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❉②♥❛♠✐❝ ❡❝♦♥♦♠✐❝s ✐♥ Pr❛❝t✐❝❡

▼♦♥✐❝❛ ❈♦st❛ ❉✐❛s ❛♥❞ ❈♦r♠❛❝ ❖✬❉❡❛

▼♦t✐✈❛t✐♦♥

◮ ▼❛♥② ❡❝♦♥♦♠✐❝ ❞❡❝✐s✐♦♥s ✭❡✳❣✳ ❡❞✉❝❛t✐♦♥ t❛❦❡✲✉♣✱ s❛✈✐♥❣s ♦r ✐♥✈❡st♠❡♥ts✮ ❛r❡ ❞✐✣❝✉❧t t♦ r❛t✐♦♥❛❧✐s❡ ✐♥ ❛ st❛t✐❝ s❡tt✐♥❣ ◮ ❚❤❡② ❛❧❧ ✐♥✈♦❧✈❡ s♦♠❡ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ ♣r❡s❡♥t ❝♦sts ❛♥❞ ❢✉t✉r❡ r❡t✉r♥s✱ s♦♠❡t✐♠❡s ✐♥ ✉♥❝❡rt❛✐♥ ❡♥✈✐r♦♥♠❡♥ts ◮ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦♠❡ ♠❛r❦❡ts ✭❝r❡❞✐t✱ ✐♥s✉r❛♥❝❡✳✳✳✮ ❤✐♥❣❡s ♦♥ t❤❡ ❞②♥❛♠✐❝ ♥❛t✉r❡ ♦❢ s♦♠❡ ❞❡❝✐s✐♦♥s ◮ ❚❤❡✐r ❡①✐st❡♥❝❡ ♠❛② r❡✐♥❢♦r❝❡ t❤❡ ❞②♥❛♠✐❝ ♥❛t✉r❡ ♦❢ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❝❡ss

❚❤❡ ♣r♦❜❧❡♠ ■

❉②♥❛♠✐❝ ♠✐❝r♦❡❝♦♥♦♠✐❝ ♣r♦❜❧❡♠s ❛r❡ ♥♦t❛❜❧② ❞✐✣❝✉❧t t♦ s♦❧✈❡ ◮ ❱❡r② ❤✐❣❤ ❞✐♠❡♥s✐♦♥❛❧

◮ Pr❡s❡♥t ❝♦st ♦❢ ❛ ❞❡❝✐s✐♦♥ ❞❡♣❡♥❞s ♦♥ ♣r❡s❡♥t ❝✐r❝✉♠st❛♥❝❡s✱ ❛♥❞ t❤❡s❡ ❛r❡ ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ♣❛st ❝✐r❝✉♠st❛♥❝❡s ❛♥❞ ❞❡❝✐s✐♦♥s ◮ ❋✉t✉r❡ r❡t✉r♥s ♠❛② ❛❧s♦ ❞❡♣❡♥❞ ♦♥ ♣r❡s❡♥t ❝✐r❝✉♠st❛♥❝❡s ❛♥❞ ❜❡ r❡❛❧✐s❡❞ ✐♥ ♠❛♥② ♣❡r✐♦❞s ❛♥❞ ✐♥ ❞✐✛❡r❡♥t ✇❛②s✱ ♣♦ss✐❜❧② ✐♥✢✉❡♥❝✐♥❣ ❢✉t✉r❡ ❞❡❝✐s✐♦♥s

◮ ■♥ ♠♦st ❝❛s❡s✱ ❞②♥❛♠✐❝ ♣r♦❜❧❡♠s ❛r❡ ♥♦t tr❛❝t❛❜❧❡ ❛♥❛❧②t✐❝❛❧❧② ◮ P♦ss✐❜❧❡ s♦❧✉t✐♦♥✿ ❜r❡❛❦ t❤❡ ❜✐❣ ♣r♦❜❧❡♠ ✐♥t♦ ❛ s❡q✉❡♥❝❡ ♦❢ s✐♠✐❧❛r s♠❛❧❧❡r ♣r♦❜❧❡♠s t❤❛t ✇❡ ❝❛♥ s♦❧✈❡ ✲ ✉s❡ ❘❡❝✉rs✐✈❡ ▼❡t❤♦❞s

❚❤❡ ♣r♦❜❧❡♠ ■■

◮ ❙♦❧✉t✐♦♥ ✇❡ ❡①♣❧♦r❡✿ ❜r❡❛❦ t❤❡ ❜✐❣ ♣r♦❜❧❡♠ ✐♥t♦ ❛ s❡q✉❡♥❝❡ ♦❢ s✐♠✐❧❛r s♠❛❧❧❡r ♣r♦❜❧❡♠s t❤❛t ✇❡ ❝❛♥ s♦❧✈❡ ◮ ❚❤✐s ✐s ✇❤❛t ❛ r❡❝✉rs✐✈❡ ♠❡t❤♦❞ ❝❛❧❧❡❞ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❞♦❡s

◮ ❉❡s❝r✐❜❡ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ❛t ❛ ♠♦♠❡♥t ✐♥ t✐♠❡✿ t❤❡ st❛t❡ ♦❢ t❤❡ ✇♦r❧❞ t♦❞❛② ✕ ✐t s✉♠♠❛r✐s❡s ❛❧❧ t❤❡ ❝✉rr❡♥t ✐♥❢♦r♠❛t✐♦♥ r❡❧❡✈❛♥t ❢♦r ❞❡❝✐s✐♦♥✲♠❛❦✐♥❣ ◮ ❲❤❡r❡ ✐t ♠✐❣❤t ❜❡ t♦♠♦rr♦✇✿ t❤❡ st❛t❡ ♦❢ t❤❡ ✇♦r❧❞ t♦♠♦rr♦✇ ◮ ❆♥❞ ❤♦✇ t❤❡ ❛❣❡♥ts ❝❛r❡ ❛❜♦✉t t♦♠♦rr♦✇ ✈✐s✲❛✲✈✐s t♦❞❛②

◮ ❉P ❛❧❧♦✇s ✉s t♦ ❝❤❛r❛❝t❡r✐s❡ t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ t✇♦ ❢✉♥❝t✐♦♥s

◮ ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ ♠❛♣s t❤❡ st❛t❡ t♦❞❛② ✐♥t♦ t❤❡ st❛t❡ t♦♠♦rr♦✇ ◮ ❈❤♦✐❝❡ ❢✉♥❝t✐♦♥✿ ♠❛♣s t❤❡ st❛t❡ t♦❞❛② ✐♥t♦ t❤❡ ❡♥❞♦❣❡♥♦✉s ❝❤♦✐❝❡s

❚❤✐s ❝♦✉rs❡ ■

◮ ●❡♥t❧❡ ❛♥❞ ♣r❛❝t✐❝❛❧ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❞②♥❛♠✐❝ ♦♣t✐♠✐s❛t✐♦♥

◮ ❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ◮ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ◮ ❈♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞s

◮ ▼❛✐♥ ❣♦❛❧s

◮ ■♥tr♦❞✉❝❡ st❛♥❞❛r❞ t♦♦❧s t♦ st✉❞② ❛♥❞ s♦❧✈❡ ❞②♥❛♠✐❝ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s ✐♥ ♠✐❝r♦❡❝♦♥♦♠✐❝s ◮ ❉❡♠♦♥str❛t❡ ♣r❛❝t✐❝❛❧❧② ❤♦✇ t❤❡s❡ t♦♦❧s ❛r❡ ✉s❡❞ ◮ ❉✐s❝✉ss t❤❡✐r ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡s ◮ ❋♦❝✉s ♦♥ ♠❡t❤♦❞s ❛♥❞ t♦♦❧s t❤❛t ❝❛♥ ❜❡ ❡❛s✐❧② ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❣❡♥❡r❛❧ ❛♥❞ ❝♦♠♣❧❡① s❡t✉♣s

❚❤✐s ❝♦✉rs❡ ■■

◮ ❲♦r❦❤♦rs❡✿ t❤❡ ❝♦♥s✉♠♣t✐♦♥✲s❛✈✐♥❣s ♠♦❞❡❧

◮ ■♥t❡r❡st✐♥❣ ♣❡r✲s❡✿ ❛ ❦❡② ♠♦❞❡❧ ✐♥ ❡❝♦♥♦♠✐❝s✱ ✉♥❞❡r❧②✐♥❣ t❤❡ ♣❡r♠❛♥❡♥t ✐♥❝♦♠❡ t❤❡♦r② ❛♥❞ ❛❧❧ ❞❡✈❡❧♦♣♠❡♥ts t❤❛t ❤✐♥❣❡ ♦♥ ✐t ◮ ■♥❤❡r❡♥t❧② ❞②♥❛♠✐❝ ◮ ▼❛♥② ✐♥t❡r❡st✐♥❣ ✈❛r✐❛t✐♦♥s ✉s❡❢✉❧ t♦ ✐❧❧✉str❛t❡ ❤♦✇ t♦ t❛❝❦❧❡ ❛❧t❡r♥❛t✐✈❡ ❞②♥❛♠✐❝ ♣r♦❜❧❡♠s✿ ✉♥❝❡rt❛✐♥t②✱ r✐s❦ ❛✈❡rs✐♦♥✱ ❧✐❢❡✲❝②❝❧❡✴✐♥✜♥✐t❡ ❤♦r✐③♦♥✱ ❤❛❜✐t ❢♦r♠❛t✐♦♥✱ ♠❛♥② ❝❤♦✐❝❡ ♦r st❛t❡ ✈❛r✐❛❜❧❡s✱ ✳✳✳ ◮ ❱❛r✐♦✉s ❛❧t❡r♥❛t✐✈❡ s♣❡❝✐✜❝❛t✐♦♥s r❡✢❡❝t ✉♥❞❡r❧②✐♥❣ ❛ss✉♠♣t✐♦♥s ❛❜♦✉t ♠❛r❦❡t str✉❝t✉r❡ ◮ ❈r✉❝✐❛❧ t♦♦❧ ❢♦r ♣♦❧✐❝② ❛♥❛❧②s✐s

❚❤✐s ❝♦✉rs❡ ■■■

◮ Pr❛❝t✐❝❛❧ ❢♦❝✉s

◮ ❉✐s❝✉ss t❤❡ ❛♣♣r♦❛❝❤❡s ❛♥❞ ♣r♦❝❡❞✉r❡s ✇❡ ❢♦✉♥❞ ✉s❡❢✉❧ ◮ ❲❤✐❧❡ ❦❡❡♣✐♥❣ ❛♥ ❡②❡ ♦♥ ❡✣❝✐❡♥❝② ✭❜✉t ✐t ✇✐❧❧ ♥♦t ❜❡ ❝❡♥tr❛❧✮ ◮ ❲❡ ♠❛❦❡ ♥♦ ❛tt❡♠♣t t♦ ❞✐s❝✉ss ❝♦♠♣r❡❤❡♥s✐✈❡❧② t❤❡ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥s ♦❢ t❤❡ ♣r♦❜❧❡♠ ♦r s♦❧✉t✐♦♥ ◮ ●♦❛❧✿ t♦ s♦❧✈❡ ✐♥❝r❡❛s✐♥❣❧② ♠♦r❡ r❡❛❧✐st✐❝ ✭❜✉t ❛❧s♦ ♠♦r❡ ❝♦♠♣❧❡①✮ ♠♦❞❡❧s s❤♦✇✐♥❣ ♠❡t❤♦❞s t❤❛t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ❛♥❞ ❛♣♣❧✐❡❞ t♦ ♦t❤❡r s❡tt✐♥❣s

❖✉t❧✐♥❡ ♦❢ t❤✐s ❝♦✉rs❡

✶✳ ❚❤❡ s✐♠♣❧❡st ❝♦♥s✉♠♣t✐♦♥✲s❛✈✐♥❣s ♣r♦❜❧❡♠✿ t❤❡ ❝❛❦❡✲❡❛t✐♥❣ ♣r♦❜❧❡♠

❚❤❡ ♣r♦❜❧❡♠❀ ❙✐♠♣❧❡ ❡①❛♠♣❧❡❀ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥

✷✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣

❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥❀ ❘❡❝✉rs✐✈❡ s♦❧✉t✐♦♥❀ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s❀ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥❀ Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥

✸✳ ▲✐❢❡✲❝②❝❧❡ ✐♥❝♦♠❡ ♣r♦❝❡ss

❈r❡❞✐t ♠❛r❦❡ts❀ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥❀ Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥

✹✳ ❙t♦❝❤❛st✐❝ ♦♣t✐♠✐s❛t✐♦♥

▼❛r❦♦✈ ♣r♦❝❡ss❡s❀ ■✐❞ ✐♥❝♦♠❡ ♣r♦❝❡ss❀ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥❀ Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥❀ ❆✉t♦❝♦rr❡❧❛t❡❞ ✐♥❝♦♠❡ ♣r♦❝❡ss❀ Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥

✺✳ ■♥✜♥✐t❡ ❤♦r✐③♦♥

❚❤❡ ♣r♦❜❧❡♠❀ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥❀ ❙✐♠♣❧❡ ❡①❛♠♣❧❡❀ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥❀ Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥

❚❤❡ ❝❛❦❡✲❡❛t✐♥❣ ♣r♦❜❧❡♠ ❙❡t✉♣ ❛♥❞ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥

❚❤❡ ❝❛❦❡✲❡❛t✐♥❣ ♣r♦❜❧❡♠

❙✐♠♣❧❡st ♣♦ss✐❜❧❡ ❧✐❢❡✲❝②❝❧❡ ❝♦♥s✉♠♣t✐♦♥✲s❛✈✐♥❣s ♣r♦❜❧❡♠ ◮ ■♥t❡rt❡♠♣♦r❛❧ ♣r♦❜❧❡♠ ♦❢ ❛ ❝♦♥s✉♠❡r ❧✐✈✐♥❣ ❢♦r T ♣❡r✐♦❞s ❛♥❞ ❡♥❞♦✇❡❞ ✇✐t❤ ✐♥✐t✐❛❧ ✇❡❛❧t❤ a✶ ✐♥ ♣❡r✐♦❞ t = ✶ ◮ ❍❡r ❣♦❛❧✿ t♦ ❛❧❧♦❝❛t❡ t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ t❤✐s ✇❡❛❧t❤ ♦✈❡r ❤❡r T ♣❡r✐♦❞s ♦❢ ❧✐❢❡ ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐s❡ ❤❡r ❧✐❢❡t✐♠❡ ✇❡❧❧❜❡✐♥❣ ◮ ❈♦♥s✉♠♣t✐♦♥ ✐s ❞✐✈✐s✐❜❧❡✿ ❛ ❝♦♥t✐♥✉♦✉s ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡ ◮ ❆♥② r❡♠❛✐♥✐♥❣ ✇❡❛❧t❤ ✐♥ ♣❡r✐♦❞ t ✐s ♣r♦❞✉❝t✐✈❡✱ ❣❡♥❡r❛t✐♥❣ k(a) ✉♥✐ts ♦❢ ✇❡❛❧t❤ t♦ ❝♦♥s✉♠❡ ✐♥ t❤❡ ❢✉t✉r❡ ◮ ◆♦ ♦✉tst❛♥❞✐♥❣ ❞❡❜ts ❛r❡ ❛❧❧♦✇❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❧✐❢❡ ◮ ❆♥❞ ❛♥② r❡♠❛✐♥✐♥❣ ✇❡❛❧t❤ ❛t t❤❡ ❡♥❞ ♦❢ ❧✐❢❡ ✐s ♦❢ ♥♦ ✈❛❧✉❡

❋♦r♠❛❧ ♠♦❞❡❧

max

(c✶,...,cT )∈CT T

• t=✶

βt−✶u(ct) s✳t at+✶ = k(at − ct) ❢♦r t = ✶, . . . , T aT+✶ ≥ ✵ a✶ (∈ A) ❣✐✈❡♥ ◮ P❡r✲♣❡r✐♦❞ ✇❡❧❧❜❡✐♥❣ u✿ ✐♥❝r❡❛s✐♥❣ ✐♥ ❝♦♥s✉♠♣t✐♦♥ ◮ ❈♦♥s✉♠♣t✐♦♥✿ ❝❤♦✐❝❡ ✈❛r✐❛❜❧❡✱ ✇✐t❤ ❞♦♠❛✐♥ C ✭❤❡r❡ R+

✵ ♦r R+✱

❞❡♣❡♥❞✐♥❣ ♦♥ u✮ ◮ ❆ss❡ts ✐s t❤❡ st❛t❡ ✈❛r✐❛❜❧❡✱ ✇✐t❤ ❞♦♠❛✐♥ A ✭❤❡r❡ R+

✵ ♦r R+✮

◮ k✿ ❧❛✇ ♦❢ ♠♦t✐♦♥ ❢♦r ❛ss❡ts✱ ❛ ♣♦s✐t✐✈❡ ❛♥❞ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ✐♥ A k (at − ct) = R (at − ct) ✇❤❡r❡ R = ✶ + r ✐s t❤❡ ✐♥t❡r❡st ❢❛❝t♦r

❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥

◮ ❖❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s C✶ ✭❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✮✿ ✐♥t❡r✐♦r ♦♣t✐♠✉♠ s❛t✐s✜❡s ❢♦❝ ◮ ❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥✿ ❛tt❛❝❦ ♣r♦❜❧❡♠ ❞✐r❡❝t❧② ❜② s♦❧✈✐♥❣ ❛❧❧ ✐ts ❢♦❝✬s ◮ ❯s❡❢✉❧ t♦ ✇r✐t❡ ♠♦❞❡❧ r❡str✐❝t✐♦♥s ♠♦r❡ ❝♦♠♣❛❝t❧② ❜② ♥♦t✐♥❣ t❤❛t t❤❡ ❧❛✇ ♦❢ ♠♦t✐♦♥ ❢♦r ❛ss❡ts t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐♠♣❧② aT+✶ = RTa✶ −

T

• t=✶

RT−t+✶ct ◮ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦♥s✉♠❡r✬s ♣r♦❜❧❡♠ ❢♦r ❛ ❣✐✈❡♥ a✶ ≥ ✵ ✐s max

(c✶,...,cT )∈(C)T T

• t=✶

βt−✶u(ct) s✳t

T

• t=✶

R✶−tct ≤ a✶

❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥✿ ❊✉❧❡r ❡q✉❛t✐♦♥ ■

◮ ▲❛❣r❛♥❣✐❛♥ ❢♦r t❤✐s ♣r♦❜❧❡♠ L =

T

• t=✶

βt−✶u(ct) − λ T

• t=✶

R✶−tct − a✶

• ◮ ❲✐t❤ ♥❡❝❡ss❛r② ❢♦❝✬s ✇✐t❤ r❡s♣❡❝t t♦ ct✱ ❢♦r t = ✶, . . . , T✿

βt−✶u′ (ct) = λR✶−t ◮ P✉tt✐♥❣ t♦❣❡t❤❡r t✇♦ s✉❜s❡q✉❡♥t ❝♦♥❞✐t✐♦♥s ②✐❡❧❞s u′ (ct) = βRu′ (ct+✶) ❢♦r t = ✶, . . . , T − ✶ ✭✶✮ ◮ ❚❤❡s❡ ❛r❡ t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥s ❢♦r t❤✐s ♣r♦❜❧❡♠

❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥✿ ❊✉❧❡r ❡q✉❛t✐♦♥ ■■

u′ (ct) = βRu′ (ct+✶) ❢♦r t = ✶, . . . , T − ✶ ◮ ❊✉❧❡r ❡q✉❛t✐♦♥✿ ❡st❛❜❧✐s❤❡s r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❝♦♥s✉♠♣t✐♦♥ ✐♥ s✉❜s❡q✉❡♥t ♣❡r✐♦❞s ◮ ❇✉t ♥♦t t❤❡ ❝♦♥s✉♠♣t✐♦♥ ❧❡✈❡❧ ◮ ❋♦r t❤❛t ✇❡ ♥❡❡❞ t❤❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t ◮ ❚❤❡ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s ❞♦ ❥✉st t❤❛t

❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥✿ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s

◮ ❚❤❡ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s ❢♦r t❤✐s ♣r♦❜❧❡♠✿ λ T

• t=✶

R✶−tct − a✶

• = ✵,

λ ≥ ✵,

T

• t=✶

R✶−tct ≤ a✶ ◮ ■❢ u str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✭u′ > ✵✮✿

◮ λ > ✵✿ ✰✈❡ ♠❛r❣✐♥❛❧ ✈❛❧✉❡ ♦❢ r❡❧❛①✐♥❣ t❤❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t ◮

t=✶,...,T R✶−tct = a✶✿ ❝♦♥s✉♠❡r ❜❡tt❡r ♦✛ ❜② ❝♦♥s✉♠✐♥❣ ❛❧❧ a✶

◮ ❚❤❡♥ aT+✶ = ✵ ✭✷✮

◮ ❚♦❣❡t❤❡r✱ t❤❡ T ❝♦♥❞✐t✐♦♥s ✭❄❄✮ ❛♥❞ ✭❄❄✮ ❞❡t❡r♠✐♥❡ t❤❡ T ✐♥t❡r✐♦r ♦♣t✐♠❛❧ ❝♦♥s✉♠♣t✐♦♥ ❝❤♦✐❝❡s

❈♦r♥❡r s♦❧✉t✐♦♥s

◮ ❯♣ t♦ ❤❡r❡ ✇❡ ❛ss✉♠❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ ✐s ✐♥t❡r✐♦r ◮ ❚❤❡ ❊✉❧❡r ❝♦♥❞✐t✐♦♥s ❛❧❧♦✇✐♥❣ ❢♦r ❝♦r♥❡r s♦❧✉t✐♦♥s ❛r❡ u′(ct) ≤ βRu′(ct+✶) ❢♦r t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ct = ✵ ♦r u′(ct) ≥ βRu′(ct+✶) ❢♦r t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ct = at ◮ ❚②♣✐❝❛❧ ❝❤♦✐❝❡s ♦❢ ✉t✐❧✐t② ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ✐♥ R+✱ ✇✐t❤ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿ lim

ct→✵+ u(ct) = −∞

❛♥❞ lim

ct→✵+ u′(ct) = +∞

■♥ t❤✐s ❝❛s❡ ❛ s♦❧✉t✐♦♥✱ ✐❢ ✐t ❡①✐sts✱ ✐s ✐♥t❡r✐♦r

❚❤❡ ❝❛❦❡✲❡❛t✐♥❣ ♣r♦❜❧❡♠ ❙✐♠♣❧❡ ❡①❛♠♣❧❡✿ ❈❘❘❆ ✉t✐❧✐t②

❈❘❘❆ ✉t✐❧✐t②

◮ ❆ ❝♦♥✈❡♥✐❡♥t ❛♥❞ ♣♦♣✉❧❛r s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ✭γ > ✵✮ u(c) = c✶−γ ✶ − γ γ−✶ ✐s t❤❡ ❡❧❛st✐❝✐t② ♦❢ ✐♥t❡rt❡♠♣♦r❛❧ s✉❜st✐t✉t✐♦♥ ◮ ■t ✐s ❣❡♥❡r❛❧❧② ❛❝❝❡♣t❡❞ t❤❛t γ ≥ ✶✱ ✐♥ ✇❤✐❝❤ ❝❛s❡✱ ❢♦r c ∈ R+ u(c) < ✵, limc→✵ u(c) = −∞, limc→+∞ u(c) = ✵ u′(c) > ✵, limc→✵ u′(c) = +∞, limc→+∞ u′(c) = ✵

❈❘❘❆ ✉t✐❧✐t②✿ s♦❧✉t✐♦♥ ■

◮ ❚❤❡ ♣r♦❜❧❡♠ ✐s max

(c✶,...,cT )∈(R+)T T

• t=✶

βt−✶ c✶−γ

t

✶ − γ s✳t

T

• t=✶

R✶−tct ≤ a✶ ◮ ❊✉❧❡r ❡q✉❛t✐♦♥s✿ c−γ

t

= βRc−γ

t+✶

⇒ ct = (βR)− ✶

γ ct+✶

❢♦r t = ✶, . . . , T − ✶ ◮ ❇② s✉❝❝❡ss✐✈❡ s✉❜st✐t✉t✐♦♥✿ ct = (βR)

t−✶ γ c✶

❈❘❘❆ ✉t✐❧✐t②✿ s♦❧✉t✐♦♥ ■■

◮ ❚❤❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t ❛♥❞ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐♠♣❧② a✶ =

• t=✶,...,T

R✶−tct = c✶

• t=✶,...,T
• β

✶ γ R ✶−γ γ

t−✶ = c✶

• t=✶,...,T

αt−✶ ✇❤❡r❡ α = β

✶ γ R ✶−γ γ

◮ ❚❤❡ s♦❧✉t✐♦♥ ❢♦r t = ✶, . . . , T✿ c✶ = ✶ − α ✶ − αT a✶ ❛♥❞ ct = ✶ − α ✶ − αT (βR)

t−✶ γ a✶

❈❘❘❆ ✉t✐❧✐t②✿ s♦❧✉t✐♦♥ ■■■

■♥ ❣❡♥❡r❛❧✱ ✐❢ t❤❡ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠ st❛rts ❛t t✐♠❡ t ❛s ❢♦❧❧♦✇s max

(ct,...,cT )∈(R+)T−t+✶ T

• τ=t

βτ−t c✶−γ

τ

✶ − γ s✳t

T

• τ=t

Rτ−tcτ ≤ aτ t❤❡ s♦❧✉t✐♦♥ ❢♦r ct ✐s ct = ✶ − α ✶ − αT−t+✶ at ❚❤✐s ✐s t❤❡ ❝♦♥s✉♠♣t✐♦♥ ❢✉♥❝t✐♦♥✱ ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ♦❢ ❛ss❡ts ✐❢ ✉t✐❧✐t② ✐s ❈❘❘❆

❈❘❘❆ ✉t✐❧✐t②✿ ❝♦♥s✉♠♣t✐♦♥ ♦✈❡r t❤❡ ❧✐❢❡✲❝②❝❧❡

βR ❞❡t❡r♠✐♥❡s t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✿ ct =

✶−α ✶−αT (βR)

t−✶ γ a✶ .02 .03 .04 .05 .06 consumption 20 30 40 50 60 age r=4% r=2.5% r=1%

β = ✶.✵✷✺−✶ ❛♥❞ ✐♥✐t✐❛❧ ❛ss❡ts ❛r❡ a✷✵ = ✶✳

❚❤❡ ❝❛❦❡✲❡❛t✐♥❣ ♣r♦❜❧❡♠ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥

❲❤❡♥ ❝❛♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ♦♣t✐♠✉♠ ❜❡ ❣✉❛r❛♥t❡❡❞❄

◮ ❋❡❛s✐❜✐❧✐t② s❡t✿ s♣❛❝❡ ♦❢ ❝❤♦✐❝❡s s❛t✐s❢②✐♥❣ t❤❡ ♣r♦❜❧❡♠ ❝♦♥str❛✐♥ts

C✶:T(a✶) =   (c✶, . . . , cT) ∈ CT :

• t=✶,...,T

R✶−tct ≤ a✶   

✇❤❡r❡ t②♣✐❝❛❧❧② C = R+ ◮ ❆♣♣❧② ❲❡✐❡rstr❛ss t❤❡♦r❡♠ t♦ ❡♥s✉r❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥✿ ▲❡t u : C → R ❜❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ s✉♣♣♦s❡ C✶:T(a✶) ⊂ CT ✐s ♥♦♥✲❡♠♣t② ❛♥❞ ❝♦♠♣❛❝t✳ ❚❤❡♥ t❤❡ ❝♦♥s✉♠❡r✬s ♣r♦❜❧❡♠ max

(c✶,...,cT )∈C✶:T (a✶)

• t=✶,...,T

βt−✶u(ct) ❤❛s ❛ s♦❧✉t✐♦♥ ✐♥ C✶:T(a✶) ❢♦r ❛♥② a✶ ∈ A✳

❲❤❡♥ ✐s t❤❡ ♦♣t✐♠✉♠ ✐♥t❡r✐♦r ❛♥❞ ✉♥✐q✉❡❄

◮ ❚②♣✐❝❛❧ ❝♦♥s✉♠❡r✬s ♣r♦❜❧❡♠✿ u ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱ ❝♦♥❝❛✈❡ ❛♥❞ C✶

◮ ❚❤❡♥ t❤❡ s✉♠ ♦❢ ♣❡r✲♣❡r✐♦❞ ✉t✐❧✐t✐❡s ✐s ❛❧s♦ str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱ ❝♦♥❝❛✈❡ ❛♥❞ C✶

◮ ❆❧s♦ ❛ss✉♠❡ t❤❛t t❤❡ ❢❡❛s✐❜✐❧✐t② s❡t C✶:T(a✶) ✐s ♥♦♥✲❡♠♣t② ❛♥❞ ❝♦♠♣❛❝t ◮ ❯♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s t❤❡ s♦❧✉t✐♦♥ ✐s ✉♥✐q✉❡ ◮ ■t ✐s ❛❧s♦ ✐♥t❡r✐♦r ✭T > ✶✮ ◮ ❇✉t ✐❢ ✇❡ ❤❛❞ ❛ ❝♦♥✈❡① u✿ ❝♦r♥❡r s♦❧✉t✐♦♥

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❚❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣

◮ ❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ s♣❧✐ts t❤❡ ❜✐❣ ♣r♦❜❧❡♠ ✐♥t♦ s♠❛❧❧❡r ♣r♦❜❧❡♠s t❤❛t ❛r❡ ♦❢ s✐♠✐❧❛r str✉❝t✉r❡ ❛♥❞ ❡❛s✐❡r t♦ s♦❧✈❡ ◮ ❚❤❡ tr✐❝❦ ✐s t♦ ✜♥❞ t❤❡ ❧✐♠✐t❡❞ s❡t ♦❢ ✈❛r✐❛❜❧❡s t❤❛t ❝♦♠♣❧❡t❡❧② ❞❡s❝r✐❜❡ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✐♥ ❡❛❝❤ ♣❡r✐♦❞ ✕ t❤❡ st❛t❡ ◮ ❚❤❡♥ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s ♦✈❡r ❛ s♠❛❧❧ st❛t❡✲s♣❛❝❡ ❞❡t❡r♠✐♥❡s ❛ s❡t ♦❢ ♣♦❧✐❝② ❢✉♥❝t✐♦♥s✿ ♦♣t✐♠❛❧ ❝♦♥s✉♠♣t✐♦♥ ✐s ht(at) ❢♦r t = ✶, . . . T ◮ ❉P r❡t✉r♥s ❛ ❣❡♥❡r❛❧ s♦❧✉t✐♦♥✿ ✐t s♦❧✈❡s t❤❡ ❡♥t✐r❡ ❢❛♠✐❧② ♦❢ ♣r♦❜❧❡♠s ♦❢ t❤❡ s❛♠❡ t②♣❡ ◮ ❚❤❡ s♣❡❝✐✜❝ s♦❧✉t✐♦♥ t♦ ♦✉r ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ r❡❝✉rs✐✈❡❧②✱ ❜② ✐t❡r❛t✐♥❣ ct = ht(at) at+✶ = R(at − ct) st❛rt✐♥❣ ❢r♦♠ t❤❡ ❣✐✈❡♥ a✶

Pr♦❜❧❡♠ s♣❡❝✐✜❝❛t✐♦♥ ■

◮ ■♥ ♦✉r ♣r♦❜❧❡♠✱ t❤❡ ❧❡✈❡❧ ♦❢ ❛ss❡ts ❛t t❤❡ st❛rt ♦❢ ♣❡r✐♦❞ t s✉♠♠❛r✐s❡s ❛❧❧ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♥❡❡❞❡❞ t♦ s♦❧✈❡ ❢♦r ❝♦♥s✉♠♣t✐♦♥ ◮ ❚❤❡ ❢❡❛s✐❜✐❧✐t② s❡t ❛t t✐♠❡ t ❢♦r t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣r❡s❡♥t ❛♥❞ ❢✉t✉r❡ ❝♦♥s✉♠♣t✐♦♥ ❝❤♦✐❝❡s ❣✐✈❡♥ at ∈ A ✐s Ct:T(at) =   (ct, . . . , cT) ∈ CT−t+✶ :

• τ=t,...,T

Rt−τcτ ≤ at    ◮ ■❢ ❝♦♥s✉♠♣t✐♦♥ ♠✉st ❜❡ ♣♦s✐t✐✈❡ ✐♥ ❡✈❡r② ♣❡r✐♦❞✱ t❤❡♥ C = A = R+ ❛♥❞ t❤❡ ❢❡❛s✐❜✐❧✐t② s❡t ❛t t✐♠❡ t ✐s Ct(at) =    {ct > ✵ : at+✶ = R (at − ct) > ✵} ✐❢ t < T {ct > ✵ : at+✶ = R (at − ct) ≥ ✵} ✐❢ t = T

Pr♦❜❧❡♠ s♣❡❝✐✜❝❛t✐♦♥ ■■

◮ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❛ ❝♦♥s✉♠❡r ✇✐t❤ ❛ss❡ts at ❛t t✐♠❡ t ✐s Vt(at) = max

(ct,...,cT )∈Ct:T (at)

• τ=t,...,T

βτ−tu(cτ) ◮ Vt ✐s t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥

◮ ■♥❞✐r❡❝t ❧✐❢❡t✐♠❡ ✉t✐❧✐t②✿ ♠❡❛s✉r❡s ♠❛① ✉t✐❧✐t② t❤❛t ❛ss❡ts at ❝❛♥ ❞❡❧✐✈❡r ◮ ■t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ at ❛❧♦♥❡ ◮ ❉❡♣❡♥❞❡♥❝❡ ♦♥ at ❛r✐s❡s t❤r♦✉❣❤ t❤❡ ❢❡❛s✐❜✐❧✐t② s❡t

Pr♦❜❧❡♠ s♣❡❝✐✜❝❛t✐♦♥ ■■■

❚❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❞❡✜♥❡❞ r❡❝✉rs✐✈❡❧② Vt(at) = max

(ct,...,cT )∈Ct:T (at)

• τ=t,...,T

βτ−tu(cτ) = max

ct∈Ct(at)

             u(ct) + β    max

(ct+✶,...,cT )∈Ct+✶:T (at+✶) T

• τ=t+✶

βτ−(t+✶)u(cτ)

• 

 

Vt+✶(at+✶)

             = max

ct∈Ct(at) {u(ct) + βVt+✶ (R[at − ct])}

❚❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ■

Vt(at) = max

ct∈Ct(at) {u(ct) + βVt+✶ (R[at − ct])}

◮ ❚❤✐s ✐s ❛ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✿ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛t✐♦♥ ◮ ❇r❡❛❦s t❤❡ ❧❛r❣❡ ❧✐❢❡❝②❝❧❡ ♣r♦❜❧❡♠ ✐♥ s♠❛❧❧❡r st❛t✐❝ ♣r♦❜❧❡♠s

◮ ❑❡②✿ ♠❡♠♦r②❧❡ss ♣r♦❝❡ss ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s ❛t t❤❡ t✐♠❡ ♦❢ ❞❡❝✐s✐♦♥

◮ Pr✐♥❝✐♣❧❡ ♦❢ ❖♣t✐♠❛❧✐t②✿ ✐❢ t❤❡ ❝♦♥s✉♠❡r ❜❡❤❛✈❡s ♦♣t✐♠❛❧❧② ✐♥ t❤❡ ❢✉t✉r❡✱ ❛❧❧ t❤❛t ♠❛tt❡rs ❢♦r t❤❡ s♦❧✉t✐♦♥ ❛t t✐♠❡ t ✐s t❤❡ ❞❡❝✐s✐♦♥ ♦❢ ❤♦✇ ♠✉❝❤ t♦ ❝♦♥s✉♠❡ t♦❞❛② ◮ Vt+✶ ❡①✐sts ✭❜② r❡❝✉rs✐♦♥✮ ❜✉t ✐s ✉♥❦♥♦✇♥✦

❚❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ■■

❖❢t❡♥ ✉s❡❢✉❧ t♦ r❡❢♦r♠✉❧❛t❡ t❤❡ ♣r♦❜❧❡♠ ✐♥ t❡r♠s ♦❢ s❛✈✐♥❣s ❞❡❝✐s✐♦♥s ◮ ❉❡✜♥❡ t❤❡ ♣❛②♦✛ ❢✉♥❝t✐♦♥ ❛s f (at, at+✶) = u

• at − at+✶

R

• = u (ct)

◮ ❚❤❡♥ t❤❡ ❝♦♥s✉♠♣t✐♦♥✴s❛✈✐♥❣s ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t❧② s♣❡❝✐✜❡❞ ❛s Vt(at) = max

at+✶∈Dt(at) {f (at, at+✶) + βVt+✶ (at+✶)}

✇❤❡r❡ t❤❡ ❢❡❛s✐❜✐❧✐t② s❡t ❛t t✐♠❡ t ✭❢♦r C = A = R+✮

Dt(at) =   

• at+✶ > ✵ : at − at+✶R−✶ > ✵,
• ✐❢ t < T
• at+✶ ≥ ✵ : at − at+✶R−✶ > ✵
• ✐❢ t = T

❚❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ■■■

◮ ❚❤❡ s♦❧✉t✐♦♥ ✐s gt(at) = arg max

at+✶∈Dt(at)

{f (at, at+✶) + βVt+✶ (at+✶)} ◮ ❊①✐sts ❛♥❞ ✐s ✉♥✐q✉❡ ✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥s ❞✐s❝✉ss❡❞ ❡❛r❧✐❡r✿

◮ f r❡❛❧✲✈❛❧✉❡❞✱ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✭❞❡❝r❡❛s✐♥❣✮ ✐♥ t❤❡ ✜rst ✭s❡❝♦♥❞✮ ❛r❣✉♠❡♥t✱ ❝♦♥❝❛✈❡ ❛♥❞ C✶ ✐♥ ❜♦t❤ ❛r❣✉♠❡♥ts ◮ D ✐s ♥♦♥✲❡♠♣t② ❛♥❞ ❝♦♠♣❛❝t

◮ ❯♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s g ✐s ❛❧s♦ ❝♦♥t✐♥✉♦✉s ◮ ▼♦r❡♦✈❡r✱ V ✐♥❤❡r✐ts s♦♠❡ ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ f

◮ ❝♦♥t✐♥✉✐t②✱ ♠♦♥♦t♦♥✐❝✐t② ❛♥❞ ❝♦♥❝❛✈✐t② ◮ ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❛t ♣♦✐♥ts a ∈ A ✐♥ ✇❤✐❝❤ t❤❡ s♦❧✉t✐♦♥ ✐s ✐♥t❡r✐♦r

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❘❡❝✉rs✐✈❡ s♦❧✉t✐♦♥

❘❡❝✉rs✐✈❡ s♦❧✉t✐♦♥

Vt(at) = max

at+✶∈Dt(at) {f (at, at+✶) + βVt+✶ (at+✶)}

❑❡② ✐♥s✐❣❤t ♦❢ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣✿ t❤❡ ✉♥❦♥♦✇♥ V ❝❛♥ ❜❡ ♣✐♥♥❡❞ ❞♦✇♥ ❜② ❜❛❝❦✇❛r❞ ✐♥❞✉❝t✐♦♥ ◮ ❚❤✐s ❤✐❣❤❧✐❣❤ts t❤❡ ✉s❡❢✉❧♥❡ss ♦❢ t❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ◮ ❆♥❞ ✐♥s♣✐r❡s t❤❡ ♥✉♠❡r✐❝❛❧ str❛t❡❣② t♦ s♦❧✈❡ ♠♦❞❡❧s ✇✐t❤ ♥♦ ❝❧♦s❡❞✲❢♦r♠ s♦❧✉t✐♦♥

▲❛st ♣❡r✐♦❞

❙♦❧✉t✐♦♥ str❛t❡❣②✿ st❛rt ❢r♦♠ ♣❡r✐♦❞ T ❛♥❞ ♠♦✈❡ ❜❛❝❦✇❛r❞s ❛s t❤❡ ❢✉t✉r❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥✱ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ✈❛❧✉❡✱ ✐s ❞❡t❡r♠✐♥❡❞ ◮ ❚❤❡ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❧❛st ♣❡r✐♦❞ ✐s VT(aT) = max

aT+✶∈DT (aT ) {f (aT, aT+✶)}

✇❤❡r❡ DT(aT) = [✵, RaT] ◮ ❚❤❡ s♦❧✉t✐♦♥ ✐s ✭❢♦r ❛♥② aT ∈ A✮ gT(aT) = ✵ ✇✐t❤ ✈❛❧✉❡ VT(aT) = f (aT, ✵) = u(aT)

▲❛st ❜✉t ♦♥❡ ♣❡r✐♦❞

◮ ❙✐♥❝❡ VT(aT) = u(aT)✱ t❤❡ ♣r♦❜❧❡♠ ❛t T − ✶ ✐s ❦♥♦✇♥ VT−✶(aT−✶) = max

aT ∈DT−✶(aT−✶) {f (aT−✶, aT) + βVT(aT)}

◮ ❯♥❞❡r ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ t❤❡ ♠❛①✐♠✐s✐♥❣ ❢✉♥❝t✐♦♥✱ ❛♥ ✐♥t❡r✐♦r ♦♣t✐♠✉♠ s❛t✐s✜❡s t❤❡ ❢♦❝✬s ✭❢♦r ❛♥② aT−✶ ∈ A gT−✶(aT−✶) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ f✷ (aT−✶, aT) + βV ′

T(aT) = ✵

◮ ❙♦ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ❛t T − ✶ ✐s ✭❢♦r ❡❛❝❤ aT−✶ ∈ A✮ VT−✶(aT−✶) = f (aT−✶, gT−✶(aT−✶)) + βVT (gT−✶(aT−✶))

P❡r✐♦❞ t

▼♦✈❡ ❜❛❝❦✇❛r❞s ✐♥ s✐♠✐❧❛r st❡♣s ◮ ❖♥❝❡ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ❢♦r ♣❡r✐♦❞ t + ✶ ❤❛s ❜❡❡♥ ❞❡t❡r♠✐♥❡❞✱ s♦❧✈❡ ✭❢♦r ❡❛❝❤ at ∈ A✮ gt(at) = arg max

at+✶∈Dt(at)

{f (at, at+✶) + βVt+✶(at+✶)} ◮ ❚❤❡ s♦❧✉t✐♦♥ ❝❛♥ t❤❡♥ ❜❡ ✉s❡❞ t♦ ❜✉✐❧❞ Vt ✭❢♦r ❡❛❝❤ at ∈ A✮✿ Vt(at) = f (at, gt(at)) + βVt+✶ (gt(at))

❙♦❧✉t✐♦♥ t♦ ♦✉r s♣❡❝✐✜❝ ♣r♦❜❧❡♠

◮ ❚❤❡ s♣❡❝✐✜❝ ♣r♦❜❧❡♠ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ✐s ❢✉❧❧② ❝❤❛r❛❝t❡r✐s❡❞ ❜② t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱ a✶ ◮ ❚♦ ❝♦♥str✉❝t t❤❡ s♦❧✉t✐♦♥✱ ✇❡ ✉s❡ t❤❡ ♣♦❧✐❝② ❢✉♥❝t✐♦♥s gt ❛♥❞ ✐t❡r❛t❡✱ ❢♦r t = ✶, . . . , T ct = at − R−✶gt(at) ❛♥❞ at+✶ = gt(at)

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s

❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ■

◮ ❚❤❡ t②♣✐❝❛❧ ♣r♦❜❧❡♠ ✐♥ ❡❝♦♥♦♠✐❝s ❛ss✉♠❡s t❤❛t t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣✱ ❝♦♥❝❛✈❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ✭✐♥ ❝♦♥s✉♠♣t✐♦♥✮✱ ❛♥❞ t❤❛t t❤❡ ❢❡❛s✐❜✐❧✐t② s♣❛❝❡ ✐s ❝❧♦s❡❞ ❛♥❞ ❜♦✉♥❞❡❞ ◮ ❯♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s t❤❡ s♦❧✉t✐♦♥ ✐s ✉♥✐q✉❡ ❛♥❞ V ✐s ❞✐✛❡r❡♥t✐❛❜❧❡ ◮ ❆♥❞ t❤❡ ✜rst ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❢♦r ❛♥ ✐♥t❡r✐♦r ♦♣t✐♠✉♠

❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ■■

◮ ❚❤❡ ♣r♦❜❧❡♠ ❛t t✐♠❡ t ✐s Vt(at) = max

at+✶∈Dt(at) {f (at, at+✶) + βVt+✶ (at+✶)}

◮ ❚❤❡ ❢♦❝ ❛t t✐♠❡ t ✐s f✷ (at, at+✶) + βV ′

t+✶ (at+✶)

= ✵ ◮ ❯s❡ t❤❡ ❡♥✈❡❧♦♣❡ ❝♦♥❞✐t✐♦♥ t♦ ✇♦r❦♦✉t V ′

t+✶ (at+✶)

V ′

t (at)

= f✶ (at, at+✶) + f✷ (at, at+✶) ∂at + ✶ ∂at + βV ′

t+✶ (at+✶) ∂at + ✶

∂at = f✶ (at, at+✶) +

• f✷ (at, at+✶) + βV ′

t+✶ (at+✶)

• ❢♦❝ ❛t t

∂at + ✶ ∂at = f✶ (at, at+✶) = u′ (ht(at))

❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ■■■

P✉t t❤❡ ❢♦❝ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❡♥✈❡❧♦♣❡ ❝♦♥❞✐t✐♦♥ t♦ ❣❡t t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ f✷ (at, at+✶) + βf✶ (at+✶, at+✷) = ✵ ⇔ u′(ct) = βRu′(ct+✶) s✐♥❝❡✿ u(ct) = f (at, at+✶) = u

• at − at+✶

R

• ❛♥❞ s♦✿ f✶ (at, at+✶) = u′(ct) ❛♥❞ f✷ (at, at+✶) = − u′(ct)

R

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥

◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥

◮ ❚❤❡ ❝❛❦❡✲❡❛t✐♥❣ ♣r♦❜❧❡♠ ✐s ❡❛s② t♦ s♦❧✈❡ ♦♥ t❤❡ ♣❛♣❡r ◮ ❇✉t ✐t ✐s ❛♥ ✐♥str✉❝t✐✈❡ ❡①❛♠♣❧❡ t♦ ♣❧❛② ✇✐t❤ ♥✉♠❡r✐❝❛❧❧②

◮ ❙♦♣❤✐st✐❝❛t❡❞ ❡♥♦✉❣❤ t♦ r❡q✉✐r❡ ♠♦st ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ tr✐❝❦s ✉s❡❞ ✐♥ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♠♦❞❡❧s ◮ ❇✉t ❡❛s② ❡♥♦✉❣❤ t♦ ❦❡❡♣ t❤❡ ❞✐s❝✉ss✐♦♥ s✐♠♣❧❡ ◮ ❈❛♥ ❜❡ ✉s❡❞ t♦ ❞❡♠♦♥str❛t❡ t❤❡ ❝♦♠♣❛r❛t✐✈❡ ❛❞✈❛♥t❛❣❡s ♦❢ ✈❛r✐♦✉s ♥✉♠❡r✐❝❛❧ ♣r♦❝❡❞✉r❡s s✐♥❝❡ t❤❡ s♦❧✉t✐♦♥ ✐s ❦♥♦✇♥✦

❈♦♠♣✉t❡rs ❞♦ ♥♦t ❦♥♦✇♥ ✐♥✜♥✐t②

✶✳ ▼♦❞❡❧ s♣❡❝✐✜❝❛t✐♦♥

◮ ❈❘❘❆ ✉t✐❧✐t② ✐s ❣r❡❛t t♦ ❡♥s✉r❡ t❤❛t ❝♦♥s✉♠❡rs ❛✈♦✐❞ ❣❡tt✐♥❣ ❝❧♦s❡ t♦ ③❡r♦ ❝♦♥s✉♠♣t✐♦♥ ◮ ❚❤❡ s❛♠❡ ❞♦❡s ♥♦t ❤♦❧❞ ❢♦r ❝♦♠♣✉t❛t✐♦♥❛❧ s♦❧✉t✐♦♥s✿ ❡①tr❡♠❡ ✈❛❧✉❡s ❝❛✉s❡ t❤❡ r♦✉t✐♥❡ t♦ ❝r❛s❤ ⇒ ❇♦✉♥❞ s♦❧✉t✐♦♥ s♣❛❝❡ t♦ ✐ts r❡❧❡✈❛♥t ♣❛rts t♦ ❛✈♦✐❞ ♣r♦❜❧❡♠s

✷✳ ❉✐s❝r❡t✐s❡ st❛t❡ s♣❛❝❡

◮ ❙❡❧❡❝t ❣r✐❞ ✐♥ ❛ss❡ts A = {ai}i=✶,...,na ◮ ❙♦❧✈❡ ♣r♦❜❧❡♠ ♦♥❧② ❢♦r ♣♦✐♥ts ✐♥ t❤❡ ❣r✐❞ ◮ ❆♣♣r♦①✐♠❛t❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ♥✉♠❡r✐❝❛❧❧② ♦✉ts✐❞❡ t❤❡ ❣r✐❞

❆❧❣♦r✐t❤♠ ❢♦r r❡❝✉rs✐✈❡ s♦❧✉t✐♦♥

✶✳ P❛r❛♠❡t❡r✐s❡ ♠♦❞❡❧ ❛♥❞ s❡❧❡❝t ❣r✐❞ ✐♥ ❛ss❡ts✿

• ai

i=✶,...,na

✷✳ ❈❤♦♦s❡ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥ ǫ > ✵ ✸✳ ❙t♦r❡ VT+✶

• ai

= ✵ ❢♦r ❛❧❧ i = ✶, . . . , na ✹✳ ▲♦♦♣ ♦✈❡r t ❜❛❝❦✇❛r❞s✿ t = T, . . . , ✶ ❋♦r ❡❛❝❤ i = ✶, . . . , na

✹✳✶ ❈♦♠♣✉t❡ g i

t = arg maxat+✶∈Dt(ai )

• u
• ai − at+✶

R

• + β

Vt+✶ (at+✶)

• ✹✳✷ ❈♦♠♣✉t❡ V i

t = u

• ai −

gi

t

R

• + β

Vt+✶

• g i

t

• ✹✳✸ ❆♣♣r♦①✐♠❛t❡ Vt ♦✈❡r ✐ts ❡♥t✐r❡ ❞♦♠❛✐♥ t♦ ❣❡t

Vt ❛♥❞ st♦r❡ ✐t ❚❤✐s st❡♣ ✐s ♦♣t✐♦♥❛❧✿ ❝❛♥ ❜❡ ❞♦♥❡ ❞✐r❡❝t❧② ✐♥ st❡♣ ✹✳✶ ♦r s❦✐♣♣❡❞ ❛❧t♦❣❡t❤❡r✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s♦❧✉t✐♦♥ ♠❡t❤♦❞ ✲ ♠♦r❡ t♦ ❢♦❧❧♦✇

❙♦❧✉t✐♦♥ ❛t ❡❛❝❤ ♣♦✐♥t

◮ ❙t❡♣ ✹✳✶ ✐s t❤❡ ✭❝♦♠♣✉t❛t✐♦♥❛❧❧②✮ ❤❡❛✈② ♣❛rt ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛❧❣♦r✐t❤♠ ◮ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✇❛②s ♦❢ ✜♥❞✐♥❣ t❤❡ ♦♣t✐♠✉♠ g i

t

◮ ❯s❡ ❛ s❡❛r❝❤ ❛❧❣♦r✐t❤♠ t♦ ❧♦♦❦ ❢♦r t❤❡ ✈❛❧✉❡ ♦❢ s❛✈✐♥❣s at+✶ t❤❛t ♠❛①✐♠✐s❡ Vt(at) ❚❤✐s ✐s t❤❡ ♣r♦❝❡❞✉r❡ ✐♠♣❧✐❝✐t ✐♥ t❤❡ ❛❧❣♦r✐t❤♠ ✇❡ ♣r❡s❡♥t❡❞ ◮ ❖r ❧♦♦❦ ❢♦r t❤❡ r♦♦t ♦❢ t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ u′(ct) = βRV ′(at+✶) ✈s♣❛❝❡✵✳✶❝♠ ❲❡ ✇✐❧❧ ❞✐s❝✉ss t❤✐s s♦❧✉t✐♦♥ ❧❛t❡r

❙♦❧✉t✐♦♥ ❛t ❡❛❝❤ ♣♦✐♥t ✉s✐♥❣ t❤❡ ❢♦❝✿ ❛ tr✐❝❦ ■

◮ ❯s❡❢✉❧ tr✐❝❦ ✉♥❞❡r ❈❘❘❆✿ s♣❡❡❞ ✉♣ ❛♥❞ ✐♠♣r♦✈❡ ❛❝❝✉r❛❝② ♦❢ s♦❧✉t✐♦♥ ◮ ❚❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✐s c−γ

t

= βRV ′

t+✶(at+✶)

⇔ ct = (βR)−✶/γ V ′

t+✶(at+✶)

−✶/γ ◮ ❇✉t s✐♥❝❡ ✭❡♥✈❡❧♦♣❡ ❝♦♥❞✐t✐♦♥✮ V ′

t+✶(at+✶) = u′ (ht+✶(at+✶)) =

• at+✶ − gt+✶(at+✶)

R −γ ◮ ❚❤❡ s♦❧✉t✐♦♥ ✐s t❤❡ ❧❡✈❡❧ ♦❢ s❛✈✐♥❣s at+✶ t❤❛t s❛t✐s✜❡s at − at+✶ R

• ct

= (βR)−✶/γ

• at+✶ − gt+✶(at+✶)

R

• ct+✶

❙♦❧✉t✐♦♥ ❛t ❡❛❝❤ ♣♦✐♥t ✉s✐♥❣ t❤❡ ❢♦❝✿ ❛ tr✐❝❦ ■■

at − at+✶ R

• ct

= (βR)−✶/γ

• at+✶ − gt+✶(at+✶)

R

• ct+✶

= (βR)−✶/γ ht+✶(at+✶) ◮ ❚❤✐s ✐s ❛ ❧✐♥❡❛r ✭✐♥ at+✶✮ ❡q✉❛t✐♦♥ ✐♥ ♥♦♥✲st♦❝❤❛st✐❝ ♣r♦❜❧❡♠s ◮ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ t❤❡ ♣♦❧✐❝② ❢✉♥❝t✐♦♥ h ✐s t②♣✐❝❛❧❧② ♥♦t ✈❡r② ♥♦♥✲❧✐♥❡❛r ◮ ❙♦ ❛❧❧ ✇❡ ♥❡❡❞ ✐s t♦✿

✶✳ ❙t♦r❡ ht(ai) ❛❢t❡r s♦❧✈✐♥❣ ❝♦♥s✉♠❡rs ♣r♦❜❧❡♠ ❛t t✐♠❡ t ✷✳ ✏❈♦♥♥❡❝t t❤❡ ♣♦✐♥ts✑ t♦ ❛♣♣r♦①✐♠❛t❡ ❢✉♥❝t✐♦♥ h ❛♥❞ ♦❜t❛✐♥ t❤❡ s♦❧✉t✐♦♥ ♦✈❡r t❤❡ ❡♥t✐r❡ ❞♦♠❛✐♥✿ ▲✐♥❡❛r ■♥t❡r♣♦❧❛t✐♦♥

◮ ◆♦t✐❝❡ t❤❛t V ✐s ♥♦t ♥❡❡❞❡❞ t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✉s✐♥❣ t❤❡ ❢♦❝

❆♣♣r♦①✐♠❛t✐♥❣ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥

◮ ❆ ❜❛❞ ✐❞❡❛✿ t♦ r❡❧② ♦♥ s✐♠♣❧❡ ✭❧✐♥❡❛r✮ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ V t♦ s♦❧✈❡ ♠♦❞❡❧ ❛s V ❝❛♥ ❜❡ ❤✐❣❤❧② ♥♦♥✲❧✐♥❡❛r ◮ ❇✉t ♦♥❡ ♠❛② st✐❧❧ ♥❡❡❞ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥✱ ❡✈❡♥ ✇❤❡♥ r❡❧②✐♥❣ ♦♥ t❤❡ ❢♦❝ ❢♦r t❤❡ s♦❧✉t✐♦♥✿

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

◮ ❚✇♦ ❛❧t❡r♥❛t✐✈❡s t♦ ❛♣♣r♦①✐♠❛t❡ V

◮ ▼♦r❡ r❡❧✐❛❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞✿ s❤❛♣❡✲♣r❡s❡r✈✐♥❣ s♣❧✐♥❡s ◮ ❘❡❞✉❝❡ ♥♦♥✲❧✐♥❡❛r✐t② ❜② ❛♣♣❧②✐♥❣ s❡❧❡❝t❡❞ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡♥ ❛♣♣r♦①✐♠❛t❡ ❜② ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ❋♦r ❛ ❈❘❘❆ ✉t✐❧✐t②✿ Ψt(at) = [(✶ − γ)Vt(at)]

✶ ✶−γ

Pr❛❝t✐❝❛❧ s❡ss✐♦♥ ✶

■♥❝♦♠❡ ♣r♦❝❡ss

❆❞❞ ✐♥❝♦♠❡ ♣r♦❝❡ss

◮ ❏✉st ❛❞❞✐♥❣ ❛♥ ✐♥❝♦♠❡ ♣r♦❝❡ss ❞♦❡s ♥♦t ♠✉❝❤ ❝❤❛♥❣❡ t❤❡ ❧✐❢❡❝②❝❧❡ ♣r♦❜❧❡♠ ◮ ❇✉t r❛✐s❡s ✐♥t❡r❡st✐♥❣ ✐ss✉❡s ♦❢ ❤♦✇ t♦ ❞❡❛❧ ✇✐t❤ t❤❡ ❝r❡❞✐t ♠❛r❦❡ts ◮ ❙✉♣♣♦s❡ t❤❡ ❝♦♥s✉♠❡r ❤❛s ❛ str❡❛♠ ♦❢ ✐♥❝♦♠❡ ♦✈❡r t✐♠❡ yt = w(at, t) ◮ ❋♦r t❤❡ ♠♦♠❡♥t✱ s✉♣♣♦s❡ {yt}t=✶,...,T ✐s ❦♥♦✇♥ ❜② t❤❡ ❝♦♥s✉♠❡r ❢r♦♠ t✐♠❡ t = ✶

■♥❝♦♠❡ ♣r♦❝❡ss ❈r❡❞✐t ▼❛r❦❡ts

❋✉♥❝t✐♦♥✐♥❣ ❝r❡❞✐t ♠❛r❦❡ts ■

◮ ■❢ ❝r❡❞✐t ♠❛r❦❡ts ❛r❡ ❝♦♠♣❧❡t❡✱ t❤❡ ❝♦♥s✉♠❡r ♠❛② ❜♦rr♦✇ t♦ ❜r✐♥❣ ✐♥❝♦♠❡ ❢♦r✇❛r❞

◮ ❆ss❡ts ❛t t✐♠❡ t ❝❛♥ ❜❡ ♥❡❣❛t✐✈❡ ◮ ❇♦rr♦✇✐♥❣ ❧✐♠✐t❡❞ ❜② ❛❜✐❧✐t② t♦ r❡♣❛② ◮ ❉♦♠❛✐♥ ♦❢ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ ❛ss❡ts ❝❤❛♥❣❡s ♦✈❡r t✐♠❡✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t✐♠❡ ❧❡❢t t♦ r❡♣❛② ❞❡❜ts ❛♥❞ t❡r♠✐♥❛❧ ❝♦♥❞✐t✐♦♥

◮ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝♦♥s✉♠❡r ❛t t✐♠❡ t ❢♦r ❛ss❡ts at Vt(at, yt) = max

at+✶ {f (at, yt, at+✶) + βVt+✶(at+✶, yt+✶)}

s✳t✳ at+✶ = R(at + yt − ct) yt+✶ = w(at+✶, t + ✶) ct > ✵ ❛♥❞ aT+✶ ≥ ✵

❋✉♥❝t✐♦♥✐♥❣ ❝r❡❞✐t ♠❛r❦❡ts ■■

◮ ❚❤❡ ❢❡❛s✐❜✐❧✐t② s♣❛❝❡ ❛t t✐♠❡ t < T ✐s

Dt(at, yt) =        at+✶ : at + yt − at+✶ R

• ct

> ✵, at+✶ +

T

• τ=t+✶

R(t+✶)−τyτ > ✵        =

• −

T

• τ=t+✶

R(t+✶)−τyτ, R (at + yt)

• ◮ ❆t t✐♠❡ T

DT(aT, yT) = [✵, R (aT + yT))

❋✉♥❝t✐♦♥✐♥❣ ❝r❡❞✐t ♠❛r❦❡ts ■■■

◮ ❚❤❡ ❝♦♠♣❛❝t s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✐s Vt(at, yt) = max

at+✶∈Dt(at,yt) {f (at, yt, at+✶) + βVt+✶ (at+✶, yt+✶)}

s✳t✳ yt = w(at, t) ❢♦r ❛❧❧ t ◮ ❋♦❝ ✐s ❊✉❧❡r ❡q✉❛t✐♦♥ u′(ct) = βRu′(ct+✶) ◮ ❚❤❡ st❛t❡ s♣❛❝❡ ✐s ♥♦✇ ✷✲❞✐♠❡♥s✐♦♥❛❧

◮ ❆❧t❤♦✉❣❤ ✐t ✐s ❡❛s② t♦ r❡❞✉❝❡ t♦ ✶ ❞✐♠❡♥s✐♦♥ ✐♥ t❤✐s ❝❛s❡ ❜② ♥♦t✐♥❣ t❤❛t at+✶ = R(at + w(at, t) − ct) ◮ ❈♦♠♣✉t❛t✐♦♥✲✇✐s❡✱ r❡❞✉❝✐♥❣ t❤❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ♦❢ t❤❡ st❛t❡ s♣❛❝❡ ✐s t❤❡ ♠♦st t✐♠❡✲s❛✈✐♥❣ ♣r♦❝❡❞✉r❡

❙✐♠♣❧❡ ❡①❛♠♣❧❡✿ ❈❘❘❆ ✉t✐❧✐t②

◮ ❲✐t❤ ❈❘❘❆ ✉t✐❧✐t② t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✐♠♣❧✐❡s ct = (βR)

t−✶ γ c✶

◮ ❚❤❡ ✈❛❧✉❡ ♦❢ t♦t❛❧ ❧✐❢❡t✐♠❡ ✇❡❛❧t❤ ❛t t = ✶ ✐s W = a✶ +

• t=✶,...,T

R✶−tyt ◮ ❚♦t❛❧ ❝♦♥s✉♠♣t✐♦♥ ✐s C =

• t=✶,...,T

R✶−tct =

• t=✶,...,T
• βR✶−γ t−✶

γ c✶

◮ ❨✐❡❧❞✐♥❣✱ ❢♦r t = ✶, . . . , T ct = (βR)

t−✶ γ

✶ − α ✶ − αT W ✇❤❡r❡ α = β

✶ γ R ✶−γ γ

❈❘❘❆ ✉t✐❧✐t②✿ ♣r♦✜❧❡s ❢♦r ❛ ♣❛t✐❡♥t ❝♦♥s✉♠❡r

−4 −2 2 20 30 40 50 60 age

increasing income

1 2 3 4 20 30 40 50 60 age

constant income consumption income assets

r = ✹% ❛♥❞ β = ✶.✵✷✺−✶✳ ■♥✐t✐❛❧ ❛ss❡ts ❛r❡ a✶ = ✶✳ ■♥❝♦♠❡ ♣r♦✜❧❡s ❛s ♣❧♦tt❡❞✳

❈❘❘❆ ✉t✐❧✐t②✿ ✐♥tr♦❞✉❝✐♥❣ r❡t✐r❡♠❡♥t

−5 5 10 15 20 30 40 50 60 70 age consumption income assets

r = ✹% ❛♥❞ β = ✶.✵✷✺−✶✳ ■♥✐t✐❛❧ ❛ss❡ts ❛r❡ a✶ = ✶✳ ■♥❝♦♠❡ ♣r♦✜❧❡s ❛s ♣❧♦tt❡❞✳

❈r❡❞✐t ❝♦♥str❛✐♥ts ■

◮ ■❢ ❝r❡❞✐t ✐s r❛t✐♦♥❡❞✱ t❤❡ ❝♦♥s✉♠❡r ♠❛② ❜❡ ✇✐❧❧✐♥❣ t♦ ❝♦♥s✉♠❡ ♠♦r❡ t❤❛♥ s❤❡ ❝❛♥ ❛✛♦r❞ ✐♥ t❤❡ s❤♦rt t❡r♠ ◮ ■♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❝r❡❞✐t✱ t❤❡ ❢❡❛s✐❜✐❧✐t② s❡t ✐s r❡str✐❝t❡❞ t♦ Dt(at, yt) =

• at+✶ : at + yt − at+✶

R > ✵, at+✶ ≥ ✵

• ◮ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❝♦♥s✉♠❡r✬s ❜❡st ❝❤♦✐❝❡ ♠❛② ❜❡ ❛ ❝♦r♥❡r

s♦❧✉t✐♦♥

❈r❡❞✐t ❝♦♥str❛✐♥ts ■■

◮ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝♦♥s✉♠❡r ❛t t✐♠❡ t ❢♦r ❛ss❡ts at ✐s ♥♦✇ Vt(at, yt) = max

at+✶ {f (at, yt, at+✶) + βVt+✶(at+✶, yt+✶)}

s✳t✳ at+✶ = R(at + yt − ct) yt+✶ = w(at+✶, t + ✶) ct > ✵ ❛♥❞ at+✶ ≥ ✵ ◮ ❚❤❡r❡ ❛r❡ T ✐♥❡q✉❛❧✐t② r❡str✐❝t✐♦♥s ✐♥ ❛ss❡ts ♥♦✇✱ s♦ ✇❡ ❤❛✈❡ T ✜rst ♦r❞❡r ❛♥❞ ❑✉❤♥ ❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s✿ f✸(at, yt, at+✶) + βf✶(at+✶, yt+✶, at+✷) = λt λtat+✶ = ✵, λt ≥ ✵, at+✶ ≥ ✵ ❢♦r t = t = ✶, . . . , T − ✶ aT+✶ = ✵ ❢♦r t = T

❈r❡❞✐t ❝♦♥str❛✐♥ts ■■■

❚❤❡ s♦❧✉t✐♦♥ ✐s ct = min {at + yt, r♦♦t ♦❢ u′(ct) = βRu′(ct+✶)} ♦r at+✶ = max {✵, r♦♦t ♦❢ f✸(at, yt, at+✶) + βf✶(at+✶, yt+✶, at+✷) = ✵}

■♥❝♦♠❡ ♣r♦❝❡ss ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥

❙♦❧✉t✐♦♥ ❛❧❣♦r✐t❤♠

❚❤❡ r❡❝✉rs✐✈❡ s♦❧✉t✐♦♥ ✐♥ ♣r❛❝t✐❝❡✿ ❛❧♠♦st ❡①❛❝t❧② ❛s ❜❡❢♦r❡ ✶✳ P❛r❛♠❡t❡r✐s❡ ♠♦❞❡❧ ❛♥❞ s❡❧❡❝t ❣r✐❞s ✐♥ at✿

• ai

t

• i=✶,...,na

✷✳ ❈❤♦♦s❡ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥ ǫ > ✵ ✸✳ ❙t♦r❡ VT+✶

• ai

T+✶

• = ✵ ❢♦r ❛❧❧ i = ✶, . . . , na

✹✳ ▲♦♦♣ ♦✈❡r t ❜❛❝❦✇❛r❞s✿ t = T, . . . , ✶ ❋♦r ❡❛❝❤ i = ✶, . . . , na

✹✳✶ ❈♦♠♣✉t❡ g i

t =

arg max

at+✶∈Dt(ai

t)

• u
• ai

t + w(ai t, t) − at+✶ R

• + β

Vt+✶ (at+✶)

• ✹✳✷ ❈♦♠♣✉t❡ V i

t = u

• ai

t + w(ai t, t) − gi

t

R

• + β

Vt+✶

• g i

t

❈♦♠♣✉t❛t✐♦♥❛❧ s♦❧✉t✐♦♥✿ ❛❞❞✐t✐♦♥❛❧ ✐ss✉❡s

✶✳ ❉✐♠❡♥s✐♦♥ ♦❢ st❛t❡ s♣❛❝❡✿ r❡❞✉❝❡ t♦ ✶ ✐♥ s♦❧✉t✐♦♥ at+✶ = R(at + w(at, t) − ct) ✷✳ P♦s✐t✐✈❡ ❝♦♥s✉♠♣t✐♦♥✿ ♠❛② ❜❡ tr✐❝❦② t♦ ❡♥s✉r❡ ✇✐t❤ ❛♣♣r♦①✐♠❛t❡❞ ❢✉♥❝t✐♦♥s ⇒ ✐♠♣♦s❡ ♠✐♥✐♠✉♠ ❝♦♥s✉♠♣t✐♦♥ cmin > ✵ ✸✳ ❋✉♥❝t✐♦♥✐♥❣ ❝r❡❞✐t ♠❛r❦❡ts✿ ❣r✐❞ ✐♥ ❛ss❡ts ❝❤❛♥❣❡s ♦✈❡r t✐♠❡

◮ ▲♦✇❡r ❜♦✉♥❞ ❛t t ❡♥s✉r❡s ❞❡❜t ❝❛♥ ❜❡ r❡♣❛✐❞ ❛♥❞ cmin ✐s ❛✛♦r❞❛❜❧❡ at +

• τ=t...,T

Rt−τyτ ≥

• τ=t...,T

Rt−τcmin ◮ ❯♣♣❡r ❜♦✉♥❞ ❛t t r❡❛❝❤❡❞ ✐❢ ❝♦♥s✉♠❡s cmin ✐♥ ❛❧❧ ♣❡r✐♦❞s t♦ t at ≤ Rt−✶a✶ +

• τ=✶...,t−✶

Rt−τ (yτ − cmin)

Pr❛❝t✐❝❛❧ s❡ss✐♦♥ ✷

❙t♦❝❤❛st✐❝ ♦♣t✐♠✐s❛t✐♦♥

❙t♦❝❤❛st✐❝ ♣r♦❜❧❡♠s

◮ ▼♦st ✐♥t❡r❡st✐♥❣ ♣r♦❜❧❡♠s ✐♥ ❡❝♦♥♦♠✐❝s ✐♥✈♦❧✈❡ s♦♠❡ s♦rt ♦❢ ✉♥✐♥s✉r❛❜❧❡ r✐s❦ ◮ ❚❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❞②♥❛♠✐❝ ♣r♦❜❧❡♠ ✇✐❧❧ ❞❡♣❡♥❞ ❝r✉❝✐❛❧❧② ♦♥

✶✳ ❤♦✇ ♠✉❝❤ r✐s❦ ❝♦♥s✉♠❡rs ❢❛❝❡ ✷✳ t❤❡✐r ❛tt✐t✉❞❡s t♦✇❛r❞s r✐s❦

◮ ❲❡ ❝♦♥s✐❞❡r ❛ st♦❝❤❛st✐❝ ✐♥❝♦♠❡ ♣r♦❝❡ss t♦ ❢♦r♠❛❧✐s❡ ✉♥❝❡rt❛✐♥t② ◮ ❆♥❞ ❞♦ s♦ ✐♥ ❛ ♣❛rs✐♠♦♥✐♦✉s ✇❛②✱ ✉s✐♥❣ ▼❛r❦♦✈ ♣r♦❝❡ss❡s

❙t♦❝❤❛st✐❝ ♦♣t✐♠✐s❛t✐♦♥ ▼❛r❦♦✈ ♣r♦❝❡ss❡s

❙✉♣❡r ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ st♦❝❤❛st✐❝ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ■

❙t♦❝❤❛st✐❝ ♣r♦❝❡ss✿ s❡q✉❡♥❝❡ {yt}t=✶,... ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✴✈❡❝t♦rs ❚❤❡ ▼❛r❦♦✈ ♣r♦♣❡rt② ◮ ❙✉♣♣♦s❡ {yt}t=✶,✷,... ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s✉♣♣♦rt Y ◮ ❚❤❡♥ {yt} s❛t✐s✜❡s t❤❡ ▼❛r❦♦✈ ♣r♦♣❡rt② ✐❢✱ ❢♦r ❛❧❧ y ∈ Y Pr♦❜ (yt+✶ = y | yt, . . . , y✶) = Pr♦❜ (yt+✶ = y | yt) ❢♦r ❞✐s❝r❡t❡ Y Pr♦❜ (yt+✶ < y | yt, . . . , y✶) = Pr♦❜ (yt+✶ < y | yt) ❢♦r ❝♦♥t✐♥✉♦✉s Y

❙✉♣❡r ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ st♦❝❤❛st✐❝ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ■■

◮ ❚❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❦♥♦✇♥ ❛s t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ Qt (yt, yt+✶) = Pr♦❜ (yt+✶ | yt) ◮ ❚✐♠❡✲✐♥✈❛r✐❛♥t ♣r♦❝❡ss✿ Qt (yt, yt+✶) = Q (yt, yt+✶) ◮ Q : Y × Y → [✵, ✶] ✐s ❛ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ✐❢ Q (yt, y) ✐s ❛ ♣❞❢✿ ❋♦r ❡❛❝❤ yt ∈ Y Q (yt, y) ≥ ✵ ❢♦r ❛❧❧ y ∈ Y ❛♥❞

• Y

Q (yt, y) dy = ✶

❙✉♣❡r ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ st♦❝❤❛st✐❝ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ■■■

◮ ▼❛r❦♦✈ ♣r♦❝❡ss✿ st♦❝❤❛st✐❝ ♣r♦❝❡ss s❛t✐s❢②✐♥❣ t❤❡ ▼❛r❦♦✈ ♣r♦♣❡rt② ◮ ❈❤❛r❛❝t❡r✐s❡❞ ❜② ✸ ♦❜❥❡❝ts

◮ t❤❡ ❞♦♠❛✐♥ Y ◮ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ Q ◮ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ y✶

◮ ❚❤❡s❡ ❢✉❧❧② ❝❤❛r❛❝t❡r✐s❡ t❤❡ ❥♦✐♥t ❛♥❞ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ♦❢ y ❛t ❛❧❧ ♣♦✐♥ts ✐♥ t✐♠❡

❙✉♣❡r ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ t♦ st♦❝❤❛st✐❝ ▼❛r❦♦✈ ♣r♦❝❡ss❡s ■❱

◮ ❚❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ yt ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✐t❡r❛t✐✈❡❧② ◮ ▲❡t πt−✶ ❜❡ t❤❡ ♣❞❢ ♦❢ y ❛t t✐♠❡ t − ✶✳ ❚❤❡♥✱ ✐❢ πt−✶ ✐s ❦♥♦✇♥ πt(yt) =

• y∈Y

Q(y, yt)πt−✶(y) dy ✇❤❡r❡ πt ❜❡ t❤❡ ♣❞❢ ♦❢ y ❛t t✐♠❡ t ◮ ❆ ▼❛r❦♦✈ ♣r♦❝❡ss ✐s st❛t✐♦♥❛r② ✐❢ πt(y) = πt′(y) = π(y) ◮ ■♥ t❤✐s ❝❛s❡✱ π ✐s t❤❡ ✜①❡❞ ♣♦✐♥t ✐♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ π(yt) =

• y∈Y

Q(y, yt)π(y) dy

❙t♦❝❤❛st✐❝ ♦♣t✐♠✐s❛t✐♦♥ ■✐❞ ✐♥❝♦♠❡ ♣r♦❝❡ss

▼❡♠♦r②❧❡ss ✐♥❝♦♠❡ ♣r♦❝❡ss ✇✐t❤ ❞✐s❝r❡t❡ s✉♣♣♦rt

◮ ❚❛❦❡ ❛ ❞✐s❝r❡t❡ ✐♥❝♦♠❡ ♣r♦❝❡ss yt ∈ Y =

• y ✶, . . . , y n

◮ ❋♦r ❛ ♠❡♠♦r②❧❡ss ♣r♦❜❧❡♠✱ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ❡q✉❛❧s t❤❡ ✉♥❝♦♥❞✐t✐♦♥❛❧ ♣❞❢✿ πi = Pr♦❜

• yt = y i

= Q

• y, y i

❢♦r ❡❛❝❤ i = ✶, . . . , n ◮ ❚❤❡ ❝♦♥s✉♠❡r✬s ♣r♦❜❧❡♠ ✐s

Vt (at, yt) = max

at+✶∈Dt(at,yt)

  f (at, yt, at+✶) + β

• yi ∈Y

Vt+✶

• at+✶, y i

πi    s✳t✳ yt ✐s ❛ r✈ ✇✐t❤ ♣❞❢ π

◮ ❚❤❡ ♣r♦❜❧❡♠ ✐s s❡t✉♣ ❛s ❛ ▼❛r❦♦✈ ♣r♦❝❡ss✿ (at+✶, yt+✶) ❞❡♣❡♥❞s ♦♥❧② ♦♥ (at, yt)

▼❡♠♦r②❧❡ss ✐♥❝♦♠❡ ♣r♦❝❡ss ✇✐t❤ ❝♦♥t✐♥✉♦✉s s✉♣♣♦rt

◮ ❚❤❡ ♣r♦❜❧❡♠ ✐s

Vt (at, yt) = max

at+✶∈Dt(at,yt)

• f (at, yt, at+✶) + β
• y∈Y

Vt+✶ (at+✶, y) π(y) dy

• ◮ ❋❡❛s✐❜✐❧✐t② s❡t✿ s❛✈✐♥❣s ❝❤♦✐❝❡s ❡♥s✉r✐♥❣ ♣♦s✐t✐✈❡ ❝♦♥s✉♠♣t✐♦♥ ✐s

❛✛♦r❞❛❜❧❡ ❡✈❡♥ ✐♥ ✇♦rst ♣♦ss✐❜❧❡ s❝❡♥❛r✐♦

Dt(at, yt) =

• at+✶ : at + yt − at+✶

R > ✵, at+✶ +

T

• τ=t+✶

R(t+✶)−τymin > ✵

❙✉♣♣♦rt ❛♥❞ ❢❡❛s✐❜✐❧✐t② s❡t ✐♥ ♣r❛❝t✐❝❡

◮ ❋❡❛s✐❜✐❧✐t② s❡t ❢♦r at+✶ ✐s Dt(at, yt)

◮ ❙❡t ♦❢ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s at+✶ ❣✐✈❡♥ ❝✉rr❡♥t ✈❛❧✉❡ ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s ◮ ❈♦♠♣✉t❛t✐♦♥❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥✿ ♦♣t✐♠❛❧ s❛✈✐♥❣s ❝❤♦s❡♥ ✐♥ Dt(at, yt)

◮ ❙✉♣♣♦rt ♦❢ at+✶ ✐s At+✶

◮ ❘❛♥❣❡ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ at+✶✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ ❝✉rr❡♥t ✈❛❧✉❡ ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s ◮ ❈♦♠♣✉t❛t✐♦♥❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥✿ ❣r✐❞ ✐♥ at+✶ ❞r❛✇♥ t♦ r❡♣r❡s❡♥t At+✶

◮ ❈❧❡❛r❧② Dt(at, yt) ⊆ At+✶ ❢♦r ❛❧❧ (at, yt) ◮ ❙✉♣♣♦s❡ ✇❡ ❜♦✉♥❞ ❝♦♥s✉♠♣t✐♦♥ ❝❤♦✐❝❡s ❢r♦♠ ❜❡❧♦✇✿ ❡♥s✉r❡ cmin ❛❧✇❛②s ❛✛♦r❞❛❜❧❡ ◮ ❆♥❞ ✉s❡ ❜♦✉♥❞❡❞ s✉♣♣♦rt ♦❢ ✐♥❝♦♠❡ ✐s Y = [ymin, ymax]

❙✉♣♣♦rt ❛♥❞ ❢❡❛s✐❜✐❧✐t② s❡t ✐♥ ♣r❛❝t✐❝❡✿ s✉♣♣♦rt

◮ ❯♣♣❡r ❜♦✉♥❞ ♦❢ At+✶✿ ♠❛①✐♠✉♠ s❛✈✐♥❣s r❡❛❝❤❡❞ ✐❢ yt = ymax ❛♥❞ ct = cmin ✐♥ t❤❡ ♣❛st

at+✶ ≤ Rta✶ +

t

• τ=✶

Rτymax −

t

• τ=✶

Rτcmin ⇒ ❯❇t+✶ = Rta✶ + R ✶ − Rt ✶ − R (ymax − cmin)

◮ ▲♦✇❡r ❜♦✉♥❞ ♦❢ At+✶✿ ❡♥s✉r❡s cmin ❛❧✇❛②s ❛✛♦r❞❛❜❧❡ ✐♥ ❢✉t✉r❡

at+✶ +

T

• τ=t+✶

R(t+✶)−τymin ≥

T

• τ=t+✶

R(t+✶)−τcmin ⇒ ▲❇t+✶ = ✶ − Rt−T ✶ − R−✶ (cmin − ymin)

◮ ❙♦ At+✶ = [▲❇t+✶, ❯❇t+✶]

❙✉♣♣♦rt ❛♥❞ ❢❡❛s✐❜✐❧✐t② s❡t ✐♥ ♣r❛❝t✐❝❡✿ ❢❡❛s✐❜✐❧✐t② s❡t

◮ ❯♣♣❡r ❜♦✉♥❞ ♦❢ Dt ❝♦♥❞✐t✐♦♥❛❧ ♦♥ (at, yt) ❡♥s✉r❡s ct ≥ cmin

at + yt − at+✶R−✶ ≥ cmin ⇒ ❯❇t+✶(at, yt) = R(at + yt − cmin)

◮ ▲♦✇❡r ❜♦✉♥❞ ♦❢ Dt ❡q✉❛❧s ❧♦✇❡r ❜♦✉♥❞ ♦❢ At+✶✿ LBt+✶ ❝❛♥ ❛❧✇❛②s ❜❡ r❡❛❝❤❡❞ ♦r ♦t❤❡r✇✐s❡ ♣r♦❜❧❡♠ ❤❛s ♥♦ s♦❧✉t✐♦♥ ◮ ❙♦ Dt(at, yy) = [▲❇t+✶, ❯❇t+✶(at, yy)]

▼❡♠♦r②❧❡ss ✐♥❝♦♠❡ ♣r♦❝❡ss✿ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s

◮ ❋♦❝ ❛t t✐♠❡ t✿ ❞❡r✐✈❛t✐✈❡ ♦❢ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛t t✐♠❡ t ✐s ③❡r♦

f✸ (at, yt, at+✶) + β

• y∈Y

∂Vt+✶ (at+✶, y) ∂at+✶ π(y) dy = ✵

◮ ❲♦r❦ ♦✉t ♠❛r❣✐♥❛❧ ✈❛❧✉❡ ♦❢ at✿

∂Vt (at, yt) ∂at = f✶ +     f✸ + β

• y∈Y

∂Vt+✶ ∂at+✶ π(y) dy

• =✵

     ∂at+✶ ∂at = f✶ (at, yt, at+✶)

◮ ❙♦ ❛♥ ✐♥t❡r✐♦r ♦♣t✐♠✉♠ s❛t✐s✜❡s

f✸(at, yt, at+✶) + β

• y∈Y

f✶(at+✶, y, at+✷)π(y) dy = ✵ ⇔ u′(ct) − βR❊t

• u′(ct+✶)
• =

✵

❙t♦❝❤❛st✐❝ ♦♣t✐♠✐s❛t✐♦♥ ■✐❞ ✐♥❝♦♠❡ ♣r♦❝❡ss✿ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠

✶✳ P❛r❛♠❡t❡r✐s❡ ♠♦❞❡❧ ❛♥❞ s❡❧❡❝t ❣r✐❞s (A, Y ) ❛♥❞ ❝♦♠♣✉t❡ ✇❡✐❣❤ts πj ✷✳ ❈❤♦♦s❡ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥ ǫ > ✵ ✸✳ ❙t♦r❡ ❊VT+✶

• ai

t+✶

• = ✵ ❢♦r ❛❧❧ i = ✶, . . . , na

✹✳ ▲♦♦♣ ♦✈❡r t ❜❛❝❦✇❛r❞s✿ t = T, . . . , ✶ ▲♦♦♣ ♦✈❡r i = ✶, . . . , na

✹✳✶ ❈♦♠♣✉t❡ ❢♦r j = ✶, . . . , ny g ij

t = arg max at+✶∈Dij

t

• u
• ai

t + y j − at+✶

R

• + β

❊V t+✶ (at+✶)

• ✹✳✷ ❈♦♠♣✉t❡ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ✈❛❧✉❡

❊V i

t =

• j=✶,...,ny
• u
• ai

t + y j − g ij t

R

• + β

❊V t+✶

• g ij

t

• πj

Pr❛❝t✐❝❛❧ ✐ss✉❡s ■

◮ ❙t❛t❡ s♣❛❝❡ ✐s ✷✲❞✐♠✿ (a, y)

◮ ❚❤❡ ✐♥❝♦♠❡ ♣r♦❝❡ss ❝♦✉❧❞ ❤❛✈❡ ❛ ❝♦♥t✐♥✉♦✉s s✉♣♣♦rt✿ ❞✐s❝r❡t✐s❡ Y ❛♥❞ s♦❧✈❡ ♣r♦❜❧❡♠ ✐♥ na × ny ♣♦✐♥ts ❢♦r ❡❛❝❤ t ◮ ❇♦✉♥❞s ✐♥ Y✿ ❡♥s✉r❡ ❢❡❛s✐❜✐❧✐t② ❛♥❞ ♠❡❛s✉r❛❜✐❧✐t②

◮ ●r✐❞ ✐♥ a t♦ ❛❝❝♦✉♥t ❢♦r t❤❡ ♠❛♥② ♣♦ss✐❜❧❡ ❢✉t✉r❡ ❝✐r❝✉♠st❛♥❝❡s

◮ ❋❡❛s✐❜✐❧✐t② ❛♠♦✉♥ts t♦ ❡♥s✉r❡ cmin r❡♠❛✐♥s ❛✛♦r❞❛❜❧❡ ◮ ■♠♣♦s❡❞ ♦♥ ✇♦rst ❝❛s❡ s❝❡♥❛r✐♦ ♦❢ ❢✉t✉r❡ ✐♥❝♦♠❡ s♦ ✐t ❤♦❧❞s ✉♥❞❡r ❛❧❧ ♣♦ss✐❜❧❡ ❢✉t✉r❡ ❝✐r❝✉♠st❛♥❝❡s

◮ ❈♦♥t✐♥✉❛t✐♦♥ ✈❛❧✉❡✿ ❊tVt+✶

◮ ▼❡❛s✉r❡❞ ❛t t ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ❡①✐st✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ◮ ❖♥❧② ❛r❣✉♠❡♥t ✐♥ ❊tVt+✶ ✐s at+✶ ◮ ❈❤♦✐❝❡ ♦❢ ❣r✐❞ ✐♥ y t♦ s✉♣♣♦rt ✐♥t❡❣r❛t✐♦♥ ◮ ◆❡❡❞ s❡t ♦❢ ✇❡✐❣❤ts t♦ ❝❛❧❝✉❧❛t❡ ✐♥t❡❣r❛❧ ♥✉♠❡r✐❝❛❧❧②✱ πj

Pr❛❝t✐❝❛❧ ✐ss✉❡s ■■

❲❡ ❝❤♦♦s❡ t♦ st♦r❡ ❊V ✐♥st❡❛❞ ♦❢ V ◮ ▼♦r❡ ❡✣❝✐❡♥t✿ s❛✈❡s ❝♦♠♣✉t❛t✐♦♥s ✐♥ s♦❧✉t✐♦♥ ◮ ❈❛♥ ❜❡ ✉s❡❞ t♦ r❡❝♦✈❡r Vt ❛t (a, y) Vt(a, y) = u

• a + y −

gt(a, y) R

• + β

❊V t+✶ ( gt(a, y)) ◮ ■❢ ❤❛❞ st♦r❡❞ Vt✱ st❡♣ ✹✳✶ ✇♦✉❧❞ ❝♦♠♣✉t❡ ✭❢♦r ❡❛❝❤ (i, j, t)✮ g ij

t = arg max at+✶∈Dij

t

• u
• ai

t + y j − at+✶

R

• + β

ny

• l=✶
• Vt+✶
• at+✶, y l

πl

• ✐♥✈♦❧✈✐♥❣ ny ✐♥t❡r♣♦❧❛t✐♦♥s ❢♦r ❡❛❝❤ at+✶ ❝❛❧❧❡❞ ❜② ♠❛①✐♠✐s❛t✐♦♥

r♦✉t✐♥❡

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ■

◮ ❙✉♣♣♦s❡ ✇❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ b

a f (y)πy(y)dy ✇❤❡r❡

◮ πy ✐s t❤❡ ♣❞❢ ♦❢ ② ◮ t❤❡ ✈❛❧✉❡ ♦❢ f ✐s ❦♥♦✇♥ ✐♥ ♣♦✐♥ts y i ✐♥ ❣r✐❞ Y

◮ ❚❤❡ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛❧ ✐s ❛ s✐♠♣❧❡ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ f ♦✈❡r ❛ ❞✐s❝r❡t❡ s❡❧❡❝t❡❞ ❣r✐❞ b

a

f (y)πy(y)dy ≃

ny

• i=✶

f (y i)w i ◮ ❚❤❡ s✐♠♣❧❡st ♣r♦❝❡❞✉r❡ ✭❚❛✉❝❤❡♥✮

✶✳ ❉✐✈✐❞❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ y ✐♥t♦ ny ❡q✉❛❧✲♣r♦❜❛❜✐❧✐t② ✐♥t❡r✈❛❧s✱ Y i ✷✳ ❈♦♠♣✉t❡ t❤❡ ❣r✐❞ ♣♦✐♥ts y i = ❊(y | Y i) ✸✳ ❚❤❡ ✇❡✐❣❤ts ❛r❡ ✉♥✐❢♦r♠✿ w i = n−✶

y

✹✳ ❚❤❡♥

• Y f (y)πy(y)dy ≃ n−✶

y

ny

i=✶ f (y i)

◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ■■

❆❧t❡r♥❛t✐✈❡ ♣r♦❝❡❞✉r❡s ◮ ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡✿ ●❛✉ss✐❛♥ ♥♦❞❡s ❛♥❞ ✇❡✐❣❤ts

• (y i, w i)
• ❛r❡

s❡❧❡❝t❡❞ t♦ ♠❛❦❡ ❡①❛❝t t❤❡ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛❧ ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ✷ny + ✶ ♦r ❧❡ss

◮ ●♦♦❞ ♦♣t✐♦♥ ✐❢ f ❝❛♥ ❜❡ ❝❧♦s❡❧② ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ◮ ❲❡✐❣❤ts ❛♥❞ ♥♦❞❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ y✿ ●❛✉ss✲▲❛❣✉❡rr❡ ❢♦r ♥♦r♠❛❧✱ ●❛✉ss✲❍❡r♠✐t❡ ❢♦r ❧♦❣✲♥♦r♠❛❧✱ ✳✳✳

◮ ▼♦♥t❡✲❈❛r❧♦ s✐♠✉❧❛t✐♦♥s✿ ❞r❛✇

• y i

r❛♥❞♦♠❧② ❢r♦♠ ✐ts ❞✐str✐❜✉t✐♦♥ ❛♥❞ ❝♦♠♣✉t❡ s✐♠♣❧❡ ❛✈❡r❛❣❡ ♦❢ f (y) ❛t r❛♥❞♦♠ ♣♦✐♥ts

Pr❛❝t✐❝❛❧ ✐ss✉❡s ■■■

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

❋✐♥❞ r♦♦t ♦❢ ❊✉❧❡r ❡q✉❛t✐♦♥✿ ❈❘❘❆ ✉t✐❧✐t②

◮ ❆t ❡❛❝❤ (ai

t, y j, t) ✜♥❞ r♦♦t ✭at+✶✮ ♦❢

u′ ai

t + y j − at+✶

R

• − βR

dV t+✶(at+✶) = ✵ ◮ ■♥✈❡rs❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t② r❡❞✉❝❡s ♥♦♥✲❧✐♥❡❛r✐t② ✐♥ ♠❛r❣✐♥❛❧ ✈❛❧✉❡ ◮ ❈❛♥ s♦❧✈❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✐♥ ✐ts q✉❛s✐✲❧✐♥❡❛r✐s❡❞ ✈❡rs✐♦♥

• ai

t + y j − at+✶

R

• − (βR)− ✶

γ

❧dV t+✶(at+✶) = ✵ ✇❤❡r❡ t❤❡ q✉❛s✐✲❧✐♥❡❛r ❡①♣❡❝t❡❞ ♠❛r❣✐♥❛❧ ✈❛❧✉❡ ✭❧dV ✮ ✐s st♦r❡❞ ❧dV i

t+✶ = (u′)−✶

dV i

t+✶

• =

 

ny

• j=✶
• ai

t+✶ + y j − g ij t+✶

R −γ πj  

− ✶

γ

❙t♦❝❤❛st✐❝ ♦♣t✐♠✐s❛t✐♦♥ ❆✉t♦❝♦rr❡❧❛t❡❞ ✐♥❝♦♠❡ ♣r♦❝❡ss

❆✉t♦❝♦rr❡❧❛t❡❞ ✐♥❝♦♠❡ ♣r♦❝❡ss

◮ ▼♦r❡ ✐♥t❡r❡st✐♥❣ ♠♦❞❡❧ ♦❢ ✐♥❝♦♠❡✿ ❆❘✭✶✮ ♣r♦❝❡ss ◮ ❲❡ ❛ss✉♠❡ ln yt = α + ρ ln yt−✶ + et ◮ yt ✐s ❛ ▼❛r❦♦✈ ♣r♦❝❡ss✿ ▼❛r❦♦✈ str✉❝t✉r❡ ♦❢ ❞②♥❛♠✐❝ ♣r♦❜❧❡♠ ♥♦t ❝♦♠♣r♦♠✐s❡❞ ◮ ❙t❛t✐♦♥❛r✐t② r❡q✉✐r❡s t❤❛t ✉♥❝♦♥❞✐t✐♦♥❛❧ ♣❞❢ ♦❢ y ✐s t✐♠❡✲✐♥✈❛r✐❛♥t ◮ ❙t❛t✐♦♥❛r✐t② ✉♥❞❡r ❧♦❣✲♥♦r♠❛❧✐t② r❡q✉✐r❡s |ρ| < ✶ ❛♥❞✱ ❢♦r ❛❧❧ t

◮ ❊(ln yt) = α(✶ − ρ)−✶ ◮ ❱❛r(ln yt) = σ✷

e

• ✶ − ρ✷−✶

❆✉t♦❝♦rr❡❧❛t❡❞ ✐♥❝♦♠❡ ♣r♦❝❡ss✿ ♠♦❞❡❧

❚❤❡ ❝♦♥s✉♠♣t✐♦♥✲s❛✈✐♥❣s ♣r♦❜❧❡♠ ✐s ✭Dt(a, y) ❛s ❞❡✜♥❡❞ ❡❛r❧✐❡r✮

Vt(at, yt) = max

at+✶∈Dt

• f (at, yt, at+✶) + β
• Vt+✶ (at+✶, y ρ

t exp {α + e}) dFe(e)

• ◮ ●❡♥❡r❛❧❧② ♥❡❡❞ t♦ ❜♦✉♥❞ ❞♦♠❛✐♥ ♦❢ e t♦ ❡♥s✉r❡ ❢❡❛s✐❜✐❧✐t② ❛♥❞

♠❡❛s✉r❛❜✐❧✐t② ❛t ❛❧❧ ♣♦✐♥ts ◮ ❚❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✐s u′(ct) = βR❊t [u′(ct+✶) | yt]

❙✐♠♣❧❡ ❡①❛♠♣❧❡ ■

◆♦t ♠♦st ❛♣♣❡❛❧✐♥❣ ✷✲♣❡r✐♦❞ ♠♦❞❡❧✳✳✳ ❜✉t ❝❛♥ ❜❡ s♦❧✈❡❞ ❡①♣❧✐❝✐t❧② ◮ P❡r✐♦❞ ✶✿ ❝♦♥s✉♠❡r ❡♥❞♦✇❡❞ ✇✐t❤ (a✶, y✶)✱ ❝♦♥s✉♠❡s c✶ ◮ P❡r✐♦❞ ✷✿

◮ a✷ = R(a✶ + y✶ − c✶) ◮ y✷ = ρy✶ + e✷ ◮ c✷ = R(a✶ + y✶ − c✶) + (ρy✶ + e✷)

✇❤❡r❡ e✷ ✐s ❛ r✈ ♦❢ ♠❡❛♥ ③❡r♦✱ ✉♥❦♥♦✇♥ ❢r♦♠ ♣❡r✐♦❞ ✶ ❛♥❞ ✉♥r❡❧❛t❡❞ t♦ ♦t❤❡r ♠♦❞❡❧ ✈❛r✐❛❜❧❡s ◮ ❯t✐❧✐t② ❢✉♥❝t✐♦♥✿ u(c) = δ✵ + δ✶c + δ✷c✷ ◮ ❈♦♥s✉♠❡rs ♣r♦❜❧❡♠✿ max

c✶

{u (c✶) + β❊✶u [R(a✶ + y✶ − c✶) + (ρy✶ + e✷)]}

❙✐♠♣❧❡ ❡①❛♠♣❧❡ ■■

◮ ❚❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✐s ✭✇✐t❤ βR = ✶✮ δ✶ + δ✷c✶ = δ✶ + δ✷❊ [R(a✶ + y✶ − c✶) + (ρy✶ + e✷)] = δ✶ + δ✷ [R(a✶ + y✶ − c✶) + ρy✶] ◮ ❲✐t❤ s♦❧✉t✐♦♥ c✶ = R ✶ + R a✶ + ρ + R ✶ + R y✶ ◮ ■❢ ρ = ✵✿ ✐♥❝♦♠❡ s❤♦❝❦s ❞♦ ♥♦t ♣❡rs✐st ❛♥❞ ❝♦♥s✉♠♣t✐♦♥ r❡s♣♦♥❞s ❧❡ss t♦ s❤♦❝❦s ◮ ■❢ ρ = ✶✿ ♣❡r♠❛♥❡♥t ✐♥❝♦♠❡ s❤♦❝❦s ❛♥❞ ❝♦♥s✉♠♣t✐♦♥ r❡s♣♦♥❞s ❢✉❧❧② t♦ s❤♦❝❦s

❙♦❧✉t✐♦♥ ❛❧❣♦r✐t❤♠

✶✳ P❛r❛♠❡t❡r✐s❡ ♠♦❞❡❧ ❛♥❞ s❡❧❡❝t ❣r✐❞s (A, Y ) ❛♥❞ ❝♦♠♣✉t❡ ✇❡✐❣❤ts Qjl ✷✳ ❈❤♦♦s❡ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥ ǫ > ✵ ✸✳ ❙t♦r❡ ❊VT+✶

• ai

t+✶, y j

= ✵ ❢♦r ❛❧❧ i = ✶, . . . , na ❛♥❞ j = ✶, . . . , ny ✹✳ ▲♦♦♣ ♦✈❡r t ❜❛❝❦✇❛r❞s✿ t = T, . . . , ✶ ▲♦♦♣ ♦✈❡r i = ✶, . . . , na

✹✳✶ ❈♦♠♣✉t❡ ❢♦r j = ✶, . . . , ny g ij

t = arg max at+✶∈Dij

t

• u
• ai

t + y j − at+✶

R

• + β

❊V t+✶

• at+✶, y j

✹✳✷ ❈♦♠♣✉t❡ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ✈❛❧✉❡ ❛t ♣♦✐♥t (at, yt−✶) = (ai

t, y l)

❊V il

t

=

• j=✶,...,ny
• u
• ai

t + y j − g ij t

R

• + β

❊V t+✶

• g ij

t , y j

Qlj

Pr❛❝t✐❝❛❧ ✐ss✉❡s

◮ ❚❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ✈❛❧✉❡ ❛t t✐♠❡ t ✐s ❊t [Vt+✶(at+✶, yt+✶) | yt]✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ (at+✶, yt) ◮ ■❢ t❤❡ ❢♦❝ ✇❡r❡ t♦ ❜❡ ✉s❡❞ ✐♥ t❤❡ s♦❧✉t✐♦♥✱ t❤❡ ❧✐♥❡❛r✐s❡❞ ❡①♣❡❝t❡❞ ♠❛r❣✐♥❛❧ ✈❛❧✉❡ ✐♥ t✐♠❡ t ❊✉❧❡r ❡q✉❛t✐♦♥ ✇♦✉❧❞ ❛❧s♦ ❜❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ (at+✶, yt) ◮ P❡rs✐st❡♥❝② ✐♥ yt ✐♠♣❧✐❡s t❤❛t t❤❡ ✐♥t❡❣r❛t✐♦♥ ✇❡✐❣❤ts Q ♥❡❡❞ t♦ ❜❡ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ t❤❡ ♣❛st r❡❛❧✐s❛t✐♦♥ ♦❢ y

❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ t♦ ❞❡t❡r♠✐♥❡ Qji

◮ ❈♦♥s✐❞❡r ❛ st❛t✐♦♥❛r② ▼❛r❦♦✈ ♣r♦❝❡ss xt = α + ρxt−✶ + et ✇❤❡r❡ e ∼ N(✵, σ✷) ◮ ❆ s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ t♦ ❝♦♠♣✉t❡ Qjl

✶✳ ❉✐✈✐❞❡ t❤❡ ❞♦♠❛✐♥ X ✐♥ nx ✐♥t❡r✈❛❧s {X i =

• xi, xi

} ✷✳ ❈♦♠♣✉t❡ t❤❡ ❣r✐❞ ♣♦✐♥ts xi = ❊(xi | xi ∈ X i) ✸✳ ❚❤❡♥ Qji = Pr♦❜

• xt ∈ X i | xt−✶ = xj

= Pr♦❜

• xi ≤ α + ρxj + et ≤ xi

= Pr♦❜

• xi − α − ρxj ≤ et ≤ xi − α − ρxj

= Φ xi − α − ρxj σ

• − Φ

xi − α − ρxj σ

Pr❛❝t✐❝❛❧ s❡ss✐♦♥ ✸

■♥✜♥✐t❡ ❤♦r✐③♦♥ ❚❤❡ ♣r♦❜❧❡♠

❈♦♥s✉♠♣t✐♦♥✲s❛✈✐♥❣s ✇✐t❤ ✐♥✜♥✐t❡ ❤♦r✐③♦♥

◮ ❖❢t❡♥ ✉s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❞②♥❛♠✐❝ ♣r♦❜❧❡♠s ✐♥ ✐♥✜♥✐t❡ ❤♦r✐③♦♥

◮ ❙❤♦rt t✐♠❡ ♣❡r✐♦❞s ◮ ❊♥❞ ♣❡r✐♦❞ ✈❡r② ❢❛r ❛✇❛② ◮ ❊♥❞ ♣❡r✐♦❞ ✉♥❝❡rt❛✐♥ ❛♥❞ ♥♦t ❜❡❝♦♠✐♥❣ ♠♦r❡ ❧✐❦❡❧② ♦✈❡r t✐♠❡

◮ ■♥❤❡r✐ts ♠❛♥② ♦❢ t❤❡ ❢❡❛t✉r❡s ♦❢ ✜♥✐t❡ ❤♦r✐③♦♥ ♣r♦❜❧❡♠ ❜✉t ❝♦♥❝❡♣t✉❛❧❧② ♠♦r❡ ❝♦♠♣❧❡① ◮ ▼❛r❦♦✈ str✉❝t✉r❡ ♦❢ ♣r♦❜❧❡♠ ✐s ❦❡②✿ ❝❛♥♥♦t ❞❡❛❧ ✇✐t❤ ❞❡♣❡♥❞❡♥❝✐❡s ♦♥ ✐♥✜♥✐t❡ ♣❛st ◮ ❙t❛t✐♦♥❛r✐t② ✭❛t ❧❡❛st ✐♥ ❧✐♠✐t✮ ✐s ❛❧s♦ ❝r✉❝✐❛❧✿ ❞✐♠❡♥s✐♦♥❛❧✐t② ♣r♦❜❧❡♠✱ ❛♥❞ ♣♦ss✐❜❧② ♠❡❛s✉r❡♠❡♥t ♣r♦❜❧❡♠s ❛s ✇❡❧❧

❚❤❡ ♣r♦❜❧❡♠ ❛t t✐♠❡ t

❚❤✐s ✐s Vt (at, yt) = ❊t

• max

Dt:∞(at,yt) ∞

• τ=t

βτ−tf (aτ, yτ, aτ+✶) | at, yt

• ◮ ❚❤❡ ❤♦r✐③♦♥ ✐s ❛❧✇❛②s ✐♥✜♥✐t❡✱ ✇❤✐❝❤❡✈❡r t

◮ ❈♦♥❞✐t✐♦♥❛❧ ♦♥ (a, y)✱ t❤❡ ❢❡❛s✐❜✐❧✐t② s❡t ✐s ❛❧✇❛②s t❤❡ s❛♠❡✱ D∞(a, y) ◮ ❈♦♥❞✐t✐♦♥❛❧ ♦♥ (a, y)✱ t❤❡ ♣r♦❜❧❡♠ ✐s ❛❧✇❛②s t❤❡ s❛♠❡✱ V (a, y)

◮ ●✐✈❡♥ st❛t✐♦♥❛r✐t② t❤❡ ✐♥✜♥✐t❡ ❤♦r✐③♦♥ ♣r♦❜❧❡♠ ✐s t✐♠❡✲✐♥✈❛r✐❛♥t ◮ ❍❡♥❝❡ ❝❛♥ ❞r♦♣ t✐♠❡ ✐♥❞❡①❡s

❘❡❝✉rs✐✈❡ ❢♦r♠ ■

❚❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥

V (a, y) = ❊

• max

D∞(a,y) ∞

• t=✵

βtf (at, yt, at+✶)

• a, y
• =

max

a′∈D(a,y)

             f (a, y, a′) + β❊y′|y

• ❊
• max

D∞(a′,y′) ∞

• t=✵

βtf (at, yt, at+✶)

• a′, y ′
• ❊①♣❡❝t❡❞ ✈❛❧✉❡ t♦❞❛② ♦❢ V ′ t♦♠♦rr♦✇✱ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ (a, y)

             = max

a′∈D(a,y)

• f (a, y, a′) + β❊y′|y
• V
• a′, y ′

❘❡❝✉rs✐✈❡ ❢♦r♠ ■■

V (a, y) = max

a′∈D(a,y)

       f (a, y, a′) + β ❊y ′|y [V (a′, y ′)]

• Y V (a′,y ′)Q(y,y ′)dy ′

       ◮ ❚❤✐s ✐s t❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ◮ ❚❤❡ s♦❧✉t✐♦♥ ✐s ❛ ✜①❡❞ ♣♦✐♥t V ♦❢ t❤✐s ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ◮ ❑❡② t♦ t❤❡ s♣❡❝✐✜❝❛t✐♦♥✿ st❛t✐♦♥❛r✐t② ♦❢ t❤❡ ▼❛r❦♦✈ ♣r♦❝❡ss

❋❡❛s✐❜✐❧✐t② s❡t

◮ ❉❡t❡r♠✐♥❡❞ ❜② ❛ s❡t ♦❢ ❝♦♥❞✐t✐♦♥s a′ = R(a + y − c) ln y ′ = α + ρ ln y + e′ e ∼ N

• ✵, σ✷

e

• ln y✵ ∼ N
• µln y, σ✷

ln y

• (a✵, y✵) ∈ A × Y

❛ ❜♦✉♥❞✐♥❣ ❝♦♥❞✐t✐♦♥ ◮ ❙t❛t✐♦♥❛r✐t② r❡q✉✐r❡s µln y = α ✶ − ρ ❛♥❞ σ✷

ln y =

σ✷

e

✶ − ρ✷

❇♦✉♥❞✐♥❣ ❝♦♥❞✐t✐♦♥

◮ ❚②♣✐❝❛❧ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t tr❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐s s❛t✐s✜❡❞ lim

t→∞ βt❊

∂f (at, yt, at+✶) ∂at at

• =

✵ ◮ ❚❤✐s ✐s s✐♠✐❧❛r t♦ t❤❡ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s

◮ ■♥ t❤❡ ❧✐♠✐t✱ ❡✐t❤❡r t❤❡ ♣r❡s❡♥t ♠❛r❣✐♥❛❧ ✈❛❧✉❡ ♦❢ ❛ss❡ts ✐s ✵ ♦r t❤❡ ❛❣❡♥t ❝♦♥s✉♠❡s ❛❧❧ ❤❡r ✇❡❛❧t❤ ◮ ❊♥s✉r❡s t❤❛t ❝♦♥s✉♠❡r ❝❛♥♥♦t ❜♦rr♦✇ t♦♦ ♠✉❝❤✿ ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ ❛ss❡ts ✐♥ ❢❛r ❢✉t✉r❡ ✐s ③❡r♦

■♥✜♥✐t❡ ❤♦r✐③♦♥ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥

❈♦♥tr❛❝t✐♦♥ ▼❛♣♣✐♥❣ r❡s✉❧t ■

◮ C(X) ✐s s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✇✐t❤ s✉♣♣♦rt X ◮ T : C(X) → C(X) ✐s ❛ tr❛♥s❢♦r♠❛t✐♦♥ ✭♠❛♣♣✐♥❣✮✿ Tw(x) = v(x) ◮ T s❛t✐s❢②✐♥❣ ❇❧❛❝❦✇❡❧❧ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣ ▼♦♥♦t♦♥✐❝✐t② ✐❢ v, w ∈ C(X) ❛♥❞ v(x) ≤ w(x) ❢♦r ❛❧❧ x ∈ X✱ t❤❡♥ Tv(x) ≤ Tw(x) ❉✐s❝♦✉♥t✐♥❣ t❤❡r❡ ✐s β ∈ (✵, ✶) s✉❝❤ t❤❛t T(v + k)(x) ≤ Tv(x) + βk ❢♦r ❛❧❧ k > ✵✱ ✇❤❡r❡ (v + k)(x) = v(x) + k

❈♦♥tr❛❝t✐♦♥ ▼❛♣♣✐♥❣ r❡s✉❧t ■■

◮ ❋✐①❡❞ ♣♦✐♥t ♦❢ T ✐s ❛ ❢✉♥❝t✐♦♥ v : T(v(x)) = v(x) ◮ ❈♦♥tr❛❝t✐♦♥ ▼❛♣♣✐♥❣ ❚❤❡♦r❡♠ ■❢ T ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣ ✇✐t❤ ♠♦❞✉❧✉s β✱ t❤❡♥

✶✳ T ❤❛s ❡①❛❝t❧② ✶ ✜①❡❞ ♣♦✐♥t ✷✳ ❋✐①❡❞ ♣♦✐♥t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ✐t❡r❛t✐✈❡❧② ❢r♦♠ ❛♥② v✵ ∈ C(X)

◮ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ❞❡✜♥❡s ❛ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣ ✇✐t❤ ♠♦❞✉❧✉s β TV (a, y) = max

a′∈D(a,y)

• f (a, y, a′) + β
• Y

V (a′, y ′)Q(y, y ′)dy ′

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss r❡s✉❧t

❚❤❡ ❞②♥❛♠✐❝ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠ ✐s V (a, y) = max

a′∈D(a,y)

• f (a, y, a′) + β
• Y

V (a′, y ′)Q(y, y ′)dy ′

• ◮ β ∈ (✵, ✶)❀

◮ f ✿ r❡❛❧✲✈❛❧✉❡❞✱ ❝♦♥t✐♥✉♦✉s✱ str✐❝t❧② ❝♦♥❝❛✈❡ ✐♥ a ❛♥❞ ❜♦✉♥❞❡❞❀ ◮ y✿ ▼❛r❦♦✈ ♣r♦❝❡ss ✐♥ t❤❡ ❝♦♠♣❛❝t s❡t Y ❀ ◮ D(a, y)✿ ♥♦♥✲❡♠♣t②✱ ❝♦♠♣❛❝t ❛♥❞ ❝♦♥✈❡①✳ ❚❤❡♥✿ ✶✳ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❢✉♥❝t✐♦♥ V t❤❛t s♦❧✈❡s t❤✐s ♣r♦❜❧❡♠❀ ✷✳ ❱ ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ str✐❝t❧② ❝♦♥❝❛✈❡ ✐♥ a❀ ✸✳ g(a, y) ❡①✐sts ❛♥❞ ✐s ❛ ✭✉♥✐q✉❡✮ ❝♦♥t✐♥✉♦✉s✱ s✐♥❣❧❡✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥✳

❖t❤❡r ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣r♦❜❧❡♠

❖t❤❡r ♣r♦♣❡rt✐❡s ♦❢ f ❛r❡ tr❛♥s❢❡rr❡❞ t♦ V t❤r♦✉❣❤ t❤❡ ♠❛♣♣✐♥❣ T✿ ✶✳ f ❛❧s♦ C✶ ✐♥ (a, a′) ∈ ✐♥t(A)✷ ❛♥❞ g(a, y) ∈ ✐♥t(D(a, y)) ⇒ V ✐s C✶ ✐♥ a ❛♥❞ V✶(a, y) = f✶(a, y, g(a, y)) ✷✳ f ❛❧s♦ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✐♥ a ❛♥❞ D(a, y) ≥ D(a′, y) ❢♦r a ≥ a′ ⇒ V ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✐♥ a ✸✳ ✷ ❛❧s♦ tr✉❡ ❢♦r y ✐❢ f ❛♥❞ D(a, y) ❛r❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✐♥ y

❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ✐♥t❡r✐♦r s♦❧✉t✐♦♥

◮ ❊✉❧❡r ❡q✉❛t✐♦♥ ✉♥❞❡r t❤❡ ❝♦♥t✐♥✉♦✉s ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s f✸(a, y, a′) + β

• Y

f✶(a′, y ′, g(a′, y ′))Q(y, y ′)dy ′ = ✵ ⇔ f✸(a, y, a′) + β❊y ′|y [f✶(a′, y ′, g(a′, y ′))] = ✵ ❖r ✐♥ t❡r♠s ♦❢ t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ d u(c) d c = βR❊y ′|y d u(c′) d c′

• ◮ ❊✉❧❡r ❛♥❞ tr❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥s✿ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❢♦r t❤❡

✐♥t❡r✐♦r ♦♣t✐♠✉♠ a′ = g(a, y) lim

t→∞ βt❊

∂f (at, yt, at+✶) ∂at at

• =

✵

■♥✜♥✐t❡ ❤♦r✐③♦♥ ❙✐♠♣❧❡ ❡①❛♠♣❧❡

❙✐♠♣❧❡ ❡①❛♠♣❧❡ ■

◮ ❈♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ V (a) = max

c>✵ {ln(c) + βV (R(a − c))}

◮ ❚❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✐s ✶ ct = βR ✶ ct+✶ ⇔ ct+✶ = βRct = (βR)tc✵ ◮ ❚❤❡ tr❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s lim

t→∞ βtu′(ct)at

= lim

t→∞

βtat (βR)tc✵ = lim

t→∞

at Rtc✵ = ✵

❙✐♠♣❧❡ ❡①❛♠♣❧❡ ■■

❲♦r❦ ♦✉t t❤❡ ✈❛❧✉❡ ♦❢ at at = R(at−✶ − ct−✶) = R(R[at−✷ − ct−✷] − ct✶) . . . = Rta✵ −

t−✶

• τ=✵

Rt−τcτ = Rta✵ −

t−✶

• τ=✵

Rt−τ(βR)τc✵ = Rta✵ − Rtc✵

t−✶

• τ=✵

βτ = Rt

• a✵ − c✵

✶ − βt ✶ − β

❙✐♠♣❧❡ ❡①❛♠♣❧❡ ■■■

◮ ❲❡ ❣♦t at = Rt a✵ − c✵

✶−βt ✶−β

• ◮ ❘❡♣❧❛❝❡ ✐♥ tr❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥ t♦ ②✐❡❧❞

lim

t→∞

at Rtc✵ = lim

t→∞

Rt a✵ − c✵

✶−βt ✶−β

• Rtc✵

= a✵ − c✵ ✶ ✶ − β = ✵ ◮ ❍❡♥❝❡ t❤❡ s♦❧✉t✐♦♥ ✐s c✵ = a✵(✶ − β) ◮ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ct = a✵(✶ − β)(βR)t

■♥✜♥✐t❡ ❤♦r✐③♦♥ ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥

❘❡❝✉rs✐♦♥

V (a, y) = max

a′∈D(a,y)

• f (a, y, a′) + β❊y ′|y [V (a′, y ′)| a, y]
• ◮ ❈♦♥tr❛❝t✐♦♥ ▼❛♣♣✐♥❣ ❚❤❡♦r❡♠

✶✳ ❚❤❡ ♣r♦❜❧❡♠ ❤❛s ❛ ✉♥✐q✉❡ ✜①❡❞ ♣♦✐♥t V ✷✳ ■t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ✐t❡r❛t✐✈❡❧② ❢r♦♠ ❛♥② st❛rt✐♥❣ ❢✉♥❝t✐♦♥ V✵

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ ✐t❡r❛t✐♦♥

✶✳ ✜♥❞ ♦♣t✐♠❛❧ s❛✈✐♥❣s gn(a, y) = arg max

a′∈D(a,y)

• f (a, y, a′) + β❊y′|y
• Vn−✶(a′, y ′)
• ✷✳ ❝♦♠♣✉t❡ ♥❡✇ ✈❛❧✉❡ ❢✉♥❝t✐♦♥

Vn(a, y) = f (a, y, gn(a, y)) + β❊y′|y

• Vn−✶(gn(a, y), y ′)

❙♦❧✉t✐♦♥ ❛❧❣♦r✐t❤♠✿ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✐t❡r❛t✐♦♥

✶✳ P❛r❛♠❡t❡r✐s❡ ♠♦❞❡❧ ❛♥❞ s❡❧❡❝t ❣r✐❞s (A, Y ) ❛♥❞ ❝♦♠♣✉t❡ ✇❡✐❣❤ts Qjl ✷✳ ❈❤♦♦s❡ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥ ǫ > ✵ ✸✳ ❙❡❧❡❝t ✐♥✐t✐❛❧ ❣✉❡ss ❊V✵

• ai, y j

❢♦r ❛❧❧

• ai, y j

∈ A × Y ✹✳ ■t❡r❛t❡ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡✱ ❢♦r n = ✶, . . .

✹✳✶ ❋♦r ❛❧❧ ai ✐♥ ❣r✐❞ A ❝♦♠♣✉t❡ (a) g ij

n = arg max a′∈Dij

• f
• ai, y j, a′

+ β ❊V n−✶

• a′, y j

❢♦r ❛❧❧ y = y j (b) ❊V il

n = ny

• j=✶
• f
• ai, y j, g ij

n

• + β

❊V n−✶

• g ij

n , y j

Qlj ❢♦r ❛❧❧ y−✶ = y l ✹✳✷ ❈❤❡❝❦ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❊Vn−✶ ❛♥❞ ❊Vn

◮ ■❢ ❧❛r❣❡r t❤❛♥ ǫ t❤❡♥ ❣♦ ❜❛❝❦ t♦ st❡♣ ✹✳✶ ◮ ❊❧s❡ ❛❝❝❡♣t s♦❧✉t✐♦♥ (gn, Vn) ❛♥❞ st♦♣

Pr❛❝t✐❝❛❧ ✐ss✉❡s

◮ ❚✐♠❡ s✉❜s❝r✐♣t ❞r♦♣♣❡❞ ❛♥❞ ❧♦♦♣ ✐s ♥♦✇ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ V t♦ ✜①❡❞ ♣♦✐♥t ◮ ■♥✐t✐❛❧ ❣✉❡ss

◮ s❤♦✉❧❞ ❜❡ C✶ ◮ ❝♦✉❧❞ ❜❡ ❊V✵ = ✵✿ ✐♠♣❧②✐♥❣ ❝♦♥s✉♠❡r s❛✈❡s ♥♦t❤✐♥❣ t♦ ♥❡①t ♣❡r✐♦❞ ◮ ❜❡tt❡r s♦❧✉t✐♦♥ ✐s ❊V✵ = u(c)

◮ ❙♦❧✉t✐♦♥ ✉s✐♥❣ ❊✉❧❡r ❡q✉❛t✐♦♥✿ st♦r❡ dV ✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ (a, y−✶) ◮ ❉✐st❛♥❝❡ ✐♥ ❝♦♥t✐♥✉❛t✐♦♥ ✈❛❧✉❡✿ ♠❛① ❛❜s♦❧✉t❡ ❞✐✛❡r❡♥❝❡ ✭❧❡✈❡❧s✴r❡❧❛t✐✈❡✮

❋❡❛s✐❜✐❧✐t② s❡t

◮ ❚r❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥ ♥♦t ♣r❛❝t✐❝❛❧ lim

t→∞ βt❊

• f✶(at, yt, a′

t+✶)at

• =

✵ ◮ ■♠♣❧✐❝❛t✐♦♥✿ ❝♦♥s✉♠❡r ❛✈♦✐❞s ❧♦✇ ❛ss❡ts✱ ✇❤❡r❡ f✶ ❛r❜✐tr❛r✐❧② ❧❛r❣❡ ◮ ⇒ ❡♥s✉r❡ cmin ❛❧✇❛②s ❛✛♦r❞❛❜❧❡ ✐♥ ✇♦rst ♣♦ss✐❜❧❡ s❝❡♥❛r✐♦ a +

∞

• t=✵

R−tymin ≥

∞

• t=✵

R−tcmin ⇔ a + ✶ ✶ − R−✶ (ymin − cmin) ≥ ✵ ◮ ■❢ ♣r❡s❡♥t st❛t❡ ✐s (a, y) ⇒ ♦♣t✐♠❛❧ s❛✈✐♥❣s a′ ♠✉st ❧✐❡ ✐♥ ✐♥t❡r✈❛❧ D(a, y) =

• −

✶ ✶ − R−✶ (ymin − cmin) , R (a + y − cmin)

◆♦✇ ❢♦r t❤❡ ✜♥❛❧ ♣r❛❝t✐❝❛❧ ❡①❛♠♣❧❡✦