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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 ❳ ( t ) = ❳ ✵ + σ ( s , ❳ ( s )) ❞❇ ( s )+ ❜ ( 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 ✵ σ ( s , ❳ · ( ❙ + s )) ❞ ˜ ◮ ❳ · ( ❙ + t ) = ❳ · ( ❙ ) + ❇ ( 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❡ ◮ F t ✐s t❤❡ σ ✲✜❡❧❞ σ { ❇ s , ✵ ≤ s ≤ t } ✳ ▲❡t ❤ ❜❡ ❛ ✜①❡❞ ❝♦♥t✐♥✉♦✉s ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ ♦♥ R ♥ s✉❝❤ t❤❛t � � R ♥ | ① | ✷ ❤ ( ① ) ❞ ① < + ∞ . R ♥ ❤ ( ① ) ❞ ① = ✶ ❛♥❞ ❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

Ω = R ♥ × Ω ✱ ◮ ˆ ◮ ˆ F ❞❡♥♦t❡s t❤❡ ❇♦r❡❧ σ ✲✜❡❧❞ ♦♥ ˆ Ω ◮ ˆ P = ❤ ( ① ) ❞ ① × P ✳ ❋♦r s✐♠♣❧✐❝✐t② ✇❡ ❝❤♦♦s❡ ❤ t❤❡ ❢♦r♠ ♦❢ ♥ ✶ ✐ / ✷ = ✷ π ) ♥ ❡ − � ♥ ✐ = ✶ ① ✷ � ❤ ( ① ) = ❤ ( ① ✶ , . . . , ① ♥ ) = √ ❤ ( ① ✐ ) ( ✐ = ✶ ✶ ❡ − ① ✷ ✐ / ✷ ✳ √ ✇❤❡r❡ ❤ ( ① ✐ ) = ✷ π ❨♦✉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 , ✇ ) = ∇ ✐ ˆ ✭❥❥✮ ❋ ( ① , ✇ ) ❞ 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 ❦ ❦ = ✶ ,..., ♥ , ❥ = ✶ ,..., ❞ ∈ ❈ ❜ ([ ✵ , + ∞ ) × R → R ♥ ⊗ R ❞ ) , � � σ = ❥ ❦ = ✶ ,..., ♥ ∈ ❈ ❜ ([ ✵ , + ∞ ) × R ♥ → R ♥ ) , ❜ ❦ � � ❜ = ❲❡ ❛ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ � ≤ ❑ | ① − ① ′ | R ♥ , � � ❜ ❦ ( t , ① ) − ❜ ❦ ( t , ① ′ ) � ❍✳✶ ♠❛① ❦ ❍✳✷ ❊q✉❛t✐♦♥ ✭✷✮ ❤❛s P❯✳ ❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st

❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t ✭ ❍✳✶ ✮✲✭ ❍✳✷ ✮ ❤♦❧❞✳ ❚❤❡♥✱ t❤❡ s♦❧✉t✐♦♥ ❳ ❦ ( t ) ✐s ✐♥ ❱ ( R ♥ × Ω) ❢♦r ❛❧❧ t ∈ [ ✵ , + ∞ ) ❛♥❞ ❡❛❝❤ ˆ ❦ = ✶ , ✷ , ..., ♥✳ ❨♦✉ss❡❢ ❖✉❦♥✐♥❡ ❋❛❝✉❧té ❞❡s ❙❝✐❡♥❝❡s ❙❡♠❧❛❧✐❛ ▼❛r❝❤ ✶✽✲✷✸✱ ✷✵✵✾✱ ❇r❡st