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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●

✵ (❳) = s✉♣ P∈풫● ❊ P[❳],

❳ ∈ ▲✵(ℱ∘

❚) 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●

✵ (❳) = s✉♣ P∈풫● ❊ P[❳],

❳ ∈ ▲✵(ℱ∘

❚) 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❡

−✉t − ●(✉①①) = ✵, ✉(❚, ①) = ❢ ;

• (①) := ✶

✷ s✉♣

②∈❉

①②. ❊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 ℰt(❳) ✏ = ✑ s✉♣P′∈풫 ❊ P′[❳∣ℱ∘

t ]✳

❲❡ s❤❛❧❧ ❝♦♥str✉❝t ℰt(❳) s✉❝❤ t❤❛t ℰt(❳) = ❡ss s✉♣

P′∈풫(t,P) P❊ P′[

❳

• ℱ∘

t

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫, ✇❤❡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✿ ℰt(❳)(휔) := s✉♣

P∈풫(t,휔)

❊ P[❳ t,휔], 휔 ∈ Ω. ❇❡♥❡✜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 ✼ ✴ ✶✼

❖✉r ❆♣♣r♦❛❝❤

❆t t > ✵✿ ✇❛♥t ℰt(❳) ✏ = ✑ s✉♣P′∈풫 ❊ P′[❳∣ℱ∘

t ]✳

❲❡ s❤❛❧❧ ❝♦♥str✉❝t ℰt(❳) s✉❝❤ t❤❛t ℰt(❳) = ❡ss s✉♣

P′∈풫(t,P) P❊ P′[

❳

• ℱ∘

t

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫, ✇❤❡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✿ ℰt(❳)(휔) := s✉♣

P∈풫(t,휔)

❊ P[❳ t,휔], 휔 ∈ Ω. ❇❡♥❡✜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 ✼ ✴ ✶✼

❖✉r ❆♣♣r♦❛❝❤

❆t t > ✵✿ ✇❛♥t ℰt(❳) ✏ = ✑ s✉♣P′∈풫 ❊ P′[❳∣ℱ∘

t ]✳

❲❡ s❤❛❧❧ ❝♦♥str✉❝t ℰt(❳) s✉❝❤ t❤❛t ℰt(❳) = ❡ss s✉♣

P′∈풫(t,P) P❊ P′[

❳

• ℱ∘

t

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫, ✇❤❡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✿ ℰt(❳)(휔) := s✉♣

P∈풫(t,휔)

❊ P[❳ t,휔], 휔 ∈ Ω. ❇❡♥❡✜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 ✼ ✴ ✶✼

❙tr♦♥❣ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ❱♦❧❛t✐❧✐t② ❯♥❝❡rt❛✐♥t②

P✵ ❲✐❡♥❡r ♠❡❛s✉r❡✱ 픽∘ r❛✇ ✜❧tr❛t✐♦♥ ♦❢ ❇✳ 풫❙ = { P훼 := P✵ ∘ ( ∫ 훼✶/✷❞❇)−✶, 훼 > ✵, ∫ ❚

✵ 훼 ❞t < ∞

} ✳ ❉❡✜♥❡ ⟨❇⟩ ❛♥❞ ˆ ❛ = ❞⟨❇⟩/❞t s✐♠✉❧t❛♥❡♦✉s❧② ✉♥❞❡r ❛❧❧ P ∈ 풫❙✳ ✭❊✳❣✳ ❜② ❋ö❧❧♠❡r✬s ✭✽✶✮ ♣❛t❤✇✐s❡ ❝❛❧❝✉❧✉s✳✮ 풫 := { P ∈ 풫❙ : ˆ ❛ ∈ ■♥t훿 ❉ ❞s × P✲❛✳❡✳ ❢♦r s♦♠❡ 훿 > ✵ } ✱ ✇❤❡r❡ ■♥t훿 ❉ := [❛ + 훿, ❜ − 훿] ❢♦r 훿 > ✵✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✽ ✴ ✶✼

❙tr♦♥❣ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ❱♦❧❛t✐❧✐t② ❯♥❝❡rt❛✐♥t②

P✵ ❲✐❡♥❡r ♠❡❛s✉r❡✱ 픽∘ r❛✇ ✜❧tr❛t✐♦♥ ♦❢ ❇✳ 풫❙ = { P훼 := P✵ ∘ ( ∫ 훼✶/✷❞❇)−✶, 훼 > ✵, ∫ ❚

✵ 훼 ❞t < ∞

} ✳ ❉❡✜♥❡ ⟨❇⟩ ❛♥❞ ˆ ❛ = ❞⟨❇⟩/❞t s✐♠✉❧t❛♥❡♦✉s❧② ✉♥❞❡r ❛❧❧ P ∈ 풫❙✳ ✭❊✳❣✳ ❜② ❋ö❧❧♠❡r✬s ✭✽✶✮ ♣❛t❤✇✐s❡ ❝❛❧❝✉❧✉s✳✮ 풫 := { P ∈ 풫❙ : ˆ ❛ ∈ ■♥t훿 ❉ ❞s × P✲❛✳❡✳ ❢♦r s♦♠❡ 훿 > ✵ } ✱ ✇❤❡r❡ ■♥t훿 ❉ := [❛ + 훿, ❜ − 훿] ❢♦r 훿 > ✵✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✽ ✴ ✶✼

❈♦♥❞✐t✐♦♥✐♥❣ ❛♥❞ ❘❡❣✉❧❛r✐t②

❚♦ ❝♦♥❞✐t✐♦♥ ❳ t♦ 휔 ✉♣ t♦ t✱ s❡t ❳ t,휔(⋅) := ❳(휔 ⊗t ⋅)✱ ✇❤❡r❡ ⊗t ✐s t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ❛t t✳ ❳ t,휔 ✐s ❛♥ r✳✈✳ ♦♥ t❤❡ s♣❛❝❡ Ωt ♦❢ ♣❛t❤s st❛rt✐♥❣ ❛t t✐♠❡ t✳ ❖♥ Ωt ✇❡ ❤❛✈❡ ❇t✱ Pt

✵✱ ˆ

❛t✱ 풫

t ❙✱ . . . ❛s ❢♦r t = ✵✳

풫(t, 휔) := { P ∈ 풫

t ❙ : ˆ

❛t ∈ ■♥t훿 ❉t,휔 ❞s×P✲❛✳❡✳ ♦♥ [t, ❚]×Ωt, 훿 > ✵ } ✳ ❉❡✜♥❡ ℰt(❳) ❛s t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ℰt(❳)(휔) := s✉♣

P∈풫(t,휔)

❊ P[❳ t,휔], 휔 ∈ Ω. ❘❡❣✉❧❛r✐t②✿ ❳ ∈ ❯❈❜(Ω) ❛♥❞ ❉ ✉♥✐❢♦r♠❧② ❝♦♥t✐♥✉♦✉s✿ ❢♦r ❛❧❧ 훿 > ✵ ❛♥❞ (t, 휔) ∈ [✵, ❚] × Ω t❤❡r❡ ❡①✐sts 휀 = 휀(t, 휔, 훿) > ✵ s✳t✳ ∥휔 − 휔′∥t ≤ 휀 ⇒ ■♥t훿 ❉t,휔

s

(˜ 휔)⊆ ■♥t휀 ❉t,휔′

s

(˜ 휔) ∀ (s, ˜ 휔) ∈ [t, ❚] × Ωt.

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✾ ✴ ✶✼

❈♦♥❞✐t✐♦♥✐♥❣ ❛♥❞ ❘❡❣✉❧❛r✐t②

❚♦ ❝♦♥❞✐t✐♦♥ ❳ t♦ 휔 ✉♣ t♦ t✱ s❡t ❳ t,휔(⋅) := ❳(휔 ⊗t ⋅)✱ ✇❤❡r❡ ⊗t ✐s t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ❛t t✳ ❳ t,휔 ✐s ❛♥ r✳✈✳ ♦♥ t❤❡ s♣❛❝❡ Ωt ♦❢ ♣❛t❤s st❛rt✐♥❣ ❛t t✐♠❡ t✳ ❖♥ Ωt ✇❡ ❤❛✈❡ ❇t✱ Pt

✵✱ ˆ

❛t✱ 풫

t ❙✱ . . . ❛s ❢♦r t = ✵✳

풫(t, 휔) := { P ∈ 풫

t ❙ : ˆ

❛t ∈ ■♥t훿 ❉t,휔 ❞s×P✲❛✳❡✳ ♦♥ [t, ❚]×Ωt, 훿 > ✵ } ✳ ❉❡✜♥❡ ℰt(❳) ❛s t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ℰt(❳)(휔) := s✉♣

P∈풫(t,휔)

❊ P[❳ t,휔], 휔 ∈ Ω. ❘❡❣✉❧❛r✐t②✿ ❳ ∈ ❯❈❜(Ω) ❛♥❞ ❉ ✉♥✐❢♦r♠❧② ❝♦♥t✐♥✉♦✉s✿ ❢♦r ❛❧❧ 훿 > ✵ ❛♥❞ (t, 휔) ∈ [✵, ❚] × Ω t❤❡r❡ ❡①✐sts 휀 = 휀(t, 휔, 훿) > ✵ s✳t✳ ∥휔 − 휔′∥t ≤ 휀 ⇒ ■♥t훿 ❉t,휔

s

(˜ 휔)⊆ ■♥t휀 ❉t,휔′

s

(˜ 휔) ∀ (s, ˜ 휔) ∈ [t, ❚] × Ωt.

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✾ ✴ ✶✼

❈♦♥s❡q✉❡♥❝❡s ♦❢ ❯♥✐❢♦r♠ ❈♦♥t✐♥✉✐t②

휔 → ℰt(❳)(휔) ✐s ℱ∘

t ✲♠❡❛s✉r❛❜❧❡ ❛♥❞ ▲❙❈ ❢♦r ❳ ∈ ❯❈❜(Ω)✳

❚❤❡♦r❡♠ ✭❉PP✱ t✐♠❡ ❝♦♥s✐st❡♥❝②✮

▲❡t ❳ ∈ ❯❈❜(Ω) ❛♥❞ ✵ ≤ s ≤ t ≤ ❚✳ ❚❤❡♥ ℰs(❳)(휔) = s✉♣

P∈풫(s,휔)

❊ P[ ℰt(❳)s,휔] ❢♦r ❛❧❧ 휔 ∈ Ω✱ ℰs(❳) = ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

ℰt(❳)

• ℱ∘

s

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫✱ ✇❤❡r❡ 풫(s, P) := {P′ ∈ 풫 : P′ = P ♦♥ ℱ∘

s }✳

❖♥ t❤❡ ♣r♦♦❢✿ ▼❛✐♥ ♣r♦❜❧❡♠ ❞✉❡ t♦ st♦❝❤❛st✐❝ ❉✿ ❛❞♠✐ss✐❜✐❧✐t② ♦❢ ♣❛st✐♥❣s✳ ❘❡❣✉❧❛r✐t② ♦❢ ℰt(❳) t✉r♥s ♦✉t ♥♦t t♦ ❜❡ ❛ ♣r♦❜❧❡♠✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✵ ✴ ✶✼

❈♦♥s❡q✉❡♥❝❡s ♦❢ ❯♥✐❢♦r♠ ❈♦♥t✐♥✉✐t②

휔 → ℰt(❳)(휔) ✐s ℱ∘

t ✲♠❡❛s✉r❛❜❧❡ ❛♥❞ ▲❙❈ ❢♦r ❳ ∈ ❯❈❜(Ω)✳

❚❤❡♦r❡♠ ✭❉PP✱ t✐♠❡ ❝♦♥s✐st❡♥❝②✮

▲❡t ❳ ∈ ❯❈❜(Ω) ❛♥❞ ✵ ≤ s ≤ t ≤ ❚✳ ❚❤❡♥ ℰs(❳)(휔) = s✉♣

P∈풫(s,휔)

❊ P[ ℰt(❳)s,휔] ❢♦r ❛❧❧ 휔 ∈ Ω✱ ℰs(❳) = ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

ℰt(❳)

• ℱ∘

s

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫✱ ✇❤❡r❡ 풫(s, P) := {P′ ∈ 풫 : P′ = P ♦♥ ℱ∘

s }✳

❖♥ t❤❡ ♣r♦♦❢✿ ▼❛✐♥ ♣r♦❜❧❡♠ ❞✉❡ t♦ st♦❝❤❛st✐❝ ❉✿ ❛❞♠✐ss✐❜✐❧✐t② ♦❢ ♣❛st✐♥❣s✳ ❘❡❣✉❧❛r✐t② ♦❢ ℰt(❳) t✉r♥s ♦✉t ♥♦t t♦ ❜❡ ❛ ♣r♦❜❧❡♠✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✵ ✴ ✶✼

❊①t❡♥s✐♦♥ t♦ ❈♦♠♣❧❡t✐♦♥ ♦❢ ❯❈❜(Ω)

▲✶

풫 = s♣❛❝❡ ♦❢ r✳✈✳ ❳ s✉❝❤ t❤❛t ∥❳∥▲✶

풫 := s✉♣P∈풫 ∥❳∥▲✶(P) < ∞✳

핃✶

풫 ❂ ❝❧♦s✉r❡ ♦❢ ❯❈❜ ⊂ ▲✶ 풫 ✭❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❡①♣❧✐❝✐t❧②✮✳

❉PP ✐♠♣❧✐❡s t❤❛t ℰt ✐s ✶✲▲✐♣s❝❤✐t③ ✇rt✳ ∥ ⋅ ∥▲✶

풫✱ ❤❡♥❝❡ ❡①t❡♥❞s t♦

ℰt : 핃✶

풫 → ▲✶ 풫(ℱ∘ t ).

❚❤❡♦r❡♠✳ ❋♦r ❳ ∈ 핃✶

풫 t❤❡ ❉PP ❤♦❧❞s✿

ℰs(❳) = ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

ℰt(❳)

• ℱ∘

s

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫. ■♥ ♣❛rt✐❝✉❧❛r✱ ℰs(❳) ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ℰs(❳) = ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

❳

• ℱ∘

s

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫.

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✶ ✴ ✶✼

❊①t❡♥s✐♦♥ t♦ ❈♦♠♣❧❡t✐♦♥ ♦❢ ❯❈❜(Ω)

▲✶

풫 = s♣❛❝❡ ♦❢ r✳✈✳ ❳ s✉❝❤ t❤❛t ∥❳∥▲✶

풫 := s✉♣P∈풫 ∥❳∥▲✶(P) < ∞✳

핃✶

풫 ❂ ❝❧♦s✉r❡ ♦❢ ❯❈❜ ⊂ ▲✶ 풫 ✭❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❡①♣❧✐❝✐t❧②✮✳

❉PP ✐♠♣❧✐❡s t❤❛t ℰt ✐s ✶✲▲✐♣s❝❤✐t③ ✇rt✳ ∥ ⋅ ∥▲✶

풫✱ ❤❡♥❝❡ ❡①t❡♥❞s t♦

ℰt : 핃✶

풫 → ▲✶ 풫(ℱ∘ t ).

❚❤❡♦r❡♠✳ ❋♦r ❳ ∈ 핃✶

풫 t❤❡ ❉PP ❤♦❧❞s✿

ℰs(❳) = ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

ℰt(❳)

• ℱ∘

s

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫. ■♥ ♣❛rt✐❝✉❧❛r✱ ℰs(❳) ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ℰs(❳) = ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

❳

• ℱ∘

s

] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫.

▼❛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 ✶✶ ✴ ✶✼

❆①✐♦♠❛t✐❝ ❙❡t✉♣

❋♦r t❤❡ r❛♥❞♦♠ ●✲❡①♣❡❝t❛t✐♦♥s✱ ✇❡ ❤❛❞✿ ❛ s❡t 풫 ⊆ 풫❙ ✇✐t❤ ❛ t✐♠❡ ❝♦♥s✐st❡♥❝② ✐s ❛ ♣r♦♣❡rt②✱ ❛♥ ❛❣❣r❡❣❛t❡❞ r✳✈✳ ❢♦r ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

❳

• ℱ∘

s

] ✱ P ∈ 풫✱ ❢♦r ❳ ✐♥ ❛ s✉❜s♣❛❝❡ 핃✶

풫 ⊆ ▲✶ 풫✳

❆①✐♦♠❛t✐❝ ❛♣♣r♦❛❝❤✿ st❛rt ✇✐t❤ s♦♠❡ s❡t 풫 ⊆ 풫❙✳ 풫 ✐s ❛ss✉♠❡❞ t♦ ❜❡ st❛❜❧❡ ✉♥❞❡r 픽∘✲♣❛st✐♥❣ ✭≈ t✐♠❡ ❝♦♥s✐st❡♥❝②✮✿ ❢♦r ❛❧❧ P ∈ 풫 ❛♥❞ P✶, P✷ ∈ 풫(ℱ∘

t , P) ❛♥❞ Λ ∈ ℱ∘ t ✱

¯ P(⋅) := ❊ P[ P✶( ⋅ ∣ℱ∘

t )✶Λ + P✷( ⋅ ∣ℱ∘ t )✶Λ❝]

∈ 풫. ❛❣❣r❡❣❛t❡❞ r✳✈✳ ℰ∘

s (❳) =

❡ss s✉♣

P′∈풫(ℱ∘

s ,P)

P❊ P′[

❳

• ℱ∘

s

] P✲❛✳s✳✱ P ∈ 풫 ❢♦r ❛❧❧ ❳ ✐♥ s♦♠❡ s✉❜s♣❛❝❡ ℋ ⊆ ▲✶

풫✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✷ ✴ ✶✼

❆①✐♦♠❛t✐❝ ❙❡t✉♣

❋♦r t❤❡ r❛♥❞♦♠ ●✲❡①♣❡❝t❛t✐♦♥s✱ ✇❡ ❤❛❞✿ ❛ s❡t 풫 ⊆ 풫❙ ✇✐t❤ ❛ t✐♠❡ ❝♦♥s✐st❡♥❝② ✐s ❛ ♣r♦♣❡rt②✱ ❛♥ ❛❣❣r❡❣❛t❡❞ r✳✈✳ ❢♦r ❡ss s✉♣

P′∈풫(s,P) P❊ P′[

❳

• ℱ∘

s

] ✱ P ∈ 풫✱ ❢♦r ❳ ✐♥ ❛ s✉❜s♣❛❝❡ 핃✶

풫 ⊆ ▲✶ 풫✳

❆①✐♦♠❛t✐❝ ❛♣♣r♦❛❝❤✿ st❛rt ✇✐t❤ s♦♠❡ s❡t 풫 ⊆ 풫❙✳ 풫 ✐s ❛ss✉♠❡❞ t♦ ❜❡ st❛❜❧❡ ✉♥❞❡r 픽∘✲♣❛st✐♥❣ ✭≈ t✐♠❡ ❝♦♥s✐st❡♥❝②✮✿ ❢♦r ❛❧❧ P ∈ 풫 ❛♥❞ P✶, P✷ ∈ 풫(ℱ∘

t , P) ❛♥❞ Λ ∈ ℱ∘ t ✱

¯ P(⋅) := ❊ P[ P✶( ⋅ ∣ℱ∘

t )✶Λ + P✷( ⋅ ∣ℱ∘ t )✶Λ❝]

∈ 풫. ❛❣❣r❡❣❛t❡❞ r✳✈✳ ℰ∘

s (❳) =

❡ss s✉♣

P′∈풫(ℱ∘

s ,P)

P❊ P′[

❳

• ℱ∘

s

] P✲❛✳s✳✱ P ∈ 풫 ❢♦r ❛❧❧ ❳ ✐♥ s♦♠❡ s✉❜s♣❛❝❡ ℋ ⊆ ▲✶

풫✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✷ ✴ ✶✼

• ❡tt✐♥❣ P❛t❤ ❘❡❣✉❧❛r✐t②

∙ ˆ ❋ = { ˆ ℱt}✵≤t≤❚✱ ✇❤❡r❡ ˆ ℱt := ℱ∘

t+ ∨ 풩 풫 ❛♥❞ 풩 풫 = 풫✲♣♦❧❛r s❡ts✳

❚❛❦❡ r✐❣❤t ❧✐♠✐ts ♦❢ {ℰ∘

t (❳), t ∈ [✵, ❚]}✿

❚❤❡♦r❡♠

❋♦r ❳ ∈ ℋ✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝à❞❧à❣ ˆ ❋✲❛❞❛♣t❡❞ ♣r♦❝❡ss ❨ ✱ ❨t = ℰ∘

t+(❳) 풫✲q✳s✳ ❢♦r ❛❧❧ t✳

❨ ✐s t❤❡ ♠✐♥✐♠❛❧ (ˆ ❋, 풫)✲s✉♣❡r♠❛rt✐♥❣❛❧❡ ✇✐t❤ ❨❚ = ❳✳ ❨t = ❡ss s✉♣

P′∈풫( ˆ ℱt,P) P❊ P′[❳∣ ˆ

ℱt] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫✳ ∙ ❨ ✐s ❛ 풫✲♠♦❞✐✜❝❛t✐♦♥ ♦❢ {ℰ∘

t (❳), t ∈ [✵, ❚]} ✐♥ r❡❣✉❧❛r ❝❛s❡s

❜✉t t❤❡r❡ ❛r❡ ❝♦✉♥t❡r❡①❛♠♣❧❡s✳ ∙ ❚❤❡ ♣r♦❝❡ss ℰ(❳) := ❨ ✐s ❝❛❧❧❡❞ t❤❡ ✭❝à❞❧à❣✮ ℰ✲♠❛rt✐♥❣❛❧❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❳ ∈ ℋ✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✸ ✴ ✶✼

❙t♦♣♣✐♥❣ ❚✐♠❡s ❛♥❞ ❖♣t✐♦♥❛❧ ❙❛♠♣❧✐♥❣

❚②♣✐❝❛❧❧②✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ℰ∘ ✐s ♥♦t ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ st♦♣♣✐♥❣ t✐♠❡s ✭❡✳❣✳ ●✲❡①♣❡❝t❛t✐♦♥✮✳ ❇✉t ✇❡ ❝❛♥ ❡❛s✐❧② ❞❡✜♥❡ ℰ ❛t ❛ st♦♣♣✐♥❣ t✐♠❡✳

❚❤❡♦r❡♠

▲❡t ✵ ≤ 휎 ≤ 휏 ≤ ❚ ❜❡ ˆ ❋✲st♦♣♣✐♥❣ t✐♠❡s ❛♥❞ ❳ ∈ ℋ✳ ❚❤❡♥ ℰ휎(❳) = ❡ss s✉♣

P′∈풫( ˆ ℱ휎,P) P❊ P′[❳∣ ˆ

ℱ휎] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫; ℰ휎(❳) = ❡ss s✉♣

P′∈풫( ˆ ℱ휎,P) P❊ P′[ℰ휏(❳)∣ ˆ

ℱ휎] P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫.

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✹ ✴ ✶✼

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ℰ✲▼❛rt✐♥❣❛❧❡s

❚❤❡♦r❡♠

▲❡t ❳ ∈ ℋ✳ ❚❤❡r❡ ❡①✐st ❛♥ ˆ ❋✲♣r♦❣r❡ss✐✈❡ ♣r♦❝❡ss ❩ ❳ ❛ ❢❛♠✐❧② (❑ P)P∈풫 ♦❢ 픽

P✲♣r❡❞✳ ✐♥❝r❡❛s✐♥❣ ♣r♦❝❡ss❡s✱ ❊ P[∣❑ P ❚ ∣] < ∞✱

s✉❝❤ t❤❛t ℰt(❳) = ℰ✵(❳) +

(P)

∫ t

✵

❩ ❳

s ❞❇s − ❑ P t ,

P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫. ∙ ❩ ❳ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ P✱ ❜✉t t❤❡ ✐♥t❡❣r❛❧ ♠❛② ❞♦✳ ∙ ❈❢✳ ♦♣t✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❊❧ ❑❛r♦✉✐✱ ◗✉❡♥❡③ ✭✾✺✮✱ ❑r❛♠❦♦✈ ✭✾✻✮✳ ∙ ❈♦♥str✉❝t✐♦♥ ❛s ✐♥ t❤❡ t❤❡♦r② ♦❢ ✷❇❙❉❊s✿ ❙♦♥❡r✱ ❚♦✉③✐✱ ❩❤❛♥❣ ✭✶✵✮ ∙ ❍❡r❡ ✇❡ ♦♥❧② ♥❡❡❞ ❉♦♦❜✲▼❡②❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✰ ♠❛rt✐♥❣❛❧❡ r❡♣r❡s❡♥t✳ ✰ ♣❛t❤✇✐s❡ ✐♥t❡❣r❛t✐♦♥ ✭❇✐❝❤t❡❧❡r ✽✶✮✳ ∙ ▼♦r❡ ♣r❡❝✐s❡ r❡s✉❧ts ❢♦r ●✲❡①♣❡❝t❛t✐♦♥✿ P❡♥❣ ✭✵✼✮✱ ❳✉✱ ❇✳ ❩❤❛♥❣ ✭✵✾✮✱ ❙♦♥❡r✱ ❚♦✉③✐✱ ❩❤❛♥❣ ✭✶✵✮✱ ❙♦♥❣ ✭✶✵✮✱ ❨✳ ❍✉✱ P❡♥❣ ✭✶✵✮✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✺ ✴ ✶✼

❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ℰ✲▼❛rt✐♥❣❛❧❡s

❚❤❡♦r❡♠

▲❡t ❳ ∈ ℋ✳ ❚❤❡r❡ ❡①✐st ❛♥ ˆ ❋✲♣r♦❣r❡ss✐✈❡ ♣r♦❝❡ss ❩ ❳ ❛ ❢❛♠✐❧② (❑ P)P∈풫 ♦❢ 픽

P✲♣r❡❞✳ ✐♥❝r❡❛s✐♥❣ ♣r♦❝❡ss❡s✱ ❊ P[∣❑ P ❚ ∣] < ∞✱

s✉❝❤ t❤❛t ℰt(❳) = ℰ✵(❳) +

(P)

∫ t

✵

❩ ❳

s ❞❇s − ❑ P t ,

P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫. ∙ ❩ ❳ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ P✱ ❜✉t t❤❡ ✐♥t❡❣r❛❧ ♠❛② ❞♦✳ ∙ ❈❢✳ ♦♣t✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❊❧ ❑❛r♦✉✐✱ ◗✉❡♥❡③ ✭✾✺✮✱ ❑r❛♠❦♦✈ ✭✾✻✮✳ ∙ ❈♦♥str✉❝t✐♦♥ ❛s ✐♥ t❤❡ t❤❡♦r② ♦❢ ✷❇❙❉❊s✿ ❙♦♥❡r✱ ❚♦✉③✐✱ ❩❤❛♥❣ ✭✶✵✮ ∙ ❍❡r❡ ✇❡ ♦♥❧② ♥❡❡❞ ❉♦♦❜✲▼❡②❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✰ ♠❛rt✐♥❣❛❧❡ r❡♣r❡s❡♥t✳ ✰ ♣❛t❤✇✐s❡ ✐♥t❡❣r❛t✐♦♥ ✭❇✐❝❤t❡❧❡r ✽✶✮✳ ∙ ▼♦r❡ ♣r❡❝✐s❡ r❡s✉❧ts ❢♦r ●✲❡①♣❡❝t❛t✐♦♥✿ P❡♥❣ ✭✵✼✮✱ ❳✉✱ ❇✳ ❩❤❛♥❣ ✭✵✾✮✱ ❙♦♥❡r✱ ❚♦✉③✐✱ ❩❤❛♥❣ ✭✶✵✮✱ ❙♦♥❣ ✭✶✵✮✱ ❨✳ ❍✉✱ P❡♥❣ ✭✶✵✮✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✺ ✴ ✶✼

❙✉♣❡r❤❡❞❣✐♥❣

■♥t❡r♣r❡t❛t✐♦♥ ❢♦r ❞❡❝♦♠♣♦s✐t✐♦♥ ❳ = ℰ❚(❳) = ℰ✵(❳) +

(P)

∫ ❚

✵

❩ ❳

s ❞❇s − ❑ P ❚ :

ℰ✵(❳) = ˆ ℱ✵✲s✉♣❡r❤❡❞❣✐♥❣ ♣r✐❝❡✱ ❩ ❳ = s✉♣❡r❤❡❞❣✐♥❣ str❛t❡❣②✱ ❑ P

❚ = ♦✈❡rs❤♦♦t ❢♦r t❤❡ s❝❡♥❛r✐♦ P

▼✐♥✐♠❛❧✐t② ♦❢ t❤❡ ♦✈❡rs❤♦♦t✿ ❡ss ✐♥❢

P′∈풫( ˆ ℱt,P) P❊ P′[

❑ P′

❚ − ❑ P′ t

• ˆ

ℱt ] = ✵ P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫. r❡♣❧✐❝❛❜❧❡ ❝❧❛✐♠s ❝♦rr❡s♣♦♥❞ t♦ ❑ P ≡ ✵ ❢♦r ❛❧❧ P ∈ 풫✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✻ ✴ ✶✼

✷❇❙❉❊ ❢♦r ℰ(❳)

(❨ , ❩) ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✷❇❙❉❊ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② (❑ P)P∈풫 ♦❢ 픽

P✲❛❞❛♣t❡❞ ✐♥❝r❡❛s✐♥❣ ♣r♦❝❡ss❡s s❛t✐s❢②✐♥❣ ❊ P[∣❑ P ❚ ∣] < ∞ s✉❝❤ t❤❛t

❨t = ❳ −

(P)

∫ ❚

t

❩s ❞❇s + ❑ P

❚ − ❑ P t ,

✵ ≤ t ≤ ❚, P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫 ❛♥❞ s✉❝❤ t❤❛t ❡ss ✐♥❢

P′∈풫( ˆ ℱt,P) P❊ P′[

❑ P′

❚ − ❑ P′ t

• ˆ

ℱt ] = ✵ P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫.

❚❤❡♦r❡♠ ✭❳ ∈ ℋ✮

(ℰ(❳), ❩ ❳) ✐s t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✷❇❙❉❊✳ ■❢ (❨ , ❩) ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✷❇❙❉❊ s✉❝❤ t❤❛t ❨ ✐s ♦❢ ❝❧❛ss ✭❉✱풫✮✱ t❤❡♥ (❨ , ❩) = (ℰ(❳), ❩ ❳)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ❳ ∈ ℋ♣ ❢♦r s♦♠❡ ♣ ∈ (✶, ∞)✱ t❤❡♥ (ℰ(❳), ❩ ❳) ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✷❇❙❉❊ ✐♥ t❤❡ ❝❧❛ss ✭❉✱풫✮✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✼ ✴ ✶✼

P❛st✐♥❣ ❛♥❞ ❚✐♠❡ ❈♦♥s✐st❡♥❝②

풫 ✐s ♠❛①✐♠❛❧❧② ❝❤♦s❡♥ ❢♦r ℋ ✐❢ 풫 ❝♦♥t❛✐♥s ❛❧❧ P ∈ 풫❙ s✉❝❤ t❤❛t ❊ P[❳] ≤ s✉♣P′∈풫 ❊ P′[❳] ❢♦r ❛❧❧ ❳ ∈ ℋ✳ 풫 ✐s t✐♠❡✲❝♦♥s✐st❡♥t ♦♥ ℋ ✐❢ ❡ss s✉♣P

P′∈풫(ℱ∘

s ,P)

❊ P′[ ❡ss s✉♣P′

P′′∈풫(ℱ∘

t ,P′)

❊ P′′[❳∣ℱ∘

t ]

• ℱ∘

s

] = ❡ss s✉♣P

P′∈풫(ℱ∘

s ,P)

❊ P′[❳∣ℱ∘

s ]

P✲❛✳s✳ ❢♦r ❛❧❧ P ∈ 풫✱ ✵ ≤ s ≤ t ≤ ❚ ❛♥❞ ❳ ∈ ℋ✳

❚❤❡♦r❡♠

st❛❜✐❧✐t② ✉♥❞❡r ♣❛st✐♥❣ ⇒ t✐♠❡ ❝♦♥s✐st❡♥❝②✳ ■❢ 풫 ✐s ♠❛①✐♠❛❧❧② ❝❤♦s❡♥✿ t✐♠❡ ❝♦♥s✐st❡♥❝② ⇒ st❛❜✐❧✐t② ✉♥❞❡r ♣❛st✐♥❣ ∙ ❙✐♠✐❧❛r r❡s✉❧ts ❜② ❉❡❧❜❛❡♥ ✭✵✻✮ ❢♦r ❝❧❛ss✐❝❛❧ r✐s❦ ♠❡❛s✉r❡s✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✽ ✴ ✶✼

❚✐♠❡ ❈♦♥s✐st❡♥❝② ♦❢ ▼❛♣♣✐♥❣s

❈♦♥s✐❞❡r ❛ ❢❛♠✐❧② (ℰt)✵≤t≤❚ ♦❢ ♠❛♣♣✐♥❣s ℰt : ℋ → ▲✶

풫(ℱ∘ t )✳

ℋt := ℋ ∩ ▲✶

풫(ℱ∘ t )✳

❉❡✜♥✐t✐♦♥

(ℰt)✵≤t≤❚ ✐s ❝❛❧❧❡❞ t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ ℰs(❳) ≤ (≥) ℰs(휑) ❢♦r ❛❧❧ 휑 ∈ ℋt s✉❝❤ t❤❛t 피t(❳) ≤ (≥) 휑 ❛♥❞ ✭ℋt✲✮ ♣♦s✐t✐✈❡❧② ❤♦♠♦❣❡♥❡♦✉s ✐❢ ℰt(❳휑) = ℰt(❳)휑 ❢♦r ❛❧❧ ❜♦✉♥❞❡❞ ♥♦♥♥❡❣❛t✐✈❡ 휑 ∈ ℋt ❢♦r ❛❧❧ ✵ ≤ s ≤ t ≤ ❚ ❛♥❞ ❳ ∈ ℋ✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✶✾ ✴ ✶✼

▼♦r❡ ♦♥ 핃✶

풫

❇② ❛r❣✉♠❡♥ts ♦❢ ❉❡♥✐s✱ ❍✉✱ P❡♥❣ ✭✶✵✮✿ 핃✶

풫 =

{ ❳ ∈ ▲✶

풫

• ❳ ✐s 풫✲q✉❛s✐ ✉♥✐❢♦r♠❧② ❝♦♥t✐♥✉♦✉s✱

❧✐♠♥ ∥❳✶{∣❳∣≥♥}∥▲✶

풫 = ✵

} ■❢ ❉ ✐s ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞✱ ✇❡ r❡tr✐❡✈❡ t❤❡ s♣❛❝❡ ♦❢ ❉❡♥✐s✱ ❍✉✱ P❡♥❣✿

▶ 핃✶

풫 ✐s ❛❧s♦ t❤❡ ❝❧♦s✉r❡ ♦❢ ❈❜ ⊂ ▲✶ 풫✱

▶ ❵q✉❛s✐ ✉♥✐❢♦r♠❧② ❝♦♥t✐♥✉♦✉s✬ ❂ ❵q✉❛s✐ ❝♦♥t✐♥✉♦✉s✬✳

■❢ ❉ ✐s ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞✱ ℰt ♠❛♣s 핃✶

풫 ✐♥t♦ 핃✶ 풫(ℱ∘ t )✳

❍❡♥❝❡ t✐♠❡ ❝♦♥s✐st❡♥❝② ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ℰs ∘ ℰt = ℰs✳

▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘❛♥❞♦♠ ●✲❊①♣❡❝t❛t✐♦♥s ✷✵ ✴ ✶✼