Dynamic competition and intellectual property rights in a model of product development

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Munich Personal RePEc Archive

Dynamic competition and intellectual property rights in a model of product development

Billette de Villemeur, Etienne and Ruble, Richard and Versaevel, Bruno

LEM, Université de Lille, F-59655, France., Emlyon business School, F-69134 (primary affiliation) and CNRS, GATE Lyon Saint-Etienne, Ecully F-69130, France., Emlyon business School, F-69134 (primary affiliation) and CNRS, GATE Lyon Saint-Etienne, Ecully F-69130, France.

September 2017

Online at https://mpra.ub.uni-muenchen.de/90355/

MPRA Paper No. 90355, posted 05 Dec 2018 09:11 UTC

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❉②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ ❛♥❞ ✐♥t❡❧❧❡❝t✉❛❧ ♣r♦♣❡rt② r✐❣❤ts ✐♥ ❛ ♠♦❞❡❧

♦❢ ♣r♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t

❊t✐❡♥♥❡ ❇✐❧❧❡tt❡ ❞❡ ❱✐❧❧❡♠❡✉r✱ ❘✐❝❤❛r❞ ❘✉❜❧❡ ❛♥❞ ❇r✉♥♦ ❱❡rs❛❡✈❡❧^∗

◆♦✈❡♠❜❡r ✶✷✱ ✷✵✶✽

❆❜str❛❝t

❲❡ st✉❞② ✐♥♥♦✈❛t✐♦♥ t✐♠✐♥❣ ❛♥❞ s♦❝✐❛❧❧② ♦♣t✐♠❛❧ ✐♥t❡❧❧❡❝t✉❛❧ ♣r♦♣❡rt② r✐❣❤ts ✭■P❘s✮ ✇❤❡♥

✜r♠s ❢❛❝✐♥❣ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t② ✐♥✈❡st str❛t❡❣✐❝❛❧❧② ✐♥ ♣r♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t✳ ■❢ ❞❡♠❛♥❞ ❣r♦✇t❤

❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ ❤✐❣❤✱ ❛ttr✐t✐♦♥ ♦❝❝✉rs ❛♥❞ ■P❘s s❤♦✉❧❞ ❡♥s✉r❡ t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ ❛tt❛✐♥s

❛ ❧♦✇❡r ❜♦✉♥❞ ✇❡ ✐❞❡♥t✐❢②✳ ■❢ ❞❡♠❛♥❞ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ ❧♦✇ t❤❡♥ ♣r♦✈✐❞❡❞ t❤❛t t❤❡

♣r✐✈❛t❡ ✐♠✐t❛t✐♦♥ ✐♥❝❡♥t✐✈❡ ✐s s♦❝✐❛❧❧② ❡①❝❡ss✐✈❡✱ ■P❘s s❤♦✉❧❞ s❡t t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ ❤✐❣❤

❡♥♦✉❣❤ t♦ ✐♥❞✉❝❡ ♣r❡❡♠♣t✐♦♥✱ ❛♥❞ ♣♦ss✐❜❧② ✇✐♥♥❡r✲t❛❦❡✲❛❧❧ ♣r❡❡♠♣t✐♦♥✳ ▼♦r❡♦✈❡r✱ t❤❡ ✇❡❧❢❛r❡

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

♣r❡❞❡t❡r♠✐♥❡❞✱ ✐❧❧✉str❛t✐♥❣ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ❢♦st❡r✐♥❣ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥✳ ■♥ ❡①t❡♥s✐♦♥s

✇❡ s❤♦✇ ❤♦✇ ✜r♠s ❜❡♥❡✜t ❢r♦♠ ♦♣❡♥ st❛♥❞❛r❞s✱ t❛❦❡♦✈❡rs ❤❛✈❡ ❛♠❜✐❣✉♦✉s ✇❡❧❢❛r❡ ❡✛❡❝ts ❛♥❞

s✐♠♣❧❡ ❧✐❝❡♥s✐♥❣ s❝❤❡♠❡s ❛r❡ ✇❡❧❢❛r❡ ✐♠♣r♦✈✐♥❣✳

❏❊▲ ❝❧❛ss✐✜❝❛t✐♦♥✿ ●✸✶✱ ▲✶✸✱ ❖✸✸

❑❡②✇♦r❞s✿ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥✱ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥✱ ♣❛t❡♥t ♣♦❧✐❝②✱ ✇✐♥♥❡r✲t❛❦❡✲❛❧❧ ♣r❡❡♠♣✲

t✐♦♥

∗❇✐❧❧❡tt❡ ❞❡ ❱✐❧❧❡♠❡✉r✿ ▲❊▼✱ ❯♥✐✈❡rs✐té ❞❡ ▲✐❧❧❡✱ ❋✲✺✾✻✺✺✱ ❋r❛♥❝❡✳ ❘✉❜❧❡ ❛♥❞ ❱❡rs❛❡✈❡❧✿ ❊▼ ▲②♦♥ ❇✉s✐♥❡ss

❙❝❤♦♦❧✱ ●❆❚❊ ▲✲❙❊ ❯▼❘ ✺✽✷✹✱ ✻✾✶✸✹ ❊❝✉❧❧②✱ ❋r❛♥❝❡✳ ❈♦rr❡s♣♦♥❞✐♥❣ ❛✉t❤♦r✿ r✉❜❧❡❅❡♠✲❧②♦♥✳❝♦♠✳ ❲❡ ❛r❡ ❣r❛t❡❢✉❧

t♦ t✇♦ ❛♥♦♥②♠♦✉s r❡❢❡r❡❡s✱ ❳❛✈✐❡r ❇♦✉t✐♥✱ ❨❛♥♥ ❇r❛♦✉❡③❡❝✱ ❇❡♥♦ît ❈❤❡✈❛❧✐❡r✲❘♦✐❣♥❛♥t✱ ❉❛✈✐❞ ❊♥❝❛♦✉❛✱ ❚❤✐❧♦

▼❡②❡r✲❇r❛♥❞✐s✱ ❉❡❛♥ P❛①s♦♥✱ ❘é❣✐s ❘❡♥❛✉❧t✱ ❏❛❝❝♦ ❚❤✐❥ss❡♥✱ ▼✐❤❦❡❧ ❚♦♠❜❛❦✱ ❛♥❞ ▲❡♥♦s ❚r✐❣❡♦r❣✐s ❛s ✇❡❧❧ ❛s s❡♠✐♥❛r ♣❛rt✐❝✐♣❛♥ts ❛t P❛r✐s✲❉❛✉♣❤✐♥❡ ✉♥✐✈❡rs✐t②✱ ■❊❙❊● ❜✉s✐♥❡ss s❝❤♦♦❧✱ ▲▼❯ ❛♥❞ ❚❯ ▼ü♥❝❤❡♥✱ ❛♥❞ ♣❛rt✐❝✐♣❛♥ts

❛t t❤❡ ✷✵✶✽ ❊❊❆ ♠❡❡t✐♥❣✱ ✷✵✶✻ ❆❋❙❊ ♠❡❡t✐♥❣ ❛♥❞ ◆♦rt❤✇❡st❡r♥ ❆♥♥✉❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ■♥♥♦✈❛t✐♦♥ ❊❝♦♥♦♠✐❝s✱

✷✵✶✺ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❘❡❛❧ ❖♣t✐♦♥s ❛♥❞ ✷✵✶✹ ❊❆❘■❊ ♠❡❡t✐♥❣ ❢♦r ✈❛❧✉❛❜❧❡ s✉❣❣❡st✐♦♥s✳

✶

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✶ ■♥tr♦❞✉❝t✐♦♥

❲❤❡♥ ❞❡✈❡❧♦♣✐♥❣ ❛♥ ✐♥✈❡♥t✐♦♥ ✐♥t♦ ❛ ❝♦♠♠❡r❝✐❛❧ ♣r♦❞✉❝t r❡q✉✐r❡s s✉❜st❛♥t✐❛❧ r❡s♦✉r❝❡s✱ ♦♥❧② ❛

❢❡✇ ✜r♠s ♠❛② ❜❡ ❝♦♥t❡♥❞❡rs ❢♦r ♣♦s✐t✐♦♥s ❛s ✜rst✲♠♦✈❡r ♦r s❡❝♦♥❞ ❡♥tr❛♥t ✐♥ ❛♥ ✐♥❞✉str②✳ ❆❧♦♥❣

✇✐t❤ ♦t❤❡r ❦❡② ♠❛r❦❡t ❝❤❛r❛❝t❡r✐st✐❝s ❧✐❦❡ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t②✱ t❤❡ ❝♦st ♦❢ ✐♥♥♦✈❛t✐♦♥ r❡❧❛t✐✈❡ t♦

✐♠✐t❛t✐♦♥ ✐s t❤❡♥ ❛ ❦❡② ❞❡t❡r♠✐♥❛♥t ♦❢ ✐♥✈❡st♠❡♥t str❛t❡❣✐❡s ❛♥❞ ❤❡♥❝❡ ♦❢ t❤❡ t✐♠✐♥❣ ♦❢ ♣r♦❞✉❝t

✐♥tr♦❞✉❝t✐♦♥s✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❡①✐sts s✉❜st❛♥t✐❛❧ ✈❛r✐❛t✐♦♥ ❢r♦♠ ♦♥❡ ♠❛r❦❡t t♦ ❛♥♦t❤❡r ❜♦t❤ ✐♥

❞②♥❛♠✐❝ ❝❤❛r❛❝t❡r✐st✐❝s ❛♥❞ ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ✐♠✐t❛t✐♦♥✱ ✇❤✐❝❤ t❤❡♠s❡❧✈❡s ♠❛② ❤✐♥❣❡ ♦♥ t❤❡ ✉s❡

t❤❛t ✐♥♥♦✈❛t✐♥❣ ✜r♠s ♠❛❦❡ ♦❢ ✐♥t❡❧❧❡❝t✉❛❧ ♣r♦♣❡rt② r✐❣❤ts ✭■P❘s✮✳ ❋✐r♠s ❛❝❝♦r❞✐♥❣❧② r❡s♣♦♥❞ t♦

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

❝❤♦✐❝❡ ❜② r❡❣✉❧❛t♦rs ♠✉st t❤❡r❡❢♦r❡ ❛❝❝♦✉♥t ❢♦r t❤❡ ❞②♥❛♠✐❝s ♦❢ ✐♥❞✉str② ✐♥✈❡st♠❡♥ts✳

❖♥❡ ♠❛r❦❡t ✐♥ ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦❢ ✐♠✐t❛t✐♦♥ ❛♥❞ ■P❘s ❤❛✈❡ ♣❧❛②❡❞ ❛ ♣r♦♠✐♥❡♥t r♦❧❡ ✐s t❤❡

s♠❛rt♣❤♦♥❡ ♠❛r❦❡t✳ ■♥ t❤❡ ❞❡❝❛❞❡ ❢♦❧❧♦✇✐♥❣ t❤❡ r❡❧❡❛s❡ ♦❢ t❤❡ ✜rst ✐P❤♦♥❡ ✐♥ ✷✵✵✼✱ ❛ ❜r♦❛❞ ✐♥t❡r✲

♣r❡t❛t✐♦♥ ♦❢ ■P❘s ❛❧❧♦✇❡❞ t❤❡ ❡rst✇❤✐❧❡ ❧❡❛❞❡r ❆♣♣❧❡ t♦ ♣❛t❡♥t ♥✉♠❡r♦✉s ❛s♣❡❝ts ♦❢ ✐ts ✐♥♥♦✈❛t✐✈❡

♣r♦❞✉❝ts ❛♥❞ str❛t❡❣✐❝❛❧❧② ❡♥❣❛❣❡ ✐♥ ❧✐t✐❣❛t✐♦♥ ❛❣❛✐♥st ❛❧❧❡❣❡❞ ✐♠✐t❛t✐♦♥ ❜② ✐ts ❝❤✐❡❢ r✐✈❛❧✱ ❙❛♠s✉♥❣

✭●❛❧❛ss♦ ❬✶✺❪✮✳ ❇② ✇❛② ♦❢ ❝♦♥tr❛st✱ ✐♥ t❤❡ ♠♦r❡ ✉♥❝❡rt❛✐♥ ♠❛r❦❡t ❢♦r ❧✐t❤✐✉♠ ✐♦♥ ❜❛tt❡r✐❡s✱ ❡❧❡❝tr✐❝✲

✈❡❤✐❝❧❡ t❡❝❤♥♦❧♦❣② ♣✐♦♥❡❡r ❚❡s❧❛ ▼♦t♦rs ❛♥♥♦✉♥❝❡❞ ✐♥ ✷✵✶✹ t❤❛t ✐t ✇♦✉❧❞ ♦♣❡♥ ✐ts ❜❛s❡ ♦❢ s❡✈❡r❛❧

❤✉♥❞r❡❞s ♦❢ ❛♣♣r♦✈❡❞ ♣❛t❡♥ts t♦ ❝♦♠♣❡t✐t♦rs ❛t ♥♦ ❝♦st✱ ❛♥ ❛♣♣r♦❛❝❤ ✇❤✐❝❤ ✐s ♥♦t s♦ r❛r❡ ❤✐st♦r✐✲

❝❛❧❧② ❢♦r t❡❝❤♥♦❧♦❣✐❡s ✐♥ ❡❛r❧② st❛❣❡s ♦❢ ❞❡✈❡❧♦♣♠❡♥t ✭❇❡ss❡♥ ❬✶❪✮✳ ■♥ t❤❡ ♣❤❛r♠❛❝❡✉t✐❝❛❧ ✐♥❞✉str②✱

❝♦♥s✐❞❡r❛❜❧❡ ✈❛r✐❛t✐♦♥ ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ❛❝r♦ss ❜✉s✐♥❡ss s❡❣♠❡♥ts ✐♥ ❜♦t❤ ❞②♥❛♠✐❝ ❝❤❛r❛❝t❡r✐st✐❝s

♦❢ ♣r♦❞✉❝t ♠❛r❦❡ts ❛♥❞ ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ✐♠✐t❛t✐♦♥✱ s❤❛♣✐♥❣ t❤❡ str❛t❡❣✐❡s ♦❢ t❤❡ ✐♥❞✉str②✬s

♠❛❥♦r ✜r♠s✳ ●❧❛①♦❙♠✐t❤❑❧✐♥❡ ❢♦r ✐♥st❛♥❝❡ ❛♥♥♦✉♥❝❡❞ ✐♥ ✷✵✶✻ t❤❛t ✐t ✇♦✉❧❞ ♣✉rs✉❡ ❛ ✏❣r❛❞✉❛t❡❞

❛♣♣r♦❛❝❤✑ t♦ ■P❘s✱ ✉♥❞❡r ✇❤✐❝❤ ✐t ✇♦✉❧❞ ❞❡❢❡♥❞ ✐ts ♣❛t❡♥ts ✐♥ ❤✐❣❤✲✐♥❝♦♠❡ ❡❝♦♥♦♠✐❡s ❜✉t ✇♦✉❧❞

❛❧❧♦✇ ❣❡♥❡r✐❝ ♠❛♥✉❢❛❝t✉r❡rs t♦ ♣r♦❞✉❝❡ ❧♦✇✲❝♦st ❝♦♣✐❡s ♦❢ ✐ts ❞r✉❣s ✐♥ ❧♦✇✲✐♥❝♦♠❡ ♠❛r❦❡ts ✇❤❡r❡

❞❡♠❛♥❞ t②♣✐❝❛❧❧② ❣r♦✇s ❢❛st❡r ❜✉t ✐s ❛❧s♦ ♠♦r❡ ✈♦❧❛t✐❧❡ ✭P❤✐❧❧✐♣s ❬✷✽❪✮✳ ❙❡✈❡r❛❧ r❡❝❡♥t ❡♠♣✐r✐❝❛❧

✐♥✈❡st✐❣❛t✐♦♥s ❤❛✈❡ st✉❞✐❡❞ ✐♥♥♦✈❛t✐♦♥ t✐♠✐♥❣ ✐♥ t❤✐s ✐♥❞✉str②✱ s✉❝❤ ❛s ❲❛❣♥❡r ❛♥❞ ❲❛❦❡♠❛♥ ❬✸✻❪✱

✇❤♦ ✜♥❞ t❤❛t t❤❡ ❝♦st ♦❢ ✏✐♥✈❡♥t✐♥❣ ❛r♦✉♥❞✑ ✭❛s ♠❡❛s✉r❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ♣❛t❡♥t ❛♣♣❧✐❝❛t✐♦♥s✮

❢♦r ❛ ❣✐✈❡♥ ❞r✉❣ ❝❛♥❞✐❞❛t❡ ✐s ♣♦s✐t✐✈❡❧② ❝♦rr❡❧❛t❡❞ ✇✐t❤ t❤❡ s♣❡❡❞ ♦❢ ♣r♦❞✉❝t ❧❛✉♥❝❤✱ ❛♥❞ ❈♦❝❦✲

❜✉r♥✱ ▲❛♥❥♦✉✇ ❛♥❞ ❙❝❤❛♥❦❡r♠❛♥ ❬✼❪✱ ✇❤♦ ✜♥❞ t❤❛t ♠♦r❡ ❡①t❡♥s✐✈❡ ♣❛t❡♥t ♣r♦t❡❝t✐♦♥ s✉❜st❛♥t✐❛❧❧②

❛❝❝❡❧❡r❛t❡s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ♥❡✇ ❞r✉❣s ♦✈❡r ❛ ❜r♦❛❞ s❛♠♣❧❡ ♦❢ ❝♦✉♥tr✐❡s✳ ❚❤❡ ♣❤❛r♠❛❝❡✉t✐❝❛❧

s❡❝t♦r ✐s ❛♥ ✐♠♣♦rt❛♥t ❛♥❞ ✇❡❧❧✲❞♦❝✉♠❡♥t❡❞ ✐♥❞✉str② ❡①❛♠♣❧❡ ✇✐t❤✐♥ ✇❤✐❝❤ t❤❡r❡ ✐s s✉❜st❛♥t✐❛❧

✈❛r✐❛t✐♦♥ ✐♥ ❜♦t❤ ■P❘ ❛♥❞ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t② ♣❛r❛♠❡t❡rs s♦ t❤❛t ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ r❡❢❡rr✐♥❣ t♦ ✐t t❤r♦✉❣❤♦✉t t❤❡ ❛rt✐❝❧❡ ✐♥ ❛ s❡r✐❡s ♦❢ ❞❡t❛✐❧❡❞ ❢♦♦t♥♦t❡s t♦ t❤❡ ♠❛✐♥ t❡①t✳

■♥ ❛ ♠♦❞❡❧ ♦❢ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ t♦ ❞❡✈❡❧♦♣ ❛ ♥❡✇ ♣r♦❞✉❝t ✐♥ t❤❡ ❢❛❝❡ ♦❢ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t②✱

✷

(4)

✇❡ ♣r♦♣♦s❡ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❡✛❡❝t ♦❢ t❤❡ r❡❧❛t✐✈❡ ❝♦sts ♦❢ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ ✐♠✐t❛t✐♦♥ ♦♥ t❤❡

✐♥✈❡st♠❡♥t str❛t❡❣✐❡s ♦❢ ✜r♠s ✇❤❡♥ t❤❡✐r r♦❧❡s ❛s ✐♥♥♦✈❛t♦r ♦r ✐♠✐t❛t♦r ❛r❡ ❡♥❞♦❣❡♥♦✉s ❛♥❞ st✉❞② t❤❡ r❡❣✉❧❛t♦r✬s ❝❤♦✐❝❡ ♦❢ ♦♣t✐♠❛❧ ■P❘ ❧❡✈❡❧s✳ ❇② ✐❞❡♥t✐❢②✐♥❣ t❤❡ r♦❧❡ ♣❧❛②❡❞ ❜② t❤❡ ❞r✐❢t ❛♥❞

✈♦❧❛t✐❧✐t② ♦❢ ♣r♦❞✉❝t ♠❛r❦❡t ❞❡♠❛♥❞ ♦♥ t❤❡ t✐♠✐♥❣ ♦❢ ✐♥♥♦✈❛t✐✈❡ ❛♥❞ ✐♠✐t❛t✐✈❡ ✐♥✈❡st♠❡♥ts ❛♥❞

❤❡♥❝❡ ♦♥ ❡❝♦♥♦♠✐❝ ✇❡❧❢❛r❡✱ ✇❡ ❝♦♥tr✐❜✉t❡ ♥♦✈❡❧ ✐♥s✐❣❤ts ❝♦♥❝❡r♥✐♥❣ ♠✐♥✐♠✉♠ ■P❘ ❧❡✈❡❧s✱ ❝♦♥❞✐t✐♦♥s

❢♦r str♦♥❣ ■P❘s t♦ ❜❡ ♦♣t✐♠❛❧✱ ❛♥❞ t❤❡ r♦❧❡ ♦❢ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ✜r♠s✳

■❢ ✐♥♥♦✈❛t✐♦♥ ❤❛s ♣♦s✐t✐✈❡ s♣✐❧❧♦✈❡rs ❢♦r ❛♥ ✐♠✐t❛t♦r ❛ttr✐t✐♦♥ ♠❛② ❛r✐s❡✱ ❛♥❞ ✐t ✐s ❛❧❧ t❤❡ ♠♦r❡

❧✐❦❡❧② ✇❤❡♥ ❞❡♠❛♥❞ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ ❤✐❣❤✳ ❆ s♦❝✐❛❧❧② ♦♣t✐♠❛❧ ❧❡✈❡❧ ♦❢ ■P❘s ✐♥✈♦❧✈❡s ❛

♠✐♥✐♠✉♠ ❝♦st ♦❢ ✐♠✐t❛t✐♥❣ t❤❛t ✇❡ ✐❞❡♥t✐❢②✳ ■❢ ❞❡♠❛♥❞ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ ❧♦✇✱ ✇❡ s❤♦✇

t❤❛t ❛ ❤✐❣❤ ❧❡✈❡❧ ♦❢ ■P❘ ♣r♦t❡❝t✐♦♥ ✇❤✐❝❤ ✐♥❞✉❝❡s ❛ ♣r❡❡♠♣t✐♦♥ r❛❝❡ ❝♦♥st✐t✉t❡s ❛ ❝♦♥str❛✐♥❡❞

✇❡❧❢❛r❡ ♦♣t✐♠✉♠ ✐♥ ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ s✐t✉❛t✐♦♥s✳ ■❢ ❞❡♠❛♥❞ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ s✉✣❝✐❡♥t❧②

❧♦✇✱ ✐t ✐s ❡✈❡♥ s♦❝✐❛❧❧② ❞❡s✐r❛❜❧❡ t♦ ♣r♦✈✐❞❡ ✐♥♥♦✈❛t♦rs ✇✐t❤ ❝♦♠♣❧❡t❡ ♣r♦t❡❝t✐♦♥ ❛♥❞ ❤❛✈❡ str❛t❡❣✐❝

✐♥✈❡st♠❡♥t t❛❦❡ t❤❡ ❢♦r♠ ♦❢ ✇✐♥♥❡r✲t❛❦❡✲❛❧❧ ♣r❡❡♠♣t✐♦♥✱ s♦ t❤❛t ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ ✐s ✐♥t❡♥s❡

❡♥♦✉❣❤ t❤❛t t❤❡ ✜r♠s ✐♥✈❡st ❛t t❤❡ ♥❡t ♣r❡s❡♥t ✈❛❧✉❡ t❤r❡s❤♦❧❞✳ ❚❤❡ ❡♥❞♦❣❡♥❡✐t② ♦❢ ✐♥♥♦✈❛t✐♦♥

t✐♠✐♥❣ ♣❧❛②s ❛ ❦❡② r♦❧❡ ✐♥ ❡st❛❜❧✐s❤✐♥❣ t❤❡s❡ r❡s✉❧ts✱ ❛♥❞ ✇❡ s❤♦✇ t❤❛t ❛ r❡❣✉❧❛t♦r ✇❤♦ ❞✐❞ ♥♦t

❛❝❝♦✉♥t ❢♦r t❤❡ ❢✉❧❧ r❛♥❣❡ ♦❢ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ ✇♦✉❧❞ r✉♥ t❤❡ r✐s❦ s❡tt✐♥❣ t♦♦ ✇❡❛❦ ❛ ❧❡✈❡❧ ♦❢

■P❘s✳

❚♦ ❡st❛❜❧✐s❤ t❤❡s❡ ❝♦♥❝❧✉s✐♦♥s✱ ✇❡ st✉❞② t❤❡ ❡①❡r❝✐s❡ ♦❢ str❛t❡❣✐❝ ❣r♦✇t❤ ♦♣t✐♦♥s ❜② t✇♦ ✐♥✐t✐❛❧❧②

✐❞❡♥t✐❝❛❧ ✜r♠s ♣✉rs✉✐♥❣ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ♣r♦❞✉❝t ❢♦r ❛ ♥❡✇ ♠❛r❦❡t ✐♥ ✇❤✐❝❤ t❤❡② ❛r❡ ♣♦t❡♥✲

t✐❛❧ ❤♦r✐③♦♥t❛❧ ❝♦♠♣❡t✐t♦rs✳^✶ ❉❡✈❡❧♦♣♠❡♥t ♦❢ ✐♥♥♦✈❛t✐✈❡ ❛♥❞ ✐♠✐t❛t✐✈❡ ♣r♦❞✉❝ts r❡q✉✐r❡s ❞✐✛❡r✐♥❣

st❛t✐♦♥❛r② ❧❡✈❡❧s ♦❢ ✐rr❡✈❡rs✐❜❧❡ ✐♥✈❡st♠❡♥t ✇❤✐❝❤ ♠❛② ♦r ♠❛② ♥♦t ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ❡①♣❧♦✐t❛t✐♦♥

♦❢ ❛ s✐♥❣❧❡ ♣❛t❡♥t✱ ❛♥❞ ♦❝❝✉rs ✐♥ ❛ ❝♦♥t❡①t ♦❢ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t② ❛s t❤❡ s❝❛❧❡ ♦❢ ❞❡♠❛♥❞ ❢♦❧❧♦✇s

❛ ❣❡♦♠❡tr✐❝ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳^✷ ❇♦t❤ ✜r♠s ✐♥❞❡♣❡♥❞❡♥t❧② ❝❤♦♦s❡ t❤r❡s❤♦❧❞s t❤❛t ❞❡t❡r♠✐♥❡ t❤❡

t✐♠✐♥❣ ♦❢ t❤❡✐r ✐♥✈❡st♠❡♥t ✐♥ ♣r♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t✱ ✇❤✐❝❤ ♦♥❝❡ ♣❡r❢♦r♠❡❞ ②✐❡❧❞s ❛♥ ✐♠♠❡❞✐❛t❡

❛♥❞ ♣❡r♣❡t✉❛❧ ♣r♦✜t ✢♦✇ ✇❤♦s❡ ❧❡✈❡❧ ❛t ❛♥② ♠♦♠❡♥t ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ ✜r♠s✳ ❲❡

t❤✉s st✉❞② ❛ r❡❛❧ ♦♣t✐♦♥ ❣❛♠❡✱^✸ ❜✉t ❞❡♣❛rt ❢r♦♠ t❤❡ st❛♥❞❛r❞ ♠♦❞❡❧ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛♥ ❡①✲♣♦st

❛s②♠♠❡tr② t❤r♦✉❣❤ t❤❡ ❞✐✛❡r✐♥❣ ✜①❡❞ ❝♦sts ♦❢ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ ✐♠✐t❛t✐♦♥ t❤❛t ✜r♠s ❢❛❝❡ ❛♥❞ ❜②

❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢✉❧❧ r❛♥❣❡ ♦❢ r❡❧❛t✐✈❡ ✜①❡❞ ❝♦sts✳ ❖✉r ♠♦❞❡❧ t❤❡r❡❢♦r❡ ❛❧❧♦✇s ❥✉st ❛s ✇❡❧❧ ❢♦r ❜r♦❛❞

■P❘s ✐♠♣❧②✐♥❣ ❛ r❡❧❛t✐✈❡❧② ❤✐❣❤ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ ❛♥❞ ♣r❡❡♠♣t✐♦♥ ❜❡t✇❡❡♥ ✜r♠s ❛s ❢♦r s✐❣♥✐✜❝❛♥t

✶❖✉r ❢♦❝✉s ♦♥ t❤❡s❡ ✐♥❞✉str✐❡s ✐s t❤❡r❡❢♦r❡ ❝♦♠♣❧❡♠❡♥t❛r② t♦ r❡s❡❛r❝❤ ♦♥ ❝✉♠✉❧❛t✐✈❡ ✐♥♥♦✈❛t✐♦♥ s✉❝❤ ❛s ●r❡❡♥

❛♥❞ ❙❝♦t❝❤♠❡r ❬✶✽❪✳

✷❲❡ ❢♦❝✉s ♦♥ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t② r❛t❤❡r t❤❛♥ ❘✫❉ ✉♥❝❡rt❛✐♥t②✱ ✇❤✐❝❤ ❤❛s ❜❡❡♥ ❡①t❡♥s✐✈❡❧② st✉❞✐❡❞ ❜② t❤❡ ♣❛t❡♥t r❛❝❡ ❧✐t❡r❛t✉r❡ ✭s❡❡ ❡✳❣✳ ❉❡♥✐❝♦❧ò ❬✾❪✮✳

✸❙❡❡ ❈❤❡✈❛❧✐❡r✲❘♦✐❣♥❛♥t ❛♥❞ ❚r✐❣❡♦r❣✐s ❬✻❪ ❢♦r ❛ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡s❡ ❣❛♠❡s ✇❤❡r❡ ✜r♠s ❜❛❧❛♥❝❡ t❤❡ ✈❛❧✉❡ ♦❢

r❡t❛✐♥✐♥❣ ✢❡①✐❜✐❧✐t② ✐♥ t❤❡ ❢❛❝❡ ♦❢ ✉♥❝❡rt❛✐♥t② ✇✐t❤ t❤❡ str❛t❡❣✐❝ ✐♥❝❡♥t✐✈❡ t♦ ✐♥✈❡st ❡❛r❧②✳

✸

(5)

s♣✐❧❧♦✈❡rs r❡s✉❧t✐♥❣ ✐♥ ❛ ❝♦♠♣❛r❛t✐✈❡❧② ❧♦✇ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ ❛♥❞ ❛ttr✐t✐♦♥ ❜❡t✇❡❡♥ ✜r♠s✳

❚❤❡ ✐♠✐t❛t✐♦♥ ❝♦st ♣r♦✈✐❞❡s ✉s ✇✐t❤ ❛ ✇❛② t♦ ♣❛r❛♠❡tr✐③❡ ✜rst✲ ❛♥❞ s❡❝♦♥❞✲♠♦✈❡r ❛❞✈❛♥t❛❣❡

♣❛rs✐♠♦♥✐♦✉s❧② ❛♥❞ t♦ ♥❡st ✇✐t❤✐♥ ❛ s✐♥❣❧❡ ❢r❛♠❡✇♦r❦ t✇♦ ✐♠♣♦rt❛♥t t✐♠✐♥❣ ❣❛♠❡s✱ t❤❡ ✇❛r ♦❢

❛ttr✐t✐♦♥ ✭❍❡♥❞r✐❝❦s✱ ❲❡✐ss ❛♥❞ ❲✐❧s♦♥ ❬✷✵❪✮ ❛♥❞ ♣r❡❡♠♣t✐♦♥ ✭❋✉❞❡♥❜❡r❣ ❛♥❞ ❚✐r♦❧❡ ❬✶✹❪✮✳^✹ ❚❤❡

t✐♠✐♥❣ ❣❛♠❡ ✇❡ st✉❞② ✐♥✈♦❧✈❡s ✜r♠s ❝❤♦♦s✐♥❣ ✐♥✈❡st♠❡♥t t❤r❡s❤♦❧❞s✱ ♦r ❤✉r❞❧❡ r❛t❡s✱ t❤❛t ❞❡t❡r✲

♠✐♥❡ st♦❝❤❛st✐❝ ✐♥✈❡st♠❡♥t t✐♠❡s✱ ❛♥❞ ❤❛s ❛ str❛✐❣❤t❢♦r✇❛r❞ ♥♦r♠❛❧ ❢♦r♠✳ ❲❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡

✉♥✐q✉❡ s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐✉♠ ✐♥ ✐♥✈❡st♠❡♥t t❤r❡s❤♦❧❞ ❝❤♦✐❝❡s✳ ❚❤✐s ❛♥❛❧②s✐s ♣r♦✈✐❞❡s t❤❡ ❢♦✉♥❞❛✲

t✐♦♥ ❢♦r s✉❜s❡q✉❡♥t ✇❡❧❢❛r❡ r❡s✉❧ts ✇❤✐❝❤ ❛r❡ t❤❡ ♠❛✐♥ ❢♦❝✉s ♦❢ t❤✐s ❛rt✐❝❧❡✱ ❛♥❞ ✇❡ ❝♦♠♣❧❡♠❡♥t ✐t

✇✐t❤ ❛ ♠♦r❡ t❡❝❤♥✐❝❛❧ ❞✐s❝✉ss✐♦♥ ♦❢ ❝❧♦s❡❞✲❧♦♦♣ str❛t❡❣✐❡s ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡ ✐♥ t❤❡ ❛♣♣❡♥❞✐①✳

❆❢t❡r ❝❤❛r❛❝t❡r✐③✐♥❣ ✐♥✈❡st♠❡♥t t✐♠✐♥❣ ❛♥❞ ❞✐s❝✉ss✐♥❣ ♦♣t✐♠❛❧ ✐♠✐t❛t✐♦♥ ❝♦st ❧❡✈❡❧s ✐♥ ❛ ❜❡♥❝❤✲

♠❛r❦ ❝❛s❡ ✇✐t❤ ❛ ♣r❡❞❡t❡r♠✐♥❡❞ s❡q✉❡♥❝❡ ♦❢ ✐♥✈❡st♠❡♥t ❞❡❝✐s✐♦♥s ✐♥st❡❛❞ ♦❢ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥

✭Pr♦♣♦s✐t✐♦♥ ✶✮✱ ✇❡ ❞❡r✐✈❡ t❤❡ s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐✉♠ t✐♠✐♥❣ ♦❢ ✐♥♥♦✈❛t✐♦♥ ✇✐t❤ ❡♥❞♦❣❡♥♦✉s ✜r♠

r♦❧❡s ❢♦r ❞✐✛❡r❡♥t ❧❡✈❡❧s ♦❢ t❤❡ ✐♠✐t❛t✐♦♥ ❝♦st ❛♥❞ ✐❞❡♥t✐❢② ❛ ❝r✐t✐❝❛❧ ✐♠✐t❛t✐♦♥ ❝♦st✱ K✱ ✇❤✐❝❤ ❞❡✲b t❡r♠✐♥❡s ✇❤❡t❤❡r str❛t❡❣✐❝ ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ✜r♠s t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ❛ttr✐t✐♦♥ ♦r ♣r❡❡♠♣t✐♦♥

✭Pr♦♣♦s✐t✐♦♥ 2✮✳ ❋♦r ❡①tr❡♠❡ ✈❛❧✉❡s ♦❢ t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥✱ ✇❡ ✜♥❞ t❤❛t ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥

❤❛s t❤❡ ❢♦r♠ ♦❢ ❛ st❛♥❞❛r❞ t✐♠✐♥❣ ❣❛♠❡✳ ❆ ✈❡r② ❧♦✇ ✐♠✐t❛t✐♦♥ ❝♦st ✐♥❞✉❝❡s ✐♠♠❡❞✐❛t❡ ✐♠✐t❛t✐♦♥

❛♥❞ ❧❡❛❞s t♦ ❛ s✐t✉❛t✐♦♥ ♦❢ ❛ttr✐t✐♦♥ ❛s ✜r♠s s❡❡❦ t♦ ❡♥t❡r s❡❝♦♥❞✱ ❞❡❧❛②✐♥❣ ♣r♦❞✉❝t ✐♥tr♦❞✉❝t✐♦♥✳^✺

❈♦♥✈❡rs❡❧② ❛ ✈❡r② ❤✐❣❤ ✐♠✐t❛t✐♦♥ ❝♦st ❧❡❛❞s t♦ ❛ s✐t✉❛t✐♦♥ ♦❢ ♣r❡❡♠♣t✐♦♥ ❛s ✜r♠s s❡❡❦ t♦ ❡♥t❡r

✜rst ❛♥❞ ❡♥❥♦② ❛ ♣❤❛s❡ ♦❢ ♠♦♥♦♣♦❧② ♣r♦✜t ❜❡❢♦r❡ ✐♠✐t❛t✐♦♥ ♦❝❝✉rs✳ ■♥t❡r♠❡❞✐❛t❡ ✈❛❧✉❡s ♦❢ t❤❡

✐♠✐t❛t✐♦♥ ❝♦st r❡s✉❧t ✐♥ ❤②❜r✐❞ ❢♦r♠s ♦❢ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥✿ ❛ ✇❛✐t✐♥❣ ❣❛♠❡ ✐♥ ✇❤✐❝❤ ✜r♠ ✐♥✲

✈❡st♠❡♥t t❤r❡s❤♦❧❞s ❛r❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐str✐❜✉t❡❞ ♦✈❡r ❛ ❞✐s❝♦♥♥❡❝t❡❞ s✉♣♣♦rt ✐❢ t❤❡ ✐♠✐t❛t✐♦♥ ❝♦st

✐s ♠♦❞❡r❛t❡❧② ❧♦✇✱ ❛♥❞ ❛ ♣r❡❡♠♣t✐♦♥ r❛❝❡ ✐♥ ✇❤✐❝❤ ❛♥ ❛ttr✐t✐♦♥ ♣❤❛s❡ ♦❝❝✉rs ♦✛ t❤❡ ❡q✉✐❧✐❜r✐✉♠

♣❛t❤ ✐❢ t❤❡ ✐♠✐t❛t✐♦♥ ❝♦st ✐s ♠♦❞❡r❛t❡❧② ❤✐❣❤✳

Pr♦✈✐❞❡❞ t❤❛t ✐♥♥♦✈❛t✐♦♥ ❤❛s ♣♦s✐t✐✈❡ s♣✐❧❧♦✈❡rs ❛ttr✐t✐♦♥ ♠❛② ♦❝❝✉r✱ ❛♥❞ ✐t ✐s ♠♦r❡ ❧✐❦❡❧② ✐❢

t❤❡r❡ ✐s ❛ ❧♦✇ ❞❡❣r❡❡ ♦❢ ♣r♦❞✉❝t ♠❛r❦❡t ❝♦♠♣❡t✐t✐♦♥ ♦r ✐❢ ♠❛r❦❡t ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ ❤✐❣❤

✭Pr♦♣♦s✐t✐♦♥ ✸✮✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ❤✐❣❤ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t② r❛✐s❡ t❤❡ ♦♣t✐♦♥ ✈❛❧✉❡ ♦❢ ❞❡❧❛②✐♥❣

✐♥✈❡st♠❡♥t✱ ❡✈❡♥t✉❛❧❧② ❝♦♠♣❡♥s❛t✐♥❣ ❢♦r t❤❡ ❧♦st ♠♦♥♦♣♦❧② ♣r♦✜t ♣❤❛s❡ ✐❢ ❛ ✜r♠ ❡♥t❡rs s❡❝♦♥❞

❛♥❞ ✐♠✐t❛t❡s ✐♥st❡❛❞ ♦❢ ✐♥♥♦✈❛t✐♥❣✳ ❆ ❦❡② ❛❞❞✐t✐♦♥❛❧ r❡s✉❧t ❝♦♥❝❡r♥s t❤❡ ♦♣t✐♠❛❧ ❜❛❧❛♥❝❡ ❜❡t✇❡❡♥

✜rst✲ ❛♥❞ s❡❝♦♥❞✲♠♦✈❡r ❛❞✈❛♥t❛❣❡ ❢r♦♠ t❤❡ st❛♥❞♣♦✐♥t ♦❢ t❤❡ ✐♥❞✉str②✳ ❯♥❞❡r ❜♦t❤ ❛ttr✐t✐♦♥ ❛♥❞

✹❚❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡s❡ ❣❛♠❡s t♦ t❤❡ st♦❝❤❛st✐❝ ❝❛s❡ ✐ts❡❧❢ ♣r❡s❡♥ts ❛ ♥✉♠❜❡r ♦❢ ❝❤❛❧❧❡♥❣❡s ✭❚❤✐❥ss❡♥✱ ❍✉✐s♠❛♥

❛♥❞ ❑♦rt ❬✸✹❪✱ ❙t❡❣ ❛♥❞ ❚❤✐❥ss❡♥ ❬✸✷❪✮✳

✺❖✉r ♠♦❞❡❧ t❤✉s ❡♥❝♦♠♣❛ss❡s t❤❡ ❞②♥❛♠✐❝s ❞❡s❝r✐❜❡❞ ❜② ❙❝❤❡r❡r ✭q✉♦t❡❞ ✐♥ ❋✉❞❡♥❜❡r❣ ❛♥❞ ❚✐r♦❧❡ ❬✶✹❪✮ ❛s ✏❡❛❝❤

✐♥❞✉str② ♠❡♠❜❡r ❤♦❧❞✐♥❣ ❜❛❝❦ ✐♥✐t✐❛t✐♥❣ ✐ts ❘✫❉ ❡✛♦rt ✐♥ t❤❡ ❢❡❛r t❤❛t r❛♣✐❞ ✐♠✐t❛t✐♦♥ ❜② ♦t❤❡rs ✇✐❧❧ ❜❡ ❡♥❝♦✉r❛❣❡❞✱

♠♦r❡ t❤❛♥ ✇✐♣✐♥❣ ♦✉t ✐ts ✐♥♥♦✈❛t✐✈❡ ♣r♦✜ts✳✑

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♣r❡❡♠♣t✐♦♥✱ ♣♦s✐t✐♦♥❛❧ r❡♥ts ❛r❡ ❞✐ss✐♣❛t❡❞ ✐♥ t❤❡ s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐✉♠ ❛♥❞ ❡①♣❡❝t❡❞ ✐♥❞✉str②

✈❛❧✉❡ ✐s t❤❡r❡❢♦r❡ ♠❛①✐♠✐③❡❞ ✐❢ t❤❡ ✐♠✐t❛t✐♦♥ ❝♦st ❛tt❛✐♥s t❤❡ ❝r✐t✐❝❛❧ ❧❡✈❡❧ Kb ❛t ✇❤✐❝❤ t❤❡r❡ ✐s

♥❡✐t❤❡r ❛ ✇❛r ♦❢ ❛ttr✐t✐♦♥ ♥♦r ❛ ♣r❡❡♠♣t✐♦♥ r❛❝❡✱ s♦ t❤❛t ✜r♠s ❞♦ ♥♦t ❝♦♠♣❡t❡ ❢♦r ♣♦s✐t✐♦♥❛❧ r❡♥ts

❜② ❡✐t❤❡r ✉♥❞✉❧② ✇❛✐t✐♥❣ ♦r r✉s❤✐♥❣ t♦ ✐♥♥♦✈❛t❡ ✭Pr♦♣♦s✐t✐♦♥ 4✮✳

❇❡❝❛✉s❡ ♦❢ t❤❡ tr❛❝t❛❜✐❧✐t② ♦❢ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✇❡ ❝❤❛r❛❝t❡r✐③❡✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ st✉❞② s♦❝✐❛❧❧②

♦♣t✐♠❛❧ ■P❘ ❧❡✈❡❧s ✐❢ ❛ r❡❣✉❧❛t♦r ❛❞❥✉sts t❤❡ ❝♦st ✐♥❝✉rr❡❞ ❜② ❛♥ ✐♠✐t❛t♦r t❤r♦✉❣❤ ❡✐t❤❡r ❧❡❣✐s❧❛t✐✈❡

♠❡❛s✉r❡s ♦r ❡♥❢♦r❝❡♠❡♥t✳^✻ ❲✐t❤ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ t❤❡ ✇❡❧❢❛r❡ tr❛❞❡✲♦✛ ❛ss♦❝✐❛t❡❞ ✇✐t❤ r❛✐s✐♥❣

t❤❡ ✐♠✐t❛t✐♦♥ ❝♦st ✐s ♠♦r❡ ✐♥✈♦❧✈❡❞ t❤❛♥ ❛ str❛✐❣❤t ❜❛❧❛♥❝✐♥❣ ♦❢ t❤❡ ✐♥❝❡♥t✐✈❡ t♦ ✐♥♥♦✈❛t❡ ❛❣❛✐♥st t❤❡ ❞❡❛❞✇❡✐❣❤t ❧♦ss ♦❢ ♠♦♥♦♣♦❧②✱ ❛s t❤❡ ❡✛❡❝t ♦❢ ❤✐❣❤❡r ✐♠✐t❛t✐♦♥ ❝♦st ♦♥ t❤❡ t✐♠✐♥❣ ♦❢ ✐♠✐t❛✲

t✐♦♥ ✐s ❛♠❜✐❣✉♦✉s ✉♥❞❡r ❛ttr✐t✐♦♥✳ ❲❡ ✐❞❡♥t✐❢② ❛ ❧♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ s♦❝✐❛❧❧② ♦♣t✐♠❛❧ ✐♠✐t❛t✐♦♥

❝♦st ✭Pr♦♣♦s✐t✐♦♥ ✺✮✱ ✇❤✐❝❤ ♠✉st ♣r♦✈✐❞❡ s✉✣❝✐❡♥t q✉❛s✐✲r❡♥ts ❢♦r ✜r♠s t♦ ❛✈♦✐❞ t❤❡ ❙❝❤❡r❡r✐❛♥

❞②♥❛♠✐❝s ❞❡s❝r✐❜❡❞ ✐♥ ❢♦♦t♥♦t❡ ✺ ❛❜♦✈❡✳

❊✈❡♥ ✐❢ ✐t ✐s ❝❤❛❧❧❡♥❣✐♥❣ t♦ ❞r❛✇ ❣❡♥❡r❛❧ ❝♦♥❝❧✉s✐♦♥s r❡❣❛r❞✐♥❣ ♦♣t✐♠❛❧ ■P❘ ❧❡✈❡❧s✱ ✇❡ ❛r❡ ❛❜❧❡

t♦ s❤♦✇ t❤❛t ✐❢ t❤❡ st❛t✐❝ ✐♥❝❡♥t✐✈❡ t♦ ✐♠✐t❛t❡ ✐s s♦❝✐❛❧❧② ❡①❝❡ss✐✈❡✱ ❛s ♦❝❝✉rs ✐♥ ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢

♦❧✐❣♦♣♦❧② ♠♦❞❡❧s✱ ❛♥ ✐♠✐t❛t✐♦♥ ❝♦st t❤❛t ✐♥❞✉❝❡s ♣r❡❡♠♣t✐♦♥ ✐s ♦♣t✐♠❛❧ ✇❤❡♥ ♠❛r❦❡t ❣r♦✇t❤ ❛♥❞

✈♦❧❛t✐❧✐t② ❛r❡ s✉✣❝✐❡♥t❧② ❧♦✇ ✭Pr♦♣♦s✐t✐♦♥6✮✳ ❚❤❡ ♠♦❞❡❧ t❤❡r❡❢♦r❡ ♣r♦✈✐❞❡s ❛♥ ❛r❣✉♠❡♥t ❢♦r str♦♥❣

■P❘s ✐♥ s✉❝❤ ✐♥❞✉str✐❡s ❜❛s❡❞ ♦♥ ♦❜❥❡❝t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t②✳ ■♥ ♣❛ss✐♥❣ ✇❡

♦❜t❛✐♥ ❝❧♦s❡❞✲❢♦r♠ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ ♦♣t✐♠❛❧ t❤r❡s❤♦❧❞ ❢♦r ✐♥♥♦✈❛t✐♦♥ ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❧❡✈❡❧ ♦❢

✇❡❧❢❛r❡ ✉♥❞❡r ♣r❡❡♠♣t✐♦♥ ✭▲❡♠♠❛ ✶✮✱ ❡st❛❜❧✐s❤✐♥❣ t❤❛t ❛ ❧✐♠✐t ✐♠✐t❛t✐♦♥ ❝♦st ❧❡✈❡❧ ✇❤✐❝❤ r❡s✉❧ts

✐♥ ✇✐♥♥❡r✲t❛❦❡✲❛❧❧ ♣r❡❡♠♣t✐♦♥ ✐s s♦❝✐❛❧❧② ♦♣t✐♠❛❧ ✇❤❡♥ t❤❡r❡ ✐s s✉✣❝✐❡♥t ❞✐s❝♦✉♥t✐♥❣✳ ▼♦r❡♦✈❡r

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

✇✐t❤ ❛ttr✐t✐♦♥ ♦r ♣r❡❡♠♣t✐♦♥✱ s✉❝❤ ❛s ❛ ❧♦✇ ❝♦♥s✉♠❡r s✉r♣❧✉s ❢r♦♠ ✐♥♥♦✈❛t✐♦♥ ♦r ❝♦❧❧✉s✐♦♥ ✐♥ t❤❡

♣r♦❞✉❝t ♠❛r❦❡t ✭Pr♦♣♦s✐t✐♦♥ ✼✮✳

❚❤❡s❡ ✇❡❧❢❛r❡ r❡s✉❧ts t❛❦❡ ♦♥ ❡✈❡♥ ❣r❡❛t❡r r❡❧❡✈❛♥❝❡ ✇❤❡♥ t❤❡② ❛r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ♦♣t✐♠❛❧

✇❡❧❢❛r❡ ❧❡✈❡❧s ❞❡r✐✈❡❞ ✇✐t❤♦✉t ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥✱ ❛s ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ ✐♥ t❤❡ ❡❝♦♥♦♠✐❝ ❧✐t❡r❛t✉r❡

✻❘❡❣✉❧❛t♦rs ❤❛✈❡ ❛ ❜r♦❛❞ ❛rr❛② ♦❢ s♣❡❝✐✜❝ ♠❡❛s✉r❡s ❛t t❤❡✐r ❞✐s♣♦s❛❧ ✇❤✐❝❤ ❛✛❡❝t t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥✳ ■♥ t❤❡

♣❤❛r♠❛❝❡✉t✐❝❛❧ ❝♦♥t❡①t ❢♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❜✐♦❧♦❣✐❝✲❜❛s❡❞ ❞r✉❣s ✭✐✳❡✳✱ ❧❛r❣❡ ❛♥❞ ❝♦♠♣❧❡① ♠♦❧❡❝✉❧❡s t❤❛t

❛r❡ t②♣✐❝❛❧❧② ♣r♦❞✉❝❡❞ ✇✐t❤ r❡❝♦♠❜✐♥❛♥t ❉◆❆ t❡❝❤♥♦❧♦❣②✮✱ r❡❝❡♥t ❯❙ ❧❡❣✐s❧❛t✐♦♥ ✐♥tr♦❞✉❝❡❞ ✉♥❞❡r t❤❡ ❖❜❛♠❛

❆❞♠✐♥✐str❛t✐♦♥ ❣✉❛r❛♥t❡❡s ✶✷ ②❡❛rs ♦❢ ❞❛t❛ ❡①❝❧✉s✐✈✐t②✳ ❉✉r✐♥❣ t❤✐s ♣❡r✐♦❞✱ ❛♥ ✐♠✐t❛t✐♥❣ ✜r♠ ❝❛♥♥♦t r❡❢❡r t♦ t❤❡

♦r✐❣✐♥❛t♦r✬s ❝❧✐♥✐❝❛❧ ❞❛t❛ ✐♥ t❤❡ r❡❣✉❧❛t♦r② ✜❧✐♥❣s ❢♦r ❛ ❣❡♥❡r✐❝ ✈❡rs✐♦♥ ♦❢ ❛♥ ✐♥♥♦✈❛t✐✈❡ ♣r♦❞✉❝t✳ ■♥ t❤❡ ✇♦r❞s ♦❢

❇r❛♥st❡tt❡r✱ ❈❤❛tt❡r❥❡❡ ❛♥❞ ❍✐❣❣✐♥s ❬✹❪✱ ✭♣✳ ✼✮✿

✏❲✐t❤ ♥♦ ✇❛② t♦ ❡st❛❜❧✐s❤ ❜✐♦❡q✉✐✈❛❧❡♥❝❡✱ ❛♥② ❣❡♥❡r✐❝ ✈❡rs✐♦♥ ♦❢ ❛ ❜✐♦❧♦❣✐❝✲❜❛s❡❞ ❞r✉❣ ✇♦✉❧❞ ❤❛✈❡

t♦ ✉♥❞❡r❣♦ s❡♣❛r❛t❡ ❝❧✐♥✐❝❛❧ tr✐❛❧s t♦ r❡❝❡✐✈❡ ❋❉❆ ❛♣♣r♦✈❛❧ ✕ ❛ ❜❛rr✐❡r t♦ ❡♥tr② s♦ ❞❛✉♥t✐♥❣ t❤❛t ♥♦

❜✐♦s✐♠✐❧❛r ❤❛s ②❡t ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❯✳❙✳ ♠❛r❦❡t✳✑

✺

(7)

♦♥ ♣❛t❡♥ts✳ ❆ r❡❣✉❧❛t♦r ❢♦❧❧♦✇✐♥❣ t❤✐s ❦✐♥❞ ♦❢ ❛♣♣r♦❛❝❤ ❝♦✉❧❞ ❜❡ ❧❡❞ t♦ s❡t t❤❡ ❧❡✈❡❧ ♦❢ ■P❘

♣r♦t❡❝t✐♦♥ ♠✉❝❤ t♦♦ ❧♦✇✱ ✇❤❡♥ ✐♥ ❢❛❝t ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ✜r♠s t♦ ✐♥♥♦✈❛t❡ ♣❧❛②s ❛ ✈✐t❛❧ r♦❧❡

❛♥❞ ✐s ❜❡st ✐♥❝❡♥t✐✈✐③❡❞ ✇✐t❤ ❧❡✈❡❧s ♦❢ ✐♠✐t❛t✐♦♥ ❝♦st ✐♥ t❤❡ ♣r❡❡♠♣t✐✈❡ r❛♥❣❡ ✭Pr♦♣♦s✐t✐♦♥ ✽✮✳

❋✐♥❛❧❧② ✇❡ ❞✐s❝✉ss s❡✈❡r❛❧ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ♠♦❞❡❧✳ ❋✐rst✱ ✇❡ ❡♥❞♦❣❡♥✐③❡ t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ ❜②

❛❧❧♦✇✐♥❣ t❤❡ ✐♥♥♦✈❛t♦r t♦ ♣✉rs✉❡ ♣❛t❡♥t ♣r♦t❡❝t✐♦♥ ♠♦r❡ ❛❣❣r❡ss✐✈❡❧② ♦r t♦ ♠❛❦❡ r❡✈❡rs❡ ❡♥❣✐♥❡❡r✐♥❣

♦❢ ✐ts ♣r♦❞✉❝t ♠♦r❡ ❞✐✣❝✉❧t✳ ❆ ❤✐❣❤❡r ❜❛s❡❧✐♥❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ r❡❞✉❝❡s t❤❡ ❡✛♦rt ❡①❡rt❡❞ ❜②

✐♥♥♦✈❛t♦rs t♦ r❛✐s❡ ❡♥tr② ❜❛rr✐❡rs✱ ❛♥❞ ✜r♠s ❛r❡ s❤♦✇♥ t♦ ❣❛✐♥ ❢r♦♠ ❝♦♦r❞✐♥❛t✐♥❣ ❡①✲❛♥t❡ ♥♦t t♦

✐♥tr♦❞✉❝❡ s✉❜s❡q✉❡♥t ❝♦♠♣❧❡①✐t②✱ ❛ ♣♦❧✐❝② t❤❛t ♠❛② ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛♥ ♦♣❡♥ st❛♥❞❛r❞ ✭Pr♦♣♦s✐t✐♦♥

9✮✳ ❲❡ ❛❧s♦ ❞✐s❝✉ss ❝♦♥tr❛❝t✐♥❣ ❜❡t✇❡❡♥ ✐♥♥♦✈❛t♦r ❛♥❞ ✐♠✐t❛t♦r t❤❛t ❝❛♥ t❛❦❡ t❤❡ ❢♦r♠ ❡✐t❤❡r ♦❢

❛ t❛❦❡♦✈❡r ♦r ♦❢ ❛ ❧✐❝❡♥s❡ ❛❣r❡❡♠❡♥t✱ ❛♥❞ s❤♦✇ t❤❛t ❡✣❝✐❡♥❝② ❛❧✇❛②s ✐♥❝r❡❛s❡s ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡

✭Pr♦♣♦s✐t✐♦♥ 10✮✳

❖✉r ♣❛♣❡r ✐s r❡❧❛t❡❞ t♦ ❡❛r❧② r❡s❡❛r❝❤ ♦♥ ✐♥♥♦✈❛t✐♦♥ ✐♥❝❡♥t✐✈❡s ❛♥❞ ♦♣t✐♠❛❧ ♣❛t❡♥ts✱ ❛♥❞ ✐♥

♣❛rt✐❝✉❧❛r t♦ ●❛❧❧✐♥✐ ❬✶✻❪ ✇❤♦ ✐♥tr♦❞✉❝❡s ❛ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ t❤❛t t❤❡ r❡❣✉❧❛t♦r ♠❛② ✉s❡ ❛s ❛ ♣♦❧✐❝②

✐♥str✉♠❡♥t✳ ❲❡ s✐♠✐❧❛r❧② ❡♠♣❤❛s✐③❡ t❤❡ r♦❧❡ ♦❢ ♠❡❛s✉r❡s ❧✐❦❡ ♣❛t❡♥t ❜r❡❛❞t❤ ✐♥ ❞❡t❡r♠✐♥✐♥❣ t❤❡

❝♦st ♦❢ ✐♥✈❡♥t✐♥❣ ❛r♦✉♥❞ ❛♥ ❡①✐st✐♥❣ ✐♥♥♦✈❛t✐♦♥✱ ❜✉t ✐♥ ❝♦♥tr❛st ✇✐t❤ t❤✐s ❡❛r❧✐❡r ✇♦r❦ ✇❡ ❛❝❝♦✉♥t

❢♦r t❤❡ ❡♥❞♦❣❡♥♦✉s t✐♠✐♥❣ ♦❢ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ t❤✉s ❛❧❧♦✇ ✜r♠s t♦ ✇❛✐t ❜❡❢♦r❡ ✐♥✈❡st✐♥❣ r❛t❤❡r t❤❛♥

❛ss✉♠✐♥❣ t❤❛t ♣r♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t ♦❝❝✉rs ✇❤❡♥ ✐ts ♥❡t ♣r❡s❡♥t ✈❛❧✉❡ ✐s ♣♦s✐t✐✈❡✳ ❉❡♥✐❝♦❧ò ❬✾❪✬s

♠♦❞❡❧ ♦❢ ♦♣t✐♠❛❧ ■P❘ ♣r♦t❡❝t✐♦♥ ✐♥ ❛ ♣❛t❡♥t r❛❝❡ ✐s t❤❡r❡❢♦r❡ ❝❧♦s❡r t♦ ♦✉r ✇♦r❦✱ ❛s ✐t ❢♦r♠❛❧✐③❡s

✐♥♥♦✈❛t✐♦♥ ❛♥❞ ✐♠✐t❛t✐♦♥ ❛s t❤❡ ♦✉t❝♦♠❡ ♦❢ ❛ ♥♦♥✲❝♦♦♣❡r❛t✐✈❡ ✐♥t❡r❛❝t✐♦♥ t❤❛t ♣r❡❝❡❞❡s ♠❛r❦❡t

❝♦♠♣❡t✐t✐♦♥✱ t❤♦✉❣❤ ✐♥ ❝♦♥tr❛st ♦✉r ♠♦❞❡❧ ❛❧❧♦✇s ❢♦r s❡❝♦♥❞✲♠♦✈❡r ❛❞✈❛♥t❛❣❡ ❛♥❞ ❛ttr✐t✐♦♥✱ ✇❤✐❝❤

❧✐❦❡❧② ❛r✐s❡s ✐♥ ✐♥❞✉str✐❡s ✇✐t❤ ❤✐❣❤ ❣r♦✇t❤ ❛♥❞ ✈♦❧❛t✐❧✐t②✳

❖✉r ✇♦r❦ ✐s t❤❡r❡❢♦r❡ ❛❧s♦ r❡❧❛t❡❞ t♦ ♣❛♣❡rs ✇❤✐❝❤ st✉❞② t❤❡ ❡✛❡❝t ♦❢ s❡❝♦♥❞✲♠♦✈❡r ❛❞✈❛♥t❛❣❡

♦♥ ✐♥✈❡st♠❡♥t ❞❡❝✐s✐♦♥s✱ ♦❢t❡♥ ❛s ❛ r❡s✉❧t ♦❢ ❡①♣❧✐❝✐t ✐♥❢♦r♠❛t✐♦♥❛❧ s♣✐❧❧♦✈❡rs✳ ❍♦♣♣❡ ❬✷✸❪ ❛❧❧♦✇s

❢♦r ✉♥❝❡rt❛✐♥t② r❡❣❛r❞✐♥❣ t❤❡ s✉❝❝❡ss ♦❢ ♥❡✇ t❡❝❤♥♦❧♦❣② ❛❞♦♣t✐♦♥ t♦ ❜❡♥❡✜t ❛ r✐✈❛❧✬s ✐♥♥♦✈❛t✐♦♥

❞❡❝✐s✐♦♥ ✇❤❡r❡❛s ✐♥ ❚❤✐❥ss❡♥✱ ❍✉✐s♠❛♥ ❛♥❞ ❑♦rt ❬✸✸❪ ✐♥❢♦r♠❛t✐♦♥ r❡❣❛r❞✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ ❛ ♣r♦❥❡❝t

❛rr✐✈❡s ❝♦♥t✐♥✉♦✉s❧② ♦✈❡r t✐♠❡✳ ❋❡♠♠✐♥✐s ❛♥❞ ▼❛rt✐♥✐ ❬✶✸❪ ❛❧❧♦✇ ❢♦r ❛ ❞✐s❝❧♦s✉r❡ ❧❛❣ ♦❢ r❛♥❞♦♠

❞✉r❛t✐♦♥ ❜❡❢♦r❡ t❤❡ ❢♦❧❧♦✇❡r r❡❝❡✐✈❡s t❤❡ ✐♥❢♦r♠❛t✐♦♥✳ ■♥ t❤❡s❡ ♠♦❞❡❧s✱ ❜♦t❤ ♣r❡❡♠♣t✐♦♥ ❛♥❞

❛ttr✐t✐♦♥ ❝❛♥ ♦❝❝✉r ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❧❡✈❡❧ ♦❢ s♣✐❧❧♦✈❡rs✱ ❜✉t t❤❡ ✇❡❧❢❛r❡ ❛♥❛❧②s✐s ✐s ❡✐t❤❡r ❜❛s❡❞ ♦♥

♣✉r❡ str❛t❡❣② ❡q✉✐❧✐❜r✐✉♠ ♦r r❡str✐❝t❡❞ t♦ ♣r❡❡♠♣t✐♦♥ r❡❣✐♠❡s✳ ❲❡ ❛♥❛❧②③❡ t❤❡ ✇❡❧❢❛r❡ ♣r♦♣❡rt✐❡s

♦❢ s②♠♠❡tr✐❝ ♠✐①❡❞ str❛t❡❣② ❡q✉✐❧✐❜r✐✉♠ ♦✈❡r ❛ ❝♦♠♣❧❡t❡ r❛♥❣❡ ♦❢ r❡❣✐♠❡s✱ ♣r♦✈✐❞✐♥❣ ✐♥t✉✐t✐✈❡

❛♥❛❧②t✐❝ r❡s✉❧ts r❡❣❛r❞✐♥❣ ♦♣t✐♠❛❧ ■P❘s t❤❛t ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ❞②♥❛♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❞❡♠❛♥❞✳

❖✉r ✇❡❧❢❛r❡ r❡s✉❧ts ❝❛♥ ❜❡ r❡❧❛t❡❞ t♦ s❡✈❡r❛❧ ♣❛♣❡rs t❤❛t ❝♦♠♣❛r❡ ✇❡❧❢❛r❡ ❛❝r♦ss t✇♦ ❦❡②

❛❧t❡r♥❛t✐✈❡ ♣♦❧✐❝② r❡❣✐♠❡s✱ ❛ str✐❝t ✇✐♥♥❡r✲t❛❦❡✲❛❧❧ r❡❣✐♠❡ ✇❤❡r❡ ♦♥❧② t❤❡ ✜rst ✜r♠ t♦ ✐♥♥♦✈❛t❡

✻

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r❡❝❡✐✈❡s ❛ ♣❛t❡♥t✱ ❛♥❞ ❛ ♠♦r❡ ♣❡r♠✐ss✐✈❡ r❡❣✐♠❡ ✇❤❡r❡ ❧❛t❡ ✐♥✈❡st♦rs ❛r❡ ❛❧❧♦✇❡❞ t♦ ❝♦♠♣❡t❡ ✇✐t❤

t❤❡ ✜rst ❜❡❢♦r❡ ✐ts ♣❛t❡♥t ❡①♣✐r❡s✳ ■♥ ▲❛ ▼❛♥♥❛✱ ▼❛❝❧❡♦❞✱ ❛♥❞ ❞❡ ▼❡③❛ ❬✷✺❪ ✜r♠s s♣❡♥❞ ❛ ✜①❡❞

✐♥✐t✐❛❧ ❛♠♦✉♥t ✐♥ ❘✫❉ t❤❛t ❞❡t❡r♠✐♥❡s ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ ✐♥✈❡♥t✐♥❣ ❛t ❛ ❢✉t✉r❡ ❞❛t❡✳ ❙✐♠♣❧❡ ❝♦st

❛♥❞ ❞❡♠❛♥❞ ❝♦♥❞✐t✐♦♥s✱ s✉❝❤ ❛s ❝♦♥st❛♥t r❡t✉r♥s t♦ s❝❛❧❡ ❛♥❞ ❛ ❧✐♥❡❛r ❞❡♠❛♥❞✱ ❛r❡ ✐❞❡♥t✐✜❡❞

❢♦r t❤❡ ♣❡r♠✐ss✐✈❡ r❡❣✐♠❡ t♦ ❜❡ ✇❡❧❢❛r❡ s✉♣❡r✐♦r✳ ❍❡♥r② ❬✷✶❪ ✐♥tr♦❞✉❝❡s ❛ ♠❡❝❤❛♥✐s♠ ✇❤❡r❡❜② ❛

❧❛t❡ ✐♥✈❡♥t♦r ❝❛♥ s❤❛r❡ t❤❡ ♣❛t❡♥t ✇✐t❤ t❤❡ ✐♥♥♦✈❛t♦r ✇✐t❤✐♥ ❛ ❣✐✈❡♥ t✐♠❡ ♣❡r✐♦❞✳ ❲❤❡♥ ❛❞❥✉st❡❞✱

t♦❣❡t❤❡r ✇✐t❤ ♦t❤❡r ♣♦❧✐❝② ✐♥str✉♠❡♥ts✱ t❤✐s ♠❡❝❤❛♥✐s♠ ✐s s♦❝✐❛❧❧② ❜❡♥❡✜❝✐❛❧ ✉♥❞❡r ♠✐❧❞ ❝♦♥❞✐t✐♦♥s✱

♥♦t❛❜❧② ✇✐t❤ ❛ ❧✐♥❡❛r ❞❡♠❛♥❞ ❛♥❞ q✉❛♥t✐t② ❝♦♠♣❡t✐t✐♦♥✳ ❍♦✇❡✈❡r✱ ✐♥ ❛ ♠♦❞❡❧ ✇❤❡r❡ ✜r♠s ✐♥❝✉r ❛

✢♦✇ ❝♦st✱ ✐♥ ❉❡♥✐❝♦❧ò ❛♥❞ ❋r❛♥③♦♥✐ ❬✶✶❪ ✐t ✐s t❤❡ str✐❝t ♣❛t❡♥t r❡❣✐♠❡ t❤❛t ✐s ❢♦✉♥❞ t♦ ❜❡ ♦♣t✐♠❛❧

✐♥ ❛ ❜r♦❛❞ s❡t ♦❢ ❝✐r❝✉♠st❛♥❝❡s✱ ✐♥ ♣❛rt✐❝✉❧❛r ✇❤❡♥ ❞❡♠❛♥❞ ✐s ❧✐♥❡❛r✱ ♣r♦❞✉❝t ♠❛r❦❡t ❝♦♠♣❡t✐t✐♦♥

✐s ✇❡❛❦✱ ❛♥❞ ❞✉♣❧✐❝❛t✐♦♥ ✢♦✇ ❝♦sts ❛r❡ ❧❛r❣❡✳ ❖✉r ❛♣♣r♦❛❝❤ ✐s ❜r♦❛❞❧② ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡s❡

❝♦♥tr✐❜✉t✐♦♥s✱ ❜✉t ✇❡ ❝❤❛r❛❝t❡r✐③❡ ❛♥ ♦♣t✐♠❛❧ ❞❡❣r❡❡ ♦❢ ■P❘s ✇✐t❤ t❤❡ ✇✐♥♥❡r✲t❛❦❡✲❛❧❧ r❡❣✐♠❡

♦❝❝✉rr✐♥❣ ❛s ❛ ❧✐♠✐t ❝❛s❡ r❛t❤❡r t❤❛♥ ❡✈❛❧✉❛t✐♥❣ ❛ ❞✐s❝r❡t❡ s❡t ♦❢ r❡❣✐♠❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ♠♦❞❡❧ ♦❢

✐♥✈❡st♠❡♥t ✉♥❞❡r ♠❛r❦❡t ✉♥❝❡rt❛✐♥t② ❛❧❧♦✇s ✉s t♦ ✐❞❡♥t✐❢② ❞❡t❡r♠✐♥❛♥ts ♦❢ ♦♣t✐♠❛❧ ♣r♦t❡❝t✐♦♥ t❤❛t

❛r❡ ♥♦t ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s str❡❛♠ ♦❢ ❧✐t❡r❛t✉r❡ r❡❧❛t❡❞ t♦ ♠❡❛s✉r❛❜❧❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❞②♥❛♠✐❝s ♦❢

♠❛r❦❡ts✳

❙❡❝t✐♦♥ ✷ ❞❡s❝r✐❜❡s t❤❡ ♠♦❞❡❧✳ ❙❡❝t✐♦♥ ✸ st✉❞✐❡s ❛ ❜❡♥❝❤♠❛r❦ ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡ r♦❧❡s ♦❢

✜r♠s ❛r❡ ♣r❡❞❡t❡r♠✐♥❡❞✳ ❙❡❝t✐♦♥ ✹ ❝❤❛r❛❝t❡r✐③❡s t❤❡ s②♠♠❡tr✐❝ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡♥ ✜r♠s ❡♥❣❛❣❡ ✐♥

❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥✳ ❙❡❝t✐♦♥ ✺ st✉❞✐❡s ✇❡❧❢❛r❡ ✇❤❡♥ ❛ r❡❣✉❧❛t♦r ❞❡t❡r♠✐♥❡s t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥✳

❙❡❝t✐♦♥ ✻ ❞✐s❝✉ss❡s t✇♦ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ♠♦❞❡❧✳ ❙❡❝t✐♦♥ ✼ ❝♦♥❝❧✉❞❡s✳

✷ ❆ ♠♦❞❡❧ ♦❢ ♥❡✇ ♣r♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t

❚❤✐s s❡❝t✐♦♥ s❡ts ✉♣ ❛ ♠♦❞❡❧ ♦❢ str❛t❡❣✐❝ ✐♥✈❡st♠❡♥t ✐♥ ♣r♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t t❤❛t r❡✢❡❝ts t❤❡

❝❤❛r❛❝t❡r✐st✐❝ ❢❡❛t✉r❡s ♦❢ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ ✐♠✐t❛t✐♦♥ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✳ ❚❤❡ ❛ss✉♠♣t✐♦♥s

❝♦♥❝❡r♥✐♥❣ t❤❡ ❡❝♦♥♦♠✐❝ ❡♥✈✐r♦♥♠❡♥t ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✶ ❛♥❞ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♣❛②♦✛s t❤❛t ✜r♠s ♦❜t❛✐♥ ♦♥❝❡ ✐♥♥♦✈❛t✐♦♥ ♦❝❝✉rs ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✳✷✳

✷✳✶ ❆ss✉♠♣t✐♦♥s

❚✇♦ ✐❞❡♥t✐❝❛❧ ✜r♠s ❡♥❣❛❣❡ ✐♥ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ t♦ ✐♥tr♦❞✉❝❡ s✉❜st✐t✉t❛❜❧❡ ✈❡rs✐♦♥s ♦❢ ❛ ♥♦✈❡❧

♣r♦❞✉❝t ✐♥ ❛♥ ❡✈♦❧✈✐♥❣ ♠❛r❦❡t✳ ❖r❣❛♥✐③❛t✐♦♥❛❧ ❝♦♥str❛✐♥ts ♣r❡✈❡♥t ❛ ✜r♠ ❢r♦♠ ❞❡✈❡❧♦♣✐♥❣ t✇♦

✈❛r✐❛♥ts ♦❢ t❤❡ ♥♦✈❡❧ ♣r♦❞✉❝t ❛♥❞ ❡♥tr② ❜❛rr✐❡rs s❤✐❡❧❞ ❜♦t❤ ✜r♠s ❢r♦♠ ♦t❤❡r ❝♦♠♣❡t✐t♦rs✳

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■♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❞✉❝t ✐♠♠❡❞✐❛t❡❧② ❣❡♥❡r❛t❡s ❛ ♣❡r♣❡t✉❛❧ ❜❛s❡❧✐♥❡ ♣r♦✜t ✢♦✇ ❢♦r t❤❡ ✜r♠✱

✇❤♦s❡ ✈❛❧✉❡ ✐sπ^M ✐❢ ✐t ♦♣❡r❛t❡s ❛s ❛ ♠♦♥♦♣♦❧② ♦rπ^D ✐❢ ❜♦t❤ ✜r♠s ❛r❡ ❛❝t✐✈❡✱ ✇✐t❤0<2π^D < π^M

✇❤✐❝❤ ♠❡❛♥s t❤❛t ✐♠✐t❛t✐♦♥ r❡❞✉❝❡s ✐♥❞✉str② ♣r♦✜ts✳ ❆ ✜r♠ t❤❛t ✐♥tr♦❞✉❝❡s ✐ts ♣r♦❞✉❝t ✇❤❡♥ ♥♦

♦t❤❡r ✜r♠ ❤❛s ②❡t ❞♦♥❡ s♦ ✐s r❡❢❡rr❡❞ t♦ ❛s ❛♥ ✐♥♥♦✈❛t♦r✱ ❛♥❞ ✐❢ ❛ ✜r♠ ✐♥tr♦❞✉❝❡s ✐ts ♣r♦❞✉❝t ❛❢t❡r

✐ts r✐✈❛❧ ✐t ✐s r❡❢❡rr❡❞ t♦ ❛s t❤❡ ✐♠✐t❛t♦r✳

❚❤❡ ❜❛s❡❧✐♥❡ ♣r♦✜t ✢♦✇ ✐s s❝❛❧❡❞ ❜② ❛ ♠❡❛s✉r❡ ♦❢ ♠❛r❦❡t s✐③❡ Y(t)✱ t ≥ 0✳ ❚❤❡ t♦t❛❧ ✢♦✇

♣r♦✜ts ❛❝t✐✈❡ ✜r♠s ♦❜t❛✐♥ ❛t ❛ ❣✐✈❡♥ t✐♠❡ ❛r❡ ❡✐t❤❡rπ^MY(t)♦r π^DY(t)✳ ❚♦ ❝❛♣t✉r❡ t❤❡ ✐❞❡❛ t❤❛t

❞❡♠❛♥❞ ❢♦r t❤❡ ♥❡✇ ♣r♦❞✉❝t ❡✈♦❧✈❡s ✐♥ ❛ ❝♦♥t❡①t ♦❢ ♠❛r❦❡t ✉♥❝❡rt❛✐♥t②✱ t❤❡ ♠❡❛s✉r❡ ♦❢ ♠❛r❦❡t s✐③❡

✐s ❛ss✉♠❡❞ t♦ ❢♦❧❧♦✇ ❛ ❣❡♦♠❡tr✐❝ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥dY(t) =αY(t)dt+σY(t)dW(t)✇❤❡r❡W(t)✐s

❛ st❛♥❞❛r❞ ❲✐❡♥❡r ♣r♦❝❡ss✱ ❛♥❞α❛♥❞σ≥0❛r❡ t❤❡ ❞r✐❢t ❛♥❞ ✈♦❧❛t✐❧✐t②✳ ❇♦t❤ ✜r♠s ❤❛✈❡ t❤❡ s❛♠❡

❞✐s❝♦✉♥t r❛t❡r✳ ■♥ ♦r❞❡r ❢♦r t❤❡ ✐♥✈❡st♠❡♥t ♣r♦❜❧❡♠s ✇❡ st✉❞② t♦ ❜❡ ❡❝♦♥♦♠✐❝❛❧❧② ♠❡❛♥✐♥❣❢✉❧ ✇❡

❛ss✉♠❡ t❤❛t α < r✳

Pr♦❞✉❝t ❞❡✈❡❧♦♣♠❡♥t ✐♥✈♦❧✈❡s ❛♥ ✐rr❡✈❡rs✐❜❧❡ ✐♥✈❡st♠❡♥t ✇❤✐❝❤ ❡♥❝♦♠♣❛ss❡s st❛♥❞❛r❞ s❡t✉♣

❝♦sts ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❜r✐♥❣✐♥❣ ❛ ♣r♦❞✉❝t t♦ ♠❛r❦❡t s✉❝❤ ❛s ❞❡❞✐❝❛t❡❞ ♣❧❛♥t✱ ❡q✉✐♣♠❡♥t ❛♥❞ ♠❛r✲

❦❡t✐♥❣ ❡①♣❡♥s❡s ❛s ✇❡❧❧ ❛s t❤❡ ❝♦st ♦❢ ❞❡✈❡❧♦♣✐♥❣ t❤❡ ✜r♠✬s ♣r♦❞✉❝t ✈❛r✐❛♥t✳ ❚❤❡ ✜①❡❞ ❝♦sts ♦❢

✐♥♥♦✈❛t✐♦♥ ❛♥❞ ✐♠✐t❛t✐♦♥ ❛r❡ r❡s♣❡❝t✐✈❡❧② ❞❡♥♦t❡❞ I ❛♥❞ K✳ ❲❤✐❧❡ I ✐s ❛ss✉♠❡❞ t♦ ❜❡ ♣♦s✐t✐✈❡

❛♥❞ ✜♥✐t❡✱ t❤❡ ❡①tr❡♠❡ ❝❛s❡s K = 0 ❛♥❞ K = ∞ ❛r❡ ❛❧❧♦✇❡❞ ❛♥❞ ✇❡ ❛r❡ ❛❣♥♦st✐❝ ❛❜♦✉t t❤❡

r❡❧❛t✐✈❡ ♠❛❣♥✐t✉❞❡s ♦❢I ❛♥❞K✳^✼ ■❢ t❤❡ s❡❝♦♥❞ ✜r♠ ❝❛♥ ❞❡✈❡❧♦♣ ❛♥ ❡q✉✐✈❛❧❡♥t ♣r♦❞✉❝t ❝♦♠♣❧❡t❡❧②

✐♥❞❡♣❡♥❞❡♥t❧② t❤❡♥ K =I✱ ✇❤❡r❡❛s ✐❢ t❤❡r❡ ❛r❡ ♣♦s✐t✐✈❡ s♣✐❧❧♦✈❡rsK < I, ❛♥❞ ✇✐t❤ s❝❛r❝❡ ✐♥♣✉ts

♦r ■P❘ ♣r♦t❡❝t✐♦♥ t❤❛t ❝♦♠♣❡❧s ✐♠✐t❛t♦rs t♦ ✐♥✈❡♥t ❛r♦✉♥❞ ❛♥② ✐♥t❡❧❧❡❝t✉❛❧ ♣r♦♣❡rt② ❤❡❧❞ ❜② t❤❡

✐♥♥♦✈❛t♦r K > I ❝❛♥ ❤♦❧❞✳^✽

✼❲✐t❤✐♥ t❤❡ ♣❤❛r♠❛❝❡✉t✐❝❛❧ ✐♥❞✉str② ❢♦r ✐♥st❛♥❝❡✱ t❤❡ ❝♦st ♦❢ ✐♠✐t❛t✐♦♥ ✈❛r✐❡s ❣r❡❛t❧② ❛❝r♦ss ❜✉s✐♥❡ss s❡❣♠❡♥ts✳

❚❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ✐♠✐t❛t✐♦♥ ❢♦r ❞r✉❣s ❛r❡ ❡♥t✐r❡❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤♦s❡ ❢♦r ✈❛❝❝✐♥❡s✳ P❤❛r♠❛❝❡✉t✐❝❛❧ ✜r♠s r❡❧② ♦♥

✐♥t❡❧❧❡❝t✉❛❧ ♣r♦♣❡rt② r✐❣❤ts ✐♥ ♦r❞❡r t♦ ✐♥❝r❡❛s❡ t❤❡ ❝♦sts ♦❢ ✐♠✐t❛t♦rs ❢♦r ♥❡✇ ❞r✉❣s ✏✇❤✐❝❤ ♦t❤❡r✇✐s❡ ❝♦✉❧❞ ❜❡

❝♦♣✐❡❞ ♠♦r❡ ❡❛s✐❧② t❤❛♥ ♣r♦❞✉❝ts ✇❤♦s❡ ♣r♦❞✉❝t✐♦♥ ♣r♦❝❡ss❡s ❝❛♥ ❜❡ ❦❡♣t s❡❝r❡t✱ ♦r ❢♦r ✇❤✐❝❤ t❤❡ t✐♠❡ ❛♥❞ r❡❧❛t✐✈❡

❡①♣❡♥s❡ ♥❡❡❞❡❞ t♦ ❝♦♣② t❤❡ ✐♥✈❡♥t✐♦♥ ❛r❡ ♠✉❝❤ ❤✐❣❤❡r✑ ✭❙❝❤❡r❡r ❛♥❞ ❲❛t❛❧ ❬✸✺❪✱ ♣✳ ✹✮✳ ■❢ s✉❝❤ ♣❛t❡♥t ♣r♦t❡❝t✐♦♥

✐s ♥♦t ❛✈❛✐❧❛❜❧❡✱ ❛ ❣❡♥❡r✐❝ ♣r♦❞✉❝t ❝❛♥ ❜❡ ✐♥tr♦❞✉❝❡❞ ❛t ❛ ♠✉❝❤ ❧♦✇❡r ✜①❡❞ ❝♦st t❤❛♥ ✐♥❝✉rr❡❞ ❜② t❤❡ ❜r❛♥❞❡❞

♣r♦❞✉❝t s✉♣♣❧✐❡r✳ ❍♦✇❡✈❡r✱ t❤✐s ❡❛s❡ ♦❢ ✐♠✐t❛t✐♦♥ ✐s ♥♦t ❢♦✉♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✈❛❝❝✐♥❡s✱ ✇❤✐❝❤ ❛r❡ ♠❛❞❡ ❢r♦♠ ❧✐✈✐♥❣

♠✐❝r♦✲♦r❣❛♥✐s♠s✱ ❛♥❞ ✉♥❧✐❦❡ ❞r✉❣s ✏❛r❡ ♥♦t ❡❛s✐❧② r❡✈❡rs❡✲❡♥❣✐♥❡❡r❡❞✱ ❛s t❤❡ ❣r❡❛t❡st ❝❤❛❧❧❡♥❣❡s ♦❢t❡♥ ❧✐❡ ✐♥ ❞❡t❛✐❧s

♦❢ ♣r♦❞✉❝t✐♦♥ ♣r♦❝❡ss❡s t❤❛t ❝❛♥♥♦t ❜❡ ✐♥❢❡rr❡❞ ❢r♦♠ t❤❡ ✜♥❛❧ ♣r♦❞✉❝t✱✑ ✐♠♣❧②✐♥❣ t❤❛t ✏t❤❡r❡ ✐s t❡❝❤♥✐❝❛❧❧② ♥♦ s✉❝❤

t❤✐♥❣ ❛s ❛ ❣❡♥❡r✐❝ ✈❛❝❝✐♥❡✑ ✭❲✐❧s♦♥ ❬✸✼❪✱ ♣✳ ✶✸✮✳

✽■♠♣❡r❢❡❝t ❝♦♠♣❡t✐t✐♦♥ ✐♥ t❤❡ ✐♥♣✉t ♠❛r❦❡t ❝❛♥ ❛❧s♦ ❧❡❛❞ t♦ ❛s②♠♠❡tr✐❝ ✜①❡❞ ❝♦sts ❢♦r ✐♥✐t✐❛❧❧② ✐❞❡♥t✐❝❛❧ ✜r♠s✳

❇✐❧❧❡tt❡ ❞❡ ❱✐❧❧❡♠❡✉r ❡t ❛❧✳ ❬✷❪ s❤♦✇ ❢♦r ✐♥st❛♥❝❡ t❤❛t ✐❢ t❤❡ ❝♦st ♦❢ ✐♥✈❡st♠❡♥t ✐s ❞❡t❡r♠✐♥❡❞ ❡♥❞♦❣❡♥♦✉s❧② ❜② ❛

♠♦♥♦♣♦❧② ✐♥♣✉t s✉♣♣❧✐❡r✱ t❤❡ ✜①❡❞ ❝♦st ✐s ❧♦✇❡r ❢♦r t❤❡ ✜rst ✜r♠ t❤❛t ✐♥✈❡sts✳

❚❤❡ ✜①❡❞ ❝♦st ❛s②♠♠❡tr② ✇❡ ♣♦s✐t ❝❛♥ ❛❧s♦ ❜❡ ❝♦♥tr❛st❡❞ ✇✐t❤ P❛✇❧✐♥❛ ❛♥❞ ❑♦rt ❬✷✼❪✬s ♠♦❞❡❧ ♦❢ ❞②♥❛♠✐❝

❝♦♠♣❡t✐t✐♦♥ ✐♥ ❛♥ ❛s②♠♠❡tr✐❝ ❞✉♦♣♦❧②✳ ❙✐♠✐❧❛r❧② t♦ ♦✉r ❡q✉✐❧✐❜r✐✉♠ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✐♥ ❙❡❝t✐♦♥ ✹✱ t❤❡s❡ ❛✉t❤♦rs

✐❞❡♥t✐❢② s❡✈❡r❛❧ t②♣❡s ♦❢ ❡q✉✐❧✐❜r✐❛ ✇❤✐❝❤ ❛r❡ ♣❛r❛♠❡tr✐③❡❞ ♣❛rs✐♠♦♥✐♦✉s❧② ❜② t❤❡ ❞❡❣r❡❡ ♦❢ ❡①✲❛♥t❡ ✭r❛t❤❡r t❤❛♥

✽

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❲❡ ❝♦♥s✐❞❡r ❛♥ ✐♥❞✉str② t❤❛t ✐s ❛t ❛♥ ❡❛r❧② st❛❣❡ ♦❢ ♠❛r❦❡t ❞❡✈❡❧♦♣♠❡♥t✱ ✇✐t❤ Y(0) ≤ (r−α)I/π^M✳ ❆t s✉❝❤ ❧♦✇ ♠❛r❦❡t s✐③❡✱ ❛ ♠♦♥♦♣♦❧② ✜r♠ ✇♦✉❧❞ ✐♥✐t✐❛❧❧② ♣r❡❢❡r t♦ ❞❡❧❛② ✐♥✈❡st✲

♠❡♥t✱ ❛♥❞ t❤❡ s❛♠❡ ❤♦❧❞s tr✉❡ ❢♦r t❤❡ ❞✉♦♣♦❧② ✜r♠s ✇❤♦s❡ ✐♥✈❡st♠❡♥t ❞❡❝✐s✐♦♥s ✇❡ st✉❞② ✐♥ t❤❡

s✉❜s❡q✉❡♥t s❡❝t✐♦♥s✳^✾

❋✐♥❛❧❧②✱ ✐♥ ♦r❞❡r t♦ ❞❡r✐✈❡ ♦✉r ✇❡❧❢❛r❡ r❡s✉❧ts ✇❡ s✉♣♣♦s❡ t❤❛t ❧✐❦❡ ♣r♦✜ts✱ t❤❡ ❜❛s❡❧✐♥❡ ❝♦♥s✉♠❡r s✉r♣❧✉s ✢♦✇s ✉♥❞❡r ♠♦♥♦♣♦❧② ❛♥❞ ❞✉♦♣♦❧②✱ s^M ❛♥❞ s^D✱ ✇✐t❤ 0 < s^M ≤ s^D✱ ❛r❡ s❝❛❧❡❞ ❜② t❤❡

♠❛r❦❡t s✐③❡ ♣❛r❛♠❡t❡rY(t)✱ ❛♥❞ t❤❛t t❤❡ s♦❝✐❛❧ ❞✐s❝♦✉♥t r❛t❡ ✐s ❡q✉❛❧ t♦r✳ ❚❤❡ st❛t✐❝ ✇❡❧❢❛r❡ ❣❛✐♥

❢r♦♠ ✐♠✐t❛t✐♦♥✱ s^D + 2π^D

− s^M +π^M

✱ ♣❧❛②s ❛ ❦❡② r♦❧❡ ✐♥ ♦✉r ♥♦r♠❛t✐✈❡ ❛♥❛❧②s✐s✳ ❲❡ ❛ss✉♠❡

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

❢♦r ✐♠✐t❛t✐♦♥ π^D✱ ✐♠♣❧②✐♥❣ t❤❛t s^D +π^D < s^M +π^M✳ ❆ st❛♥❞❛r❞ r❡s✉❧t ✐♥ ✐♥❞✉str✐❛❧ ❡❝♦♥♦♠✐❝s

✐s t❤❛t t❤✐s ❛ss✉♠♣t✐♦♥ ❝❤❛r❛❝t❡r✐③❡s ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ ♦❧✐❣♦♣♦❧② ♠♦❞❡❧s ✭▼❛♥❦✐✇ ❛♥❞ ❲❤✐♥st♦♥

❬✷✻❪✮✳ ❆ s❡t ♦❢ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ♦♥ ♣r♦❞✉❝t ♠❛r❦❡t ❝♦♠♣❡t✐t✐♦♥ ❢♦r ✐t t♦ ❤♦❧❞ ✐s t❤❛t ❡♥tr② r❛✐s❡ ✐♥❞✉str② ♦✉t♣✉t✱ t❤❛t ✐t ❞❡❝r❡❛s❡ t❤❡ ♣r♦✜t ♦❢ ❡①✐st✐♥❣ ✜r♠s ❛♥❞ t❤❛t ✜r♠s ♥♦t ♣r✐❝❡ ❜❡❧♦✇

♠❛r❣✐♥❛❧ ❝♦st✳^✶✵

✷✳✷ ❈♦♥t✐♥✉❛t✐♦♥ ♣❛②♦✛s

❋✐r♠s ♦❜t❛✐♥ ❝♦♥t✐♥✉❛t✐♦♥ ♣❛②♦✛s ♦♥❝❡ ✐♥♥♦✈❛t✐♦♥ ♦❝❝✉rs ✇❤✐❝❤ ❞❡♣❡♥❞ ♦♥ t❤❡✐r ♣♦s✐t✐♦♥ ✐♥ t❤❡

✐♥✈❡st♠❡♥t s❡q✉❡♥❝❡✳ ❚❤❡s❡ ♣❛②♦✛s ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛ ❣✐✈❡♥ ✈❛❧✉❡ y = Y(0) ♦❢ t❤❡ ♠❛r❦❡t s✐③❡

♣r♦❝❡ss ❛s ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♠❛r❦❡t s✐③❡ ❛t ✇❤✐❝❤ ✐♥♥♦✈❛t✐♦♥ ♦❝❝✉rs✱ ✇❤✐❝❤ ✐s ❞❡♥♦t❡❞Y ❛♥❞ s❛t✐s✜❡s Y ≥y✳ ❚❤❡② t❤✉s r❡♣r❡s❡♥t t❤❡ ❝✉rr❡♥t ✈❛❧✉❡s ♦❢ ❛♥t✐❝✐♣❛t❡❞ r❛t❤❡r t❤❛♥ ✐♥st❛♥t❛♥❡♦✉s ♣❛②♦✛s✱

❛♥❞ ❛s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ t❤❡② ❛r❡ ❞❡♥♦t❡❞ F✱ L ❛♥❞ M ❛❝❝♦r❞✐♥❣ t♦ ✇❤❡t❤❡r ❛ ✜r♠ ✐♥✈❡sts ❛s ❛

❢♦❧❧♦✇❡r✱ ❛s ❛ ❧❡❛❞❡r ♦r ✐❢ ✐♥✈❡st♠❡♥ts ❛r❡ s✐♠✉❧t❛♥❡♦✉s✳ ■♥ t❤❡ t❡r♠✐♥♦❧♦❣② ♦❢ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱

t❤❡ ✜rst ❝❛s❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✐♠✐t❛t♦r ✇❤❡r❡❛s t❤❡ ❧❛st t✇♦ ❝♦rr❡s♣♦♥❞ t♦ ✐♥♥♦✈❛t♦r ✜r♠s✱ ❛s ✇❡

r✉❧❡ ♦✉t ✐♠✐t❛t✐♦♥ ✐❢ ✐♥✈❡st♠❡♥ts ❛r❡ s✐♠✉❧t❛♥❡♦✉s✳

❚❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♣❛②♦✛ ♦❢ ❛ ❢♦❧❧♦✇❡r ✐s ♦❜t❛✐♥❡❞ ❜② st✉❞②✐♥❣ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ♦❢ ❛ ✜r♠

♦♥❝❡ ✐ts r✐✈❛❧ ❤❛s ✐♥♥♦✈❛t❡❞✳ ■t t❤❡♥ ❤♦❧❞s ❛ ❣r♦✇t❤ ♦♣t✐♦♥ ♦♥ ❛ ❞✉♦♣♦❧② ♠❛r❦❡t ❛♥❞ ✐ts ♦♣t✐♠❛❧

❡①✲♣♦st✮ ❛s②♠♠❡tr②✳

✾■❢ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s ♥♦t s❛t✐s✜❡❞✱ t❤❡♥ ✐♠♠❡❞✐❛t❡ ✐♥✈❡st♠❡♥t ❛t Y(0) ❝❛♥ ❛r✐s❡ ❛s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❞❡❝✐s✐♦♥

♣r♦❜❧❡♠s st✉❞✐❡❞ ✐♥ ❙❡❝t✐♦♥s ✸ ❛♥❞ ✹✱ ❛♥❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♠✉st ❜❡ ❡①♣❛♥❞❡❞ t♦ ❛❝❝♦✉♥t ❢♦r s✉❝❤

❝❛s❡s✳

✶✵❚❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐s ❦♥♦✇♥ ❛s t❤❡ ❜✉s✐♥❡ss✲st❡❛❧✐♥❣ ❡✛❡❝t✱ ✇❤✐❝❤ ❤♦❧❞s ❜② ❛ss✉♠♣t✐♦♥ ✐♥ ♦✉r ♠♦❞❡❧ s✐♥❝❡

π^D< π^M❛♥❞ ❝❤❛r❛❝t❡r✐③❡s ❝♦♠♣❡t✐t✐♦♥ ❜❡t✇❡❡♥ ✜r♠s ♣r♦❞✉❝✐♥❣ s✉❜st✐t✉t❡ ♣r♦❞✉❝ts ✐♥ ❛ ❜r♦❛❞ r❛♥❣❡ ♦❢ ✐♥❞✉str✐❡s✳

■♥ ❛♥ ❡♠♣✐r✐❝❛❧ st✉❞② ♦❢ t❤❡ ❤②♣❡rt❡♥s✐♦♥ ❞r✉❣ ♠❛r❦❡t ❢♦r ❡①❛♠♣❧❡✱ ❇r❛♥st❡tt❡r✱ ❈❤❛tt❡r❥❡❡ ❛♥❞ ❍✐❣❣✐♥s ❬✺❪ ❡st✐♠❛t❡

t❤❡ ✐♠♣❛❝t ♦❢ ✐♠✐t❛t✐♦♥ ✭❣❡♥❡r✐❝ ❡♥tr②✮ ♦♥ ♣r♦❞✉❝❡r ♣r♦✜ts ❛♥❞ ♦♥ ❝♦♥s✉♠❡r s✉r♣❧✉s ✜♥❞✐♥❣ ♦♥❧② ❛ ♠♦❞❡st ♣♦s✐t✐✈❡

❡✛❡❝t ♦♥ s♦❝✐❛❧ ✇❡❧❢❛r❡ ♦✇✐♥❣ t♦ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❜✉s✐♥❡ss st❡❛❧✐♥❣ ❡✛❡❝t ✭❣❡♥❡r✐❝ s❛❧❡s ❞✐s♣❧❛❝❡ ❜r❛♥❞❡❞ s❛❧❡s✮✳

✾

(11)

♣♦❧✐❝② ✐s t♦ ❞❡✈❡❧♦♣ t❤❡ ✐♠✐t❛t✐✈❡ ♣r♦❞✉❝t ✇❤❡♥❡✈❡r t❤❡ ♠❛r❦❡t s✐③❡ r❡❛❝❤❡s ❛♥ ❡①❡r❝✐s❡ t❤r❡s❤♦❧❞

✇❡ ❞❡♥♦t❡ ❜② Y^F✳ ❙t❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✭s❡❡ ❆♣♣❡♥❞✐① ❆✳✶✮ ❡st❛❜❧✐s❤ t❤❛t ✐❢ t❤❡ ❝✉rr❡♥t ♠❛r❦❡t s✐③❡ ✐s Y(t) =Y t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈❛❧✉❡ ♦❢ t❤✐s ♦♣t✐♦♥ ✐s

V^D(Y) = sup

τ≥tEY

Z ∞ τ

π^DY(s)e^−rsds−e^−rτK

=

( AY^β✱ Y < Y^F

π^D

r−αY −K✱ Y ≥Y^F ✱ ✭✶✮

✇❤❡r❡

Y^F := β β−1

r−α π^D K

✐s t❤❡ ❡①❡r❝✐s❡ t❤r❡s❤♦❧❞✱ β ✐s s❤♦rt❤❛♥❞ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs β(α, σ, r) := 1

2 − α σ² +

s1 2 − α

σ² 2

+ 2r

σ² ✭✷✮

❛♥❞ A:= (K/(β−1)) Y^F−β

✐s ❛ ❝♦♥st❛♥t✳ ❚❤❡ ❞✐s❝♦✉♥t✐♥❣ ♣❛r❛♠❡t❡r ✭✷✮ s❛t✐s✜❡s β > 1 ❛♥❞

∂β/∂α, ∂β/∂σ, −∂β/∂r < 0✳ ❚❤❡ ❡①♣❡❝t❡❞ ❞✐s❝♦✉♥t❡❞ ✈❛❧✉❡ ❛t t✐♠❡ t = 0 ♦❢ ♦❜t❛✐♥✐♥❣ t❤❡

❞✉♦♣♦❧② ♦♣t✐♦♥V^D(Y) ❛t Y ✇❤❡♥ r✐✈❛❧ ✐♥♥♦✈❛t✐♦♥ ♦❝❝✉rs ✐s ❛❝❝♦r❞✐♥❣❧② F(Y) = Ey

"Z _∞

τ(max{Y,Y^F})

π^DY(s)e^−rsds−e^−rτ(^max{^Y,Y^F})K

#

=





Ay^β✱ Y < Y^F π^D

r−αY −K

y Y

β✱ Y ≥Y^F ✭✸✮

✇❤❡r❡ ✐♥ t❤❡ ✜rst ❧✐♥❡ τ(Z) := inf{t≥0|Y(t)≥Z} ✐s t❤❡ st♦❝❤❛st✐❝ t✐♠❡ ❛t ✇❤✐❝❤ t❤❡ ♠❛r❦❡t s✐③❡ ♣r♦❝❡ss ✜rst ❤✐ts ❛ t❤r❡s❤♦❧❞ Z ❛♥❞ Y^F ♥♦✇ r❡♣r❡s❡♥ts t❤❡ ✐♠✐t❛t♦r ✭♦r ❢♦❧❧♦✇❡r✮ ✐♥✈❡st♠❡♥t t❤r❡s❤♦❧❞✳^✶✶

●✐✈❡♥ t❤❛t t❤❡ ♦♣t✐♠❛❧ ♣♦❧✐❝② ♦❢ t❤❡ ❢♦❧❧♦✇❡r ✐s t♦ ✐♥✈❡st ♦♥❝❡ Y^F ✐s r❡❛❝❤❡❞✱ t❤❡ ❧❡❛❞❡r ♣❛②♦✛ ❛t t✐♠❡t= 0 t❤❛t ❛ ✜r♠ r❡❝❡✐✈❡s ❢r♦♠ ✐♥♥♦✈❛t✐♥❣ ✜rst ❛t ❛ t❤r❡s❤♦❧❞Y ✐s

L(Y) = Ey

"Z _τ(max{Y,Y^F})

τ(Y)

π^MY(s)e^−rsds−e^−rτ(Y⁾I+ Z _∞

τ(max{Y,Y^F})

π^DY(s)e^−rsds

#

=

π^M

r−αY −I y Y

β

−π^M −π^D r−α

y^β

[max{Y, Y^F}]^β−1 ✭✹✮

✶✶❆ st❛♥❞❛r❞ ♣r♦♣❡rt② ♦❢β ✇❤✐❝❤ ✐s ❛♣♣❛r❡♥t ✐♥ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♣❛②♦✛s ✐s t❤❛t t❤❡ ❡①♣❡❝t❡❞ ❞✐s❝♦✉♥t❡❞ ✈❛❧✉❡

♦❢ ❛ ♠♦♥❡t❛r② ✉♥✐t r❡❝❡✐✈❡❞ ❛t t❤❡ ✜rst ❤✐tt✐♥❣ t✐♠❡ τ(Y) ♦❢ t❤r❡s❤♦❧❞Y ≥y✐sEye⁻^rτ(Y⁾ = (y/Y)^β✳ ◆♦t❡ t❤❛t limσ→0β=r/α✳

✶✵

(12)

✭s❡❡ ❆♣♣❡♥❞✐① ❆✳✷✮✳ ❚❤❡ ✜rst s✉♠♠❛♥❞ ✐♥ ✭✹✮ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❞✐s❝♦✉♥t❡❞ ✈❛❧✉❡ ♦❢ ♣❡r♣❡t✉❛❧

♠♦♥♦♣♦❧② ♣r♦✜ts ❢r♦♠ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ t❤❡ s❡❝♦♥❞ ❝♦rr❡❝ts ❢♦r t❤❡ ❛♥t✐❝✐♣❛t❡❞ r❡❞✉❝t✐♦♥ ✐♥ ♣r♦✜t

✢♦✇ st❡♠♠✐♥❣ ❢r♦♠ ✐♠✐t❛t✐♦♥✳ ▼♦r❡♦✈❡r✱ t❤❡ ✜rst s✉♠♠❛♥❞ ✐s ❛ str✐❝t❧② q✉❛s✐❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢

Y✱ ✇❤✐❝❤ ❛tt❛✐♥s ✐ts ♠❛①✐♠✉♠ ❛t t❤❡ st❛♥❞❛❧♦♥❡ ♠♦♥♦♣♦❧② t❤r❡s❤♦❧❞

Y^L:= β β−1

r−α π^M I✳

■❢ ❜♦t❤ ✜r♠s ✐♥✈❡st t♦❣❡t❤❡r ✜♥❛❧❧②✱ ❛t ❛ ♠❛r❦❡t s✐③❡ t❤r❡s❤♦❧❞Y✱ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♣❛②♦✛ ❢r♦♠

s✐♠✉❧t❛♥❡♦✉s ✐♥✈❡st♠❡♥ts ✐s

M(Y) = Ey

"Z ∞ τ(Y)

π^DY(s)e^−rsds−e^−rτ(Y⁾I

#

=

π^D

r−αY −I y Y

β

. ✭✺✮

❚❤✐s ♣❛②♦✛ ✐s ❛ str✐❝t❧② q✉❛s✐❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ Y ✇❤✐❝❤ ❛tt❛✐♥s ✐ts ♠❛①✐♠✉♠ ❛t t❤❡ t❤r❡s❤♦❧❞

Y^M := β β−1

r−α π^D I✱

✇❤✐❝❤ s❛t✐s✜❡s Y^M > Y^L✳ ❚❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢M(Y) ✐♥❝♦r♣♦r❛t❡s s✐♠✉❧t❛♥❡♦✉s ✐♥♥♦✈❛t✐♦♥ s♦ t❤❛t t❤❡ ✜①❡❞ ❝♦st ✐sI ❢♦r ❜♦t❤ ✜r♠s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❞✉❡✱ s❛②✱ t♦ ❛ ❧❛❣ ❢♦r ✐♥❢♦r♠❛t✐♦♥ s♣✐❧❧♦✈❡rs t♦ t❛❦❡

♣❧❛❝❡ r✉❧✐♥❣ ♦✉t t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❛♥② ✐♠✐t❛t✐♦♥✳^✶✷

✸ Pr❡❞❡t❡r♠✐♥❡❞ ✐♥✈❡st♠❡♥t s❡q✉❡♥❝❡

❚♦ ❣❡t ❛ ✜rst ✐♥s✐❣❤t ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝s ♦❢ ✐♥♥♦✈❛t✐♦♥ ❛♥❞ ✐♠✐t❛t✐♦♥ ✐♥ ♦✉r ♠♦❞❡❧✱ ✇❡ st✉❞②

✐♥✈❡st♠❡♥t ❞❡❝✐s✐♦♥s ✇❤❡♥ ✜r♠ r♦❧❡s ❛s ❧❡❛❞❡r ♦r ❢♦❧❧♦✇❡r ❛r❡ ❡①♦❣❡♥♦✉s✳ ❖♥❡ ✇❛② t❤✐s s✐t✉❛t✐♦♥

❝❛♥ ❛r✐s❡ ✐s t❤r♦✉❣❤ ❝❡rt❛✐♥ ✐♥❞✉str✐❛❧ ♣♦❧✐❝✐❡s✱ ❢♦r ❡①❛♠♣❧❡ ✐❢ ❛ ❢♦r♠❡r st❛t❡✲r✉♥ ♠♦♥♦♣♦❧✐st ❡♥❥♦②s

♣r✐♦r✐t② ♠❛r❦❡t ❛❝❝❡ss ✐♥ ❛ ❞❡r❡❣✉❧❛t❡❞ ✐♥❞✉str②✳ ❇❡s✐❞❡s ✐ts ✐♥❤❡r❡♥t ✐♥t❡r❡st t❤✐s ✐s ❛ ✉s❡❢✉❧ ❝❛s❡

t♦ ❡①❛♠✐♥❡ ❢r♦♠ ❛ t❤❡♦r❡t✐❝❛❧ st❛♥❞♣♦✐♥t✱ ✜rst ❜❡❝❛✉s❡ ✐t ❧❡❛❞s ✉s t♦ ✐❞❡♥t✐❢② ❛ ❝r✐t✐❝❛❧ ✐♠✐t❛t✐♦♥

❝♦st ❧❡✈❡❧✱ Ke✱ ✇❤✐❝❤ ✐s r❡❢❡rr❡❞ t♦ t❤r♦✉❣❤♦✉t t❤❡ r❡st ♦❢ ♦✉r ❛♥❛❧②s✐s✱ ❛♥❞ s❡❝♦♥❞ ❜❡❝❛✉s❡ t❤❡

❡q✉✐❧✐❜r✐✉♠ ✇✐t❤♦✉t ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ t❤❛t ✇❡ ❞❡r✐✈❡ ✐♥ t❤✐s s❡❝t✐♦♥ ♣r♦✈✐❞❡s ❛ ❜❡♥❝❤♠❛r❦ t♦

❛ss❡ss t❤❡ ❡✛❡❝t ♦❢ ❞②♥❛♠✐❝ ❝♦♠♣❡t✐t✐♦♥ ❢✉rt❤❡r ❜❡❧♦✇ ✇❤❡♥ ✇❡ ❡♥❞♦❣❡♥✐③❡ ✜r♠ r♦❧❡s✳

✶✷❆♥ ❛❧t❡r♥❛t✐✈❡ ❛ss✉♠♣t✐♦♥ ✇♦✉❧❞ ❜❡ t❤❛t ✜①❡❞ ❝♦sts ❛r❡ ❛ttr✐❜✉t❡❞ r❛♥❞♦♠❧② ✇❤❡♥ ✐♥✈❡st♠❡♥ts ❛r❡ s✐♠✉❧t❛♥❡♦✉s s♦ t❤❡ ❡①♣❡❝t❡❞ ✜①❡❞ ❝♦st ❛t t❤❡ ♠♦♠❡♥t ♦❢ ✐♥✈❡st♠❡♥t ✐s(I+K)/2✳ ❲✐t❤ t❤✐s ❛ss✉♠♣t✐♦♥✱ t❤❡ ✜rst ✐♥t❡rs❡❝t✐♦♥

♦❢L(Y)❛♥❞M(Y)✐s ♥♦ ❧♦♥❣❡r ❛tY^F ❛♥❞ t❤❡s❡ t✇♦ ❢✉♥❝t✐♦♥s ♥♦ ❧♦♥❣❡r ♦✈❡r❧❛♣ ✐❢I6=K✳

✶✶