T, XT), νT
σ(Oi
T, XT), ¯
νT
T
0 qt, ei νt, ei dt
ˆ aij(k + 1) = ˆ aij(k)
T
0 qt, ei
T
0 qt, ei νt, ei dt
, ❛♥❞ ˆ λi(k + 1) =
H
qT, ei ei, ¯ qT −
T
0 Nt h, ei ei, ¯
νt
H
qt
T
0 qt, ei νt, ei dt
−1
st❡♣s ✐♥ t❤❡ s♠♦♦t❤❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠✿ ✶✳ ❙t❡♣ ■✿ ❙❡❧❡❝t ˆ aij(0) ❛♥❞ ˆ λi(0)✳ ✷✳ ❙t❡♣ ■■✿ ❈♦♠♣✉t❡ t❤❡ ▼▲❊s✱ ˆ aij(k + 1) ❛♥❞ ˆ λi(k + 1)✱ r❡s♣❡❝t✐✈❡❧②✳ ✸✳ ❙t❡♣ ■■■✿ ❙t♦♣ ♦r ❝♦♥t✐♥✉❡ ❢r♦♠ ❙t❡♣ ■■✳
➓✻✳ ❙✉♠♠❛r②
❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♠♦❞❡❧ ❛♥❞ ♠❛r❦❡❞ ♣♦✐♥t ♣r♦❝❡ss ❢♦r ❛❝✲ t✉❛r✐❛❧ ✉s❡ ❛♥❞ ♦t❤❡r ♣♦t❡♥t✐❛❧ ❛♣♣❧✐❝❛t✐♦♥s✳
♦✈❡r t✐♠❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ st❛t❡ ♦❢ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✜♥✐t❡✲ st❛t❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳
s✐t② ❛♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❥✉♠♣ s✐③❡ ❞❡♣❡♥❞ ♦♥ t❤❡ st❛t❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳
✐♥ ❜♦t❤ ❝❛s❡s✱ ✇❤✐❝❤ ♣r♦✈✐❞❡ ❛ ♠❡t❤♦❞ t♦ s❡❧❡❝t ♦r ❡st✐♠❛t❡ r✐s❦ ♠♦❞❡❧ ✐♥ t❤❡ ✏♠❡❛♥✲sq✉❛r❡✲❡rr♦r✑ s❡♥s❡✳
♠♦❞✉❧❛t❡❞ ♠❛r❦❡❞ ♣♦✐♥t ♣r♦❝❡ss ❜❛s❡❞ ♦♥ t❤❡ r♦❜✉st ✜❧t❡r✲ ❜❛s❡❞ ❛♥❞ s♠♦♦t❤❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠s
❘❡❢❡r❡♥❝❡s
♠❡♥t✳ ❙❝❛♥❞✐♥❛✈✐❛♥ ❆❝t✉❛r✐❛❧ ❏♦✉r♥❛❧ ✷✱ ✻✾✲✶✵✵✳
❣❛♣♦r❡✳
t♦ t❤❡ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❢♦r ◆♦♥❧✐♥❡❛r ❋✐❧t❡r✲ ✐♥❣✳ ■♥✿ ❙❦✇✐r③②♥s❦✐✱ ❏✳ ❑✳ ✭❊❞✳✮✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ❙②st❡♠s ❛♥❞ ❘❛♥❞♦♠ Pr♦❝❡ss ❚❤❡♦r②✳ ❉❛r❧✐♥❣t♦♥✱ ❆❧♣❤❡♥ ❛♥❞ ❞❡♥ ❘✐❥♥✱ ❚❤❡ ◆❡t❤❡r❧❛♥❞s✱ ❙✐❥t❤♦✛ ❛♥❞ ◆♦♦r❤♦✛✱ ✼✷✶✲✼✸✹✳
P❛rt✐❛❧❧② ❖❜s❡r✈❡❞ ❈♦♥t✐♥✉♦✉s ❚✐♠❡ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ✈✐❛ t❤❡ ❊▼ ❆❧❣♦r✐t❤♠✳ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ❛♥❞ ❆♣♣❧✐❝❛✲ t✐♦♥s ✷✸✭✶✮✱ ✾✶✲✶✶✸✳
❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳
▼❛r❦♦✈ ▼♦❞❡❧s✿ ❊st✐♠❛t✐♦♥ ❛♥❞ ❈♦♥tr♦❧✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦✳
❘♦❜✉st ❊▼ ❆❧✲ ❣♦r✐t❤♠s ❢♦r ▼❛r❦♦✈✲▼♦❞✉❧❛t❡❞ P♦✐ss♦♥ Pr♦❝❡ss❡s✳ ■❊❊❊ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❉❡❝✐s✐♦♥ ❛♥❞ ❈♦♥tr♦❧✳ ❙②❞♥❡②✱ ❆✉str❛❧✐❛✱ ❉❡✲ ❝❡♠❜❡r✱ ✷✵✵✵✳
✐♥❣ ❋♦r♠✉❧❛s ❢♦r ▼❛r❦♦✈✲▼♦❞✉❧❛t❡❞ P♦✐ss♦♥ ❖❜s❡r✈❛t✐♦♥s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❆✉t♦♠❛t✐❝ ❈♦♥tr♦❧ ✺✵✭✽✮✱ ✶✶✷✸✲✶✶✸✹✳
▼♦❞❡❧s✿ ❉❛t❛✱ ❉❡❝✐s✐♦♥s✱ ❛♥❞ ❘✐s❦s✳ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ◆❡✇ ❨♦r❦✳
❖♥ ❛ ❈❧❛ss ♦❢ ❙❡♠✐✲▼❛r❦♦✈ ❘✐s❦ ▼♦❞❡❧s ❖❜t❛✐♥❡❞ ❛s ❈❧❛ss✐❝❛❧ ❘✐s❦ ▼♦❞❡❧s ✐♥ ❛ ▼❛r❦♦✈✐❛♥ ❡♥✈✐r♦♥♠❡♥t✳ ❆❙❚■◆ ❇✉❧❧❡t✐♥ ✸✶✭✷✮✱ ✷✺✺✲✷✼✸✳
❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ❢♦r ❋✐♥❛♥❝❡ ❛♥❞ ■♥s✉r❛♥❝❡✳ ❲✐❧❡②✱ ◆❡✇ ❨♦r❦✳