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r ts r s st stst - - PowerPoint PPT Presentation

Feb 20, 2024 178 likes •598 views

r ts r s st stst tt s sst ts r rst

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SLIDE 1

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r s♦♠❡ ❢❛st st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❜② ✈✐s❝♦s✐t② ♠❡t❤♦❞s

❉❛r✐❛ ●❤✐❧❧✐ ❯♥✐✈❡rs✐t② ♦❢ P❛❞✉❛ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼❛rt✐♥♦ ❇❛r❞✐ ❛♥❞ ❆♥♥❛❧✐s❛ ❈❡s❛r♦♥✐ ❚❯ ❇❡r❧✐♥✱ ❙❛t✉r❞❛② 26 ❖❝t♦❜❡r✱ ✷✵✶✹

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

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SLIDE 2

P❧❛♥

✶ ❙t♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ✷ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❜② ✈✐s❝♦s✐t② ♠❡t❤♦❞s ✸ ❆s ❛♣♣❧✐❝❛t✐♦♥s✱ ❛s②♠♣t♦t✐❝ ❡st✐♠❛t❡s ❢♦r ❊✉r♦♣❡❛♥ ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ♦♣t✐♦♥ ♣r✐❝❡s ♥❡❛r ♠❛t✉r✐t② ❛♥❞ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ❢♦r ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✳ ✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ♥♦♥✲❝♦♠♣❛❝t ❝❛s❡ ✭✐✳❡✳ ✇❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ st♦❝❤❛st✐❝ s②st❡♠ ❛r❡ ♥♦t ♣❡r✐♦❞✐❝✮✱ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

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SLIDE 3

❙t♦❝❤❛st✐❝ s②st❡♠ ✇✐t❤ ❢❛st ♦s❝✐❧❧❛t✐♥❣ r❛♥❞♦♠ ♣❛r❛♠❡t❡r

❲❡ ❝♦♥s✐❞❡r ❛ st♦❝❤❛st✐❝ s②st❡♠ ✐♥ Rn ✇✐t❤ r❛♥❞♦♠ ❝♦❡✣❝✐❡♥ts✱ ✐♥ ♣❛rt✐❝✉❧❛r ✇✐t❤ ❝♦❡✣❝✐❡♥ts ❞❡♣❡♥❞❡♥t ♦♥ r❛♥❞♦♠ ♣❛r❛♠❡t❡r Yt✳ dXt = φ(Xt, Yt)dt + √ 2σ(Xt, Yt)dWt, X0 = x0 ∈ Rn. ❆ss✉♠♣t✐♦♥✿ ✇❡ ♠♦❞❡❧ t❤✐s ♥❡✇ ♣❛r❛♠❡t❡r ❛s ❛ ♠❛r❦♦✈ ♣r♦❝❡ss ❡✈♦❧✈✐♥❣ ♦♥ ❛ ❢❛st❡r t✐♠❡ s❝❛❧❡ τ = t

δ✿

dYt = 1 δ b(Yt)dt +

  • 2

δ τ(Yt)dWt, Y0 = y0 ∈ Rm. ◆♦t❛t✐♦♥✿ Xt ❛r❡ t❤❡ s❧♦✇ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ s②st❡♠✱ ❛♥❞ Yt ❛r❡ t❤❡ ❢❛st ❝♦♠♣♦♥❡♥ts✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

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SLIDE 4

❆ss✉♠♣t✐♦♥s

❚②✐❝❛❧ ❆ss✉♠♣t✐♦♥s✿ ❚❤❡ ❢❛st ✈❛r✐❛❜❧❡ ❛r❡ ❝♦♥str❛✐♥❡❞ ✐♥ ❛ ❝♦♠♣❛❝t s❡t✱ s❛②✿ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♣r♦❝❡ss❡s ❛r❡ Zm✲♣❡r✐♦❞✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈❛r✐❛❜❧❡ y✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✿ Yt ✐s ❛ r❡❝✉rr❡♥t ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❝❛♥ ❣❡♥❡r❛❧✐③❡ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s t❤❛t Yt ✐s ❡r❣♦❞✐❝✱ t❤✐s ♠❡❛♥s t❤❛t Y ❢♦r❣❡ts t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r ❧❛r❣❡ t✐♠❡ ✭✐✳❡✳ ❛s δ → 0✮ ❛♥❞ ✐ts ❞✐str✐❜✉t✐♦♥ ❜❡❝♦♠❡s st❛t✐♦♥❛r②✳ ❋♦r t❡❝❤♥✐❝❛❧ s✐♠♣❧✐❝✐t② ❢r♦♠ ♥♦✇ ♦♥ ✇❡ ❛ss✉♠❡ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♣❡r✐♦❞✐❝✐t② ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts✳ ❚❤✐s ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ r❡❧❛①❡❞ t♦ ❡r❣♦❞✐❝✐t② ❛♥❞ ✇✐❧❧ ❜❡ tr❡❛t❡❞ ✐♥ ❛♥ ❛rt✐❝❧❡ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❋✉rt❤❡r ❛ss✉♠♣t✐♦♥✿ t❤❡ ❞✐✛✉s✐♦♥ ♠❛tr✐① τ ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

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SLIDE 5

▼♦t✐✈❛t✐♦♥✿✿ ❛♥❛❧②s✐s ♦❢ ❢✐♥❛♥❝✐❛❧ ♠♦❞❡❧s ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②

❇❧❛❝❦✲❙❝❤♦❧❡s ♠♦❞❡❧✿ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r✐❝❡ ♦❢ ❛ st♦❝❦ ❙ ✐s ❞❡s❝r✐❜❡❞ ❜② dlogSt = γdt + σdWt, t❂t✐♠❡, Wt = ❲✐❡♥❡r ♣r♦❝✳, ❛♥❞ t❤❡ ❝❧❛ss✐❝❛❧ ❇❧❛❝❦✲❙❝❤♦❧❡s ❢♦r♠✉❧❛ ❢♦r ♦♣t✐♦♥ ♣r✐❝✐♥❣ ✐s ❞❡r✐✈❡❞ ❛ss✉♠✐♥❣ ♣❛r❛♠❡t❡rs ❛r❡ ❝♦♥st❛♥ts✳ ■♥ r❡❛❧✐t② t❤❡ ♣❛r❛♠❡t❡rs ♦❢ s✉❝❤ ♠♦❞❡❧s ❛r❡ ♥♦t ❝♦♥st❛♥ts✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ✈♦❧❛t✐❧✐t② σ✱ ❛ ♠❡❛s✉r❡ ❢♦r ✈❛r✐❛t✐♦♥ ♦❢ ♣r✐❝❡ ♦✈❡r t✐♠❡✱ ✐s ♥♦t ❝♦♥st❛♥t ❜✉t ❡①❤✐❜✐ts r❛♥❞♦♠ ❜❡❤❛✈✐♦✉r✳ ❚❤❡r❡❢♦r❡ ✐t ❤❛s ❜❡❡♥ ♠♦❞❡❧❡❞ ❛s ❛ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥ σ = σ(Yt) ♦❢ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss Yt ✇✐t❤ ✶ ♥❡❣❛t✐✈❡ ❝♦rr❡❧❛t✐♦♥ ✭♣r✐❝❡s ❣♦ ✉♣ ✇❤❡♥ ✈♦❧❛t✐❧✐t② ❣♦❡s ❞♦✇♥✮ ✷ ♠❡❛♥ r❡✈❡rs✐♦♥ ✭t❤❡ t✐♠❡ ✐t t❛❦❡s ❢♦r ❛❣❡♥ts t♦ ❛❞❥✉st t❤❡✐r t❤r❡s❤♦❧❞s t♦ ❝✉rr❡♥t ♠❛r❦❡t ❝♦♥❞✐t✐♦♥s✮ ❘❡❢s✳✿ ❍✉❧❧✲❲❤✐t❡ ✽✼✱ ❍❡st♦♥ ✾✸✱ ❋♦✉q✉❡✲P❛♣❛♥✐❝♦❧❛♦✉✲❙✐r❝❛r ✷✵✵✵✱✳✳✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

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SLIDE 6

Multiscale stochastic volatility

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SLIDE 7

❋❛st st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②

■t ✐s ❛r❣✉❡❞ ✐♥ t❤❡ ❜♦♦❦ ❋♦✉q✉❡✱ P❛♣❛♥✐❝♦❧❛♦✉✱ ❙✐r❝❛r✿ ❉❡r✐✈❛t✐✈❡s ✐♥ ✜♥❛♥❝✐❛❧ ♠❛r❦❡ts ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②✱ ✷✵✵✵✱ t❤❛t Yt ❛❧s♦ ❡✈♦❧✈❡s ♦♥ ❛ ❢❛st❡r t✐♠❡ s❝❛❧❡ t❤❛♥ t❤❡ st♦❝❦ ♣r✐❝❡s✱ ♠♦❞❡❧❧✐♥❣ ❜❡tt❡r t❤❡ t②♣✐❝❛❧ ❜✉rst② ❜❡❤❛✈✐♦r ♦❢ ✈♦❧❛t✐❧✐t②✱ s❡❡ ♣r❡✈✐♦✉s ♣✐❝t✉r❡✳ ❋♦r t❤✐s r❡❛s♦♥ ✇❡ ♣✉t ♦✉rs❡❧✈❡s ✐♥t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♠✉❧t✐♣❧❡ t✐♠❡ s❝❛❧❡ s②st❡♠s ❛♥❞ s✐❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ❛♥❞ ✇❡ ♠♦❞❡❧ Yt ✇✐t❤ t❤❡ ❢❛st st♦❝❤❛st✐❝ ♣r♦❝❡ss ❢♦r δ > 0 dYt = 1 δ b(Yt)dt +

  • 2

δ τ(Yt)dWt Y0 = y0 ∈ Rm. P❛ss✐♥❣ t♦ t❤❡ ❧✐♠✐t ❛s δ → 0 ✐s ❛ ❝❧❛ss✐❝❛❧ s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♣r♦❜❧❡♠✱ ✐ts s♦❧✉t✐♦♥ ❧❡❛❞s t♦ t❤❡ ❡❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ Yt ❛♥❞ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ❛✈❡r❛❣❡❞ s②st❡♠ ❞❡✜♥❡❞ ✐♥ Rn ♦♥❧②✳ ❚❤❡r❡ ✐s ❛ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✭❇❡♥s♦✉ss❛♥✱ ❑✉s❤♥❡r✱ ❍❛s♠✐♥s❦✐✐✱ P❛r❞♦✉①✱ ❇♦r❦❛r✱

  • ❛❧ts❣♦r②✱ ❆❧✈❛r❡①✱ ❇❛r❞✐✳✳✳✮

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

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SLIDE 8

❋❛st st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②

■t ✐s ❛r❣✉❡❞ ✐♥ t❤❡ ❜♦♦❦ ❋♦✉q✉❡✱ P❛♣❛♥✐❝♦❧❛♦✉✱ ❙✐r❝❛r✿ ❉❡r✐✈❛t✐✈❡s ✐♥ ✜♥❛♥❝✐❛❧ ♠❛r❦❡ts ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②✱ ✷✵✵✵✱ t❤❛t Yt ❛❧s♦ ❡✈♦❧✈❡s ♦♥ ❛ ❢❛st❡r t✐♠❡ s❝❛❧❡ t❤❛♥ t❤❡ st♦❝❦ ♣r✐❝❡s✱ ♠♦❞❡❧❧✐♥❣ ❜❡tt❡r t❤❡ t②♣✐❝❛❧ ❜✉rst② ❜❡❤❛✈✐♦r ♦❢ ✈♦❧❛t✐❧✐t②✱ s❡❡ ♣r❡✈✐♦✉s ♣✐❝t✉r❡✳ ❋♦r t❤✐s r❡❛s♦♥ ✇❡ ♣✉t ♦✉rs❡❧✈❡s ✐♥t♦ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ♠✉❧t✐♣❧❡ t✐♠❡ s❝❛❧❡ s②st❡♠s ❛♥❞ s✐❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ❛♥❞ ✇❡ ♠♦❞❡❧ Yt ✇✐t❤ t❤❡ ❢❛st st♦❝❤❛st✐❝ ♣r♦❝❡ss ❢♦r δ > 0 dYt = 1 δ b(Yt)dt +

  • 2

δ τ(Yt)dWt Y0 = y0 ∈ Rm. P❛ss✐♥❣ t♦ t❤❡ ❧✐♠✐t ❛s δ → 0 ✐s ❛ ❝❧❛ss✐❝❛❧ s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♣r♦❜❧❡♠✱ ✐ts s♦❧✉t✐♦♥ ❧❡❛❞s t♦ t❤❡ ❡❧✐♠✐♥❛t✐♦♥ ♦❢ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ Yt ❛♥❞ t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛♥ ❛✈❡r❛❣❡❞ s②st❡♠ ❞❡✜♥❡❞ ✐♥ Rn ♦♥❧②✳ ❚❤❡r❡ ✐s ❛ ❧❛r❣❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t ✭❇❡♥s♦✉ss❛♥✱ ❑✉s❤♥❡r✱ ❍❛s♠✐♥s❦✐✐✱ P❛r❞♦✉①✱ ❇♦r❦❛r✱

  • ❛❧ts❣♦r②✱ ❆❧✈❛r❡①✱ ❇❛r❞✐✳✳✳✮

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-9
SLIDE 9

❙♠❛❧❧ t✐♠❡ ❛s②♠♣t♦t✐❝s ❢♦r t❤❡ s②st❡♠

❲❡ st✉❞② t❤❡ s♠❛❧❧ t✐♠❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ s②st❡♠✱ s♦ ✇❡ r❡s❝❛❧❡ t✐♠❡ ❛s t → εt. ❲❡ st✉❞② t❤❡ ❛s②♠♣t♦t✐❝s ✇❤❡♥ ❜♦t❤ ♣❛r❛♠❡t❡rs ❣♦ t♦ 0 ❛♥❞ ✇❡ ❡①♣❡❝t ❞✐✛❡r❡♥t ❧✐♠✐t ❜❡❤❛✈✐♦rs ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r❛t❡ ε/δ✳ ❚❤❡r❡❢♦r❡ ✇❡ ♣✉t δ = εα, ✇✐t❤ α > 1. ❲❡ ❝♦♥s✐❞❡r t❤❡ ❧✐♠✐t ♦❢ t❤❡ s②st❡♠ ❢♦r ε → 0

  • dXε

t = εφ(Xε t , Y ε t )dt +

√ 2εσ(Xε

t , Y ε t )dWt,

Xε

0 = x0 ∈ Rn

dY ε

t = 1 εα−1 b(Y ε t )dt +

  • 2

εα−1 τ(Y ε t )dWt,

Y ε

0 = y0 ∈ Rm.

✭✶✮

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-10
SLIDE 10

▼♦t✐✈❛t✐♦♥✿ ❛s②♠♣t♦t✐❝ ❡st✐♠❛t❡s ❢♦r ✈♦❧❛t✐❧✐t② ♦❢ ♦♣t✐♦♥ ♣r✐❝❡s ♥❡❛r ♠❛t✉r✐t②

❆✈❡❧❧❛♥❡❞❛ ❛♥❞ ❝♦❧❧❛❜♦r❛t♦rs ✭2002, 2003✮ ✉s❡❞ t❤❡ t❤❡♦r② ♦❢ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s t♦ ❣✐✈❡ ❛s②♠♣t♦t✐❝ ❡st✐♠❛t❡s ❢♦r t❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ♦❢ ♦♣t✐♦♥ ♣r✐❝❡s ♥❡❛r ♠❛t✉r✐t② ✭s♠❛❧❧ t✐♠❡✮ ✐♥ ♠♦❞❡❧s ✇✐t❤ ❝♦♥st❛♥t ✭❧♦❝❛❧✮ ✈♦❧❛t✐❧✐t②✳ ❲❡ ❝❛rr② ♦♥ t❤❡ s❛♠❡ t②♣❡ ♦❢ ❛♥❛❧②s✐s ✐♥ ♠♦❞❡❧s ✇✐t❤ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t②✳ ■♥ t❤✐s ❝❛s❡ ✜♥❞✐♥❣ ❡①♣❧✐❝✐t ❡st✐♠❛t❡s ❤❛♣♣❡♥s t♦ ❜❡ ♠♦r❡ ❞✐✣❝✉❧t ❛♥❞ ✇❡ ♥❡❡❞ t♦ ❛ss✉♠❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♣❡r✐♦❞✐❝✐t②✴❡r❣♦❞✐❝✐t② ♦♥ t❤❡ ❢❛st ♣r♦❝❡ss✳

❘❡♠❛r❦

■♥ t❤✐s ♠♦❞❡❧✿ ε ✿ s❤♦rt ♠❛t✉r✐t② ♦❢ t❤❡ ♦♣t✐♦♥ δ = εα ✿ r❛t❡ ♦❢ ♠❡❛♥ r❡✈❡rs✐♦♥ ♦❢ t❤❡ ✈♦❧❛t✐❧✐t②✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-11
SLIDE 11

❋✉rt❤❡r r❡❢❡r❡♥❝❡s

✶ ❏✳ ❋❡♥❣✱ ❏✳✲P✳ ❋♦✉q✉❡✱ ❘✳ ❑✉♠❛r ✭✷✵✶✷✮ st✉❞✐❡❞ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r s②st❡♠s ♦❢ t❤❡ ❢♦r♠ t❤❛t ✇❡ ❞❡✜♥❡❞ ❢♦r α = 2, 4 ✐♥ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ n = m = 1✱ ❛ss✉♠✐♥❣ t❤❛t Yt ✐s ❛♥ ❖r♥st❡✐♥✲❯❤❧❡♥❜❡❝❦ ♣r♦❝❡ss ❛♥❞ t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❢♦r Xt ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ Xt✳ ❚❤❡ ♠❡t❤♦❞s ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♠♦♥♦❣r❛♣❤ ❜② ❋❡♥❣ ❛♥❞ ❑✉rt③✱ ▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✷✵✵✻✳ ✷ ❘❡❧❛t❡❞ ✇♦r❦s ❜② P✳ ❉✉♣✉✐s✱ ❑✳ ❙♣✐❧✐♦♣♦✉❧♦s✱ ❑✳ ❙♣✐❧✐♦♣♦✉❧♦s ✭✷✵✶✷✱ ✷✵✶✸✮ ❞❡❛❧ ✇✐t❤ ❞✐✛❡r❡♥t s❝❛❧✐♥❣ ❛♥❞ ✉s❡ ❞✐✛❡r❡♥t ♠❡t❤♦❞s ❜❛s❡❞ ♦♥ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-12
SLIDE 12

▲❛r❣❡ ❉❡✈✐❛t✐♦♥ Pr✐♥❝✐♣❧❡

▲❡t {µε} ❜❡ ❛ ❢❛♠✐❧② ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s✳ ❆ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ♣r✐♥❝✐♣❧❡ ✭▲❉P✮ ❝❤❛r❛❝t❡r✐③❡s t❤❡ ❧✐♠✐t✐♥❣ ❜❡❤❛✈✐♦r✱ ❛s ǫ → 0✱ ♦❢ {µǫ} ✐♥ t❡r♠s ♦❢ ❛ r❛t❡ ❢✉♥❝t✐♦♥ t❤r♦✉❣❤ ❛s②♠♣t♦t✐❝ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❡①♣♦♥❡♥t✐❛❧ ❜♦✉♥❞s ♦♥ t❤❡ ✈❛❧✉❡s t❤❛t µǫ ❛ss✐❣♥s t♦ ♠❡❛s✉r❛❜❧❡ s✉❜s❡ts ♦❢ Rn✳ ❘♦✉❣❤❧② s♣❡❛❦✐♥❣✱ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ t❤❡♦r② ❝♦♥❝❡r♥s ✐ts❡❧❢ ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❞❡❝❧✐♥❡ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ♦❢ ❝❡rt❛✐♥ ❦✐♥❞s ♦❢ ❡①tr❡♠❡ ♦r t❛✐❧ ❡✈❡♥ts✳ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ✜♥❛♥❝✐❛❧ ♠❛t❤❡♠❛t✐❝s✱ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s t❤❡♦r② ❛r✐s❡s ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ♠❛t✉r✐t② ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ♦♣t✐♦♥ ♣r✐❝❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-13
SLIDE 13

▼❛✐♥ r❡s✉❧ts✲▲❛r❣❡ ❉❡✈✐❛t✐♦♥ ♣r✐♥❝✐♣❧❡

❲❡ ♣r♦✈❡ ❛ ▲❛r❣❡ ❉❡✈✐❛t✐♦♥ Pr✐♥❝✐♣❧❡ ✭▲❉P✮ ❢♦r t❤❡ ♣r♦❝❡ss Xε

t ✭✐✳❡✳ ❢♦r

♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❧❛✇s ♦❢ Xε

t ✮✳

■♥ ♦t❤❡r ✇♦r❞s ✇❡ ♣r♦✈❡ t❤❛t t❤❡♥ ❢♦r ❡✈❡r② t > 0 ❛♥❞ ❢♦r ❛♥② ♦♣❡♥ s❡t B ⊆ Rn P(Xε

t ∈ B) = e− infx∈B

I(x;x0,t) ε

+o( 1

ε ), ❛s ε → 0.

❢♦r s♦♠❡ ✭❣♦♦❞✮ r❛t❡ ❢✉♥❝t✐♦♥ I✱ ♥♦♥✲♥❡❣❛t✐✈❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s✱ ✇❤✐❝❤ ✇❡ ✇✐❧❧ ❞❡✜♥❡ ✐♥ t❤❡ ♥❡①t s❧✐❞❡s✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-14
SLIDE 14

❚❤❡ ▲❛r❣❡ ❉❡✈✐❛t✐♦♥ Pr✐♥❝✐♣❧❡

❇r②❝✬s ✐♥✈❡rs❡ ❱❛r❛❞❤❛♥ ❧❡♠♠❛

❆ss✉♠❡ t❤❛t ❢♦r ❛❧❧ t > 0 ✶ Xε

t ✐s ❡①♣♦♥❡♥t✐❛❧❧② t✐❣❤t✳

✷ ❢♦r ❡✈❡r② h ❜♦✉♥❞❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s t❤❡ ❧✐♠✐t lim

ε→0 ε log E

  • eε−1h(Xε

t ) | X0 = x0, Y0 = y0

  • := Lh(x0, t)

❡①✐sts ✜♥✐t❡✳ ❚❤❡♥ Xε

t s❛t✐s✜❡s ❛ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ♣r✐♥❝✐♣❧❡ ✇✐t❤ ❣♦♦❞ r❛t❡ ❢✉♥❝t✐♦♥

I(x, x0, t) = sup

h∈BC(Rn)

{h(x) − Lh(x0, t)}.

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-15
SLIDE 15

P❉❊ ♠❡t❤♦❞s

❲❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧♦❣❛r✐t❤♠✐❝ ♣❛②♦✛ vε(t, x0, y0) := ε log E

  • eε−1h(Xε

t ) | X0 = x0, Y0 = y0

  • , x0 ∈ Rn, y0 ∈ Rm, t ≥ 0,

✇❤❡r❡ h ✐s ❛ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ Rn✳ ❚❤❡♥✱ t♦ ♦❜t❛✐♥ t❤❡ ▲❉P ✇❡ ❤❛✈❡✿ ✶ ♣r♦✈❡ t❤❛t vε ❝♦♥✈❡r❣❡s t♦ s♦♠❡ ❢✉♥❝t✐♦♥ v(t, x) ❛♥❞ ❝❤❛r❛❝t❡r✐③❡ v❀ ✷ ❝♦♠♣✉t❡ t❤❡ r❛t❡ ❢✉♥❝t✐♦♥ I ✐♥ t❡r♠ ♦❢ t❤❡ ❧✐♠✐t ♦❢ vε✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-16
SLIDE 16

▼❛✐♥ r❡s✉❧t✲❈♦♥✈❡r❣❡♥❝❡ ❜② P❉❊ ♠❡t❤♦❞s

❚❤❡ ❛ss♦❝✐❛t❡❞ ❍❏❇ ❡q✉❛t✐♦♥ t♦ vε ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛❜♦❧✐❝ ♣❞❡ ✇✐t❤ q✉❛❞r❛t✐❝ ♥♦♥❧✐♥❡❛r✐t② ✐♥ t❤❡ ❣r❛❞✐❡♥t ✭b, τ ❝♦♠♣✉t❡❞ ✐♥ y✱ φ, σ ✐♥ (x, y)✮✳ vε

t = |σT Dxvε|2 + ε

  • tr(σσT D2

xxvε) + φ · Dxvε

+ 2ε− α

2 (τσT Dxvε) · Dyvε+

+ 2ε1− α

2 tr(στ T D2

xyvε) + ε1−α

b · Dyvε + tr(ττ T D2

yyvε)

  • + ε−α|τ T Dyvε|2.

◆♦t❡ t❤❛t ❧❡tt✐♥❣ ε → 0 ✐♥ t❤❡ P❉❊ ✐s ❛ r❡❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♣r♦❜❧❡♠✳

❘❡♠❛r❦

❚❤✐s ♣r♦❜❧❡♠ ❢❛❧❧s ✐♥ t❤❡ ❝❧❛ss ♦❢ ❛✈❡r❛❣✐♥❣✴❤♦♠♦❣❡♥✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❍❏❇ t②♣❡ ❡q✉❛t✐♦♥s ✇❤❡r❡ t❤❡ ❢❛st ✈❛r✐❛❜❧❡ ❧✐✈❡s ✐♥ ❛ ❝♦♠♣❛❝t s♣❛❝❡

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-17
SLIDE 17

▼❛✐♥ r❡s✉❧t✲❈♦♥✈❡r❣❡♥❝❡ ❜② ✈✐s❝♦s✐t② ♠❡t❤♦❞s

❚❤❡♦r❡♠

▲❡t h ❜❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞✳ ❚❤❡♥ vε(x, y, t) = ε log Ee

h(Xε t ) ε

→ v(x, t) ❧♦❝❛❧❧② ✉♥✐❢♦r♠❧② ✐♥ y ✇❤❡r❡ v ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ t♦ t❤❡ ❡✛❡❝t✐✈❡ ❡q✉❛t✐♦♥

  • vt − ¯

H(x, Dv) = 0 ✐♥ ]0, T[×Rn, v(0, x) = h(x) ✐♥ Rn. ✇❤❡r❡ ¯ H ✐s t❤❡ ❧✐♠✐t ♦r ❡✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-18
SLIDE 18

❚❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥

❲❡ ✐❞❡♥t✐❢② t❤❡ ❧✐♠✐t ♦r ❡✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✱ ❜② s♦❧✈✐♥❣ t❤r❡❡ ❞✐✛❡r❡♥t ❝❡❧❧ ♣r♦❜❧❡♠s ❞❡♣❡♥❞✐♥❣ ♦♥ α✳ ❲❡ ♣♦✐♥t ♦✉t t❤r❡❡ r❡❣✐♠❡s ❞❡♣❡♥❞✐♥❣ ♦♥ ❤♦✇ ❢❛st t❤❡ ✈♦❧❛t✐❧✐t② ♦s❝✐❧❧❛t❡s r❡❧❛t✐✈❡ t♦ t❤❡ ❤♦r✐③♦♥ ❧❡♥❣t❤✿    α > 2 s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ α = 2 ❝r✐t✐❝❛❧ ❝❛s❡✱ α < 2 s✉❜❝r✐t✐❝❛❧ ❝❛s❡. ✶ ■♥ ❛❧❧ t❤❡ ❝❛s❡s t❤❡ ❧✐♠✐t ❍❛♠✐❧t♦♥✐❛♥ ¯ H ✐s ❝♦♥t✐♥✉♦✉s ♦♥ Rn × Rn ❛♥❞ ❝♦♥✈❡① ✐♥ t❤❡ s❡❝♦♥❞ ✈❛r✐❛❜❧❡✳ ✷ ■♥ ❛❧❧ t❤❡ ❝❛s❡s ✇❡ ♣r♦✈✐❞❡ s♦♠❡ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❧✐♠✐t ❍❛♠✐❧t♦♥✐❛♥ ¯ H✳ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ❝❛s❡✿ α = 2✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-19
SLIDE 19

❚❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥

❲❡ ✐❞❡♥t✐❢② t❤❡ ❧✐♠✐t ♦r ❡✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✱ ❜② s♦❧✈✐♥❣ t❤r❡❡ ❞✐✛❡r❡♥t ❝❡❧❧ ♣r♦❜❧❡♠s ❞❡♣❡♥❞✐♥❣ ♦♥ α✳ ❲❡ ♣♦✐♥t ♦✉t t❤r❡❡ r❡❣✐♠❡s ❞❡♣❡♥❞✐♥❣ ♦♥ ❤♦✇ ❢❛st t❤❡ ✈♦❧❛t✐❧✐t② ♦s❝✐❧❧❛t❡s r❡❧❛t✐✈❡ t♦ t❤❡ ❤♦r✐③♦♥ ❧❡♥❣t❤✿    α > 2 s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ α = 2 ❝r✐t✐❝❛❧ ❝❛s❡✱ α < 2 s✉❜❝r✐t✐❝❛❧ ❝❛s❡. ✶ ■♥ ❛❧❧ t❤❡ ❝❛s❡s t❤❡ ❧✐♠✐t ❍❛♠✐❧t♦♥✐❛♥ ¯ H ✐s ❝♦♥t✐♥✉♦✉s ♦♥ Rn × Rn ❛♥❞ ❝♦♥✈❡① ✐♥ t❤❡ s❡❝♦♥❞ ✈❛r✐❛❜❧❡✳ ✷ ■♥ ❛❧❧ t❤❡ ❝❛s❡s ✇❡ ♣r♦✈✐❞❡ s♦♠❡ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❧✐♠✐t ❍❛♠✐❧t♦♥✐❛♥ ¯ H✳ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ❝❛s❡✿ α = 2✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-20
SLIDE 20

❚❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✿ α > 2

❲❤❡♥ n = 1✱ ¯ H(¯ x, ¯ p) = (¯ σ¯ p)2 ✇❤❡r❡ ¯ σ(¯ x) =

  • Tm σ2(¯

x, y)dµ(y) ❛♥❞ µ ✐s t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ t❤❡ ♣r♦❝❡ss Yt ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✱ ✐✳❡✳ dYt = b(Yt)dt + √ 2τ(Yt)dWt,

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-21
SLIDE 21

❚❤❡ r❛t❡ ❢✉♥❝t✐♦♥

❚❤r♦✉❣❤♦✉t t❤❡ s❡❝t✐♦♥ ✇❡ s✉♣♣♦s❡ t❤❛t σ ✐s ✉♥✐❢♦r♠❧② ♥♦♥ ❞❡❣❡♥❡r❛t❡✱ t❤❛t ✐s✱ ❢♦r s♦♠❡ ν > 0 ❛♥❞ ❢♦r ❛❧❧ x, p ∈ Rn |σT (x, y)p|2 > ν|p|2. ✭✷✮ ◆♦t❡ t❤❛t ✉♥❞❡r t❤❡ ♣r❡✈✐♦✉s ❛ss✉♠♣t✐♦♥✱ t❤❡ ❡✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥ ✐s ❝♦❡r❝✐✈❡✳ ▲❡t ¯ L ❜❡ t❤❡ ❡✛❡❝t✐✈❡ ▲❛❣r❛♥❣✐❛♥ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❡✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥ ¯ H ✈✐❛ ❝♦♥✈❡① ❞✉❛❧✐t②✱ ✐✳❡✳ ❢♦r x ∈ Rn ¯ L(x, q) = max

p∈Rn{p · q − ¯

H(x, p)}. ◆♦t❡ t❤❛t ¯ L(x, ·) ✐s ❛ ❝♦♥✈❡① ♥♦♥♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ¯ L(x, 0) = 0 ❢♦r ❛❧❧ x ∈ Rn✱ s✐♥❝❡ ¯ H(x, ·) ✐s ❝♦♥✈❡① ♥♦♥♥❡❣❛t✐✈❡ ❛♥❞ ¯ H(x, 0) = 0 ❢♦r ❛❧❧ x ∈ R✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-22
SLIDE 22

❚❤❡ r❛t❡ ❢✉♥❝t✐♦♥

❚❤❡♥ t❤❡ r❛t❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s I(x; x0, t) := inf t ¯ L(ξ(s), ˙ ξ(s)) ds

  • ξ ∈ AC(0, t), ξ(0) = x0, ξ(t) = x
  • .

■ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ✈♦❧❛t✐❧✐t② σ ❛♥❞ ♦♥ t❤❡ ❢❛st ♣r♦❝❡ss Y ε

t ❀

I ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❞r✐❢t φ ♦❢ t❤❡ ❧♦❣✲♣r✐❝❡ Xε

t ❛♥❞ ♦♥ t❤❡ ✐♥✐t✐❛❧

✈❛❧✉❡ y0 ♦❢ t❤❡ ♣r♦❝❡ss Yt✳ I s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥ ❢♦r s♦♠❡ ν, C > 0 ❛♥❞ ❛❧❧ x, x0 ∈ Rn 1 4C |x − x0|2 t ≤ I(x; x0, t) ≤ 1 4ν |x − x0|2 t ; ✐❢ σ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ x✱ ✐✳❡✳ ¯ H = ¯ H(p)✱ t❤❡ r❛t❡ ❢✉♥❝t✐♦♥ ✐s I(x; x0, t) = t¯ L x − x0 t

  • .

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-23
SLIDE 23

❚❤❡ r❛t❡ ❢✉♥❝t✐♦♥

■❢ α > 2 ❛♥❞ n = 1 ❛♥❞ ¯ H = ¯ H(p)✱ t❤❡♥ I(x; x0, t) = |x − x0|2 4¯ σ2t ✭✸✮ ✇❤❡r❡ ¯ σ =

  • Tm σ(y)2dµ(y)

❛♥❞ µ ✐s t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ t❤❡ ♣r♦❝❡ss Yt ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡s✱ ✐✳❡✳ dYt = b(Yt)dt + √ 2τ(Yt)dWt,

❘❡♠❛r❦

❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ r❛t❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ✐♥ ✭✸✮ ✐s t❤❡ s❛♠❡ ❛s t❤❡ r❛t❡ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s ♠♦❞❡❧ ✇✐t❤ ❝♦♥st❛♥t ✈♦❧❛t✐❧✐t② ¯ σ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐♥ t❤❡ ✉❧tr❛ ❢❛st r❡❣✐♠❡✱ t♦ t❤❡ ❧❡❛❞✐♥❣ ♦r❞❡r✱ ✐t ✐s t❤❡ s❛♠❡ ❛s ❛✈❡r❛❣✐♥❣ ✜rst ❛♥❞ t❤❡♥ t❛❦✐♥❣ t❤❡ s❤♦rt ♠❛t✉r✐t② ❧✐♠✐t✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-24
SLIDE 24

❆♣♣❧✐❝❛t✐♦♥s✲❖✉t✲♦❢✲t❤❡✲♠♦♥❡② ♦♣t✐♦♥ ♣r✐❝✐♥❣

▲❡t Sε

t ❜❡ t❤❡ ❛ss❡t ♣r✐❝❡✱ ❡✈♦❧✈✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ st♦❝❤❛st✐❝

❞✐✛❡r❡♥t✐❛❧ s②st❡♠ dSε

t = εξ(Sε t , Y ε t )Sε t dt +

√ 2εζ(Sε

t , Y ε t )Sε t dWt

Sε

0 = S0 ∈ R+

dY ε

t = ε1−αb(Y ε t )dt +

√ 2ε1−ατ(Y ε

t )dWt

Y ε

0 = y0 ∈ Rm,

✭✹✮ ✇❤❡r❡ α > 1✱ τ, b ❛r❡ Zm✲♣❡r✐♦❞✐❝ ✐♥ y ✇✐t❤ τ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❛♥❞ ξ : R+ × Rm → R✱ ζ : R+ × Rm → M1,r ❛r❡ ▲✐♣s❝❤✐t③ ❝♦♥t✐♥✉♦✉s ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s✱ ♣❡r✐♦❞✐❝ ✐♥ y✳ ❖❜s❡r✈❡ t❤❛t Sε

t > 0 ❛❧♠♦st s✉r❡❧② ✐❢ S0 > 0✳

❲❡ ❝♦♥s✐❞❡r ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ❝❛❧❧ ♦♣t✐♦♥ ✇✐t❤ str✐❦❡ ♣r✐❝❡ K ❛♥❞ s❤♦rt ♠❛t✉r✐t② t✐♠❡ T = ǫt✱ ❜② t❛❦✐♥❣ S0 < K ♦r x0 < log K. ❙✐♠✐❧❛r❧②✱ ❜② ❝♦♥s✐❞❡r✐♥❣ ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ♣✉t ♦♣t✐♦♥s✱ ♦♥❡ ❝❛♥ ♦❜t❛✐♥ t❤❡ s❛♠❡ ❢♦r S0 > K✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-25
SLIDE 25

❖✉t✲♦❢✲t❤❡✲♠♦♥❡② ♦♣t✐♦♥ ♣r✐❝✐♥❣

❆s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ▲❛r❣❡ ❉❡✈✐❛t✐♦♥ Pr✐♥❝✐♣❧❡✱ ✇❡ ♣r♦✈❡

❈♦r♦❧❧❛r②

❋♦r ✜①❡❞ t > 0 lim

ε→0+ ε log E

  • (Sε

t − K)+

= − inf

y>log K I (y; x0, t) .

❲❤❡♥ ζ(s, y) = ζ(y)✱ t❤❡ ♦♣t✐♦♥ ♣r✐❝❡ ❡st✐♠❛t❡ r❡❛❞s lim

ε→0+ ε log E

  • (Sε

t − K)+

= −I (log K; x0, t)

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-26
SLIDE 26

■♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②

❲❡ r❡❝❛❧❧ t❤❛t ❣✐✈❡♥ ❛♥ ♦❜s❡r✈❡❞ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ♣r✐❝❡ ❢♦r ❛ ❝♦♥tr❛❝t ✇✐t❤ str✐❦❡ ♣r✐❝❡ K ❛♥❞ ❡①♣✐r❛t✐♦♥ ❞❛t❡ T✱ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② σ ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈♦❧❛t✐❧✐t② ♣❛r❛♠❡t❡r t❤❛t ♠✉st ❣♦ ✐♥t♦ t❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s ❢♦r♠✉❧❛ t♦ ♠❛t❝❤ t❤❡ ♦❜s❡r✈❡❞ ♣r✐❝❡✳ ❲❡ ❝♦♥s✐❞❡r ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥✱ ✇✐t❤ str✐❦❡ ♣r✐❝❡ K✱ ❛♥❞ ✇❡ ❞❡♥♦t❡ ❜② σε(t, log K, x0) t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-27
SLIDE 27

❆♣♣❧✐❝❛t✐♦♥s✲❆♥ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ❢♦r ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②

❆s ❛ ❢✉rt❤❡r ❛♣♣❧✐❝❛t✐♦♥✱ ✇❡ ♣r♦✈❡

❈♦r♦❧❧❛r②

lim

ε→0+ σ2 ε(t, log K, x0) =

(log K − x0)2 2 infy>log K I(y; x0, t)t. ◆♦t❡ t❤❛t t❤❡ ✐♥✜♠✉♠ ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✐s ❛❧✇❛②s ♣♦s✐t✐✈❡ ❜② t❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ S0 ❛♥❞ ❜② t❤❡ ❣r♦✇t❤ ♦❢ t❤❡ r❛t❡ ❢✉♥❝t✐♦♥✳

❘❡♠❛r❦

❲❤❡♥ α > 2✱ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ✐s ¯ σ t❤❛t ✐s ¯ σ =

  • Tm σ2(y)dµ(y).

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-28
SLIDE 28

❚❤❛♥❦ ②♦✉ ❢♦r t❤❡ ❛tt❡♥t✐♦♥✦

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-29
SLIDE 29

▼❛✐♥ r❡s✉❧t✲❈♦♥✈❡r❣❡♥❝❡ ❜② P❉❊ ♠❡t❤♦❞s

❚❤❡♦r❡♠

▲❡t h ❜❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❜♦✉♥❞❡❞✳ ❚❤❡♥ vε(x, y, t) = ε log Ee

h(Xε t ) ε

→ v(x, t) ❧♦❝❛❧❧② ✉♥✐❢♦r♠❧② ✐♥ y ✇❤❡r❡ v ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ t♦ t❤❡ ❡✛❡❝t✐✈❡ ❡q✉❛t✐♦♥

  • vt − ¯

H(x, Dv) = 0 ✐♥ ]0, T[×Rn, v(0, x) = h(x) ✐♥ Rn. ✇❤❡r❡ ¯ H ✐s t❤❡ ❧✐♠✐t ♦r ❡✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✳

❘❡♠❛r❦

vε ✐s ✉♥✐❢♦r♠❧② ❜♦✉♥❞❡❞ ✐♥ ε✳ ❚♦ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✇❡ ✉s❡ r❡❧❛①❡❞ s❡♠✐❧✐♠✐ts ❇❛r❧❡s✲P❛rt❤❛♠❡ ♣r♦❝❡❞✉r❡ ❛♥❞ t❤❡ t❡❝❤♥✐q✉❡s ✉s❡❞ t♦ tr❡❛t s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♣r♦❜❧❡♠s ✐♥ ❆❧✈❛r❡③✱ ❇❛r❞✐ ✭✷✵✵✸✮ ❛❞❛♣t❡❞ t♦ r❡❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥s ♦❢ s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♣r♦❜❧❡♠s ✐♥ ❆❧✈❛r❡③✱ ❇❛r❞✐✱ ▼❛r❝❤✐ ✭✷✵✵✼✮✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-30
SLIDE 30

❚❤❡ ❝❡❧❧ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✱ α = 2

P❧✉❣❣✐♥ ✐♥ t❤❡ ❡q✉❛t✐♦♥ t❤❡ ❢♦r♠❛❧ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ vε(t, x, y) = v0(t, x) + εw(t, x, y). ✇❡ ♦❜t❛✐♥ v0

t −|σT Dxv0|2−2(τσT Dxv0)·Dyw−b·Dyw−|τ T Dyw|2−tr(ττ T D2 yyw) = O(ε).

Pr♦♣♦s✐t✐♦♥

❋♦r ❛♥② ✜①❡❞ (¯ x, ¯ p)✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ¯ H(¯ x, ¯ p) ❢♦r ✇❤✐❝❤ t❤❡ ✉♥✐❢♦r♠❧② ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥ ✇✐t❤ q✉❛❞r❛t✐❝ ♥♦♥❧✐♥❡❛r✐t② ✐♥ t❤❡ ❣r❛❞✐❡♥t ¯ H(¯ x, ¯ p) − |σT ¯ p|2 −

  • 2τσT ¯

p + b

  • · Dyw(y) − |τ T Dyw(y)|2 − tr(ττ T D2

yyw(y)) = 0,

❤❛s ❛ ♣❡r✐♦❞✐❝ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ w✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-31
SLIDE 31

❚❤❡ ❝❡❧❧ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✱ α = 2

P❧✉❣❣✐♥ ✐♥ t❤❡ ❡q✉❛t✐♦♥ t❤❡ ❢♦r♠❛❧ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ vε(t, x, y) = v0(t, x) + εw(t, x, y). ✇❡ ♦❜t❛✐♥ v0

t −|σT Dxv0|2−2(τσT Dxv0)·Dyw−b·Dyw−|τ T Dyw|2−tr(ττ T D2 yyw) = O(ε).

Pr♦♣♦s✐t✐♦♥

❋♦r ❛♥② ✜①❡❞ (¯ x, ¯ p)✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ¯ H(¯ x, ¯ p) ❢♦r ✇❤✐❝❤ t❤❡ ✉♥✐❢♦r♠❧② ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥ ✇✐t❤ q✉❛❞r❛t✐❝ ♥♦♥❧✐♥❡❛r✐t② ✐♥ t❤❡ ❣r❛❞✐❡♥t ¯ H(¯ x, ¯ p) − |σT ¯ p|2 −

  • 2τσT ¯

p + b

  • · Dyw(y) − |τ T Dyw(y)|2 − tr(ττ T D2

yyw(y)) = 0,

❤❛s ❛ ♣❡r✐♦❞✐❝ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ w✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-32
SLIDE 32

¯ H✿ ❋✐rst ❘❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛

¯ H ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ t❤r♦✉❣❤ st♦❝❤❛st✐❝ ❝♦♥tr♦❧ ❛s ¯ H(¯ x, ¯ p) = lim

δ→0 sup β(·)

δE ∞

  • |σ(¯

x, Zt)T ¯ p|2 − |β(t)|2 e−δtdt | Z0 = z

  • ❛♥❞

¯ H(¯ x, ¯ p) = lim

t→∞ sup β(·)

1 t E t (|σT (¯ x, Zs)¯ p|2 − |β(s)|2)ds | Z0 = z

  • ,

✇❤❡r❡ β(·) ✐s ❛♥ ❛❞♠✐ss✐❜❧❡ ❝♦♥tr♦❧ ♣r♦❝❡ss t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ Rr ❢♦r t❤❡ st♦❝❤❛st✐❝ ❝♦♥tr♦❧ s②st❡♠ dZt =

  • b(Zt) + 2τ(Zt)σT (¯

x, Zt)¯ p − 2τ(Zt)β(t)

  • dt +

√ 2τ(Zt)dWt; ✭✺✮

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-33
SLIDE 33

¯ H✿ ❙❡❝♦♥❞ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛

▼♦r❡♦✈❡r ¯ H =

  • Tm
  • |σ(¯

x, z)T ¯ p|2 − |τ(z)T Dw(z)|2 dµ(z), ✇❤❡r❡ w = w(·; ¯ x, ¯ p) ✐s t❤❡ s♠♦♦t❤ s♦❧✉t✐♦♥ t♦ ¯ H(¯ x, ¯ p) − tr(ττ T D2

yyw) − |τ T Dyw|2+

− (2τσT ¯ p + b) · Dyw − |σT ¯ p|2 = 0 ✐♥ Rm ❛♥❞ µ = µ(·; ¯ x, ¯ p) ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ t♦r✉s Tm ♦❢ t❤❡ ♣r♦❝❡ss ✭✺✮ ✇✐t❤ t❤❡ ❢❡❡❞❜❛❝❦ β(z) = −τ T (z)Dw(z)✱ ✐✳❡✳ dZt =

  • b(Zt) + 2τ(Zt)σT (¯

x, Zt)¯ p + 2τ(Zt)τ T (Zt)Dw(Zt)

  • dt+

√ 2τ(Zt)dWt.

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-34
SLIDE 34

¯ H✿ ❚❤✐r❞ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛

▼♦r❡♦✈❡r ¯ H(¯ x, ¯ p) = lim

t→∞

1 t log E

  • e

t

0 |σT (¯

x,Ys)¯ p|2 ds | Y0 = y

  • ,

✇❤❡r❡ Yt ✐s t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss ❞❡✜♥❡❞ ❜② dYt =

  • b(Yt) + 2τ(Yt)σT (¯

x, Yt)¯ p

  • dt +

√ 2τ(Yt)dWt. ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢✿ ❚❛❦❡ v = v(t, x; ¯ x, ¯ p) ❛ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥ ♦❢ t❤❡ t✲❝❡❧❧ ♣r♦❜❧❡♠ ❛♥❞ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ f(t, y) = ev(t,y)✳ ❚❤❡♥ f s♦❧✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥

  • ∂f

∂t − f|σT ¯

p|2 − (2τσT ¯ p + b) · Df − tr(ττ T D2f) = 0 ✐♥ (0, ∞) × Rm f(0, z) = 1 ✐♥ Rm. ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡ ✉s✐♥❣ t❤❡ ❋❡②♥❛♠✲❑❛❝ ❢♦r♠✉❧❛✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-35
SLIDE 35

¯ H✿ ❚❤✐r❞ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛

▼♦r❡♦✈❡r ¯ H(¯ x, ¯ p) = lim

t→∞

1 t log E

  • e

t

0 |σT (¯

x,Ys)¯ p|2 ds | Y0 = y

  • ,

✇❤❡r❡ Yt ✐s t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss ❞❡✜♥❡❞ ❜② dYt =

  • b(Yt) + 2τ(Yt)σT (¯

x, Yt)¯ p

  • dt +

√ 2τ(Yt)dWt. ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢✿ ❚❛❦❡ v = v(t, x; ¯ x, ¯ p) ❛ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥ ♦❢ t❤❡ t✲❝❡❧❧ ♣r♦❜❧❡♠ ❛♥❞ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ f(t, y) = ev(t,y)✳ ❚❤❡♥ f s♦❧✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥

  • ∂f

∂t − f|σT ¯

p|2 − (2τσT ¯ p + b) · Df − tr(ττ T D2f) = 0 ✐♥ (0, ∞) × Rm f(0, z) = 1 ✐♥ Rm. ❛♥❞ ✇❡ ❝♦♥❝❧✉❞❡ ✉s✐♥❣ t❤❡ ❋❡②♥❛♠✲❑❛❝ ❢♦r♠✉❧❛✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-36
SLIDE 36

❚❤❡ ❝❡❧❧ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✱ α > 2

P❧✉❣❣✐♥❣ t❤❡ ❢♦r♠❛❧ ❛s②♠♣t♦t✐❝ ❡①♣❛♥s✐♦♥ vε(t, x, y) = v0(t, x) + εα−1w(t, x, y) ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✇❡ ❣❡t v0

t = |σT Dxv0|2 + b · Dyw + tr(ττ T D2 yyw) + O(ε).

Pr♦♣♦s✐t✐♦♥

❋♦r ❡❛❝❤ (¯ x, ¯ p) ✜①❡❞✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♥st❛♥t ¯ H(¯ x, ¯ p) s✉❝❤ t❤❛t t❤❡ ❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r ✉♥✐❢♦r♠❧② ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥ ¯ H(¯ x, ¯ p) − tr(ττ(y)T D2

yywδ(y)) − b(y) · Dywδ(y) − |σ(¯

x, y)T ¯ p|2 = 0 ✐♥ Rm, ❤❛s ❛ ♣❡r✐♦❞✐❝ s♠♦♦t❤ s♦❧✉t✐♦♥✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-37
SLIDE 37

¯ H✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛

¯ H ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ¯ H =

  • Tm|σ(¯

x, y)T ¯ p|2 dµ(y), ✇❤❡r❡ µ ✐s t❤❡ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ t♦r✉s Tm ♦❢ t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss dYt = b(Yt)dt + √ 2τ(Yt)dWt, t❤❛t ✐s✱ t❤❡ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥ ♦❢ −

  • i,j

∂2 ∂yi∂yj ((ττ T )ij(y))µ +

  • i

∂ ∂yi (bi(y))µ = 0 ✐♥ Rm, ✇✐t❤

  • Tnµ(y) dy = 1✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-38
SLIDE 38

¯ H✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛

❲❤❡♥ n = 1✱ ¯ H(¯ x, ¯ p) = (¯ σ¯ p)2 ✇❤❡r❡ ¯ σ(¯ x) =

  • Tm σ2(¯

x, y)dµ(y) ❛♥❞ µ ✐s t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ t❤❡ ♣r♦❝❡ss Yt ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✱ ✐✳❡✳ dYt = b(Yt)dt + √ 2τ(Yt)dWt,

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-39
SLIDE 39

❚❤❡ ❝❡❧❧ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❡❢❢❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥✱ α < 2

❲❡ ♣❧✉❣ ✐♥ t❤❡ ❡q✉❛t✐♦♥ t❤❡ ❢♦r♠❛❧ ❛s②♠♣t❤♦t✐❝ ❡①♣❛♥s✐♦♥ vε(t, x, y) = v0(t, x) + ε

α 2 w(t, x, y).

❛♥❞ ✇❡ ♦❜t❛✐♥ v0

t = |σT Dxv0|2 + 2(τσT Dxv0) · Dyw + |τ T Dyw|2 + O(ε).

Pr♦♣♦s✐t✐♦♥

❋♦r ❛♥② ✜①❡❞ (¯ x, ¯ p)✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦♥st❛♥t ¯ H(¯ x, ¯ p) s✉❝❤ t❤❛t t❤❡ ✜rst ♦r❞❡r ❝♦❡r❝✐✈❡ ❡q✉❛t✐♦♥ ¯ H(¯ x, ¯ p) − |τ T (y)Dyw(y) + σT (¯ x, y)¯ p|2 = 0 ✐♥ Rm ❛❞♠✐ts ❛ ✭▲✐♣s❝❤✐t③ ❝♦♥t✐♥✉♦✉s✮ ♣❡r✐♦❞✐❝ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ w✳

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-40
SLIDE 40

¯ H✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛s

¯ H s❛t✐s✜❡s ¯ H(¯ x, ¯ p) = lim

δ→0 sup β(·)

δ +∞

  • |σ(¯

x, y(t))T ¯ p|2 − |β(t)|2 e−δt dt, ✇❤❡r❡ β(·) ✈❛r✐❡s ♦✈❡r ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ Rr✱ y(·) ✐s t❤❡ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝♦♥tr♦❧ s②st❡♠

  • ˙

y(t) = 2τ(y(t))σT (¯ x, y(t))¯ p − 2τ(y(t))β, t > 0, y(0) = y ❛♥❞ t❤❡ ❧✐♠✐t ✐s ✉♥✐❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ y ♦❢ t❤❡ s②st❡♠❀ ▼♦r❡♦✈❡r ✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ τσT = 0 ♦❢ ♥♦♥✲❝♦rr❡❧❛t✐♦♥s ❛♠♦♥❣ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✇❤✐t❡ ♥♦✐s❡ ❛❝t✐♥❣ ♦♥ t❤❡ s❧♦✇ ❛♥❞ t❤❡ ❢❛st ✈❛r✐❛❜❧❡s ✐♥ t❤❡ s②st❡♠✱ ✇❡ ❤❛✈❡ ¯ H(x, p) = max

y∈Rm |σT (x, y)p|2.

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s

slide-41
SLIDE 41

¯ H✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛s

¯ H s❛t✐s✜❡s ¯ H(¯ x, ¯ p) = lim

δ→0 sup β(·)

δ +∞

  • |σ(¯

x, y(t))T ¯ p|2 − |β(t)|2 e−δt dt, ✇❤❡r❡ β(·) ✈❛r✐❡s ♦✈❡r ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ Rr✱ y(·) ✐s t❤❡ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❝♦♥tr♦❧ s②st❡♠

  • ˙

y(t) = 2τ(y(t))σT (¯ x, y(t))¯ p − 2τ(y(t))β, t > 0, y(0) = y ❛♥❞ t❤❡ ❧✐♠✐t ✐s ✉♥✐❢♦r♠ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥ y ♦❢ t❤❡ s②st❡♠❀ ▼♦r❡♦✈❡r ✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ τσT = 0 ♦❢ ♥♦♥✲❝♦rr❡❧❛t✐♦♥s ❛♠♦♥❣ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✇❤✐t❡ ♥♦✐s❡ ❛❝t✐♥❣ ♦♥ t❤❡ s❧♦✇ ❛♥❞ t❤❡ ❢❛st ✈❛r✐❛❜❧❡s ✐♥ t❤❡ s②st❡♠✱ ✇❡ ❤❛✈❡ ¯ H(x, p) = max

y∈Rm |σT (x, y)p|2.

▲❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s