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❍✐❣❤❡r r❛♥❦ s✐❣♥❛t✉r❡s ❛♥❞ ❛❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s P❛tr✐❝ ❇♦♥♥✐❡r ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤♦♥❣ ▲✐✉ ❛♥❞ ❍❛r❛❧❞ ❖❜❡r❤❛✉s❡r ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❯♥✐✈❡rs✐t② ♦❢ ❖①❢♦r❞ ❆❧❣❡❜r❛✱ ❈♦♠❜✐♥❛t♦r✐❝s ❛♥❞ P❡rs♣❡❝t✐✈❡s ✐♥ ▼❛t❤❡✲ ♠❛t✐❝❛❧ s❝✐❡♥❝❡s✱ ▼❛② ✷✵✷✵

❖✈❡r✈✐❡✇ ❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❚❤❡ s✐❣♥❛t✉r❡ ♦❢ ❛ r❛♥❦ r ♣r♦❝❡ss ▼❡tr✐③✐♥❣ t❤❡ r❛♥❦ r ❛❞❛♣t❡❞ t♦♣♦❧♦❣② ❖✉t❧♦♦❦✫❙✉♠♠❛r②

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❆ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s X n ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ V ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ X ✐❢ � � f ( x ) P ( X n ∈ dx ) → f ( x ) P ( X ∈ dx ) ❢♦r ❡✈❡r② f ∈ C b ( V , R ) . ❲❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✿ ❙t♦❝❤❛st✐❝ ♣r♦❝❡ss ( X t ) t ∈{ ✵ ,..., T } ♦♥ V ⇐ ⇒ ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ V { ✵ ,..., T } ■❣♥♦r❡s t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ ❛ ♣r♦❝❡ss✳ ✶✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❊①❛♠♣❧❡ ✭✶✮ p = ✶ p = ✵ . ✺ p = ✵ . ✺ p = ✶ ε p = ✵ . ✺ p = ✵ . ✺ p = ✶ ❊①❛♠♣❧❡ ✭✷✮ ❚❤❡ ♠❛♣ X �→ inf { E [ L τ ] : τ ≤ T } ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ✐♥ ✇❡❛❦ t♦♣♦❧♦❣② ❢♦r ❡✈❡r② ❛❞❛♣t❡❞ ❢✉♥❝t✐♦♥❛❧ ( L t ) t ∈{ ✵ ,..., T } t❤❛t ❞❡♣❡♥❞s ❝♦♥t✐♥✉♦✉s❧② ♦♥ t❤❡ tr❛❥❡❝t♦r② ♦❢ X ❬✶✱ ❙❡❝t✐♦♥ ✼❪✳ ✷✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❯s❡❢✉❧ t♦ ❤❛✈❡ ❛ str♦♥❣❡r t♦♣♦❧♦❣② ✐♥ ❛♣♣❧✐❝❛t✐♦♥s s✉❝❤ ❛s ◮ ♦♣t✐♠❛❧ st♦♣♣✐♥❣✱ ◮ q✉❡✉✐♥❣ t❤❡♦r②✱ ◮ st♦❝❤❛st✐❝ ♣r♦❣r❛♠♠✐♥❣ ✇❤❡r❡ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐s ✐♠♣♦rt❛♥t✳ ✸✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❊①t❡♥❞❡❞ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ = ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❝❡ss ❬✷❪ ˆ X t = P ( X ∈ ·|F t ) ∈ M ( I → M ( I → V )) ❉❡s❝r✐❜❡s ❤♦✇ ✇❡❧❧ ♦♥❡ ❝❛♥ ♣r❡❞✐❝t X ❛t t✐♠❡ t ✳ ❊①❛♠♣❧❡ ■❢ X = ( X ✶ , X ✷ , X ✸ ) ∈ V ✸ ˆ X ✶ = P ( X ✶ , X ✷ , X ✸ ∈ ·| X ✶ ) , ˆ X ✷ = P ( X ✶ , X ✷ , X ✸ ∈ ·| X ✶ , X ✷ ) , ˆ X ✸ = P ( X ✶ , X ✷ , X ✸ ∈ ·| X ✶ , X ✷ , X ✸ ) ❖t❤❡r r❡s❡❛r❝❤❡rs ❤❛✈❡ ♣r♦♣♦s❡❞ s✐♠✐❧❛r r❡♠❡❞✐❡s✳ ❚❤❡② ❞♦ ♥♦t ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❢✉❧❧ str✉❝t✉r❡ ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss✳ ✹✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❊①❛♠♣❧❡ ❚❤❡r❡ ❡①✐sts t✇♦ ♣r♦❝❡ss❡s X ε , Y ε ❞❡♣❡♥❞✐♥❣ ♦♥ ε s✉❝❤ t❤❛t ✇❤❡♥ ε → ✵✿ X ε − Y ε → ✵ , X ε − ˆ Y ε → ✵ , ˆ ❜✉t ✹ |F ✸ ] ✷ |F ✶ ] − E [ E [ Y ε ✹ |F ✸ ] ✷ |F ✶ ] �→ ✵ . E [ E [ X ε ❚❛❦❡ ❛✇❛②✿ ♠♦r❡ str✉❝t✉r❡ ❝❛♥ ❜❡ ❝❛♣t✉r❡❞ ❜② ✐t❡r❛t✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥✳ ✺✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✺ ✺ ✸ ε ✶ ✸ ε ✶ ✸ ✸ ✶ ✶ ✷ ε ✷ ε ✹ ✹ ✸ ε ✷ ✸ ε ✷ ε ✷ ε ✷ ε ✶ ε ✶ ✶ ✶ ✷ ε ✶ ✷ ε ✷ ✷ ✷ ✷ ✶ ✸ ε ✸ ε ✵ ✵ ✷ ✷ − ✷ − ✷ ✶ ✶ ✸ ε ✸ ε − ✶ − ✶ ✷ ε ✷ ✷ ε ✶ − ε − ε ✶ ✷ − ε − ε ✷ ✷ − ✹ − ✹ ✸ ε ✶ ✸ ε ✶ − ✸ − ✸ ✷ ✷ ✷ ε ✷ ε − ✺ − ✺ ✸ ε ✶ ✸ ε ✶ ✷ ✷ t = ✵ t = ✶ t = ✷ t = ✸ t = ✹ t = ✵ t = ✶ t = ✷ t = ✸ t = ✹ ✭❛✮ X ✭❜✮ Y ✻✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s f ( ✶ ) f ( ✶ ) E [ f ( X ✹ )] E [ f ( Y ✹ )] f ( ✷ ) f ( ✷ ) E [ f ( X ✹ )] E [ f ( Y ✹ )] f ( ✶ ) E [ f ( Y ✹ )] E [ f ( X ✹ )] E [ f ( Y ✹ )] E [ f ( X ✹ )] f ( ✷ ) E [ f ( Y ✹ )] E [ f ( Y ✹ )] E [ f ( X ✹ )] f ( ✶ ) E [ f ( X ✹ )] E [ f ( Y ✹ )] E [ f ( X ✹ )] E [ f ( Y ✹ )] E [ f ( X ✹ )] f ( ✷ ) E [ f ( X ✹ )] E [ f ( Y ✹ )] E [ f ( X ✹ )] E [ f ( Y ✹ )] E [ f ( X ✹ )] E [ f ( Y ✹ )] t = ✵ t = ✶ t = ✷ t = ✸ t = ✵ t = ✶ t = ✷ t = ✸ ✭❛✮ E [ f ( X ✹ ) |F ] ✭❜✮ E [ f ( Y ✹ ) |F ] ✼✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❍♦♦✈❡r✲❑❡✐s❧❡r ❬✸❪ ✐♥tr♦❞✉❝❡❞ t❤❡ ❝❧❛ss ♦❢ ❛❞❛♣t❡❞ ❢✉♥❝t✐♦♥❛❧s AF ❉❡✜♥✐t✐♦♥ ✶✳ f ∈ C b ( V n , R ) ✱ t❤❡♥ X �→ f ( X ( t ✶ ) , . . . , X ( t n )) ∈ AF ✷✳ f ✶ , . . . , f n ∈ AF ✱ f ∈ C b ( R n , R ) ✱ t❤❡♥ X �→ f ( f ✶ ( X ) , . . . , f n ( X )) ∈ AF ✸✳ f ∈ AF ✱ ✵ ≤ t ≤ T ✱ t❤❡♥ X �→ E [ f ( X ) |F t ] ∈ AF . ✽✴✷✺

❆❞❛♣t❡❞ t♦♣♦❧♦❣✐❡s ♦♥ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❚❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ✐s t❛❦❡♥ ✐s ❝❛❧❧❡❞ t❤❡ r❛♥❦ ♦❢ f ✳ � ✷ ❤❛s r❛♥❦ ✶✳ ◮ f ( X ) = � E [cos( X t ✶ X t ✷ ) |F t ✸ ] � � sin( E [ X ✸ � ◮ f ( X ) = E t ✶ |F t ✷ ] |F t ✸ ❤❛s r❛♥❦ ✷✳ ❉❡✜♥✐t✐♦♥ X ❛♥❞ Y ❤❛✈❡ t❤❡ s❛♠❡ ❛❞❛♣t❡❞ ❞✐str✐❜✉t✐♦♥ ✉♣ t♦ r❛♥❦ r ✐❢✿ E f ( Y ) = E f ( X ) ❢♦r ❛♥② f ∈ AF ♦❢ r❛♥❦ ≤ r . ❚❤❡ ❛❞❛♣t❡❞ t♦♣♦❧♦❣② ♦❢ r❛♥❦ r ✐s ❞❡✜♥❡❞ ❜② t❤❡ r❡q✉✐r❡♠❡♥t X n → r X ✐❢ E f ( X n ) → E f ( X ) ❢♦r ❛♥② f ∈ AF ♦❢ r❛♥❦ ≤ r . ◮ r❛♥❦ ✵ ✐s ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ◮ r❛♥❦ ✶ ✐s ❡①t❡♥❞❡❞ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡✳ ✾✴✷✺

❚❤❡ s✐❣♥❛t✉r❡ ♦❢ ❛ r❛♥❦ r ♣r♦❝❡ss

❚❤❡ s✐❣♥❛t✉r❡ ♦❢ ❛ r❛♥❦ r ♣r♦❝❡ss ❲✐❧❧ ✐♥tr♦❞✉❝❡✿ ◮ ❘❛♥❦ r t❡♥s♦r ❛❧❣❡❜r❛s✳ ◮ ❘❛♥❦ r ♣❛t❤s ❛♥❞ r❛♥❦ r s✐❣♥❛t✉r❡s✳ ◮ ❘❛♥❦ r st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛♥❞ r❛♥❦ r ❡①♣❡❝t❡❞ s✐❣♥❛t✉r❡s✳ ✶✵✴✷✺

❚❤❡ s✐❣♥❛t✉r❡ ♦❢ ❛ r❛♥❦ r ♣r♦❝❡ss ❉❡✜♥✐t✐♦♥ ✭❘❛♥❦ r t❡♥s♦r ❛❧❣❡❜r❛✮ � ⊗ m . t ✵ ( V ) = V , t r ( V ) = � t r − ✶ ( V ) � m ≥ ✵ ❋♦r ❡✈❡r② r ≥ ✶✱ ( t r ( V ) , ⊗ ( r ) ) ✐s ❛ ♠✉❧t✐✲❣r❛❞❡❞ ❛❧❣❡❜r❛ ♦✈❡r V ✳ V ⊗ ( ✶ ) n ◮ t ✶ ( V ) = � n ≥ ✵ V ⊗ ( ✶ ) n ✶ ⊗ ( ✷ ) · · · ⊗ ( ✷ ) V ⊗ ( ✶ ) n k ◮ t ✷ ( V ) = � n ✶ ,..., n k ≥ ✵ ✶ ⊗ ( ✷ ) · · ·⊗ ( ✷ ) V ⊗ ( ✶ ) n ✶ V ⊗ ( ✶ ) n ✶ ◮ t ✸ ( V ) = � k ✶ � � ⊗ ( ✸ ) n ✶ ✶ ,..., n ✶ k ✶ ≥ ✵ ✶ ⊗ ( ✷ ) · · · ⊗ ( ✷ ) V ⊗ ( ✶ ) n k ✷ ✳ V ⊗ ( ✶ ) n k ✷ ✳ � k ✶ � · · · ⊗ ( ✸ ) ✳ n k ✷ ✶ ,..., n k ✷ k ✶ ≥ ✵ ✶✶✴✷✺

❚❤❡ s✐❣♥❛t✉r❡ ♦❢ ❛ r❛♥❦ r ♣r♦❝❡ss ◮ Seq( S ) := ✜♥✐t❡ s❡q✉❡♥❝❡s ♦♥ S ✳ ◮ ❘❡❝✉rs✐✈❡❧②✿ Seq r ( S ) = Seq(Seq r − ✶ ( S )) ✱ t r ( V ) ❦ : = t r − ✶ ( V ) ❦ ✶ ⊗ ( r ) · · · ⊗ ( r ) t r − ✶ ( V ) ❦ l , � t r ( V ) = t r ( V ) ❦ ❦ ∈ Seq r − ✶ ( N ) ✶✷✴✷✺