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sr s t r Pt Prss rt tt s

  1. ■♥s✉r❛♥❝❡ ❈❧❛✐♠s ▼♦❞✉❧❛t❡❞ ❜② ❛ ❍✐❞❞❡♥ ▼❛r❦❡❞ P♦✐♥t Pr♦❝❡ss ❘♦❜❡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 ◆♦✳ ✼✹✷✻✴✵✻❍✮✳

  2. ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r❡s❡♥t❛t✐♦♥✿ • ❇❛❝❦❣r♦✉♥❞ • ▼♦❞❡❧ ❞②♥❛♠✐❝s ❛♥❞ ❝❤❛♥❣❡ ♦❢ ♠❡❛s✉r❡s • ❋✐❧t❡rs ❛♥❞ s♠♦♦t❤❡rs • ▼❛r❦♦✈✲s✇✐t❝❤✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ❛♥❞ ❝❧❛✐♠ s✐③❡s • P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥✿ ❊▼ ❛❧❣♦r✐t❤♠ • ❙✉♠♠❛r②

  3. ➓✶✳ ❇❛❝❦❣r♦✉♥❞ ✶✳✶✳ ❈♦♠♣♦✉♥❞ P♦✐ss♦♥ ♣r♦❝❡ss ❢♦r ❛❝t✉❛r✐❛❧ ✉s❡ • ❈♦♠♣♦✉♥❞ P♦✐ss♦♥ ♣r♦❝❡ss✿ ❙t❛♥❞❛r❞ ❛♥❞ ❝❧❛ss✐❝❛❧ ♠♦❞❡❧ ❢♦r ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s ✐♥ r✉✐♥ t❤❡♦r②✳ • ❉❡s❝r✐❜❡ ❛❣❣r❡❣❛t❡ ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s ♦✈❡r ❛ ✜①❡❞ t✐♠❡ ❤♦r✐✲ ③♦♥✿ ✶✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝❧❛✐♠s ♠♦❞❡❧❡❞ ❛s ❛ P♦✐ss♦♥ ♣r♦❝❡ss✳ ✷✳ ❚❤❡ ❛♠♦✉♥ts ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❝❧❛✐♠s ♠♦❞❡❧❡❞ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳

  4. ✸✳ ❈❧❛✐♠ ❢r❡q✉❡♥❝② ❛♥❞ ❝❧❛✐♠ ❛♠♦✉♥ts ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ • P♦♣✉❧❛r✿ ❆♥❛❧②t✐❝❛❧❧② tr❛❝t❛❜❧❡ r❡s✉❧ts ❢♦r r✉✐♥ ♣r♦❜❛❜✐❧✐t✐❡s✳ • ❘❡❢❡r❡♥❝❡s✿ ❘♦❧s❦✐ ❡t ❛❧✳ ✭✶✾✾✾✮ ❛♥❞ ❆ss♠✉ss❛♥ ✭✷✵✵✵✮✳

  5. ✶✳✷✳ ❘❡❣✐♠❡✲s✇✐t❝❤✐♥❣ ♠♦❞❡❧s ❢♦r ❛❝t✉❛r✐❛❧ ✉s❡ • ▼❛r❦♦✈✲♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♠♦❞❡❧ ❢♦r ❛❣❣r❡❣❛t❡ ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s✿ ❘❡✐♥❤❛r❞ ✭✶✾✽✹✮ ❛♥❞ ❆s♠✉ss❡♥ ✭✶✾✽✾✮✳ • ■♥❝❧✉❞❡ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✇❤♦s❡ st❛t❡s r❡♣r❡✲ s❡♥t ❞✐✛❡r❡♥t ❡❝♦♥♦♠✐❝ ❡♥✈✐r♦♥♠❡♥ts✳ • ▼♦❞❡❧ t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ st❛t❡ ♦❢ ❛♥ ❡❝♦♥♦♠② ❞✉❡ t♦ str✉❝✲ t✉r❛❧ ❝❤❛♥❣❡s ✐♥ t❤❡ ✭♠❛❝r♦✮✲❡❝♦♥♦♠✐❝ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❜✉s✐✲ ♥❡ss ❝②❝❧❡s✳ • ❖❜s❡r✈❛❜❧❡ ❈❧❛✐♠ ❢r❡q✉❡♥❝② ❛♥❞ ❝❧❛✐♠ s✐③❡s✳

  6. • ❯♥♦❜s❡r✈❛❜❧❡ ▼❛r❦♦✈ ❝❤❛✐♥ = > ▼♦❞❡❧ ✉♥❝❡rt❛✐♥t②✳ • ❚✇♦ ❦❡② ♣r♦❜❧❡♠s ✐♥ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ▼❛r❦♦✈✲ ♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♠♦❞❡❧✿ ✶✳ ❍♦✇ t♦ ❡st✐♠❛t❡ t❤❡ ❤✐❞❞❡♥ r✐s❦ st❛t❡❄ ✷✳ ❍♦✇ t♦ ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧❄

  7. ✶✳✸✳ ❑❡② ♣♦✐♥ts ♦❢ ♦✉r ✇♦r❦ • ❉❡✈❡❧♦♣ ♠❡t❤♦❞s ❢♦r ✜❧t❡r✐♥❣ ❛♥❞ s♠♦♦t❤✐♥❣ t❤❡ ❤✐❞❞❡♥ st❛t❡s ♦❢ ▼❛r❦♦✈✲♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♣r♦❝❡ss❡s ❜❛s❡❞ ♦♥ t❤❡ ♦❜s❡r✈❡❞ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❝❧❛✐♠s ❛♥❞ t❤❡ ❝❧❛✐♠ s✐③❡s✳ • ❈❛s❡ ■✿ ❙t♦❝❤❛st✐❝ ✐♥t❡♥s✐t② s✇✐t❝❤❡s ♦✈❡r t✐♠❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳ • ❈❛s❡ ■■✿ ❇♦t❤ t❤❡ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ❛♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❝❧❛✐♠ s✐③❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳

  8. • ❉❡r✐✈❡ r♦❜✉st ✜❧t❡rs ❛♥❞ s♠♦♦t❤❡rs ✐♥ t❤❡ ❢♦r♠ ♦❢ ❖✳❉✳❊✳s ✐♥ ❜♦t❤ ❝❛s❡s✳ • ❊st✐♠❛t❡ t❤❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs ✉s✐♥❣ t❤❡ r♦❜✉st ✜❧t❡r✲❜❛s❡❞ ❛♥❞ s♠♦♦t❤❡r✲❜❛s❡❞ ❊▼ ❛❧❣♦r✐t❤♠s✳

  9. ➓✷✳ ▼♦❞❡❧ ❉②♥❛♠✐❝s ❛♥❞ ❈❤❛♥❣❡ ♦❢ ▼❡❛s✉r❡s • ❈♦♥s✐❞❡r ❛ ▼❛r❦♦✈✲♠♦❞✉❧❛t❡❞ ❝♦♠♣♦✉♥❞ P♦✐ss♦♥ ♠♦❞❡❧ ✇✐t❤ ❛ ▼❛r❦♦✈✲s✇✐t❝❤✐♥❣ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ♦♥❧② ❢♦r ❛❣❣r❡❣❛t❡ ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s✳ • ❉❡s❝r✐❜❡ t❤❡ ❤✐❞❞❡♥ st❛t❡s ♦❢ ❛♥ ❡❝♦♥♦♠② ❜② t❤❡ st❛t❡s ♦❢ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥✳ • { X t } t ∈T ✿ ❆ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❤✐❞❞❡♥ ▼❛r❦♦✈ ❝❤❛✐♥ ♦♥ (Ω , F , P ) ✇✐t❤ st❛t❡ s♣❛❝❡ { e 1 , e 2 , . . . , e N } ✱ ❛ ✜♥✐t❡ s❡t ♦❢ ✉♥✐t ✈❡❝t♦rs ✇✐t❤ e i = (0 , . . . , 1 , . . . , 0) ∈ ℜ N ✳

  10. • A ✿ ❚❤❡ ❣❡♥❡r❛t♦r ♦r t❤❡ r❛t❡ ♠❛tr✐① [ a ij ] i,j =1 , 2 ,...,N ✳ • ❙❡♠✐✲♠❛rt✐♥❣❛❧❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜② ❊❧❧✐♦tt ❡t ❛❧✳ ✭✶✾✾✹✮✿ � t X t = X 0 + 0 AX s ds + M t , ✐s ❛ ℜ N ✲✈❛❧✉❡❞ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ ✇❤❡r❡ { M t } t ∈T ( F X , P ) ✳

  11. • ❈♦♥s✐❞❡r ❛ P♦✐ss♦♥ ♣r♦❝❡ss N := { N t } t ∈T ♦♥ (Ω , F , P ) ✱ ✇❤♦s❡ st♦❝❤❛st✐❝ ✐♥t❡♥s✐t② ✐s✿ K � λ t := � λ, X t � = � λ, e i � I { X t = e i } , i =1 ✇❤❡r❡ λ := ( λ 1 , λ 2 , . . . , λ K ) ∈ ℜ K ❛♥❞ λ k ≥ 0 ✱ ❢♦r ❡❛❝❤ k = 1 , 2 , . . . , K ✳ • N t ✿ t❤❡ ♥✉♠❜❡r ♦❢ ❝❧❛✐♠s ♦✈❡r t❤❡ t✐♠❡ [0 , t ] ✳ • ❈♦♥s✐❞❡r r✐❣❤t✲❝♦♥t✐♥✉♦✉s✱ ❝♦♠♣❧❡t❡ ✈❡rs✐♦♥s ♦❢ t❤❡ ✜❧tr❛✲ t✐♦♥s F X {F X F X := := σ { X u | u ∈ [0 , t ] } , t } t ∈T , t F N {F N F N := := σ { N u | u ∈ [0 , t ] } , t } t ∈T , t G t := F X ∨ F N G := {G t } t ∈T , t t

  12. • ❚❤❡ ❉♦♦❜✲▼❡②❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ❢♦r N ✐s ✭s❡❡ ❊❧❧✐♦tt ❛♥❞ ▼❛❧❝♦❧♠ ✭✷✵✵✺✮✮✿ � t N t = 0 � λ, X u � du + V t , ✇❤❡r❡ V := { V t } t ∈T ✐s ❛ ( P , σ {G u | u ∈ [0 , t ] } ) ✲♠❛rt✐♥❣❛❧❡✳ • ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ F Y ( · ) ♦♥ ( ℜ + , B ( ℜ + )) ✱ ✇❤❡r❡ F Y ( · ) ✐s ❣✐✈❡♥✳ • ❙✉♣♣♦s❡ t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ ❝❧❛✐♠s t♦ t✐♠❡ t ✐s✿ � ∞ � t Z t = ydF Y ( y ) dN u 0 0 • ❚❤❡ ❥✉♠♣ s✐③❡s ♣r♦✈✐❞❡ ♥♦ ❡①tr❛ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t X ✳

  13. • ❈♦♥s✐❞❡r ❛ r❡❢❡r❡♥❝❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P † ✉♥❞❡r ✇❤✐❝❤ N ✐s ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ✉♥✐t ✐♥t❡♥s✐t② ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ X ✳ • ❯♥❞❡r P † ✱ Q t := N t − t , ✐s ❛ ♠❛rt✐♥❣❛❧❡✳

  14. • ❉❡✜♥❡ ❛ ♣r♦❝❡ss Λ := { Λ 0 ,t } t ∈T ✿ � � t � � X u , λ � ∆ N u exp � Λ 0 ,t := 0 (1 − � X u , λ � ) du 0 <u ≤ t � t = 1 + 0 Λ 0 ,u − ( � X u − , λ � − 1) dQ u • Λ ✐s ❛ ( G , P † ) ✲♠❛rt✐♥❣❛❧❡✳ • ❉❡✜♥❡ t❤❡ r❡❛❧✲✇♦r❧❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ❜② s❡tt✐♥❣ � Λ 0 ,t := d P � � d P † � � G t • ❙✉♣♣♦s❡ γ := { γ t } t ∈T ✐s ❛♥② G ✲❛❞❛♣t❡❞ ♣r♦❝❡ss✳

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