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N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥✈❡r❣❡♥❝❡ ❢♦r ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

❆❧❡❦♦s ❈❡❝❝❤✐♥

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ✏❏✳❆✳ ❉✐❡✉❞♦♥♥é✑ ❯♥✐✈❡rs✐té ❈ôt❡ ❞✬❆③✉r

❲♦r❦s❤♦♣ ♦♥ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡s✳ ✶✽✲✶✾ ❏✉♥❡ ✷✵✷✵

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

■♥tr♦❞✉❝t✐♦♥

▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ❛s ❧✐♠✐ts ♦❢ N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥✱ ✇❤❡♥ t❤❡ ♥✉♠❜❡r N ♦❢ ❛❣❡♥ts t❡♥❞s t♦ ✐♥✜♥✐t②✳ ◮ ❲❡ ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡ ✇❤❡r❡ t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❡❛❝❤ ❛❣❡♥t ❜❡❧♦♥❣s t♦ ❛ ✜♥✐t❡ st❛t❡ s♣❛❝❡ d = {✶, . . . , d}✳ ◮ ❆❣❡♥ts ❛r❡ ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❛♥❞ ❝♦♥tr♦❧ t❤❡✐r tr❛♥s✐t✐♦♥ r❛t❡ ❢r♦♠ st❛t❡ t♦ st❛t❡ ✐♥ ♦r❞❡r t♦ ♠✐♥✐♠✐③❡ ❛ ❝♦st✳ ◮ ❙②♠♠❡tr✐❝ ❛♥❞ ♠❡❛♥ ✜❡❧❞ ✐♥t❡r❛❝t✐♦♥✿ ❡❛❝❤ ❛❣❡♥t ❦♥♦✇s ✐ts ♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ♦t❤❡r ❛❣❡♥ts ✐♥ ❛♥② ♦❢ t❤❡ d st❛t❡s✳ ◮ ◆♦t✐♦♥ ♦❢ ♦♣t✐♠❛❧✐t② ❛t ♣r❡❧✐♠✐t ❧❡✈❡❧✿ P❛r❡t♦ ❡q✉✐❧✐❜r✐✉♠✳ ❆❣❡♥ts ❛r❡ ❝♦♦♣❡r❛t✐✈❡ ❛♥❞ ❤❛✈❡ ❛ ❝♦♠♠♦♥ ❝♦st t♦ ♠✐♥✐♠✐③❡✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡

▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❛♥❛❧②③❡❞ ✐♥✿ ◮ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ♦♣❡♥✲❧♦♦♣ ❝♦♥tr♦❧s✱ ✉s❡ st♦❝❤❛st✐❝ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✱ ❣❡t ❋❇❙❉❊s ♦❢ ▼❝❑❡❛♥✲❱❧❛s♦✈ t②♣❡❀ ◮ ❬P❤❛♠✲❲❡✐ ✬✶✽❪✿ ❝❧♦s❡❞✲❧♦♦♣ ❝♦♥tr♦❧s✱ ✉s❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣✱ ❣❡t ❍❏❇ ❡q✉❛t✐♦♥✱ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s✳ ◮ ❬❈❛r❞❛❧✐❛❣✉❡t✲●r❛❜❡r✲P♦rr❡tt❛✲❚♦♥♦♥ ✬✶✺❪✿ ♣♦t❡♥t✐❛❧ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡✱ ▼❋● s②st❡♠ ❛s ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♦♣t✐♠❛❧✐t②✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ st✉❞✐❡❞ ✐♥✿ ❬▲❛❝❦❡r ✬✶✼❪✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✈✐❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥ts✱ ♥♦ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡❀ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✉s✐♥❣ str♦♥❣ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡

▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❛♥❛❧②③❡❞ ✐♥✿ ◮ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ♦♣❡♥✲❧♦♦♣ ❝♦♥tr♦❧s✱ ✉s❡ st♦❝❤❛st✐❝ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✱ ❣❡t ❋❇❙❉❊s ♦❢ ▼❝❑❡❛♥✲❱❧❛s♦✈ t②♣❡❀ ◮ ❬P❤❛♠✲❲❡✐ ✬✶✽❪✿ ❝❧♦s❡❞✲❧♦♦♣ ❝♦♥tr♦❧s✱ ✉s❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣✱ ❣❡t ❍❏❇ ❡q✉❛t✐♦♥✱ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s✳ ◮ ❬❈❛r❞❛❧✐❛❣✉❡t✲●r❛❜❡r✲P♦rr❡tt❛✲❚♦♥♦♥ ✬✶✺❪✿ ♣♦t❡♥t✐❛❧ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡✱ ▼❋● s②st❡♠ ❛s ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♦♣t✐♠❛❧✐t②✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ st✉❞✐❡❞ ✐♥✿ ◮ ❬▲❛❝❦❡r ✬✶✼❪✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✈✐❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥ts✱ ♥♦ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡❀ ◮ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✉s✐♥❣ str♦♥❣ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❖✉t❧✐♥❡

✶✳ N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V N

✷✳ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

❱❛❧✉❡ ❢✉♥❝t✐♦♥ ✐s ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ❍❏❇ ❡q✉❛t✐♦♥

✸✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t♦ ✇✐t❤ ❝♦♥✈❡r❣❡❝❡ r❛t❡✱ ✉♥❞❡r ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥✱ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♣r♦♣❡rt② ✹✳ ❯♥❞❡r ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s

❯♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ▼❋❈P ❘❡❣✉❧❛r✐t② ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s✱ ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❖✉t❧✐♥❡

✶✳ N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V N

✷✳ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ❍❏❇ ❡q✉❛t✐♦♥

✸✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t♦ ✇✐t❤ ❝♦♥✈❡r❣❡❝❡ r❛t❡✱ ✉♥❞❡r ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥✱ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♣r♦♣❡rt② ✹✳ ❯♥❞❡r ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s

❯♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ▼❋❈P ❘❡❣✉❧❛r✐t② ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s✱ ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❖✉t❧✐♥❡

✶✳ N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V N

✷✳ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ❍❏❇ ❡q✉❛t✐♦♥

✸✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ V N t♦ V ✇✐t❤ ❝♦♥✈❡r❣❡❝❡ r❛t❡✱ ✉♥❞❡r ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥✱ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♣r♦♣❡rt② ✹✳ ❯♥❞❡r ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s

❯♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ▼❋❈P ❘❡❣✉❧❛r✐t② ♦❢ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s✱ ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❖✉t❧✐♥❡

✶✳ N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V N

✷✳ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ❍❏❇ ❡q✉❛t✐♦♥

✸✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ V N t♦ V ✇✐t❤ ❝♦♥✈❡r❣❡❝❡ r❛t❡✱ ✉♥❞❡r ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥✱ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♣r♦♣❡rt② ✹✳ ❯♥❞❡r ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s

◮ ❯♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ▼❋❈P ◮ ❘❡❣✉❧❛r✐t② ♦❢ V ◮ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s✱ ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❆❣❡♥ts ❞②♥❛♠✐❝s

N ✐❞❡♥t✐❝❛❧ ❛❣❡♥ts X ✶, . . . , X N ✇✐t❤ X k

t ∈ d✱

❊✈♦❧✈❡ ✐♥ ❝♦♥t✐♥✉♦✉s t✐♠❡✱ ✜♥✐t❡ ❤♦r✐③♦♥ T✳ ❞❡♥♦t❡ ① = (x✶, . . . , xN), ❳ t = (X ✶

t , . . . , X N t ) ∈ dN✳

P❧❛②❡r k ❝❤♦♦s❡s ✐ts tr❛♥s✐t✐♦♥ r❛t❡ βk

j (t, ①) ≥ ✵ ✐♥ ▼❛r❦♦✈✐❛♥

❢❡❡❞❜❛❝❦ ❢♦r♠✿ P

• X k

t+h = j|❳ t = ①

• = βk

j (t, ①)h + o(h)

✐♥ ♦r❞❡r t♦ ♠✐♥✐♠✐③❡ t❤❡ ❝♦st Jk(β✶, . . . , βN)=E T

✵

ℓ(X k

t , βk(t, ❳ t))+f (X k t , µN t )dt+g(X k T, µN T)

• ✐✈❡♥ ① ❞❡♥♦t❡ t❤❡ ❡♠♣✐r✐❝❛❧ ♠❡❛s✉r❡ mN

① = ✶ N

N

k=✶ δxk

N · mN

i,t = k ✶{X k

t =i} ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣❧❛②❡rs ✐♥ st❛t❡ i✳ ❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❖♣t✐♠✐③❛t✐♦♥

❈♦♥s✐❞❡r ❢♦r s✐♠♣❧✐❝✐t② ℓi(βk(①)) = ℓ(i, βk(①)) = ✶ ✷

• j=i

|βk

j (t, i, ①−k)|✷

❆❣❡♥ts ❛r❡ ❝♦♦♣❡r❛t✐✈❡✿ ❝♦♠♠♦♥ r❡✇❛r❞ t♦ ♠✐♥✐♠✐③❡ JN(β) = ✶ N

N

• k=✶

Jk(β) str❛t❡❣② ✈❡❝t♦r β = (β✶, . . . , βN) ♥♦t ♥❡❝❡ss❛r✐❧② ❡①❝❤❛♥❣❡❛❜❧❡✳ ❙✐♥❣❧❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r t❤❡ ♣r♦❝❡ss ❳

✶

❞❡✜♥❡❞ ❜② t❤❡ ❣❡♥❡r❛t♦r ①

✶

① ① ①

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❖♣t✐♠✐③❛t✐♦♥

❈♦♥s✐❞❡r ❢♦r s✐♠♣❧✐❝✐t② ℓi(βk(①)) = ℓ(i, βk(①)) = ✶ ✷

• j=i

|βk

j (t, i, ①−k)|✷

❆❣❡♥ts ❛r❡ ❝♦♦♣❡r❛t✐✈❡✿ ❝♦♠♠♦♥ r❡✇❛r❞ t♦ ♠✐♥✐♠✐③❡ JN(β) = ✶ N

N

• k=✶

Jk(β) str❛t❡❣② ✈❡❝t♦r β = (β✶, . . . , βN) ♥♦t ♥❡❝❡ss❛r✐❧② ❡①❝❤❛♥❣❡❛❜❧❡✳ ❙✐♥❣❧❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r t❤❡ ♣r♦❝❡ss ❳ = (X ✶, . . . , X N) ❞❡✜♥❡❞ ❜② t❤❡ ❣❡♥❡r❛t♦r LN

t φ(①) = N

• k=✶
• j=i

βk

j (xk, ①−k)[φ(j, ①−k) − φ(①)]

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❱❛❧✉❡ ❢✉♥❝t✐♦♥

❱❛❧✉❡ ❢✉♥❝t✐♦♥ vN(t, ①) ◮ ❍❏❇ ❡q✉❛t✐♦♥ ✐s ❖❉❊✱ ✐♥❞❡①❡❞ ❜② ① ∈ dN❀ ◮ ❲❡❧❧✲♣♦s❡❞♥❡ss ♦❢ ❍❏❇❀ ◮ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ♦♣t✐♠❛❧ st❛rt❡❣② β✳ ◮ P♦ss✐❜❧❡ t♦ ❝♦♥s✐❞❡r ❛❧s♦ ♦♣❡♥✲❧♦♦♣ ❝♦♥tr♦❧s ❛♥❞ ♥♦♥✲❝♦♥✈❡① ℓ✿ ♠✉❧t✐♣❧❡ ♦♣t✐♠✐③❡rs✱ ♥♦♥ ❡①❝❤❛♥❣❡❛❜❧❡✳ ▼❡❛♥✲✜❡❧❞ ❛ss✉♠♣t✐♦♥✿ ❚❤❡r❡ ❡①✐sts ✵ ✵ s✉❝❤ t❤❛t ①

①

✵

✶

✶ ✱

✶

❆ss✉♠❡ ✵ ❢♦r ❛♥② ✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❱❛❧✉❡ ❢✉♥❝t✐♦♥

❱❛❧✉❡ ❢✉♥❝t✐♦♥ vN(t, ①) ◮ ❍❏❇ ❡q✉❛t✐♦♥ ✐s ❖❉❊✱ ✐♥❞❡①❡❞ ❜② ① ∈ dN❀ ◮ ❲❡❧❧✲♣♦s❡❞♥❡ss ♦❢ ❍❏❇❀ ◮ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ♦♣t✐♠❛❧ st❛rt❡❣② β✳ ◮ P♦ss✐❜❧❡ t♦ ❝♦♥s✐❞❡r ❛❧s♦ ♦♣❡♥✲❧♦♦♣ ❝♦♥tr♦❧s ❛♥❞ ♥♦♥✲❝♦♥✈❡① ℓ✿ ♠✉❧t✐♣❧❡ ♦♣t✐♠✐③❡rs✱ ♥♦♥ ❡①❝❤❛♥❣❡❛❜❧❡✳ ▼❡❛♥✲✜❡❧❞ ❛ss✉♠♣t✐♦♥✿ ❚❤❡r❡ ❡①✐sts αN : [✵, T] × SN

d → [✵, +∞)d×d s✉❝❤ t❤❛t

βk

j (t, xk, ①−k) = αN xk,j(t, µN ① )

Sd = {m ∈ Rd : mi ≥ ✵, d

i=✶ mi = ✶}✱ SN d = Sd ∩ ✶ N Zd

❆ss✉♠❡ ✵ ❢♦r ❛♥② ✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❱❛❧✉❡ ❢✉♥❝t✐♦♥

❱❛❧✉❡ ❢✉♥❝t✐♦♥ vN(t, ①) ◮ ❍❏❇ ❡q✉❛t✐♦♥ ✐s ❖❉❊✱ ✐♥❞❡①❡❞ ❜② ① ∈ dN❀ ◮ ❲❡❧❧✲♣♦s❡❞♥❡ss ♦❢ ❍❏❇❀ ◮ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ♦♣t✐♠❛❧ st❛rt❡❣② β✳ ◮ P♦ss✐❜❧❡ t♦ ❝♦♥s✐❞❡r ❛❧s♦ ♦♣❡♥✲❧♦♦♣ ❝♦♥tr♦❧s ❛♥❞ ♥♦♥✲❝♦♥✈❡① ℓ✿ ♠✉❧t✐♣❧❡ ♦♣t✐♠✐③❡rs✱ ♥♦♥ ❡①❝❤❛♥❣❡❛❜❧❡✳ ▼❡❛♥✲✜❡❧❞ ❛ss✉♠♣t✐♦♥✿ ❚❤❡r❡ ❡①✐sts αN : [✵, T] × SN

d → [✵, +∞)d×d s✉❝❤ t❤❛t

βk

j (t, xk, ①−k) = αN xk,j(t, µN ① )

Sd = {m ∈ Rd : mi ≥ ✵, d

i=✶ mi = ✶}✱ SN d = Sd ∩ ✶ N Zd

❆ss✉♠❡ αi,j ∈ [✵, M] ❢♦r ❛♥② N✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

▼❡❛♥ ✜❡❧❞ N✲❛❣❡♥t ♣r♦❜❧❡♠

❚❤❡ ❝♦st ❜❡❝♦♠❡s JN(αN)= ✶ N

N

• k=✶

E d

• i=✶

✶{X k

T =i}gi(µN

T)

+ T

✵ d

• i=✶

✶{X k

t =i}

• ℓ(i, αN(t, i, µN

t ))+f (i, µN t )

• dt
• = E

T

✵ d

• i=✶

µN

i,t

• ℓi(αN(t, i, µN

t ))+f i(µN t )

• dt +

d

• i=✶

µN

i,Tgi(µN T)

• ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥

✶

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

▼❡❛♥ ✜❡❧❞ N✲❛❣❡♥t ♣r♦❜❧❡♠

❚❤❡ ❝♦st ❜❡❝♦♠❡s JN(αN)= ✶ N

N

• k=✶

E d

• i=✶

✶{X k

T =i}gi(µN

T)

+ T

✵ d

• i=✶

✶{X k

t =i}

• ℓ(i, αN(t, i, µN

t ))+f (i, µN t )

• dt
• = E

T

✵ d

• i=✶

µN

i,t

• ℓi(αN(t, i, µN

t ))+f i(µN t )

• dt +

d

• i=✶

µN

i,Tgi(µN T)

• ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥ µN

t ∈ SN d

P

• µN

t+h = m + ✶

N (ej − ei)

• µN

t = m

• = NmiαN

i,j(t, m)h + o(h)

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❍❏❇✲◆ ❡q✉❛t✐♦♥

V N(t, m) ✈❛❧✉❡ ❢✉♥❝t✐♦♥✱ m ∈ SN

d ✱ ❍❏❇ ❡q✉❛t✐♦♥ ✐s ❖❉❊

− d dt V N +

• i∈d

miH(DN,iV N(t, m)) =

• i∈d

mif i(m) V N(T, m) =

• i∈d

migi(m), ✭❍❏❇✲◆✮ ✇❤❡r❡ [DN,iV N(t, m)]j := N

• V N

m + ✶

N (ej − ei)

• − V N(m)
• ✱

Hi(z)=

j=i{−a∗(−zj)zj − ✶ ✷|a∗(−zj)|✷}✱ a∗(r)=

  

✵ r ≤ ✵ r ✵ ≤ r ≤ M M r ≥ M

❍❏❇ ✇❡❧❧✲♣♦s❡❞✱ ✉♥✐q✉❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ✶ ❘❡s✉❧t✿ ❛♥❞ ①

①

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❍❏❇✲◆ ❡q✉❛t✐♦♥

V N(t, m) ✈❛❧✉❡ ❢✉♥❝t✐♦♥✱ m ∈ SN

d ✱ ❍❏❇ ❡q✉❛t✐♦♥ ✐s ❖❉❊

− d dt V N +

• i∈d

miH(DN,iV N(t, m)) =

• i∈d

mif i(m) V N(T, m) =

• i∈d

migi(m), ✭❍❏❇✲◆✮ ✇❤❡r❡ [DN,iV N(t, m)]j := N

• V N

m + ✶

N (ej − ei)

• − V N(m)
• ✱

Hi(z)=

j=i{−a∗(−zj)zj − ✶ ✷|a∗(−zj)|✷}✱ a∗(r)=

  

✵ r ≤ ✵ r ✵ ≤ r ≤ M M r ≥ M

◮ ❍❏❇ ✇❡❧❧✲♣♦s❡❞✱ ✉♥✐q✉❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ αN

i,j(t, m) = a∗

− N

• V N

m + ✶ N (ej − ei)

• − V N(m)
• ◮ ❘❡s✉❧t✿ minβ JN(β) = minαN JN(αN) ❛♥❞

vN(t, ①) = V N(t, µN

① )

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

N → ∞✿ X ✶, X ✷, . . . ✐✳✐✳❞✳✱ µN → µ = L(X)✱ limN V N ❄❄ ❖♥❡ r❡❢❡r❡♥❝❡ ♣❧❛②❡r X ❝❤♦♦s❡s ✐ts tr❛♥s✐t✐♦♥ r❛t❡ α = (αi,j)d

i,j=✶✱

αi,j(t) ∈ [✵, M] ❞❡t❡r♠✐♥✐st✐❝ ✭✐♥ ❢❡❡❞❜❛❝❦ ❢♦r♠✮ P(Xt+h = j|Xt = i) = αi,j(t)h + o(h) j = i ✐♥ ♦r❞❡r t♦ ♠✐♥✐♠✐③❡✱ L(Xt) = P ◦ X −✶

t

∈ Sd✱ J(α)=E T

✵

✶ ✷

• j=Xt

|αXt,j(t)|✷+ f (Xt, L(Xt))dt+ g(XT, L(XT))

• ❘❡✇r✐t❡ ✐♥ t❤❡ ❢♦r♠✱

✱

✵ ✶

✶ ✷

✷ ✶

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

N → ∞✿ X ✶, X ✷, . . . ✐✳✐✳❞✳✱ µN → µ = L(X)✱ limN V N ❄❄ ❖♥❡ r❡❢❡r❡♥❝❡ ♣❧❛②❡r X ❝❤♦♦s❡s ✐ts tr❛♥s✐t✐♦♥ r❛t❡ α = (αi,j)d

i,j=✶✱

αi,j(t) ∈ [✵, M] ❞❡t❡r♠✐♥✐st✐❝ ✭✐♥ ❢❡❡❞❜❛❝❦ ❢♦r♠✮ P(Xt+h = j|Xt = i) = αi,j(t)h + o(h) j = i ✐♥ ♦r❞❡r t♦ ♠✐♥✐♠✐③❡✱ L(Xt) = P ◦ X −✶

t

∈ Sd✱ J(α)=E T

✵

✶ ✷

• j=Xt

|αXt,j(t)|✷+ f (Xt, L(Xt))dt+ g(XT, L(XT))

• ❘❡✇r✐t❡ ✐♥ t❤❡ ❢♦r♠✱ µi

t = P(Xt = i)✱ µt = L(Xt)

˙ µi

t =

• j=i
• µj

tαj,i(t) − µi tαi,j(t)

• J(α) =

T

✵ d

• i=✶

µi

t

✶ ✷

• j=i

|αi,j(t)|✷ + f i(µt)

• dt +

d

• i=✶

µi

Tgi(µT)

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❍❏❇ ❡q✉❛t✐♦♥

❙✐♥❣❧❡ ❞❡t❡r♠✐♥✐st✐❝ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❍❏❇ ❡q✉❛t✐♦♥ ❢♦r ✈❛❧✉❡ ❢✉♥❝t✐♦♥ V (t, m)✿ − ∂tV +

• i∈d

miH (∂mj − ∂mi)V d

j=✶

• =
• i∈d

mif i(m) V (T, m) =

• i∈d

migi(m), ✭❍❏❇✮ ❋✐rst ♦r❞❡r P❉❊ ✐♥ ✵ ✱ ♥♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s❀

✶

❀ ♥♦ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥s❀ ❡①✐st❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s✱ ♥♦♥✲✉♥✐q✉❡♥❡ss❀ P♦t❡♥t✐❛❧ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡✿ ✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❍❏❇ ❡q✉❛t✐♦♥

❙✐♥❣❧❡ ❞❡t❡r♠✐♥✐st✐❝ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❍❏❇ ❡q✉❛t✐♦♥ ❢♦r ✈❛❧✉❡ ❢✉♥❝t✐♦♥ V (t, m)✿ − ∂tV +

• i∈d

miH (∂mj − ∂mi)V d

j=✶

• =
• i∈d

mif i(m) V (T, m) =

• i∈d

migi(m), ✭❍❏❇✮ ◮ ❋✐rst ♦r❞❡r P❉❊ ✐♥ [✵, T] × Sd✱ ♥♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s❀ ◮ N

• V N

m + ✶

N (ej − ei)

• − V N(m)
• → (∂mj − ∂mi)V (m)❀

◮ ♥♦ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥s❀ ◮ ❡①✐st❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s✱ ♥♦♥✲✉♥✐q✉❡♥❡ss❀ ◮ P♦t❡♥t✐❛❧ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡✿ f i(m) = F(m)✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❱✐s❝♦s✐t② s♦❧✉t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭❍❏❇✮ ✐♥ [✵, T) × Sd✳ ◮ ❚❡st ❢✉♥❝t✐♦♥s ✐♥ C✶([✵, T) × Sd)✳ ◮ V ✐s ▲✐♣s❝❤✐t③✲❝♦♥t✐♥✉♦✉s✳ ■♥t ✐❢

✵

■♥t ✳ ❣✐✈❡s ♣r♦♣❡rt② ♦♥ ✳

❚❤❡♦r❡♠

❱❛❧✉❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭❍❏❇✮ ✐♥ ✵ ■♥t ✳ ❆♥❛❧♦❣♦✉s r❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ❬P❤❛♠✲❲❡✐ ✬✶✽❪✱ ❬❲✉✲❩❤❛♥❣ ✬✶✾❪✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❱✐s❝♦s✐t② s♦❧✉t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭❍❏❇✮ ✐♥ [✵, T) × Sd✳ ◮ ❚❡st ❢✉♥❝t✐♦♥s ✐♥ C✶([✵, T) × Sd)✳ ◮ V ✐s ▲✐♣s❝❤✐t③✲❝♦♥t✐♥✉♦✉s✳ ◮ µt ∈ ■♥t(Sd) ✐❢ µ✵ ∈ ■♥t(Sd)✳ ◮ ❣✐✈❡s ♣r♦♣❡rt② ♦♥ DzH✳

❚❤❡♦r❡♠

❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭❍❏❇✮ ✐♥ [✵, T) × ■♥t(Sd)✳ ❆♥❛❧♦❣♦✉s r❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ❬P❤❛♠✲❲❡✐ ✬✶✽❪✱ ❬❲✉✲❩❤❛♥❣ ✬✶✾❪✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❱✐s❝♦s✐t② s♦❧✉t✐♦♥

◮ ❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭❍❏❇✮ ✐♥ [✵, T) × Sd✳ ◮ ❚❡st ❢✉♥❝t✐♦♥s ✐♥ C✶([✵, T) × Sd)✳ ◮ V ✐s ▲✐♣s❝❤✐t③✲❝♦♥t✐♥✉♦✉s✳ ◮ µt ∈ ■♥t(Sd) ✐❢ µ✵ ∈ ■♥t(Sd)✳ ◮ ❣✐✈❡s ♣r♦♣❡rt② ♦♥ DzH✳

❚❤❡♦r❡♠

❱❛❧✉❡ ❢✉♥❝t✐♦♥ V ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭❍❏❇✮ ✐♥ [✵, T) × ■♥t(Sd)✳ ◮ ❆♥❛❧♦❣♦✉s r❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ❬P❤❛♠✲❲❡✐ ✬✶✽❪✱ ❬❲✉✲❩❤❛♥❣ ✬✶✾❪✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

Pr❡✈✐♦✉s r❡s✉❧ts

❆✐♠✿ V N → V ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✳ ❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ❬▲❛❝❦❡r ✬✶✼❪✱ ❬❉❥❡t❡✲P♦ss❛♠❛✐✲❚❛♥ ✬✷✵❪ ❛♥❞ ❬❋♦r♥❛s✐❡r✲▲✐s✐♥✐ ✲❖rr✐❡r✐ ✲❙❛✈❛ré ✬✶✾❪✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✈✐❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥ts ✭❝♦♠♠♦♥ ♥♦✐s❡✱ ❞❡t❡r♠✐♥✐st✐❝✱ ✳✳✳✮❀ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✉s✐♥❣ ❝♦♥✈❡①✐t② ♦❢ ❛♥❞ ✐♥ ✳ Pr♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✿ ✭❍❏❇✲◆✮ ✐s ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ❢♦r ✭❍❏❇✮✳ r❡s✉❧ts ♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡ ❢♦r ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s ✐♥

❬❈❛♣✉③③♦ ❉♦❧❝❡tt❛✲■s❤✐✐ ✬✽✹❪✿ ❞✐s❝♦✉♥t❡❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠❀ ❬❙♦✉❣❛♥✐❞✐s ✬✽✺❪✿ ❣❡♥❡r❛❧ ❍❏ ❡q✉❛t✐♦♥✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

Pr❡✈✐♦✉s r❡s✉❧ts

❆✐♠✿ V N → V ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✳ ❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ◮ ❬▲❛❝❦❡r ✬✶✼❪✱ ❬❉❥❡t❡✲P♦ss❛♠❛✐✲❚❛♥ ✬✷✵❪ ❛♥❞ ❬❋♦r♥❛s✐❡r✲▲✐s✐♥✐ ✲❖rr✐❡r✐ ✲❙❛✈❛ré ✬✶✾❪✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✈✐❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥ts ✭❝♦♠♠♦♥ ♥♦✐s❡✱ ❞❡t❡r♠✐♥✐st✐❝✱ ✳✳✳✮❀ ◮ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✉s✐♥❣ ❝♦♥✈❡①✐t② ♦❢ f ❛♥❞ g ✐♥ (x, m)✳ Pr♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✿ ✭❍❏❇✲◆✮ ✐s ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ❢♦r ✭❍❏❇✮✳ r❡s✉❧ts ♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡ ❢♦r ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s ✐♥

❬❈❛♣✉③③♦ ❉♦❧❝❡tt❛✲■s❤✐✐ ✬✽✹❪✿ ❞✐s❝♦✉♥t❡❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠❀ ❬❙♦✉❣❛♥✐❞✐s ✬✽✺❪✿ ❣❡♥❡r❛❧ ❍❏ ❡q✉❛t✐♦♥✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

Pr❡✈✐♦✉s r❡s✉❧ts

❆✐♠✿ V N → V ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✳ ❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ◮ ❬▲❛❝❦❡r ✬✶✼❪✱ ❬❉❥❡t❡✲P♦ss❛♠❛✐✲❚❛♥ ✬✷✵❪ ❛♥❞ ❬❋♦r♥❛s✐❡r✲▲✐s✐♥✐ ✲❖rr✐❡r✐ ✲❙❛✈❛ré ✬✶✾❪✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s ✈✐❛ ❝♦♠♣❛❝t♥❡ss ❛r❣✉♠❡♥ts ✭❝♦♠♠♦♥ ♥♦✐s❡✱ ❞❡t❡r♠✐♥✐st✐❝✱ ✳✳✳✮❀ ◮ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✉s✐♥❣ ❝♦♥✈❡①✐t② ♦❢ f ❛♥❞ g ✐♥ (x, m)✳ Pr♦✈❡ ❝♦♥✈❡r❣❡♥❝❡ ✉s✐♥❣ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ V ✿ ◮ ✭❍❏❇✲◆✮ ✐s ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ❢♦r ✭❍❏❇✮✳ ◮ r❡s✉❧ts ♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡ ❢♦r ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s ✐♥

◮ ❬❈❛♣✉③③♦ ❉♦❧❝❡tt❛✲■s❤✐✐ ✬✽✹❪✿ ❞✐s❝♦✉♥t❡❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠❀ ◮ ❬❙♦✉❣❛♥✐❞✐s ✬✽✺❪✿ ❣❡♥❡r❛❧ ❍❏ ❡q✉❛t✐♦♥✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

▼❛✐♥ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧t

❚❤❡♦r❡♠

❆ss✉♠❡ m →

i∈d mif i(m) ❛♥❞ m → i∈d migi(m) ▲✐♣s❝❤✐t③✳

sup

t∈[✵,T],m∈SN

d

|V N(t, m) − V (t, m)| ≤ C √ N ◮ ❯s❡ ▲✐♣s❝❤✐t③✲❝♦♥t✐♥✉✐t② ♦❢ V ❛♥❞ V N✱ ✉♥✐❢♦r♠❧② ✐♥ N✳ ❱❛❧✐❞ ❛❧s♦ ❢♦r ❝♦st ♥♦t ❝♦♥✈❡① ✐♥ ❛♥❞ tr❛♥s✐t✐♦♥ r❛t❡ ✿ ✐♥ t❤✐s ❝❛s❡✱ ♥♦♥✲✉♥✐q✉❡♥❡ss ♦❢ ✲♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❛♥❞ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❧✐♠✐t✐♥❣ ♦♣t✐♠❛❧ ❝♦♥tr♦❧✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

▼❛✐♥ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧t

❚❤❡♦r❡♠

❆ss✉♠❡ m →

i∈d mif i(m) ❛♥❞ m → i∈d migi(m) ▲✐♣s❝❤✐t③✳

sup

t∈[✵,T],m∈SN

d

|V N(t, m) − V (t, m)| ≤ C √ N ◮ ❯s❡ ▲✐♣s❝❤✐t③✲❝♦♥t✐♥✉✐t② ♦❢ V ❛♥❞ V N✱ ✉♥✐❢♦r♠❧② ✐♥ N✳ ◮ ❱❛❧✐❞ ❛❧s♦ ❢♦r ❝♦st c(t, x, α, m) ♥♦t ❝♦♥✈❡① ✐♥ α ❛♥❞ tr❛♥s✐t✐♦♥ r❛t❡ Qi,j(t, α, µ)✿ ✐♥ t❤✐s ❝❛s❡✱ ♥♦♥✲✉♥✐q✉❡♥❡ss ♦❢ N✲♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❛♥❞ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❧✐♠✐t✐♥❣ ♦♣t✐♠❛❧ ❝♦♥tr♦❧✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❆♣♣r♦①✐♠❛t✐♦♥

❈♦r♦❧❧❛r②

▲❡t α ❜❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❢♦r ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✳ ❚❤❡♥ α ✭♥♦t ❞❡♣❡♥❞✐♥❣ ♦♥ m✮ ✐s q✉❛s✐✲♦♣t✐♠❛❧ ❢♦r N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥✿ JN(α) ≤ inf

αN JN(αN) + C

√ N ❆♣♣❧② st❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✐♥ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❝❤❛♦s t♦ ❡st✐♠❛t❡

✵

❆ss✉♠❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s

✵ ✱

✶ ✱ ✐✳✐✳❞✳ ❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ❬▲❛❝❦❡r ✬✶✼❪✱ ❬❉❥❡t❡✲P♦ss❛♠❛✐✲❚❛♥ ✬✷✵❪✿ ♥♦ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ r❛t❡✱ r❡q✉✐r❡s ❝♦♥✈❡①✐t② ♦❢ ❛♥❞ ✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❆♣♣r♦①✐♠❛t✐♦♥

❈♦r♦❧❧❛r②

▲❡t α ❜❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❢♦r ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✳ ❚❤❡♥ α ✭♥♦t ❞❡♣❡♥❞✐♥❣ ♦♥ m✮ ✐s q✉❛s✐✲♦♣t✐♠❛❧ ❢♦r N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥✿ JN(α) ≤ inf

αN JN(αN) + C

√ N ◮ ❆♣♣❧② st❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✐♥ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❝❤❛♦s t♦ ❡st✐♠❛t❡ |JN(α) − J(α)| ≤ C √ N , sup

t∈[✵,T]

E|µN

t (α) − µ(α)| ≤

C √ N ◮ ❆ss✉♠❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s X k

✵ ✱ k = ✶, . . . , N✱ ✐✳✐✳❞✳

❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ❬▲❛❝❦❡r ✬✶✼❪✱ ❬❉❥❡t❡✲P♦ss❛♠❛✐✲❚❛♥ ✬✷✵❪✿ ♥♦ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ r❛t❡✱ r❡q✉✐r❡s ❝♦♥✈❡①✐t② ♦❢ ❛♥❞ ✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❆♣♣r♦①✐♠❛t✐♦♥

❈♦r♦❧❧❛r②

▲❡t α ❜❡ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❢♦r ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✳ ❚❤❡♥ α ✭♥♦t ❞❡♣❡♥❞✐♥❣ ♦♥ m✮ ✐s q✉❛s✐✲♦♣t✐♠❛❧ ❢♦r N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥✿ JN(α) ≤ inf

αN JN(αN) + C

√ N ◮ ❆♣♣❧② st❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✐♥ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❝❤❛♦s t♦ ❡st✐♠❛t❡ |JN(α) − J(α)| ≤ C √ N , sup

t∈[✵,T]

E|µN

t (α) − µ(α)| ≤

C √ N ◮ ❆ss✉♠❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s X k

✵ ✱ k = ✶, . . . , N✱ ✐✳✐✳❞✳

❘❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✿ ◮ ❬▲❛❝❦❡r ✬✶✼❪✱ ❬❉❥❡t❡✲P♦ss❛♠❛✐✲❚❛♥ ✬✷✵❪✿ ♥♦ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ◮ ❬❈❛r♠♦♥❛✲❉❡❧❛r✉❡ ✬✶✺❪✿ r❛t❡✱ r❡q✉✐r❡s ❝♦♥✈❡①✐t② ♦❢ f ❛♥❞ g✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥✈❡①✐t②

◮ ❯♥✐q✉❡♥❡ss ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ ▼❋❈P ❄ ◮ ❊①✐st❡♥❝❡ ♦❢ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ✭❍❏❇✮❄ ◮ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❛♥❞ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s❄ ❆ss✉♠❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥✈❡① ❛♥❞ ✐♥

✶ ✶

✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥✈❡①✐t②

◮ ❯♥✐q✉❡♥❡ss ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ ▼❋❈P ❄ ◮ ❊①✐st❡♥❝❡ ♦❢ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ✭❍❏❇✮❄ ◮ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❛♥❞ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s❄ ❆ss✉♠❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s Sd ∋ m →

• i∈d

mif i(m), Sd ∋ m →

• i∈d

migi(m). ❛r❡ ❝♦♥✈❡① ❛♥❞ ✐♥ C✶,✶(Sd) ✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥

❚❤❡♦r❡♠

❆ss✉♠❡ ❝♦♥✈❡①✐t②✳ ❚❤❡♥ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ V ∈ C✶,✶([✵, T] × Sd) ❛♥❞ ✐s ❛ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ✭❍❏❇✮✳ ❯♥✐q✉❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧✱ ✐♥ ❢❡❡❞❜❛❝❦ ❢♦r♠✱ αi,j(t, m) = a∗ (∂mi − ∂mj)V (t, m)

• Pr♦✈❡ t❤❛t

✐s s❡♠✐❝♦♥❝❛✈❡ ❛♥❞ s❡♠✐❝♦♥✈❡①✳ ❊q✉✐✈❛❧❡♥t ♣r♦❜❧❡♠✱ s❡t ✱

✵ ✶

✶

✷

✷

✶

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈❧❛ss✐❝❛❧ s♦❧✉t✐♦♥

❚❤❡♦r❡♠

❆ss✉♠❡ ❝♦♥✈❡①✐t②✳ ❚❤❡♥ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ V ∈ C✶,✶([✵, T] × Sd) ❛♥❞ ✐s ❛ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ✭❍❏❇✮✳ ❯♥✐q✉❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧✱ ✐♥ ❢❡❡❞❜❛❝❦ ❢♦r♠✱ αi,j(t, m) = a∗ (∂mi − ∂mj)V (t, m)

• ◮ Pr♦✈❡ t❤❛t V ✐s s❡♠✐❝♦♥❝❛✈❡ ❛♥❞ s❡♠✐❝♦♥✈❡①✳

◮ ❊q✉✐✈❛❧❡♥t ♣r♦❜❧❡♠✱ s❡t wi,j = µiαi,j✱ ˙ µi

t =

• j=i

(wj,i(t) − wi,j(t)) J(α) = T

✵ d

• i=✶

✶ µi

t

• j=i

|wi,j(t)|✷ ✷ + µi

tf i(µt)

• dt +

d

• i=✶

µi

Tgi(µT)

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥✈❡r❣❡♥❝❡

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ (µN, αN) t♦ (µ, α)❄ ❘❡❧② ♦♥ r❡❣✉❧❛r✐t② ♦❢ V ✳ ❆♣♣r♦①✐♠❛t❡ N

• V
• m + ✶

N (ej − ei)

• − V (m)
• = (∂mj − ∂mi)V (t, m) + O

✶ N

• ❆♥❛❧♦❣♦✉s ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ❢♦r ▼❋●✿

❬❈❛r❞❛❧✐❛❣✉❡t✲❉❡❧❛r✉❡✲▲❛sr②✲▲✐♦♥s ✬✶✾❪✿ ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✳ ❬❇❛②r❛❦t❛r✲❈♦❤❡♥ ✬✶✽❪ ❛♥❞ ❬❈❡❝❝❤✐♥✲P❡❧✐♥♦❪✿ ✜♥✐t❡ st❛t❡ s♣❛❝❡✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥✈❡r❣❡♥❝❡

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ (µN, αN) t♦ (µ, α)❄ ❘❡❧② ♦♥ r❡❣✉❧❛r✐t② ♦❢ V ✳ ❆♣♣r♦①✐♠❛t❡ N

• V
• m + ✶

N (ej − ei)

• − V (m)
• = (∂mj − ∂mi)V (t, m) + O

✶ N

• ❆♥❛❧♦❣♦✉s ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ❢♦r ▼❋●✿

◮ ❬❈❛r❞❛❧✐❛❣✉❡t✲❉❡❧❛r✉❡✲▲❛sr②✲▲✐♦♥s ✬✶✾❪✿ ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✳ ◮ ❬❇❛②r❛❦t❛r✲❈♦❤❡♥ ✬✶✽❪ ❛♥❞ ❬❈❡❝❝❤✐♥✲P❡❧✐♥♦❪✿ ✜♥✐t❡ st❛t❡ s♣❛❝❡✳

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s

µN ❡♠♣✐r✐❝❛❧ ♠❡❛s✉r❡ ✉s✐♥❣ ♦♣t✐♠❛❧ N✲❝♦♥tr♦❧ αN( t,m) ❣✐✈❡♥ ❜② V N ρN ❡♠♣✐r✐❝❛❧ ♠❡❛s✉r❡ ✉s✐♥❣ ❧✐♠✐t✐♥❣ ❝♦♥tr♦❧ α(t, m)✱ ❣✐✈❡♥ ❜② V µ ✉♥✐q✉❡ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r② ♦❢ ▼❋❈P✱ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ α(t, m)

❚❤❡♦r❡♠

❆ss✉♠❡ V ∈ C✶,✶([✵, T] × Sd)✳ E

• sup

t∈[✵,T]

|µN

t − ρN t |

• ≤

C N✶/✹ , E

• sup

t∈[✵,T]

|µN

t − µt|

• ≤

C N✶/✾ .

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

❲❡ ♦❜t❛✐♥❡❞ ✶✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ V N t♦ V ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✈✐❛ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s✱ ✉♥❞❡r ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥s✳ ✷✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s µN t♦ µ ✐❢ V ✐s s♠♦♦t❤✱ ❡✳❣✳ ✉♥❞❡r ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s✳ ❋♦r ❢✉t✉r❡ ✇♦r❦✿ ◮ ❆♥❛❧♦❣♦✉s r❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✳

❚❍❆◆❑ ❨❖❯ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵

N✲❛❣❡♥t ♦♣t✐♠✐③❛t✐♦♥ ▼❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥✈❡r❣❡♥❝❡ ❯♥✐q✉❡♥❡ss

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

❲❡ ♦❜t❛✐♥❡❞ ✶✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ V N t♦ V ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ ✈✐❛ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s✱ ✉♥❞❡r ❣❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥s✳ ✷✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s µN t♦ µ ✐❢ V ✐s s♠♦♦t❤✱ ❡✳❣✳ ✉♥❞❡r ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s✳ ❋♦r ❢✉t✉r❡ ✇♦r❦✿ ◮ ❆♥❛❧♦❣♦✉s r❡s✉❧ts ❢♦r ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡✳

❚❍❆◆❑ ❨❖❯ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆

❆❧❡❦♦s ❈❡❝❝❤✐♥ ❈♦♥✈❡r❣❡♥❝❡ ✜♥✐t❡ st❛t❡ ♠❡❛♥ ✜❡❧❞ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✶✽✴✵✻✴✷✵✷✵