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❖♥ t❤❡ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ✢♦✇ ♦❢ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛✱ ▼❛rr❛❦❡❝❤✱ ▼♦r♦❝❝♦ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ▼♦❤❛♠❡❞ ❊rr❛♦✉✐✮ ❙t♦❝❤❛st✐❝ ❈♦♥tr♦❧ ❛♥❞ ❋✐♥❛♥❝❡ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾ ❇r❡st

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

▼♦t✐✈❛t✐♦♥ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ✶✲❞✐♠❡♥s✐♦♥❛❧ s❞❡ ❳(t) = ❳✵+ t

✵

σ(s, ❳(s))❞❇(s)+ t

✵

❜(s, ❳(s))❞s, ✭✶✮ ❲❡ ❛ss✉♠❡ t❤❛t ❆✳✶ σ ❛♥❞ ❜ ❛r❡ ❝♦♥t✐♥✉♦✉s ♦♥ [✵, +∞) × R s✳t |σ(t, ①)| + |❜(t, ①)| ≤ ▲(✶ + |①|), ❆✳✷ ❚❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❤❛s ♣❛t❤✇✐s❡ ✉♥✐q✉❡♥❡ss

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

◮ ❛ss✉♠♣t✐♦♥ ✭❆✳✶✮ ⇒ ✇❡❛❦ s♦❧✉t✐♦♥

{❳①(t), t ≥ ✵}✱ ∀① ∈ R✱

◮ ❛ss✉♠♣t✐♦♥ ✭❆✳✷✮ ⇒ str♦♥❣✳

❋♦r ① ≤ ② ✜①❡❞✱ ✇❡ ❞❡✜♥❡ st♦♣♣✐♥❣ t✐♠❡✱ ❙ = ✐♥❢{t > ✵ : ❳①(t) > ❳②(t)}. ❖♥ t❤❡ s❡t [❙ < +∞]

◮ {˜

❇(t) = ❇(❙ + t) − ❇(❙), t ≥ ✵} ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳

◮ ❳·(❙ + t) = ❳·(❙) +

t

✵ σ(s, ❳·(❙ + s))❞ ˜

❇(s) + t

✵ ❜(s, ❳·(❙ + s))❞s

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❙✐♥❝❡ ❳①(❙) = ❳②(❙) ♦♥ [❙ < +∞] t❤❡♥

◮ P❯ ⇒

P[❳①(❙+t)✶[❙<+∞] = ❳②(❙+t)✶[❙<+∞], ∀ t ≥ ✵] = ✶.

◮ P[❳①(t) ≤ ❳②(t), ∀ t ≥ ✵] = ✶. ◮ P✲❛❧♠♦st ❛❧❧ ✇✱ ❢♦r ❛♥② t ≥ ✵✱ ① → ❳①(t)(✇) ✐s

✐♥❝r❡❛s✐♥❣ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ❛✳❡✳ ✇✐t❤ r❡s♣❡❝t t♦ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❋✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ▲❡t ❞ ≥ ✶✱

◮ Ω = ❈✵([✵, +∞), R❞) ◮ t❤❡ ❇♦r❡❧ σ✲✜❡❧❞ F✳ ◮ ❇ = {❇t : t ≥ ✵} ✐s ❞❡✜♥❡❞ ❜② ❇t(ω) = ω(t)✱ ◮ P ♦♥ (Ω, F) ✐s t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡ ◮ Ft ✐s t❤❡ σ✲✜❡❧❞ σ{❇s, ✵ ≤ s ≤ t}✳

▲❡t ❤ ❜❡ ❛ ✜①❡❞ ❝♦♥t✐♥✉♦✉s ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ♦♥ R♥ s✉❝❤ t❤❛t

• R♥❤(①)❞① = ✶ ❛♥❞
• R♥ |①|✷ ❤(①)❞① < +∞.

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

◮ ˆ

Ω = R♥ × Ω✱

◮ ˆ

F ❞❡♥♦t❡s t❤❡ ❇♦r❡❧ σ✲✜❡❧❞ ♦♥ ˆ Ω

◮ ˆ

P = ❤(①) ❞① × P✳ ❋♦r s✐♠♣❧✐❝✐t② ✇❡ ❝❤♦♦s❡ ❤ t❤❡ ❢♦r♠ ♦❢ ❤(①) = ❤(①✶, . . . , ①♥) = ✶ ( √ ✷π)♥❡− ♥

✐=✶ ①✷ ✐ /✷ =

♥

• ✐=✶

❤(①✐) ✇❤❡r❡ ❤(①✐) = ✶ √ ✷π ❡−①✷

✐ /✷✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

◆♦✇ ✇❡ ❣✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❧❛ss ❱ ❛s ❢♦❧❧♦✇s

❉❡✜♥✐t✐♦♥

❲❡ ❞❡✜♥❡ ❱(R♥ × Ω) t❤❡ t♦t❛❧ s❡t ♦❢ ❇♦r❡❧ ❢✉♥❝t✐♦♥s ❋ ♦♥ (ˆ Ω, ˆ F, ˆ P) s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❇♦r❡❧ ❢✉♥❝t✐♦♥ ˆ ❋ ♦♥ (ˆ Ω, ˆ F, ˆ P) s❛t✐s❢②✐♥❣ t❤❛t ✭✐✮ ❋ = ˆ ❋ ˆ P✲❛✳s✳ ✭✐✐✮ ∀ ✇ ∈ ˆ Ω✱ ① → ˆ ❋(①, ✇) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♦♥ ❡❛❝❤ ❝♦♠♣❛❝t ✐♥ R♥✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❚❤❡♦r❡♠

▲❡t ♣ > ✶ ❛♥❞ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ {❋❥ : ❥ ∈ N} ✐♥ ▲♣(ˆ Ω, ˆ F, ˆ P) s✉❝❤ t❤❛t ❇✳✶ ❋❥ ❝♦♥✈❡r❣❡s t♦ ❋ ❛❧♠♦st s✉r❡❧②✱ ❇✳✷ {❋❥ : ❥ ∈ N} ❛r❡ ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ✐♥ ▲♣(ˆ Ω, ˆ F, ˆ P)✱ ❇✳✸ ❢♦r ❛❧❧ (①, ✇) ∈ ˆ Ω✱ ✐ ∈ {✶, . . . , ♥} ❛♥❞ ❥ ∈ N✱ t → ❋❥(① + t❡✐, ✇) ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ❇✳✹ {∇✐❋❥} ❛r❡ ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ✐♥ ▲✶(ˆ Ω, ˆ F, ˆ P)✳ ❚❤❡♥ ❋ ∈ ❱(R♥ × Ω)✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

Pr♦♣♦s✐t✐♦♥

■❢ ❋ ∈ ❱(R♥ × Ω) t❤❡♥ ❢♦r P✲❛❧♠♦st ❛❧❧ ✇ ∈ Ω ❛♥❞ ❢♦r ❛❧❧ ✐ ∈ {✶, · · · ..., ♥} ✇❡ ❤❛✈❡ ✭❥✮ t → ❋(① + t❡✐, ✇) ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♦♥ ❛♥② ✜♥✐t❡ ✐♥t❡r✈❛❧✳ ✭❥❥✮ ∂ ∂①✐ ❋(①, ✇) = ∇✐ ˆ ❋(①, ✇) ❞①✲❛✳❡✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❆♣♣❧✐❝❛t✐♦♥s t♦ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ▲❡t ✉s ❝♦♥s✐❞❡r    ❞❳ ❦

① (t) = ❞ ❥=✶ σ❦ ❥ (t, ❳ ❦ ① (t))❞❇❥(t) + ❜❦(t, ❳①(t))❞t

❦ ✶ ✷ ♥ ❳①(✵) = ①, ✭✷✮ σ =

• σ❦

❥

• ❦=✶,...,♥,❥=✶,...,❞ ∈ ❈❜([✵, +∞)×R → R♥⊗R❞),

❜ =

• ❜❦

❦=✶,...,♥ ∈ ❈❜([✵, +∞) × R♥ → R♥),

❲❡ ❛ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❍✳✶ ♠❛①

❦

• ❜❦(t, ①) − ❜❦(t, ①′)
• ≤ ❑ |① − ①′|R♥ ,

❍✳✷ ❊q✉❛t✐♦♥ ✭✷✮ ❤❛s P❯✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❚❤❡♦r❡♠

❆ss✉♠❡ t❤❛t ✭❍✳✶✮✲✭❍✳✷✮ ❤♦❧❞✳ ❚❤❡♥✱ t❤❡ s♦❧✉t✐♦♥ ˆ ❳ ❦(t) ✐s ✐♥ ❱(R♥ × Ω) ❢♦r ❛❧❧ t ∈ [✵, +∞) ❛♥❞ ❡❛❝❤ ❦ = ✶, ✷, ..., ♥✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❜✐❜❧✐♦❣r❛♣❤②

✶✳ ❙✳ ❑✉s✉♦❦❛✱ ❊①✐st❡♥❝❡ ♦❢ ❞❡♥s✐t✐❡s ♦❢ s♦❧✉t✐♦♥s ♦❢ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❜② ▼❛❧❧✐❛✈✐♥ ❝❛❧❝✉❧✉s✱ ❏♦✉r♥❛❧ ♦❢ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ✷✵✶✵✳ ✷✳ ◆✳ ❇♦✉❧❡❛✉ ❛♥❞ ❋✳ ❍✐rs❝❤✱ ❛✲ ❖♥ t❤❡ ❞❡r✐✈❛❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇✐t❤ ▲✐♣s❝❤✐t③ ❝♦❡✣❝✐❡♥ts✳ ❙é♠✐♥❛✐r❡ ❞❡ ❚❤é♦r✐❡ ❞✉ P♦t❡♥t✐❡❧ P❛r✐s ✶✾✽✾✳ ❜✲ Pr♦♣r✐étés ❞✬❛❜s♦❧✉❡ ❝♦♥t✐♥✉✐té ❞❛♥s ❧❡s ❡s♣❛❝❡s ❞❡ ❉✐r✐❝❤❧❡t ❡t ❛♣♣❧✐❝❛t✐♦♥ ❛✉① éq✉❛t✐♦♥s ❞✐✛ér❡♥t✐❡❧❧❡s st♦❝❤❛st✐q✉❡s✧✱ ❙é♠✐♥❛✐r❡ ❞❡ Pr♦❜❛❜✐❧✐tés✱ ✶✾✽✻✳

❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st