(LUẬN văn THẠC sĩ) hệ động lực ngẫu nhiên trên thang thời gian

➜➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❍➭ ♥é✐

❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥

➜✐♥❤ ❱➝♥ ❑❤➞♠

❚ã♠ t➽t ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ❦❤♦❛ ❤ä❝
➜Ị t➭✐✿
❍Ư ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥
tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

❍➭ ◆é✐ ✲ ✷✵✶✷

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➜➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❍➭ ♥é✐

❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥

➜✐♥❤ ❱➝♥ ❑❤➞♠

❚ã♠ t➽t ❧✉❐♥ ✈➝♥✿

❍Ö ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥
tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

❈❤✉②➟♥ ♥❣➭♥❤✿ ▲ý t❤✉②Õt ①➳❝ s✉✃t ✈➭ ❚❤è♥❣ ❦➟ t♦➳♥ ❤ä❝
▼➲ sè✿ ✻✵✳✹✻✳✶✺

◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿

●❙✳❚❙ ◆❣✉②Ơ♥ ❍÷✉ ❉➢

❍➭ ♥é✐ ✲ ✷✵✶✷

✐

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▼ô❝ ❧ô❝
▼ô❝ ❧ô❝

✐

▲ê✐ ❝➯♠ ➡♥

✐✐

▼ë ➤➬✉

✶

✶ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

✸

✶✳✶

❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✳

✳

✳

✳

✳

✳

✳

✸

✶✳✷

➜Þ♥❤ ❧ý ❦❤❛✐ tr✐Ĩ♥ ❉♦♦❜ ✲ ▼❡②❡r

✳

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✳

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✶✶

✶✳✸

❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

✳

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✶✽

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✶✳✸✳✶

❚Ý❝❤ ♣❤➞♥ t rt ì tí







í t rt ị ♣❤➢➡♥❣ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤

✷✹

✷ ❈➠♥❣ t❤ø❝ ■t➠ ✈➭ ø♥❣ ❞ô♥❣
✳

✳

✳

✳

✷✼

✷✳✶

❇✐Õ♥ ♣❤➞♥ ❜❐❝ ❤❛✐

✳

✷✳✷

❈➠♥❣ t❤ø❝ ■t➠ ✈➭ ø♥❣ ❞ô♥❣

✳

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✷✼

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✸✶

✸ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥
✸✳✶

P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

✸✳✷

❚Ý♥❤ ▼❛r❦♦✈ ❝ñ❛ ♥❣❤✐Ư♠

❑Õt ❧✉❐♥ ✈➭ ❦✐Õ♥ ♥❣❤Þ

✳

✳

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✹✸

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✹✸

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✺✷

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✺✺

✐

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▲ê✐ ❝➯♠ ➡♥
❚r♦♥❣ q✉➳ tr×♥❤ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥ ♥➭② t➠✐ ➤➲ ♥❤❐♥ ➤➢ỵ❝ sù ❣✐ó♣ ➤ì t♦ ❧í♥

❝đ❛ ❝➳❝ t❤➬② ❣✐➳♦✱ ❝➠ ❣✐➳♦✱ ❣✐❛ ➤×♥❤ ✈➭ ❜➵♥ ❜❒✳

❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❦Ý♥❤ trä♥❣ ✈➭ ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛

❤ä❝✱ ●❙✳❚❙ ◆❣✉②Ơ♥ ❍÷✉ ❉➢✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥ ✲ ➜❍◗● ❍➭ ◆é✐✳

❚❤➬② ❧➭ ♥❣➢ê✐ ➤➲ ❤➢í♥❣ ❞➱♥ t➠✐ ❧➭♠ ❦❤ã❛ ❧✉❐♥ tèt ♥❣❤✐Ư♣ ➤➵✐ ❤ä❝ ♥➝♠ ✷✵✵✵✱ ❣✐ê

t❤➬② ❧➵✐ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥✱ ❣✐ó♣ ➤ì t➠✐ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳

❚➠✐ ❝ị♥❣ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ tí✐ ❝➳❝ t❤➬②✱

❝➠ ❝đ❛ ❑❤♦❛ ❚♦➳♥ ✲ ❈➡ ✲ ❚✐♥ ❤ä❝✱

P❤ß♥❣ s❛✉ ➤➵✐ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥✱ ➜❍◗● ❍➭ ◆é✐ ➤➲ ❣✐➯♥❣

❞➵②✱ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣✱ tr ị t ữ ế tứ ề

t ủ ể ❧➭♠ ✈✐Ö❝✳ ➜➷❝ ❜✐Öt✱ t➠✐ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ◆❈❙✳ ◆❣✉②Ơ♥ ❚❤❛♥❤ ❉✐Ư✉✱

❑❤♦❛ ❚♦➳♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ➤➲ ❝ã ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ q✉ý ❜➳✉ ➤Ĩ ❜➯♥

❧✉❐♥ ✈➝♥ ❤♦➭♥ ❝❤Ø♥❤ ❤➡♥✳

❚➠✐ ❝ị♥❣ ❦❤➠♥❣ q✉➟♥ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ tí✐ ❝➳❝ ➤å♥❣ ❝❤Ý ❧➲♥❤ ➤➵♦ ❝ï♥❣ ❜➵♥

❜❒ ➤å♥❣ ♥❣❤✐Ư♣ ❚r➢ê♥❣ ❚❍P❚ ❈❤✉②➟♥ ▲➢➡♥❣ ❱➝♥ ❚ơ② ✲ ◆✐♥❤ ❇×♥❤✱ ♥➡✐ t➠✐ ❝➠♥❣

t➳❝✱ ➤➲ ❤Õt sø❝ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ợ tr sốt q trì ọ t ũ tự

ệ ❧✉❐♥ ✈➝♥ ❝ñ❛ t➠✐✳

❈✉è✐ ❝ï♥❣✱ t➠✐ ①✐♥ ❝➯♠ ➡♥ ❝❤❛ ẹ ị ì ỏ ủ t➠✐

➤➲ ❧✉➠♥ ❜➟♥ t➠✐ tr♦♥❣ ♥❤÷♥❣ ♥❣➭② ➤➲ q✉❛✳

▼➷❝

❞ï

➤➲

❤Õt

sø❝

❝è

❣➽♥❣

♥❤➢♥❣

❧✉❐♥

✈➝♥

❦❤➠♥❣

t❤Ĩ

tr➳♥❤

❦❤á✐

♥❤÷♥❣

t❤✐Õ✉ sãt✳ ▼ä✐ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ t➠✐ ①✐♥ ➤➢ỵ❝ ➤ã♥ ♥❤❐♥ ✈í✐ ❧ß♥❣ ❜✐Õt ➡♥ ❝❤➞♥ t❤➭♥❤✳

❍➭ ◆é✐✱ ♥❣➭② ✵✶ t❤➳♥❣ ✵✺ ♥➝♠ ✷✵✶✷

❍ä❝ ✈✐➟♥
➜✐♥❤ ❱➝♥ ❑❤➞♠

✐✐

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▼ë ➤➬✉
P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ ❧➭ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝ ❝❤♦ ❝➳❝ ❤Ö ➤é♥❣ ❧ù❝

tr♦♥❣ t❤ù❝ tÕ ❝ã t➳❝ ➤é♥❣ ❝ñ❛ ②Õ✉ tè ♥❣➱✉ ♥❤✐➟♥✳

❉♦ ➤ã✱ ♥ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣

tr♦♥❣ s✐♥❤ ❤ä❝✱ ② ❤ä❝✱ ✈❐t ❧ý ❤ä❝✱ ❦✐♥❤ tÕ✱ ❦❤♦❛ ❤ä❝ ①➲ ❤é✐✳✳✳✱ ✈➭ ➤➢ỵ❝ ♥❤✐Ị✉ ♥❤➭

t♦➳♥ ❤ä❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳

❑❤✐ ①➞② ❞ù♥❣ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝ ❝❤♦ ❝➳❝ ❤Ư t❤è♥❣ t✐Õ♥ tr✐Ĩ♥ t❤❡♦ t❤ê✐ ❣✐❛♥✱

♥❣➢ê✐ t❛ t❤➢ê♥❣ ❣✐➯ t❤✐Õt ❤Ư t❤è♥❣ ❤♦➵t ➤é♥❣ ❧✐➟♥ tơ❝ ❤♦➷❝ rê✐ r➵❝ ➤Ị✉✱ tø❝ ❧➭ ❝➳❝

t❤ê✐ ➤✐Ĩ♠ q✉❛♥ s➳t ❝➳❝❤ ♥❤❛✉ ♠ét ❦❤♦➯♥❣ ❝è ➤Þ♥❤✳ ❚õ ➤ã✱ ❝➳❝ ♣❤Ð♣ tÝ♥❤ ❣✐➯✐ tÝ❝❤

❧✐➟♥ tô❝ ✭♣❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥✮ ✈➭ rê✐ r➵❝ ✭♣❤Ð♣ tÝ♥❤ s❛✐ ♣❤➞♥✮ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ➤Ĩ ♠➠

t➯ ❤Ư t❤è♥❣ t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt t❤ê✐ ❣✐❛♥ ❧ý t➢ë♥❣ ➤➢ỵ❝ ➤➷t r❛✳ ◆❤➢♥❣ t❤ù❝

tÕ✱ ❤➬✉ ❤Õt ❝➳❝ ❤Ư t❤è♥❣ ❤♦➵t ➤é♥❣ ❦❤➠♥❣ ❤♦➭♥ t♦➭♥ ❧✐➟♥ tơ❝ ❝ị♥❣ ❦❤➠♥❣ ❤♦➭♥

t♦➭♥ ❝➳❝❤ ➤Ò✉ ♥❤❛✉✳ ➜➠✐ ❦❤✐ ❝➳❝ q✉❛♥ s➳t ❝ß♥ ①❡♥ ❧➱♥ ❝➳❝ ❦❤♦➯♥❣ t❤ê✐ ❣✐❛♥ ❧✐➟♥

tơ❝ ✈í✐ ❝➳❝ t❤ê✐ ➤✐Ĩ♠ rê✐ r➵❝✳ ❚❤Ý ❞ơ ♥❤➢ ♠ét ❧♦➭✐ s➞✉ ❜Ư♥❤✱ ❝❤ó♥❣ ❝❤Ø ♣❤➳t tr✐Ĩ♥

tr♦♥❣ s✉èt ♠ï❛ ❤❒ ♥❤➢♥❣ ➤Õ♥ ù tì sự t trể ủ ú ị

ì tr ề trờ ợ trì ♣❤➞♥ ❤♦➷❝ s❛✐ ♣❤➞♥ ❦❤➠♥❣

➤ñ ♠➠ t➯ ❝➳❝ t❤➠♥❣ t✐♥ ❝➬♥ t❤✐Õt ❝đ❛ ♠➠ ❤×♥❤✳

▲ý t❤✉②Õt t❤❛♥❣ t❤ê✐ ❣✐❛♥ r❛ ➤ê✐ ♥❤➺♠ ❦❤➽❝ ♣❤ơ❝ ♥❤➢ỵ❝ ➤✐Ĩ♠ ♥➭② ❝đ❛ ❣✐➯✐

tÝ❝❤ ❝ỉ ➤✐Ĩ♥✳ ▲ý t❤✉②Õt ♥➭② ➤➢ỵ❝ ➤➢❛ r❛ ❧➬♥ ➤➬✉ t✐➟♥ ♥➝♠ ✶✾✽✽ ❜ë✐ ♥❤➭ ❚♦➳♥ ❤ä❝

♥❣➢ê✐ ➜ø❝ ❙t❡❢❛♥ ❍✐❧❣❡r tr♦♥❣ ▲✉❐♥ ➳♥ t✐Õ♥ sü ❝ñ❛ ➠♥❣ ✭①❡♠ ❬✺❪✮❀

♥❤➺♠ t❤è♥❣

♥❤✃t ✈➭ ♠ë ré♥❣ ♠ét sè ✈✃♥ ➤Ị ❝đ❛ ❣✐➯✐ tÝ❝❤ rê✐ r➵❝ ✈➭ ❧✐➟♥ tơ❝✳ ❈➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥

❝ø✉ ✈Ị ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❝❤♦ ♣❤Ð♣ ①➞② ❞ù♥❣ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝ ❝đ❛

❝➳❝ ❤Ư t❤è♥❣ t✐Õ♥ tr✐Ĩ♥ t❤❡♦ t❤ê✐ ❣✐❛♥ ❦❤➠♥❣ ➤Ị✉✱ ♣❤➯♥ ➳♥❤ ➤ó♥❣ q✉② ❧✉❐t tr♦♥❣

t❤ù❝ tÕ✳ ❉♦ ➤ã✱ ❝❤đ ➤Ị t❤❛♥❣ t❤ê✐ ❣✐❛♥ t❤✉ ❤ót ➤➢ỵ❝ sù q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛

♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ tr➟♥ tế ớ ó ề trì ợ ❜è tr➟♥ ❝➳❝

t➵♣ ❝❤Ý t♦➳♥ ❤ä❝ ❝ã ✉② tÝ♥ ✭❬✶✱ ✷✱ ✳✳✳❪✮✳ ❚✉② ♥❤✐➟♥✱ ♣❤➬♥ ❧í♥ ❝➳❝ ❦Õt q✉➯ ➤➵t ➤➢ỵ❝

❝❤Ø ❞õ♥❣ ❧➵✐ ë ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❤Ư ➤é♥❣ ❧ù❝ tt ị tr t tờ

ì tế

ết q ỉ t ợ ì t trể tr ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ♠➠✐

✶

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tr➢ê♥❣ ❦❤➠♥❣ ❝ã ♥❤✐Ị✉ ❜✐Õ♥ ➤ỉ✐✳ ❍✐Ĩ♥ ♥❤✐➟♥✱ ❝➳❝ ♠➠ ❤×♥❤ t❤ù❝ tÕ ❦❤➠♥❣ ♥❤➢ ✈❐②

✈➭ t❛ ♣❤➯✐ tÝ♥❤ ➤Õ♥ ❝➳❝ ②Õ✉ tè ♥❣➱✉ ♥❤✐➟♥ t➳❝ ➤é♥❣ ✈➭♦ ♠➠✐ tr➢ê♥❣✳ ❉♦ ➤ã✱ ✈✐Ư❝

❝❤✉②Ĩ♥ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❣✐➯✐ tÝ❝❤ tr➟♥ t tờ ủ ì tt ị

s ❤×♥❤ ♥❣➱✉ ♥❤✐➟♥ ❧➭ ♠ét ♥❤✉ ❝➬✉ ❝✃♣ t❤✐Õt✳ ❚r➟♥ ❝➡ së ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥

❝ø✉ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✈➭ s❛✐ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❧ý t❤✉②Õt t❤❛♥❣ t❤ê✐ ❣✐❛♥✱

tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ ➤Ị ❝❐♣ tí✐

✧▼ét sè ✈✃♥ ➤Ị ❝đ❛ ❤Ư ➤é♥❣ ❧ù❝ ♥❣➱✉

♥❤✐➟♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✧✳

▲✉❐♥ ✈➝♥ ❣å♠ ✸ ❝❤➢➡♥❣✳

❈❤➢➡♥❣ ✶✳ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

✳

◆é✐

❞✉♥❣

❝❤➢➡♥❣ ♥➭② ❣å♠ ❝ã ụ ụ trì ữ ề ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤

t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ▼ơ❝ trì ị ý trể r

ố

ớ

srt

tr

t

tờ

ụ



trì



tí





t rt ì tí rt ị ì

tÝ❝❤ ✈➭ ♠ë ré♥❣ ➤è✐ ✈í✐ s❡♠✐♠❛rt✐♥❣❛❧❡ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳

❈❤➢➡♥❣ ✷✳ ❈➠♥❣ t❤ø❝ ■t➠ ✈➭ ø♥❣ ❞ô♥❣✳

◆é✐ ❞✉♥❣ ❈❤➢➡♥❣ ợ ết

t ụ ụ ú t trì ị ĩ ề ế ỗ ợ ủ

q tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳

■t➠ ➤è✐ ✈í✐ ❜é

▼ơ❝ ✷✳✷ ❚r×♥❤ ❜➭② ✈Ị ❝➠♥❣ t❤ø❝

d−

s❡♠✐♠❛rt✐♥❣❛❧❡ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣✳

❈❤➢➡♥❣ ✸✳ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ tr t tờ
ộ



ủ





ợ



t



ụ

ụ

r

ị

ĩ

ệ ề ệ ề sự tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝

♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ▼ơ❝ ✸✳✷✳ tr×♥❤ ❜➭② ✈Ị tÝ♥❤ ▼❛r❦♦✈ ♥❣❤✐Ư♠ ❝đ❛

♣❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳

✷

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❈❤➢➡♥❣ ✶

❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥
t❤❛♥❣ t❤ê✐ ❣✐❛♥
✶✳✶

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣
t❤ê✐ ❣✐❛♥
❈➳❝ ❦Õt q✉➯ trì tr ụ ợ t từ t ệ

tờ
tờ

ột t ó rỗ ủ t❐♣ sè t❤ù❝

❧➭

T.

❚❛

tr❛♥❣

❜Þ

❝❤♦

t❤❛♥❣

t❤ê✐

❣✐❛♥

T

♠ét

R

❚❤❛♥❣

✱ t❤➢ê♥❣ ❦ý ❤✐Ư✉ t❤❛♥❣

t➠♣➠

❝➯♠

s✐♥❤

❝đ❛

t➠♣➠

t❤➠♥❣ t❤➢ê♥❣ tr➟♥ t❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝✳

❉Ơ ❞➭♥❣ t❤✃② r➺♥❣ ❝➳❝ t❐♣ ❤ỵ♣

R, Z, N, N0 , [0, 1] ∪ [2, 3], [0, 1] ∪ N,

✈➭ t❐♣ ❈❛♥t♦r,

❧➭ ❝➳❝ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳

❚r♦♥❣ ❦❤✐ ➤ã ❝➳❝ t❐♣ ❤ỵ♣

Q, R \ Q, (0, 1),
❦❤➠♥❣ ♣❤➯✐ ❧➭ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈× ❝❤ó♥❣ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❝➳❝ t❐♣ ➤ã♥❣✳

✸

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➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✳

●✐➯ sư

T

❧➭ ♠ét t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ➳♥❤ ①➵

σ :TT



ị ở

(t) = inf{s T : s > t},
ợ ❣ä✐ ❧➭ t♦➳♥ tư ❜➢í❝ ♥❤➯② t✐Õ♥ ✭❢♦r✇❛r❞ ❥✉♠♣ ♦♣❡r❛t♦r✮ tr➟♥ t❤❛♥❣ t❤ê✐
❣✐❛♥

T. ➳♥❤ ①➵ ρ : T → T ①➳❝ ➤Þ♥❤ ❜ë✐
ρ(t) = sup{s ∈ T : s < t},

➤➢ỵ❝ ❣ä✐ ❧➭ t♦➳♥ tư ❜➢í❝ ♥❤➯② ❧ï✐ ✭❜❛❝❦✇❛r❞ ❥✉♠♣ ♦♣❡r❛t♦r✮ tr➟♥ t❤❛♥❣ t❤ê✐
❣✐❛♥

T.

◗✉② ➢í❝

inf ∅ = sup T ✭♥❣❤Ü❛ ❧➭ σ(M ) = M

❧í♥ ♥❤✃t ❧➭

♥Õ✉ t❤❛♥❣ t❤ê✐ ❣✐❛♥

T ❝ã ♣❤➬♥ tö

M ✮ ✈➭ sup ∅ = inf T ✭♥❣❤Ü❛ ❧➭ ρ(m) = m ♥Õ✉ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T ❝ã

♣❤➬♥ tư ♥❤á ♥❤✃t ❧➭

m✮✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷✳ ●✐➯ sư T ❧➭ ♠ét t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ▼ét ➤✐Ó♠ t ∈ T ➤➢ỵ❝ ❣ä✐
❧➭ trï ♠❐t ♣❤➯✐ ✭r✐❣❤t✲❞❡♥s❡✮ ♥Õ✉

σ(t) = t✱

❝➠ ❧❐♣ ♣❤➯✐ ✭r✐❣❤t✲s❝❛tt❡r❡❞✮ ♥Õ✉

σ(t) > t✱ trï ♠❐t tr➳✐ ✭❧❡❢t✲❞❡♥s❡✮ ♥Õ✉ ρ(t) = t✱ ❝➠ ❧❐♣ tr➳✐ ✭❧❡❢t✲s❝❛tt❡r❡❞✮ ♥Õ✉
ρ(t) < t ✈➭ ❧➭ ➤✐Ó♠ ❝➠ ❧❐♣ ✭✐s♦❧❛t❡❞✮ ♥Õ✉ t ✈õ❛ ❝➠ tr ừ
ớ

t

tự

ỗ

ý

a, b T



ệ



t

ý

ệ

ợ

[a, b]



t

ợ

(a, b]; (a, b); [a, b)

{t ∈ T : a
t➢➡♥❣

{t ∈ T : a < t

b}; {t ∈ T : a < t < b}; {t ∈ T : a

Ta = {t ∈ T : t

a}

kT

=

Tk =

ø♥❣

❧➭

t
❝➳❝

t < b}

✳

t❐♣

b}

✱

❤ỵ♣

❑ý ❤✐Ư✉

✈➭



T

♥Õ✉

min T = −∞


T \ [m, σ(m))

♥Õ✉

min T = m,



T

♥Õ✉

max T = +∞


T \ (ρ(M ), M ]

♥Õ✉

max T = M.

✹

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❑ý ❤✐Ö✉

I1 = {t : t ❝➠ ❧❐♣ tr➳✐}, I2 = {t : t ❝➠ ❧❐♣ ♣❤➯✐}, I = I1 I2 .

ệ ề

ợ

ủ t tờ

ị ĩ ✶✳✶✳✹✳

✭✶✳✶✮

❣å♠ t✃t ❝➯ ❝➳❝ ➤✐Ó♠ ❝➠ ❧❐♣ tr➳✐ ❤♦➷❝ ❝➠ ❧❐♣ ♣❤➯✐

I

T ❧➭ t❐♣ ❦❤➠♥❣ q✉➳ ➤Õ♠ ➤➢ỵ❝✳
●✐➯ sư

T

❧➭ t❤❛♥❣ tờ

à : Tk R+



ị ở

à(t) = σ(t) − t,
➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ❤➵t t✐Õ♥ ✭❢♦r✇❛r❞ ❣r❛✐♥✐♥❡ss ❢✉♥❝t✐♦♥✮ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥
➳♥❤

①➵

ν : T → R+

T.

①➳❝ ➤Þ♥❤ ❜ë✐

ν(t) = t − ρ(t),
➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ❤➵t ❧ï✐ ✭❜❛❝❦✇❛r❞ ❣r❛✐♥✐♥❡ss ❢✉♥❝t✐♦♥✮ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

T.

❱Ý ❞ô ✶✳✶✳✺✳ ✰✮ ◆Õ✉ T = R t❤× ρ(t) = t = σ(t), µ(t) = ρ(t) = 0;
✰✮ ◆Õ✉

T = Z t❤× ρ(t) = t − 1, σ(t) = t + 1, µ(t) = ν(t) = 1.

✰✮ ❱í✐

h ❧➭ sè t❤ù❝ ❞➢➡♥❣✳ ❈❤ó♥❣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T = hZ

①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿

hZ = {kh : k ∈ Z} = {· · · − 3h, −2h, −h, 0, h, 2h, 3h, · · · },
❦❤✐ ➤ã

ρ(t) = t − h, σ(t) = t + h, à(t) = (t) = h.

ị ĩ ✶✳✶✳✻✳ ❈❤♦ ❤➭♠ sè f : T → R✳ ❍➭♠ sè f
✐✮ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛t❡❞✮ ♥Õ✉

f

➤➢ỵ❝ ❣ä✐ ❧➭

❝ã ❣✐í✐ ❤➵♥ tr➳✐ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t tr➳✐ ✈➭

❝ã ❣✐í✐ ❤➵♥ ♣❤➯✐ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t ♣❤➯✐✳

✐✐✮

rd−❧✐➟♥

tơ❝ ✭rd−❝♦♥t✐♥✉♦✉s✮ ♥Õ✉

f

❧✐➟♥ tơ❝ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t ♣❤➯✐

✈➭ ❝ã ❣✐í✐ ❤➵♥ tr➳✐ t ữ ể trù t tr ợ rd− ❧✐➟♥
tơ❝ ❦ý ❤✐Ư✉ ❧➭

Crd

❤♦➷❝

Crd (T, R).
✺

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✐✐✐✮

ld−❧✐➟♥ tơ❝ ✭ld−❝♦♥t✐♥✉♦✉s✮ ♥Õ✉ f

❧✐➟♥ tơ❝ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t tr➳✐✱ ❝ã

❣✐í✐ ❤➵♥ ♣❤➯✐ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t ♣❤➯✐✳ ❚❐♣ ❤ỵ♣ ❝➳❝ ❤➭♠

❦ý ❤✐Ư✉ ❧➭

●✐➯ sư

fρ : T → R
t∈

k T✳

Cld

❤♦➷❝

❧➭ ♠ét ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥

❧➭ ❤➭♠ sè ①➳❝ ➤Þ♥❤ ❜ë✐

lim f (s)

❜ë✐

σ(s)↑t

r➺♥❣ ♥Õ✉

t

❧✐➟♥ tơ❝

Cld (T, R).

f :T→R

❑ý ❤✐Ư✉

ld−

f ρ = f◦ ρ

f (t− )

❧➭ ➤✐Ĩ♠ ❝➠ ❧❐♣ tr➳✐ t❤×

❤♦➷❝

T

✱ ♥❣❤Ü❛ ❧➭

ft−

✳

❑❤✐ ➤ã✱ ❝❤ó♥❣ t❛ ✈✐Õt

f ρ (t) = f (ρ(t))

✈í✐ ♠ä✐

♥Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ tr➳✐✳ ❚❛ t❤✃②

ft− = f ρ (t)

✳

➜Þ♥❤ ❧ý ✶✳✶✳✼✳ ●✐➯ sư f : T → R ❧➭ ♠ét ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥ T✳ ❑❤✐ ➤ã✱
✐✮ ◆Õ✉

f

❧➭ ❤➭♠ sè ❧✐➟♥ tơ❝ t❤×

✐✐✮ ◆Õ✉

f

❧➭ ❤➭♠ sè

✐✈✮ ❚♦➳♥ tư ❜➢í❝ ♥❤➯② ❧ï✐

f

❧➭ ❤➭♠ sè

f

➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã

➤➵♦ ❤➭♠✮ t➵✐
❧➞♥ ❝❐♥

U

t ∈ kT

❝đ❛

σ

❧➭ ❤➭♠ sè

rd− ❧✐➟♥ tô❝ ✈➭ ld− ❧✐➟♥ tô❝✳

❧➭ ❤➭♠ sè ❝❤Ý♥❤ q✉②✳

rd− ❧✐➟♥ tô❝✳

ρ ❧➭ ❤➭♠ sè ld− ❧✐➟♥ tô❝✳

ld− ❧✐➟♥ tụ tì f

ị ĩ sử f
số

❤➭♠ sè

rd− ❧✐➟♥ tơ❝ t❤× f

✐✐✐✮ ❚♦➳♥ tư ❜➢í❝ ♥❤➯② t✐Õ♥

✈✮ ◆Õ✉

f

❝ị♥❣ ❧➭ ❤➭♠ sè

❧➭ ♠ét ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥

ld− ❧✐➟♥ tơ❝✳

T✱ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ R✳

∇− ➤➵♦ ❤➭♠ ✭❝ã ➤➵♦ ❤➭♠ ❍✐❧❣❡r ❤♦➷❝ ➤➡♥ ❣✐➯♥

♥Õ✉ tå♥ t➵✐

f ∇ (t) ∈ R

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

ε>0

❝ã

tå♥ t➵✐ ♠ét

t ➤Ĩ

|f (ρ(t)) − f (s) − f ∇ (t)(ρ(t) − s)|

ε|ρ(t) − s|

f ∇ (t) ∈ R ➤➢ỵ❝ ❣ä✐ ❧➭ ∇−➤➵♦ ❤➭♠ ❝đ❛ ❤➭♠ sè f
◆Õ✉ ❤➭♠ sè

f

❝ã

✈í✐ ♠ä✐

s ∈ U.

t➵✐ t✳

∇−➤➵♦ t ọ ể t k T tì f

ợ ❣ä✐ ❧➭ ❝ã

∇−➤➵♦ ❤➭♠ tr➟♥ T✳

❱Ý ❞ô ✶✳✶✳✾✳ ✰✮ ◆Õ✉ T = R t❤× f ∇ (t) ≡ f (t) ❝❤Ý♥❤ ❧➭ ➤➵♦ ❤➭♠ t❤➠♥❣ t❤➢ê♥❣✳
✰✮ ◆Õ✉

T = Z t❤× f ∇ (t) = f (t) − f (t − 1) ❝❤Ý♥❤ ❧➭ s❛✐ ♣❤➞♥ ❧ï✐ ❝✃♣ ♠ét✳
✻

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➜Þ♥❤ ❧ý ✶✳✶✳✶✵✳ ●✐➯ sư f : T → R ❧➭ ♠ét ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥ T ✈➭ t ∈ k T✳
❑❤✐ ➤ã✱

∇− ➤➵♦ ❤➭♠ t➵✐ t t❤× f

✐✮ ◆Õ✉ ❤➭♠ sè

f

❝ã

✐✐✮ ◆Õ✉ ❤➭♠ sè

f

❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ ❝➠ ❧❐♣ tr➳✐

f ∇ (t) =

✐✐✐✮ ◆Õ✉

t ❧➭ ➤✐Ó♠ trï ♠❐t tr➳✐ t❤× f

❧➭ ❤➭♠ sè ❧✐➟♥ tơ❝ t➵✐ t✳

t t❤× f

❝ã

∇− ➤➵♦ ❤➭♠ t➵✐ t ✈➭

f (t) − f (ρ(t))
.
ν(t)

❧➭ ❤➭♠ sè ❝ã

∇−➤➵♦ ❤➭♠ t➵✐ t ♥Õ✉ ✈➭ ❝❤Ø

♥Õ✉ ❣✐í✐ ❤➵♥

f (t) − f (s)
,
s→t
t−s

lim

tå♥ t➵✐ ✈➭ ❤÷✉ ❤➵♥✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤ã✱

f (t) − f (s)
.
s→t
t−s

f ∇ (t) = lim

✐✈✮ ◆Õ✉ ❤➭♠ sè

f

❝ã

∇− ➤➵♦ ❤➭♠ t➵✐ t t❤×
f ρ (t) = f (t) − ν(t)f ∇ (t).

➜Þ♥❤ ❧ý ✶✳✶✳✶✶✳

●✐➯ sö

f, g : T → R

❧➭ ❝➳❝ ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥

T

✈➭ ❝ã

∇− ➤➵♦ ❤➭♠ t➵✐ t ∈ k T✳ ❑❤✐ ➤ã✱
✐✮ ❍➭♠ tæ♥❣

f + g : T → R ❝ã ∇− ➤➵♦ ❤➭♠ t➵✐ t ✈➭
(f + g)∇ (t) = f ∇ (t) + g ∇ (t).

✐✐✮ ❍➭♠ tÝ❝❤

f g : T → R ❝ã ∇−➤➵♦ ❤➭♠ t➵✐ t ✈➭ t❛ ❝ã q✉② t➽❝

(f g)∇ (t) = f ∇ (t)g(t) + f ρ (t)g ∇ (t) = f (t)g ∇ (t) + f ∇ (t)g ρ (t).

✼

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g(t)g ρ (t) = 0✱ t❤× ❤➭♠ sè

✐✐✐✮ ◆Õ✉

f
g

∇

f
g ❝ã

∇−➤➵♦ ❤➭♠ t➵✐ t ✈➭ t❛ ❝ã q✉② t➽❝

f ∇ (t)g(t) − f (t)g ∇ (t)
.
(t) =
g(t)g ρ (t)

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✷✳ ❍➭♠ sè p ①➳❝ ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T ợ ọ ồ
q rrss ế

1 + à(t)p(t) = 0,

ớ ♠ä✐

t ∈ Tk .

❑ý ❤✐Ö✉

R = {p : T → R : p ❧➭ rd − ❧✐➟♥ tô❝ ✈➭ 1 + µ(t)p(t) = 0}.
R+ = {p : T → R : p ❧➭ rd − ❧✐➟♥ tô❝ ✈➭ 1 + µ(t)p(t) > 0}.
❚✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ s➡ ❜é ✈Ị ➤é ➤♦ ▲❡❜❡s❣✉❡✲❙t✐❡❧t❥❡s tr➟♥ t❤❛♥❣

t❤ê✐ ❣✐❛♥✳

●✐➯

sư

A

❧➭

{(a; b] : a, b ∈ T}
❙✉② r❛

M1

❤➭♠

t➝♥❣✱

❧✐➟♥

tơ❝

♣❤➯✐✱

①➳❝

➤Þ♥❤

tr➟♥

T.

❑ý

❤✐Ư✉

M1 =

❧➭ ❤ä t✃t ❝➯ ❝➳❝ ❦❤♦➯♥❣ ♠ë ❜➟♥ tr➳✐ ✈➭ ➤ã♥❣ ❜➟♥ ♣❤➯✐ ❝đ❛

❧➭ ♥ư❛ ✈➭♥❤ ❝➳❝ t❐♣ ❝♦♥ ❝đ❛

T

✳ ▲✃②

m1

❧➭ ❤➭♠ t❐♣ ①➳❝ ➤Þ♥❤ tr➟♥

T

✳

M1

✈➭ ợ ị ở

m1 ((a, b]) = Ab Aa .
❈❤ó♥❣

t❛

t❤✃②

r➺♥❣

m1

❧➭

❤➭♠

t❐♣

❝é♥❣

❧➭ ♠ë ré♥❣ ❈❛r❛t❤Ð♦❞♦r② ❝đ❛ ❤➭♠ t❐♣

∇A −➤é

➤♦ ▲❡❜❡s❣✉❡✲ ❙t✐❡❧t❥❡s

tÝ♥❤

m1

➤Õ♠

✭✶✳✷✮
➤➢ỵ❝

❧✐➟♥ ❦Õt ✈í✐ ❤ä

❧✐➟♥ ❦Õt ✈í✐

A

M1 .

M1

✈➭ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭

tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

❑ý

T

✳

❤✐Ư✉

❉Ơ ❞➭♥❣

❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ s❛✉✳

❱í✐

t0 k T

t ột ể

{t0 }



A

ợ

àA
({t}) = At − At− .
❱í✐

a, b ∈ T

✈➭

a

µA
∇

tr➟♥

b

✱

A
A
µA
∇ ((a, b)) = Ab− − Aa ; µ∇ ([a, b)) = Ab− − Aa− ; µ∇ ([a, b]) = Ab − Aa− .
✽

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❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝❤♦ ❝➳❝ ❦Õt q✉➯ ♥➭② ❝ã tể tr



E

àA


ợ

àA


tr

ọ

E

tT

kT



ột

ý ệ

E

ợ ọ

t ó

àA

t

àA




f A

ợ



f : T R

tí ủ sè

f

➤é ➤♦ ▲❡❜❡s❣✉❡ tr➟♥

T

✈➭

E

fτ ∇τ

▲❡s❜❡s❣✉❡✳ ❚r♦♥❣ ▲✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ sư ❞ơ♥❣ ❦ý ❤✐Ư✉

♠ét

❤➭♠

sè

❧✐➟♥ ❦Õt ✈í✐ ➤é ➤♦

∇A −tÝ❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡ ✲ ❙t✐❡❧t❥❡s✳

∇−

❧➭

◆Õ✉

❧➭

A(t) = t

∇−

b
a f (τ )∇τ

✈í✐

tÝ❝❤ ♣❤➞♥

t❤❛② ❝❤♦

(a,b] f (τ )∇τ.
❙❛✉ ➤➞② ❝❤ó♥❣ t➠✐ ❧✐Ưt ❦➟ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ tÝ❝❤ ♣❤➞♥✳

➜Þ♥❤ ❧ý ✶✳✶✳✶✸✳
❤➭♠ sè

✐✮

✐✐✮

●✐➯ sư

b
a (f (τ )

+ g(τ ))∇τ =

b
a αf (τ )∇τ

=α

a
b f (τ )∇τ

=−

✐✈✮

c
a f (τ )∇τ

+

✈✐✮

✈➭

f : T → R, g : T → R

❧➭ ❝➳❝

ld− ❧✐➟♥ tô❝✳ ❑❤✐ ➤ã✱ ❝➳❝ ➤➻♥❣ t❤ø❝ s❛✉ ➤➞② ➤ó♥❣

✐✐✐✮

✈✮

a, b, c ∈ T, α ∈ R

●✐➯ sö

b
a g(τ )∇τ ;

b
a f (τ )∇τ ;

b
c f (τ )∇τ

b
∇
a f (τ )g (τ )∇τ

+

b
a f (τ )∇τ ;

b
∇
a f (ρ(τ ))g (τ )∇τ

❱Ý ❞ô ✶✳✶✳✶✹✳

b
a f (τ )∇τ

=

b
a f (τ )∇τ ;

= f (b)g(b) − f (a)g(a) −

= f (b)g(b) − f (a)g(a) −

a, b ∈ T, f : T → R

b ∇
a f (τ )g(τ )∇τ ;

b ∇
a f (τ )g(ρ(τ ))∇τ.

❧➭ ❤➭♠ sè ①➳❝ ị tr

T

ld tụ


ế

T = R tì

b

b

f ( ) =
a

f ( )dτ.
a

✾

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✈➭


✐✐✮

◆Õ✉

T ❧➭ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❣å♠ t✃t ❝➯ ❝➳❝ ➤✐Ó♠ ➤Ị✉ ❧➭ ➤✐Ĩ♠ ❝➠ ❧❐♣ t❤×



f (t)ν(t)



t∈(a,b]


b
f (τ )∇τ = 0

a






f (t)ν(t)
−

♥Õ✉

a
♥Õ✉

a=b

♥Õ✉

a > b.

t∈(b,a]

❈➳❝

∇−

❜➢í❝

①➞②

❞ù♥❣

∆−

tÝ❝❤

♣❤➞♥

▲❡❜❡s❣✉❡

t➢➡♥❣

tù

♥❤➢

①➞②

❞ù♥❣

t✐❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡ ✭①❡♠ ❬✶❪✮✳ ❚r♦♥❣ tr➢ê♥❣ ợ tổ qt ú t

ó ố q ệ ữ





tí ♣❤➞♥ ✈➭

tÝ❝❤ ♣❤➞♥✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt

❤➭♠ sè ❞➢í✐ ❞✃✉ tÝ❝❤ ♣❤➞♥ ❧✐➟♥ tơ❝ t❛ ❝ã ❜ỉ ➤Ị s❛✉✿

❇ỉ ➤Ị ✶✳✶✳✶✺✳

f : T → R

●✐➯ sư

❧➭ ❤➭♠ sè ❝❤Ý♥❤ q✉② tr➟♥

T,

❧✃②

b ∈ Tk ,

a ∈ k T, a < b. ❑❤✐ ➤ã ➤➻♥❣ t❤ø❝ s❛✉ ➤ó♥❣
b

b

f (τ− )∇τ =

f (τ )∆τ.

a

✭✶✳✸✮

a

❚õ ❇ỉ ➤Ị ✶✳✶✳✶✺ ✈➭ ❬✶✱ ❚❤❡♦r❡♠ ✷✳✸✸✱ ♣♣✳✺✾❪ s✉② r❛ ♥Õ✉

❝❤Ý♥❤ q✉② t❤×

ep (t, t0 )

p(t)

❤å✐ q✉② ✈➭

❧➭ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤

t

y(t) = 1 +

p(τ )y(τ )∆τ,
a

❝ị♥❣ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❈❛✉❝❤② s❛✉



y ∇ (t) = p(t− )y(t− ) ∀ t ∈ Ta

✭✶✳✹✮


y(a) = 1;
❱í✐ ❤➭♠ sè

hk : T ì T R; k N0

ợ ➤Þ♥❤ ❜ë✐

t

h0 (t, s) = 1

✈➭

hk+1 (t, s) =

hk (τ, s)∆τ

✈í✐

k ∈ N0 .

s
t❤×

hk (t, s)

❧➭ ❤➭♠ sè ❧✐➟♥ tơ❝ t❤❡♦

t

✳ ❉♦ ➤ã t❛ ❝ã

t

hk+1 (t, s) =

hk (τ− , s)∇τ.
s
✶✵

TIEU LUAN MOI download :


ữ t ó ợ ớ ợ s

0
ớ t ỳ

kN

ổ ➤Ị ✶✳✶✳✶✻✳
♠ä✐ ➤✐Ĩ♠

(t − s)k
,
k!

hk (t, s)

✭✶✳✺✮

t>s

✈➭

✳

●✐➯ sư

u(t)

❧➭ ♠ét ❤➭♠ sè ❧✐➟♥ tơ❝ ♣❤➯✐ ✈➭ ❝ã ❣✐í✐ ❤➵♥ tr➳✐ t➵✐

t ∈ Ta ✱ ua , p ∈ R+ ✳ ❑❤✐ ➤ã✱ ❜✃t ➤➻♥❣ t❤ø❝
t

u(t)

u(τ− )∇τ ∀ t ∈ Ta ,

ua + p
a

s✉② r❛

ua ep (t, a) ∀ t ∈ Ta .

u(t)

✶✳✷

➜Þ♥❤ ❧ý ❦❤❛✐ tr✐Ĩ♥ ❉♦♦❜ ✲ ▼❡②❡r

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✶✳ ●✐➯ sư A = {At }t∈Ta

❧➭ ♠ét q✉➳ tr×♥❤ ❧✐➟♥ tơ❝ ó

A ợ ọ q trì t ế t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿
✐✮

Aa = 0 ✈➭ A = (At ) q trì (Ft )ù ợ

ỹ ➤➵♦ ❝ñ❛

A ❧➭ ❤➭♠ sè t➝♥❣ t❤❡♦ t tr➟♥ Ta

◗✉➳ trì t

A = {At }tTa

ợ ọ



tí ♥Õ✉ EAt

< ∞, ∀ t ∈ Ta .

▼Ư♥❤ ➤Ị ✶✳✷✳✷✳ ●✐➯ sư A ❧➭ ♠ét q✉➳ tr×♥❤ t➝♥❣✱ ❦❤➯ tÝ❝❤ ✈➭ M
❜Þ ❝❤➷♥✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐

t ∈ Ta

❧➭ ♠❛rt✐♥❣❛❧❡

t❛ ❝ã

t

Mτ ∇Aτ .

EMt At = E
a
❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ♣❤➞♥ ❤♦➵❝❤

π (n)
(n)

❝ñ❛ ➤♦➵♥

(n)

[a, t]
(n)

π (n) : a = t0 < t1 < · · · < tkn = t,

✭✶✳✻✮

✶✶

TIEU LUAN MOI download :

t❤á❛ ♠➲♥

(n)

(n)

2−n .

max(ρ(ti+1 ) − ti )
i

✭✶✳✼✮

❚r♦♥❣ ▲✉❐♥ ✈➝♥ ♥➭②✱ ➤Ĩ ➤➡♥ ❣✐➯♥ ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝❤ó♥❣ t➠✐ ❜á q✉❛ ❝❤Ø sè

(n)

(n)
tr♦♥❣ ❦ý ❤✐Ö✉ ti
♥Õ✉ ❦❤➠♥❣ q✉➳ ❝➬♥ t❤✐Õt✳ ➜➷t
kn
(n)
Nsπ

:=

Mti 1(ti−1 ,ti ] (s).
i=1

❱× ♠❛rt✐♥❣❛❧❡

M

❝ã q✉ü ➤➵♦ ❝❛❞❧❛❣ ♥➟♥✱

Ms = lim Nsπ

(n)

∀ s ∈ (a, t].

n→∞

❚❤❡♦ ➤Þ♥❤ ❧ý ❤é✐ tơ ❜Þ ❝❤➷♥ t❛ ❝ã

t

t
n→∞

a

a
kn

Mti (Ati − Ati−1 )

= lim E
n→∞

(n)

Nτπ ∇Aτ

Mτ ∇Aτ = E lim

E

i=1
kn

Ati−1 (Mti − Mti−1 ) = EMt At .

= lim E Mt At +
n→∞

i=1

❱❐② t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✸✳ ●✐➯ sư A = (At )t∈Ta
A

❧➭ ♠ét q✉➳ tr×♥❤ t➝♥❣ ❦❤➯ tÝ❝❤✳ ❑❤✐ ➤ã✱

➤➢ỵ❝ ❣ä✐ ❧➭ t➝♥❣ tù ♥❤✐➟♥ ♥Õ✉ ✈í✐ ♠ä✐ ♠❛rt✐♥❣❛❧❡ ị tì tứ

s ợ tỏ

t

M ∇Aτ .

EMt At = E

✭✶✳✽✮

a

▼Ư♥❤ ➤Ị ✶✳✷✳✹✳ ●✐➯ sư (At )tTa

ột q trì t ó ị

s ➤ó♥❣✳

✶✮ ◆Õ✉

A = (At )

t ∈ I ∩ Ta

t❤×

At

❧➭ q✉➳ trì tụ

At

Ft

ợ ớ ọ

q trì t➝♥❣ tù ♥❤✐➟♥✳
✶✷

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A = At

✷✮ ◆Õ✉

❧➭ q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥ t❤×

A = (At )tTa

q trì

(Ft ) ợ
ứ ❱×

t ∈ Ta \ I1 ✳
❝❤♦

A = {At }

❧➭ q✉➳ trì tụ

ữ ớ ỗ rt

Mt = Mt

àA
{t} = 0

M = {Mt }✱

✈í✐ ♠ä✐

t❐♣ ❝➳❝ ❣✐➳ trÞ

t

s❛♦

❦❤➠♥❣ q✉➳ ➤Õ♠ ➤➢ỵ❝✳ ❙✉② r❛

(Mτ − Mτ− )∇Aτ = 0,

❤✳❝✳❝.

(a,t]\I1
❚❛ ❝ã

t

(Mτ − Mτ− )∇Aτ = E

E
a

(Mτ − Mτ− )∇Aτ
(a,t]\I1

(Mτ − Mτ− )∇Aτ

+E
I1 ∩(a,t]

(Ms − Ms− )(As − As− ) .

=E
s∈I1 ∩(a,t]

As

❚❛ ó



Fs

ợ ớ ọ trị

s I1 ∩ (a, t]✳ ❙✉② r❛

E (Ms − Ms− )(As − As− ) = E E(Ms − Ms− )(As − As− )|Fs−

= E (As − As− )E{(Ms − Ms− )|Fs− } = 0.
❉♦ ➤ã✱

t

(Mτ − Mτ− )∇Aτ = 0.

E
a
❙ư ❞ơ♥❣ ▼Ư♥❤ ➤Ò ✶✳✷✳✷ s✉② r❛

t

t

Mτ− ∇Aτ = E

E
a
♥❣❤Ü❛ ❧➭

Ft − −

a

(At ) ❧➭ q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥✳

✷✮ ●✐➯ sư
❧➭

Mτ ∇Aτ = EMt At ,

A = (At )

❧➭ q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥ ✱❝❤ó♥❣ t❛ ❝➬♥ ❝❤Ø r❛ r➺♥❣

➤♦ ➤➢ỵ❝ ✈í✐

t Ta

ớ ỗ rt

Mt

ị tr

Ta



TIEU LUAN MOI download :

At
✈➭


a

s < t✱ ➳♣ ❞ô♥❣ ✭✶✳✽✮ t❛ ❝ã
t

t

Mτ− ∇Aτ = E

E
s

s

Mτ− ∇Aτ − E
a

Mτ− ∇Aτ
a

= EMt At − EMs As .
❚❤❡♦ tÝ♥❤ ❝❤✃t ❝ñ❛ tÝ❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡✲❙t✐❡❧t❥❡s✱ t❛ ❝ã

t

Mτ− ∇Aτ = EMt− (At − At− ).

lim E

σ(s)↑t

s

❙✉② r❛

EMt− (At − At− ) = EMt At − EMt− At− ,
❤❛②

E(Mt − Mt− )At = 0.
▼➷t ❦❤➳❝✱

E(Mt − Mt− )E[At | Ft− ] = 0.
❙✉② r❛

E(Mt − Mt− )(At − E[At | Ft− ]) = 0.

✭✶✳✾✮

➜➷t

Mτ :=
❉♦ ➤ã✱

♥Õ✉

τ

At

♥Õ✉

τ

t.

(Mτ ) ❧➭ (Fτ )− ♠❛rt✐♥❣❛❧❡✳ ❚❤❛② ✈➭♦ ✭✶✳✾✮ t❛ ❝ã
E At − E[At | Ft− ]

❱❐②✱



E [At | Fτ ]

At − E[At | Ft− ] = 0

2

= E(Mt − Mt− )(At − E[At | Ft− ]) = 0.

❤✳❝✳❝✳

❱Ý ❞ơ ✶✳✷✳✺✳ ●✐➯ sư (At ) ❧➭ ♠ét q✉➳ tr×♥❤ t➝♥❣✱ ❦❤➯ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T.
❑❤✐ ➤ã✱ t❛ ❝ã✿
✐✮ ◆Õ✉
✈➭

T = N t❤× At

❧➭ q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

At

❧➭ ❞➲② t➝♥❣

Ft−1 −➤♦ ➤➢ỵ❝ ∀ t = 1, 2, . . .✳
✶✹

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✐✐✮ ◆Õ✉

T = R t❤× ♠ä✐ q✉➳ tr×♥❤ t➝♥❣ ❦❤➯ tí tụ (At ) q trì t

tự

ị ❧ý ✶✳✷✳✻ ✭➜Þ♥❤ ❧ý ❦❤❛✐ tr✐Ĩ♥ ❉♦♦❜✲▼❡②❡r✮✳
M

X = (Xt )t∈Ta

❧➭ s✉❜✲

(DL)✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ ❞✉② ♥❤✃t ♠ét ♠❛rt✐♥✲

♠❛rt✐♥❣❛❧❡ ❧✐➟♥ tơ❝ ♣❤➯✐ t❤✉é❝ ❧í♣
❣❛❧❡

●✐➯ sư

✈➭ ♠ét q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥

A s❛♦ ❝❤♦ ➤➻♥❣ t❤ø❝ s❛✉ t❤á❛ ♠➲♥
∀ t ∈ Ta

Xt = Mt + At

❤✳❝✳❝✳

❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❞✉② ♥❤✃t✳ ●✐➯ sö tå♥ t➵✐ ✷ ♠❛rt✐♥❣❛❧❡

M✱ M

✈➭ ✷ q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥

A✱ A

s❛♦ ❝❤♦

∀ t ∈ Ta

Xt = Mt + At = Mt + At

❤✳❝✳❝.

❙✉② r❛

Bt = At − At = Mt − Mt
❧➭ ♠❛rt✐♥❣❛❧❡✳
❱í✐ ỗ

(n)

ủ

[a, t] ị ở ✈➭ ✭✶✳✼✮✱ ➤➷t
kn −1

(n)
Bsπ

:= Ba 1{a} +

Bti 1(ti ,ti+1 ] .
i=0

❑❤✐ ➤ã✱

Bs− = lim Bsπ

(n)

n→∞

∀ s ∈ [a, t].

❚õ ➤➻♥❣ t❤ø❝ ✭✶✳✽✮ ✈➭ ➤Þ♥❤ ❧ý ❤é✐ tơ ❜Þ ❝❤➷♥ t❛ ❝ã

t

EBt (At − At ) = E

t

Bτ− ∇Aτ − E
a

Bτ− ∇Aτ
a

kn

Bti−1 (Bti − Bti−1 ) = 0.

= lim E
n→∞

◆❤➢ ✈❐②✱
♠ä✐

i=1

E(At − At )2 = E[Bt (At − At )] = 0

t ∈ Ta . ❚õ ➤ã s✉② r❛ At = At

s✉② r❛

❤✳❝✳❝✱ ✈í✐ ♠ä✐

At − At = 0

❤✳❝✳❝✱ ✈í✐

t ∈ Ta ✳

✶✺

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❚✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ sù tå♥ t➵✐

M

✈➭

A✳

M

❝❤ó♥❣ t❛ t❤✃② r➺♥❣ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ tå♥ t➵✐ q✉➳ tr×♥❤

[a; b]

b Ta

ớ ỗ

(n)

(n)

(n)

1
2n

max((ti+1 ) ti )
i



♠✃t tÝ♥❤ tỉ♥❣ q✉➳t✱ ❣✐➯ sư r➺♥❣

π (n) : a = t0

❞➲② ♣❤➞♥ ❤♦➵❝❤

❚õ tÝ♥❤ ❞✉② ♥❤✃t

(n)

< t1

(n)

< · · · < tkn = b

π (n) ⊂ π (n+1) ✳

▼❡②❡r ➤è✐ ✈í✐ ❞➲② s✉❜♠❛rt✐♥❣❛❧❡✱

(n)

➳♣

❝đ❛

A

tr➟♥ ➤♦➵♥

Xa = 0.

[a, b]

❳Ðt

t❤á❛ ♠➲♥

❞ơ♥❣ ➤Þ♥❤ ❧ý ❦❤❛✐ tr✐Ó♥ ❉♦♦❜ ✲

X (n) = (Xtj )tj ∈π(n)

t❛ ❝ã

(n)

Xtj = Mtj + Atj , j = 0, 1, ..., kn ,

✭✶✳✶✵✮

j

(n)
tr♦♥❣ ➤ã At
j

E[Xti − Xti−1 |Fti−1 ]

=

❧➭ ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ t➝♥❣✱

i=1
(n)

(n)

n
{Ftj }kj=0
− ❦❤➯ ➤♦➳♥ ✈➭ Mtj = Xtj − Atj

(n)

✳ ❍➡♥ ♥÷❛✱

(n)

(n)

Mtj = E(Mb |Ftj ) = E(Xb − Ab |Ftj ).
❚❛ ❧➵✐ ❝ã✱

X

t❤✉é❝ ❧í♣

(n)

(DL) ♥➟♥ {Ab }n∈N

❧ý ❉✉♥❢♦r❞ ✲ P❡tt✐s s✉② r❛ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥
②Õ✉ ➤Õ♥ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ tÝ❝❤
✈➭

❦❤➯ tÝ❝❤ ➤Ò✉✳ ➳♣ ❞ơ♥❣ ➜Þ♥❤

(nk )

(Ab

)k∈N

❝đ❛

(n)

{Ab }n∈N

❤é✐ tơ

Ab . ❚õ ➤ã✱ ❝❤ó♥❣ t ị ĩ q trì M

A t tứ
Mt = E(Xb − Ab |Ft ); At = Xt − Mt ; ∀ t ∈ [a, b].

❚❤❛② t❤Õ
✇❡❛❦

Mt

✈➭

At

(nk )

− lim Ab
k→∞

❜ë✐ ❝➳❝ ❜➯♥ s❛♦ ❧✐➟♥ tơ❝ ♣❤➯✐ t➢➡♥❣ ø♥❣ ❝đ❛ ❝❤ó♥❣✳ ❱×

(nk )

= Ab , s✉② r❛ ✇❡❛❦ − lim Mb
k→∞

= Mb .

▼➷t ❦❤➳❝✱ t❛ ❝ã

✇❡❛❦−
✈í✐

G

❧➭

(nk )

lim E(Mb

k→∞

|G) = E(Mb |G),

σ− tr➢ê♥❣ ❝♦♥ ❝ñ❛ σ− tr➢ê♥❣ F ✳

✶✻

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▲✃②

Π=

n∈N π

(n)

✈➭

a

s

b ✈í✐ s, t ∈ Π ❝è ➤Þ♥❤✳ ❙✉② r❛ r➺♥❣

t

At − As = Xt − Xs − [E(Mb |Ft ) − E(Mb |Fs )]
(nk )

= Xt − Xs − ✇❡❛❦✲ lim E(Mb
k→∞

(nk )

= ✇❡❛❦✲ lim Xt − Xs − E(Mb
k→∞

(nk )

= ✇❡❛❦✲ lim Xt − Xs − Mt
k→∞

(nk )

= ✇❡❛❦✲ lim At
k→∞

❱×

k)
− A(n
s

(nk )

|Ft ) − E(Mb

|Fs )

(nk )

|Ft ) + E(Mb

|Fs )

+ Ms(nk )
0 ❤✳❝✳❝.

Π ➤Õ♠ ➤➢ỵ❝ ✈➭ trï ♠❐t tr♦♥❣ [a, b] ✈➭ A ❧✐➟♥ tô❝ ♣❤➯✐✱ s✉② r❛ At

✈í✐ ♠ä✐

As

❤✳❝✳❝✱

t > s✳ ◆❣❤Ü❛ ❧➭ A ❧➭ q✉➳ tr×♥❤ t➝♥❣✳

❚✐Õ♣ t❤❡♦ ❝❤ó♥❣ t❛ ❦✐Ĩ♠ tr❛ tÝ♥❤ tự ủ q trì

A







rt tụ ị ❜✃t ❦ú✳ ➜➷t

kn
(n)
ξsπ

:=

ξti−1 1(ti−1 ,ti ] (s).
i=1

❚❛ ❝ã✱

ξs− = lim ξsπ

(n)

∀ s ∈ (a, b].

n→∞

❍➡♥ ♥÷❛✱

b

b

ξs− ∇As = lim E

E

n→∞

a
❱í✐ ỗ

kn
(n)
s As

E

i=1

kn
(n)
s As

ừ tí ủ

A(mk )

(mk )

= lim E
mk →∞

a

ξti−1 (Ati

(m )

− Ati−1k ) .

i=1

s✉② r❛

kn

kn
(m )
ξti−1 (Ati k
i=1

n

n ố ị ú t ó tể tì ợ mk ↑ ∞ s❛♦ ❝❤♦
b

E

ξti−1 (Ati − Ati−1 ) .

= lim E

a

−

(m )
Ati−1k )

(mk )

= Eξb

(Ati

(m )

(mk )

− Ati−1k ) = E ξb Ab

i=1

✶✼

TIEU LUAN MOI download :

.


◆➟♥

b

b
n→∞

a
❉♦ ➤ã✱

(n)

ξsπ ∇As = E ξb Ab .

ξs− ∇As = lim E

E

a

b

ξs− ∇As = E ξb Ab ,

E
a
♥❣❤Ü❛ ❧➭

A = (At ) ❧➭ q✉➳ tr×♥❤ t➝♥❣ tù ♥❤✐➟♥✳
M ∈ M2
M2

▲✃②

✳ ❱×

M = ( M t )t∈Ta

t➝♥❣ tù ♥❤✐➟♥

M

tr×♥❤ t➝♥❣ tù ♥❤✐➟♥

✶✳✸

t

s❛♦ ❝❤♦

➤➢ỵ❝ ❣ä✐ ❧➭

Mt2 − M

➤➷❝ tr➢♥❣

t

❧➭ ♠ét ♠❛rt✐♥❣❛❧❡✳

❝đ❛ ♠❛rt✐♥❣❛❧❡

M

◗✉➳

✳

❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥

✶✳✸✳✶

❚Ý❝❤ ♣❤➞♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ì tí

L

ý ệ

ị tr

srt tồ t➵✐ ❞✉② ♥❤✃t ♠ét q✉➳ tr×♥❤

P

♥❤✐➟♥

❧➭ t❐♣ t✃t ❝➯ ❝➳❝ q trì trị tự

Ta ì


tr


L



ớ qỹ tụ tr tr

trờ

ễ



t

t

r

P



ủ

ợ

Ta



Ta ì

s

ở

(F(t) )

s

ọ



ở

t

(t )tTa

ù ợ



q

trì

{(s, t] ì F :

s, t Ta , s < t, F Fs }



ị ĩ ỗ tư ❝đ❛ σ− tr➢ê♥❣ P ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét t❐♣
ột q trì
trờ



ợ ọ ♥Õ✉ ♥ã ➤♦ ➤➢ỵ❝ ➤è✐ ✈í✐

σ−

P.

❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✱ ♠ét q✉➳ tr×♥❤ ❧✐➟♥ tơ❝ tr➳✐ ❝❤➢❛ ❝❤➽❝ ➤➲ ❧➭ q✉➳

tr×♥❤ ❦❤➯ ➤♦➳♥✳

❈❤ó ý ✶✳✸✳✷✳ i)

◆Õ✉

T=N

t❤× q✉➳ tr×♥❤

φt

❧➭ ❦❤➯ ➤♦➳♥ ế

t

q trì

Ft1 ợ
ii)

ế

T = R

tì

t

q trì ➤♦➳♥ ♥Õ✉ ➤♦ ➤➢ỵ❝ ➤è✐ ✈í✐

σ− tr➢ê♥❣ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tơ❝ tr➳✐✳
✶✽

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▼Ư♥❤ ➤Ị ✶✳✸✳✸✳
♥❣➱✉ ♥❤✐➟♥
✐✮

✭❬✻❪✮ ●✐➯ sư

Φ

❧➭ ❦❤➠♥❣ ❣✐❛♥ t✉②Õ♥ tÝ♥❤ ❣å♠ ❝➳❝ q✉➳ tr×♥❤

φ : Ta × Ω → R ợ ị tỏ

ứ tt q trì ị L;

ọ ệ
tộ

lim n =

s

n

q trì ị ❝❤➷♥

Φ.

Φ ❝❤ø❛ t✃t ❝➯ ❝➳❝ q✉➳ tr×♥❤ ❦❤➯ ➤♦➳♥✳

❑❤✐ ➤ã✱

●✐➯

L2 (M )

{φn } ⊂ Φ

M ∈ M2

sư

❧➭

♠ét

♠❛rt✐♥❣❛❧❡

❜×♥❤

♣❤➢➡♥❣

❦❤➯

tÝ❝❤✳

❑ý

❤✐Ư✉

❧➭ ❦❤➠♥❣ ❣✐❛♥ t✃t q trì trị tự ❦❤➯ ➤♦➳♥

φ = {φt }t∈Ta ,

t❤á❛ ♠➲♥

T

φ

2
T,M

φ2τ ∇ M

=E

τ

< ∞, T > a.

a
ớ ỗ

b>a

ố ị ọ

(a, b]

r ❣✐❛♥

L2 ((a, b]; M )

L2 ((a, b]; M )

❧➭ ❤➵♥ ế ủ

L2 (M )

tr

ét ợ ị ❜ë✐

b

φ

2
b,M

φ2τ ∇ M τ .

=E
a

❍❛✐ q✉➳ tr×♥❤

φ, φ ∈ L2 ((a, b]; M )

▼ét q✉➳ tr×♥❤

φ

tå♥ t➵✐ ♠ét ♣❤➞♥ ❤♦➵❝❤

♥❣➱✉ ị

ợ ọ

ị tr

[a, b]

trù

ợ ọ ❧➭

π : a = t0 < t1 < · · · < tn = b

{fi }

s❛♦ ❝❤♦

fi

❧➭

Fti−1 −

φ−φ

♥Õ✉

b,M

=0

q✉➳ tr×♥❤ ➤➡♥ ❣✐➯♥✱
❝đ❛

[a, b]

➤♦ ➤➢ỵ❝ ✈í✐ ♠ä✐

✳

♥Õ✉

✈➭ ❞➲② ❝➳❝ ❜✐Õ♥

i = 1, n

✈➭

n

fi 1(ti−1 ,ti ] (t); t ∈ (a, b].

φ(t) =

✭✶✳✶✶✮

i=1
❈❤ó♥❣ t ý ệ t ợ tt q trì ➤➡♥ ❣✐➯♥ ❧➭

L0

✳

❇ỉ ➤Ị ✶✳✸✳✹✳ L0 trï ♠❐t tr♦♥❣ L2 ((a, b]; M ) ✈í✐ ♠❡tr✐❝ ①➳❝ ➤Þ♥❤ ❜ë✐
b

d(φ, ϕ)2 = φ − ϕ

2
b,M

|φτ − ϕτ |2 ∇ M τ .

=E

a
✶✾

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❈❤ø♥❣ ♠✐♥❤✳ ❘â r➭♥❣✱

L0 ⊂ L2 ((a, b]; M )✳ ▲✃② φ ∈ L2 ((a, b]; M )✳ ➜➷t

φK (t, ω) := φ(t, ω)1[−K,K] (φ(t, ω)).
❑❤✐ ➤ã✱

φK ∈ L2 ((a, b]; M ) ✈➭ φ − φK

t❛ ❝➬♥ ❝❤Ø r❛ ớ ỗ q trì

L2 ((a, b]; M )

(n) ∈ L0 , n = 1, 2, · · · ,

➤➢ỵ❝ ❞➲②

→ 0 ❦❤✐ K → +∞. ❉♦ ➤ã✱ ❝❤ó♥❣

b,M

s❛♦

ị tì ó tể ị

(n)

b,M

0



n ∞.

▲✃②

Υ = {φ ∈ L2 ((a, b]; M ) : φ ❜Þ ❝❤➷♥ ✈➭ tå♥ t➵✐ φ(n) ∈ L0
s❛♦ ❝❤♦

Υ

❧➭ ❦❤➠♥❣ ❣✐❛♥ t✉②Õ♥ tÝ♥❤ ✈➭ ♥Õ✉

♥➭♦ ➤ã ✈➭

❙✉② r❛

b,M

→0

φ(n) ∈ Υ, φ(n) < K

❦❤✐

n → ∞}.

✈í✐ ❤➺♥❣ sè

K >0

φ(n) ↑ tì . ớ ỗ L ➤➷t
φ(n) (t) := φ(σ(ti )),

tr♦♥❣ ➤ã

φ − φ(n)

{ti }

♥Õ✉

[a, b]

❧➭ ♠ét ♣❤➞♥ ❤♦➵❝❤ ❝đ❛

φ(n) ∈ L0

✈➭

φ(n) − φ

b,M

✈í✐

s❛♦ ❝❤♦

i = 0, kn − 1,

max(ρ(ti+1 ) − ti )
i

2−n .

→ 0 ❦❤✐ n → ∞.

❑Õt ❤ỵ♣ ✈í✐ ▼Ư♥❤ ➤Ị ✶✳✸✳✸✱ s✉② r❛
❝❤➷♥✳ ❉♦ ➤ã✱

t ∈ (ti , ti+1 ]

Υ

❝❤ø❛ t✃t ❝➯ q trì ị

= L2 ((a, b]; M ).

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✺✳ ●✐➯ sư φ ❧➭ ♠ét q✉➳ tr×♥❤ t❤✉é❝ L0 , ❝ã ❞➵♥❣ ✭✶✳✶✶✮✳ ❑❤✐ ➤ã✱
kn

b

φτ ∇Mτ :=
a
➤➢ỵ❝ ❣ä✐ ❧➭
❦❤➯ tÝ❝❤

M

fi (Mti − Mti−1 ),

✭✶✳✶✷✮

i=1

∇− tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ❝đ❛ φ ∈ L0 t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣
tr➟♥

(a, b].

❈❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣

➤➵✐ ❧➢ỵ♥❣ ♥❣➱✉ ♥❤✐➟♥

Fb −

∇−

tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥

b
a φτ ∇Mτ

❧➭

➤♦ ➤➢ỵ❝ ✈➭ ♠Ư♥❤ ➤Ị s❛✉ ➤➞② ➤➢ỵ❝ t❤á❛ ♠➲♥✳

▼Ư♥❤ ➤Ị ✶✳✸✳✻✳ ●✐➯ sư φ ❧➭ ♠ét q✉➳ tr×♥❤ t❤✉é❝ L0 ✈➭ α, β ❧➭ ❝➳❝ sè t❤ù❝✳ ❑❤✐
➤ã✱
✷✵

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✐✮

b
a φτ ∇Mτ

E

= 0,
2

✐✐✮

✐✐✐✮

b
a φτ ∇Mτ

E

b
a [αφτ

φ − φ(n)

b,M

→0

❦❤✐

n → ∞.

φτ(n) ∇Mτ −

ξ

2

φ(m)
τ ∇Mτ

= φ(m) − φ(n)

2
b,M ,

a

b (n)
a φ (τ )∇Mτ }

❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

{φ(n) } ⊂ L0

▼➷t ❦❤➳❝✱

b

a

{

b
a ξτ ∇Mτ ❤✳❝✳❝.

+β

✱ tõ ❇ỉ ➤Ị ✶✳✸✳✹ s✉② r❛ tå♥ t➵✐ ❞➲②

E
r❛

,

τ

φ ∈ L2 ((a, b]; M )

b

s✉②

M

b
a φτ ∇Mτ

+ βξτ ]M =

ớ ỗ

s

b 2
a

=E

tr

{

ó

b (n)
a (τ )∇Mτ }

❤é✐

tô

➤Õ♥

L2 (Ω, F, P)

✳ ❚ø❝ ❧➭

b

φ(n)
τ ∇Mτ .

ξ = L2 − lim

n→∞

●✐í✐ ❤➵♥

ξ

❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ✈✐Ư❝ ❝❤ä♥

ị ĩ
q trì


a

sử

{(n) }



L2 ((a, b]; M )✱ ∇−

tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ❝đ❛

φ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤ M ∈ M2

tr➟♥

(a, b]✱ ❦ý ❤✐Ö✉

b
a φτ M ợ ị ở
b

b

(n)
M ,

M = L2 − lim

n→∞

a
tr♦♥❣ ➤ã

{φ(n) } ❧➭ ❞➲② ❝➳❝ q✉➳ tr×♥❤ t❤✉é❝ L0

✭✶✳✶✸✮

a
s❛♦ ❝❤♦

b
2
|φτ − φ(n)
τ | ∇ M

lim E

n→∞

τ

= 0.

a

❱Ý ❞ô ✶✳✸✳✽✳ i) ◆Õ✉ T = N ✈➭ φ ∈ L2 ((a, b]; M ) t❤× (φn ) ❧➭ ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉
♥❤✐➟♥

(Fn−1 )− ➤♦ ➤➢ỵ❝ ✈➭
b

b

φτ ∇Mτ =
a

φi (Mi − Mi−1 ).
i=a+1
✷✶

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