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❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ▼❛r❝❡❧ ◆✉t③ ❊❚❍ ❩✉r✐❝❤ ◆❡✇ ❛❞✈❛♥❝❡s ✐♥ ❇❛❝❦✇❛r❞ ❙❉❊s ❢♦r ✜♥❛♥❝✐❛❧ ❡♥❣✐♥❡❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s ❚❛♠❡r③❛✱ ❚✉♥✐s✐❛✱ ✷✽✳✶✵✳✷✵✶✵ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✶ ✴ ✶✼

❖✉t❧✐♥❡ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✶ ❆①✐♦♠❛t✐❝ ❋r❛♠❡✇♦r❦ ❛♥❞ ❙✉♣❡r❤❡❞❣✐♥❣ ( ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▼❡t❡ ❙♦♥❡r ) ✷ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✷ ✴ ✶✼

❖✉t❧✐♥❡ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✶ ❆①✐♦♠❛t✐❝ ❋r❛♠❡✇♦r❦ ❛♥❞ ❙✉♣❡r❤❡❞❣✐♥❣ ( ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▼❡t❡ ❙♦♥❡r ) ✷ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✷ ✴ ✶✼

P❡♥❣✬s ● ✲❊①♣❡❝t❛t✐♦♥ ■♥t✉✐t✐♦♥✿ ❱♦❧❛t✐❧✐t② ✐s ✉♥❝❡rt❛✐♥✱ ❜✉t ♣r❡s❝r✐❜❡❞ t♦ ❧✐❡ ✐♥ ❛ ✜①❡❞ ✐♥t❡r✈❛❧ ❉ = [ ❛ , ❜ ] ✳ ℰ ● ✭❳✮ ✐s t❤❡ ✇♦rst✲❝❛s❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❳ ♦✈❡r ❛❧❧ t❤❡s❡ s❝❡♥❛r✐♦s ❢♦r t❤❡ ✈♦❧❛t✐❧✐t②✳ Ω ✿ ❝❛♥♦♥✐❝❛❧ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ♣❛t❤s ♦♥ [ ✵ , ❚ ] ✳ ❇ ✿ ❝❛♥♦♥✐❝❛❧ ♣r♦❝❡ss✳ 풫 ● = ♠❛rt✐♥❣❛❧❡ ❧❛✇s ✉♥❞❡r ✇❤✐❝❤ ❞ ⟨ ❇ ⟩ t / ❞t ∈ ❉ ✱ t❤❡♥ ℰ ● P ∈풫 ● ❊ P [ ❳ ] , ❳ ∈ ▲ ✵ ( ℱ ∘ ✵ ( ❳ ) = s✉♣ ❚ ) r❡❣✉❧❛r ❡♥♦✉❣❤ . ℰ ● ✵ ✐s ❛ s✉❜❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✸ ✴ ✶✼

P❡♥❣✬s ● ✲❊①♣❡❝t❛t✐♦♥ ■♥t✉✐t✐♦♥✿ ❱♦❧❛t✐❧✐t② ✐s ✉♥❝❡rt❛✐♥✱ ❜✉t ♣r❡s❝r✐❜❡❞ t♦ ❧✐❡ ✐♥ ❛ ✜①❡❞ ✐♥t❡r✈❛❧ ❉ = [ ❛ , ❜ ] ✳ ℰ ● ✭❳✮ ✐s t❤❡ ✇♦rst✲❝❛s❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❳ ♦✈❡r ❛❧❧ t❤❡s❡ s❝❡♥❛r✐♦s ❢♦r t❤❡ ✈♦❧❛t✐❧✐t②✳ Ω ✿ ❝❛♥♦♥✐❝❛❧ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ♣❛t❤s ♦♥ [ ✵ , ❚ ] ✳ ❇ ✿ ❝❛♥♦♥✐❝❛❧ ♣r♦❝❡ss✳ 풫 ● = ♠❛rt✐♥❣❛❧❡ ❧❛✇s ✉♥❞❡r ✇❤✐❝❤ ❞ ⟨ ❇ ⟩ t / ❞t ∈ ❉ ✱ t❤❡♥ ℰ ● P ∈풫 ● ❊ P [ ❳ ] , ❳ ∈ ▲ ✵ ( ℱ ∘ ✵ ( ❳ ) = s✉♣ ❚ ) r❡❣✉❧❛r ❡♥♦✉❣❤ . ℰ ● ✵ ✐s ❛ s✉❜❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✸ ✴ ✶✼

❈♦♥❞✐t✐♦♥❛❧ ● ✲❊①♣❡❝t❛t✐♦♥ ❊①t❡♥s✐♦♥ t♦ ❛ ❝♦♥❞✐t✐♦♥❛❧ ● ✲❡①♣❡❝t❛t✐♦♥ ℰ ● t ❢♦r t > ✵✳ ◆♦♥tr✐✈✐❛❧ ❛s ♠❡❛s✉r❡s ✐♥ 풫 ● ❛r❡ s✐♥❣✉❧❛r✳ P❡♥❣✬s ❛♣♣r♦❛❝❤✿ ❢♦r ❳ = ❢ ( ❇ ❚ ) ✇✐t❤ ❢ ▲✐♣s❝❤✐t③✱ ❞❡✜♥❡ ℰ ● t ( ❳ ) = ✉ ( t , ❇ t ) ✇❤❡r❡ ● ( ① ) := ✶ − ✉ t − ● ( ✉ ①① ) = ✵ , ✉ ( ❚ , ① ) = ❢ ; ✷ s✉♣ ①② . ② ∈ ❉ ❊①t❡♥❞ t♦ ❳ = ❢ ( ❇ t ✶ , . . . , ❇ t ♥ ) ❛♥❞ ♣❛ss t♦ ❝♦♠♣❧❡t✐♦♥✳ ❚✐♠❡✲❝♦♥s✐st❡♥❝② ♣r♦♣❡rt②✿ ℰ ● s ∘ ℰ ● t = ℰ ● s ❢♦r s ≤ t ✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✹ ✴ ✶✼

❙❤♦rt ✭❛♥❞ ■♥❝♦♠♣❧❡t❡✮ ❍✐st♦r② ❯♥❝❡rt❛✐♥ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧✿ ❆✈❡❧❧❛♥❡❞❛✱ ▲❡✈②✱ P❛rás ✭✾✺✮✱ ❚✳ ▲②♦♥s ✭✾✺✮✳ ❇❙❉❊s ❛♥❞ ❣ ✲❡①♣❡❝t❛t✐♦♥s✿ P❛r❞♦✉①✱ P❡♥❣ ✭✾✵✮✱ ✳ ✳ ✳ ❈❛♣❛❝✐t②✲❜❛s❡❞ ❛♥❛❧②s✐s ♦❢ ✈♦❧❛t✐❧✐t② ✉♥❝❡rt❛✐♥t②✿ ❉❡♥✐s✱ ▼❛rt✐♥✐ ✭✵✻✮✳ ● ✲❡①♣❡❝t❛t✐♦♥ ❛♥❞ r❡❧❛t❡❞ ❝❛❧❝✉❧✉s ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② P❡♥❣ ✭✵✼✮✳ ❉✉❛❧ ❞❡s❝r✐♣t✐♦♥ ✈✐❛ 풫 ● ✐s ❞✉❡ t♦ ❉❡♥✐s✱ ▼✳ ❍✉✱ P❡♥❣ ✭✶✵✮✳ ❈♦♥str❛✐♥t ♦♥ ❧❛✇ ♦❢ ❇ ❚ ●❛❧✐❝❤♦♥✱ ❍❡♥r②✲▲❛❜♦r❞èr❡✱ ❚♦✉③✐ ✭✶❄✮✳ ❘❡❧❛t✐♦♥s t♦ ✷❇❙❉❊s ❈❤❡r✐❞✐t♦✱ ❙♦♥❡r✱ ❚♦✉③✐✱ ❱✐❝t♦✐r ✭✵✼✮✱ ❙♦♥❡r✱ ❚♦✉③✐✱ ❏✳ ❩❤❛♥❣ ✭✶✵✮✳ ❇✐♦♥✲◆❛❞❛❧✱ ❑❡r✈❛r❡❝ ✭✶✵✮✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✺ ✴ ✶✼

❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥ ❆❧❧♦✇ ❢♦r ✉♣❞❛t❡s ♦❢ t❤❡ ✈♦❧❛t✐❧✐t② ❜♦✉♥❞s✳ ❚❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❤✐st♦r✐❝❛❧ ✈♦❧❛t✐❧✐t②✿ ♣❛t❤✲❞❡♣❡♥❞❡♥❝❡✳ ❘❡♣❧❛❝❡ ❉ = [ ❛ , ❜ ] ❜② ❛ st♦❝❤❛st✐❝ ✐♥t❡r✈❛❧ ❉ t ( 휔 ) = [ ❛ t ( 휔 ) , ❜ t ( 휔 )] ✳ ✭■♥ ❣❡♥❡r❛❧✿ ❛ ♣r♦❣r❡ss✐✈❡ s❡t✲✈❛❧✉❡❞ ♣r♦❝❡ss✳✮ ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ ● ✭♣♦ss✐❜❧② ✐♥✜♥✐t❡✮✳ ❆t t = ✵ ❞❡✜♥❡ ℰ ✵ ( ❳ ) = s✉♣ P ∈풫 ❊ P [ ❳ ] ❢♦r ❝♦rr❡s♣♦♥❞✐♥❣ s❡t 풫 ✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✻ ✴ ✶✼

❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥ ❆❧❧♦✇ ❢♦r ✉♣❞❛t❡s ♦❢ t❤❡ ✈♦❧❛t✐❧✐t② ❜♦✉♥❞s✳ ❚❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❤✐st♦r✐❝❛❧ ✈♦❧❛t✐❧✐t②✿ ♣❛t❤✲❞❡♣❡♥❞❡♥❝❡✳ ❘❡♣❧❛❝❡ ❉ = [ ❛ , ❜ ] ❜② ❛ st♦❝❤❛st✐❝ ✐♥t❡r✈❛❧ ❉ t ( 휔 ) = [ ❛ t ( 휔 ) , ❜ t ( 휔 )] ✳ ✭■♥ ❣❡♥❡r❛❧✿ ❛ ♣r♦❣r❡ss✐✈❡ s❡t✲✈❛❧✉❡❞ ♣r♦❝❡ss✳✮ ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ ● ✭♣♦ss✐❜❧② ✐♥✜♥✐t❡✮✳ ❆t t = ✵ ❞❡✜♥❡ ℰ ✵ ( ❳ ) = s✉♣ P ∈풫 ❊ P [ ❳ ] ❢♦r ❝♦rr❡s♣♦♥❞✐♥❣ s❡t 풫 ✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✻ ✴ ✶✼

❖✉r ❆♣♣r♦❛❝❤ ❆t t > ✵✿ ✇❛♥t ℰ t ( ❳ ) ✏ = ✑ s✉♣ P ′ ∈풫 ❊ P ′ [ ❳ ∣ℱ ∘ t ] ✳ ❲❡ s❤❛❧❧ ❝♦♥str✉❝t ℰ t ( ❳ ) s✉❝❤ t❤❛t P ❊ P ′ [ � ℱ ∘ � ℰ t ( ❳ ) = ❡ss s✉♣ ] P ✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫 , ❳ t P ′ ∈풫 ( t , P ) ✇❤❡r❡ 풫 ( t , P ) := { P ′ ∈ 풫 : P ′ = P ♦♥ ℱ ∘ t } ✳ ◆♦♥✲▼❛r❦♦✈ ♣r♦❜❧❡♠✿ P❉❊ ❛♣♣r♦❛❝❤ ♥♦t s✉✐t❛❜❧❡✳ P❛t❤✇✐s❡ ❞❡✜♥✐t✐♦♥✿ ❝♦♥❞✐t✐♦♥ ❉ , 풫 , ❳ ♦♥ 휔 ✉♣ t♦ t ✿ ❊ P [ ❳ t ,휔 ] , ℰ t ( ❳ )( 휔 ) := s✉♣ 휔 ∈ Ω . P ∈풫 ( t ,휔 ) ❇❡♥❡✜ts✿ ❈♦♥tr♦❧ ❛r❣✉♠❡♥ts✱ ❉ ♥❡❡❞ ♥♦t ❜❡ ❜♦✉♥❞❡❞✳ ❚✐♠❡ ❝♦♥s✐st❡♥❝② ❝♦rr❡s♣♦♥❞s t♦ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ♣r✐♥❝✐♣❧❡✳ ❆♣♣r♦❛❝❤ ❢♦❧❧♦✇s ❙♦♥❡r✱ ❚♦✉③✐✱ ❩❤❛♥❣ ✭✶✵✮✳ ❍❡r❡ 풫 ( t , 휔 ) ✐s ♣❛t❤✲❞❡♣❡♥❞❡♥t✳ ❘❡❣✉❧❛r✐t② ♦❢ 휔 �→ 풫 ( t , 휔 ) ✐s ♥❡❡❞❡❞✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ● ✲❊①♣❡❝t❛t✐♦♥s ✼ ✴ ✶✼