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✐ ✲ ✷✵✶✷ ➜➵✐ ❤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❝✿ ❍➭ ♥é✐ ✲ ✷✵✶✷ ✐ ●❙✳❚❙ ◆❣✉②Ơ♥ ❍÷✉ ❉➢ ▼ơ❝ ❧ơ❝ ▼ơ❝ ❧ơ❝ ✐ ▲ê✐ ❝➯♠ ➡♥ ✐✐ ▼ë ➤➬✉ ✶ ✶ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✸ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ➜Þ♥❤ ❧ý ❦❤❛✐ tr✐Ĩ♥ ❉♦♦❜ ✲ ▼❡②❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ❚Ý❝❤ ♣❤➞♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤ ✳ ✶✽ ✶✳✸✳✷ ❚Ý❝❤ ♣❤➞♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ị ì tí tứ ■t➠ ✈➭ ø♥❣ ❞ô♥❣ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶ ❇✐Õ♥ ♣❤➞♥ ❜❐❝ ❤❛✐ ✳ ✷✳✷ ❈➠♥❣ t❤ø❝ ■t➠ ✈➭ ø♥❣ ❞ô♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✸✳✶ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✸✳✷ ❚Ý♥❤ ▼❛r❦♦✈ ❝đ❛ ♥❣❤✐Ư♠ ❑Õt ❧✉❐♥ ✈➭ ❦✐Õ♥ ♥❣❤Þ ✳ ✳ ✳ ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✐ ▲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 ♥❣❤✐Ö♣ ➤➵✐ ❤ä❝ ♥➝♠ ✷✵✵✵✱ ❣✐ê t❤➬② ❧➵✐ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥✱ ❣✐ó♣ ➤ì t➠✐ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚➠✐ ❝ị♥❣ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ tí✐ ❝➳❝ t❤➬②✱ ❝➠ ❝ñ❛ ❑❤♦❛ ❚♦➳♥ ✲ ❈➡ ✲ ❚✐♥ ❤ä❝✱ P❤ß♥❣ s❛✉ ➤➵✐ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥✱ ➜❍◗● ❍➭ ◆é✐ ➤➲ ❣✐➯♥❣ ❞➵②✱ ❣✐ó♣ ➤ì t➠✐ tr sốt q trì ọ t tr ị t ♥❤÷♥❣ ❦✐Õ♥ t❤ø❝ ♥Ị♥ t➯♥❣ ➤đ ➤Ĩ ❧➭♠ ✈✐Ư❝✳ ➜➷❝ ❜✐Ưt✱ t➠✐ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ◆❈❙✳ ◆❣✉②Ơ♥ ❚❤❛♥❤ ❉✐Ư✉✱ ❑❤♦❛ ❚♦➳♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ➤➲ ❝ã ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ q✉ý ❜➳✉ ➤Ó ❜➯♥ ❧✉❐♥ ✈➝♥ ❤♦➭♥ ❝❤Ø♥❤ ❤➡♥✳ ❚➠✐ ❝ị♥❣ ❦❤➠♥❣ q✉➟♥ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ tí✐ ❝➳❝ ➤å♥❣ ❝❤Ý ❧➲♥❤ ➤➵♦ ❝ï♥❣ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ư♣ ❚r➢ê♥❣ ❚❍P❚ ❈❤✉②➟♥ ▲➢➡♥❣ ❱➝♥ ❚ơ② ✲ ◆✐♥❤ ❇×♥❤✱ ♥➡✐ t➠✐ ❝➠♥❣ t➳❝✱ ➤➲ ❤Õt sø❝ t➵♦ ➤✐Ị✉ ệ t ợ tr sốt q trì ọ t ũ ♥❤➢ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥ ❝đ❛ t➠✐✳ ❈✉è✐ ❝ï♥❣✱ t➠✐ ①✐♥ ❝➯♠ ➡♥ ❝❤❛ ♠Đ✱ ❝➳❝ ❛♥❤ ❝❤Þ ❡♠ ✈➭ ❣✐❛ ➤×♥❤ ♥❤á ❝đ❛ 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❝ ✈✐➟♥ ➜✐♥❤ ❱➝♥ ❑❤➞♠ ✐✐ ▼ë ➤➬✉ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ ❧➭ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝ ❝❤♦ ❝➳❝ ❤Ư ➤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 ❝➳❝ ❤Ö t❤è♥❣ ❤♦➵t ➤é♥❣ ❦❤➠♥❣ ❤♦➭♥ t♦➭♥ ❧✐➟♥ tơ❝ ❝ị♥❣ ❦❤➠♥❣ ❤♦➭♥ t♦➭♥ ❝➳❝❤ ➤Ị✉ ♥❤❛✉✳ ➜➠✐ ❦❤✐ ❝➳❝ 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♥ ❦❤✐ ♠ï❛ ➤➠♥❣ t❤× sù t trể ủ ú ị ì tr ề trờ ợ trì s ❦❤➠♥❣ ➤ñ ♠➠ t➯ ❝➳❝ t❤➠♥❣ t✐♥ ❝➬♥ t❤✐Õt ❝ñ❛ ♠➠ ❤×♥❤✳ ▲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➵❝ ✈➭ ❧✐➟♥ tô❝✳ ❈➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ✈Ò ❣✐➯✐ 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➟♥ t❤Õ ❣✐í✐ ✈➭ ➤➲ ó ề trì ợ ố tr t ❝❤Ý t♦➳♥ ❤ä❝ ❝ã ✉② tÝ♥ ✭❬✶✱ ✷✱ ✳✳✳❪✮✳ ❚✉② ♥❤✐➟♥✱ ♣❤➬♥ ❧í♥ ❝➳❝ ❦Õt q✉➯ ➤➵t ➤➢ỵ❝ ❝❤Ø ❞õ♥❣ ❧➵✐ ë ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❤Ư ➤é♥❣ ❧ù❝ t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ❱× t❤Õ ❝➳❝ ❦Õt q✉➯ ♥➭② ỉ t ợ ì t trể tr ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ♠➠✐ ✶ 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✉➯ ❝ñ❛ ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ủ ì tt ị s ì ♥❤✐➟♥ ❧➭ ♠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 ✸ ♠ô❝✳ ụ trì ữ ề ề ❣✐➯✐ tÝ❝❤ t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ▼ơ❝ ✶✳✷✳ trì ị ý trể r ố s✉❜♠❛rt✐♥❣❛❧❡ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ▼ơ❝ ✶✳✸ tr×♥❤ ❜➭② tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤✱ ♠❛rt✐♥❣❛❧❡ ị ì tí rộ ố ✈í✐ s❡♠✐♠❛rt✐♥❣❛❧❡ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ❈❤➢➡♥❣ ✷✳ ❈➠♥❣ t❤ø❝ ■t➠ ✈➭ ø♥❣ ❞ơ♥❣✳ ◆é✐ ❞✉♥❣ ❈❤➢➡♥❣ ✷ ➤➢ỵ❝ ✈✐Õt t ụ ụ ú t trì ị ĩ ề ế ỗ ợ ủ q trì ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ■t➠ ➤è✐ ✈í✐ ❜é ▼ơ❝ ✷✳✷ ❚r×♥❤ ❜➭② ✈Ị ❝➠♥❣ t❤ø❝ d− s❡♠✐♠❛rt✐♥❣❛❧❡ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈➭ ❝➳❝ ø♥❣ ❞ô♥❣✳ ❈❤➢➡♥❣ ✸✳ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ◆é✐ ❞✉♥❣ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ✷ ♠ơ❝✳ ▼ơ❝ ✸✳✶ ➤➢❛ r❛ ➤Þ♥❤ ♥❣❤Ü❛ ♥❣❤✐Ư♠ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✈Ị sù 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✐ ❣✐❛♥✳ ✷ ❈❤➢➡♥❣ ✶ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ết q trì tr ụ ợ t ❦❤➯♦ tõ t➭✐ ❧✐Ö✉ ❬✶❪✳ t❤ê✐ ❣✐❛♥ t❤ê✐ ❣✐❛♥ ❧➭ ột t ó rỗ ủ t số tự ❧➭ T ❚❛ tr❛♥❣ ❜Þ ❝❤♦ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T ♠ét R ✱ t❤➢ê♥❣ ❦ý ❤✐Ö✉ t❤❛♥❣ t➠♣➠ ❝➯♠ s✐♥❤ 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♥❣✳ ✸ ❚❤❛♥❣ ❝đ❛ t➠♣➠ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✳ ●✐➯ sö T ❧➭ ♠ét t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ➳♥❤ ①➵ σ :T→T ①➳❝ ➤Þ♥❤ ❜ë✐ σ(t) = inf{s ∈ T : s > t}, ➤➢ỵ❝ ❣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 ♣❤➬♥ tö ♥❤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➳✐ ✈õ❛ ❝➠ ❧❐♣ t tự ỗ ý a, b T ✱ ❤✐Ư✉ ❝➳❝ t❐♣ ❦ý ❤✐Ư✉ ❤ỵ♣ [a, b] ❧➭ t❐♣ ❤ỵ♣ (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 ♥Õ✉ T = −∞  T \ [m, σ(m)) ♥Õ✉ T = m,   T ♥Õ✉ max T = +∞  T \ (ρ(M ), M ] ♥Õ✉ max T = M ✹ ❧➭ t ❝➳❝ t < b} ✳ t❐♣ b} ✱ ❤ỵ♣ ❑ý ❤✐Ö✉ ❑ý ❤✐Ö✉ I1 = {t : t ❝➠ ❧❐♣ tr➳✐}, I2 = {t : t ❝➠ ❧❐♣ ♣❤➯✐}, I = I1 ∪ I2 ▼Ư♥❤ ➤Ị ✶✳✶✳✸✳ ❚❐♣ ợ ủ t tờ ị ĩ t✃t ❝➯ ❝➳❝ ➤✐Ó♠ ❝➠ ❧❐♣ tr➳✐ ❤♦➷❝ ❝➠ ❧❐♣ ♣❤➯✐ I T ❧➭ t❐♣ ❦❤➠♥❣ q✉➳ ➤Õ♠ ➤➢ỵ❝✳ ●✐➯ sư T ❧➭ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ➳♥❤ ①➵ µ : Tk R+ ị à(t) = (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 tì (t) = t = (t), à(t) = ρ(t) = 0; ✰✮ ◆Õ✉ T = Z t❤× ρ(t) = t − 1, σ(t) = t + 1, µ(t) = ν(t) = ✰✮ ❱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, µ(t) = ν(t) = h ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✻✳ ❈❤♦ ❤➭♠ 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➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t tr➳✐✳ ❚❐♣ ❤ỵ♣ ❝➳❝ ❤➭♠ rd− ❧✐➟♥ tơ❝ ❦ý ❤✐Ư✉ ❧➭ Crd ❤♦➷❝ Crd (T, R) ✺ ✐✐✐✮ ld−❧✐➟♥ tơ❝ ✭ld−❝♦♥t✐♥✉♦✉s✮ ♥Õ✉ f ❧✐➟♥ tơ❝ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t tr➳✐✱ ❝ã ❣✐í✐ ❤➵♥ ♣❤➯✐ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t ♣❤➯✐✳ ❚❐♣ ❤ỵ♣ ❝➳❝ ❤➭♠ ❦ý ❤✐Ư✉ ❧➭ ●✐➯ sö 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 ❧✐➟♥ tô❝ Cld (T, R) f :T→R ❑ý ❤✐Ö✉ 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 ❧✐➟♥ tô❝ t❤× ✐✐✮ ◆Õ✉ f ❧➭ ❤➭♠ sè ✐✈✮ ❚♦➳♥ tư ❜➢í❝ ♥❤➯② ❧ï✐ f ❧➭ ❤➭♠ sè f ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã ➤➵♦ ❤➭♠✮ t➵✐ ❧➞♥ ❝❐♥ U t ∈ kT ❝đ❛ σ ❧➭ ❤➭♠ sè rd− ❧✐➟♥ tơ❝ ✈➭ ld− ❧✐➟♥ tô❝✳ ❧➭ ❤➭♠ sè ❝❤Ý♥❤ q✉②✳ rd− ❧✐➟♥ tô❝✳ ρ ❧➭ ❤➭♠ sè ld− ❧✐➟♥ tô❝✳ ld− ❧✐➟♥ tụ tì f ị ĩ sử f ❍➭♠ sè ❧➭ ❤➭♠ 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 ➤Ó |f (ρ(t)) − f (s) − f ∇ (t)(ρ(t) − s)| ε|ρ(t) − s| f ∇ (t) ∈ R ➤➢ỵ❝ ❣ä✐ ❧➭ ∇−➤➵♦ ❤➭♠ ❝ñ❛ ❤➭♠ 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✳ ✻ ❱Ý ❞ô ✷✳✷✳✽✳ ✭✶✮ ◆Õ✉ T = R t❤× Et (M ) ❝❤Ý♥❤ ❧➭ ❤➭♠ ♠ị ❉♦❧Ð❛♥s✲ ❉❛❞❡ ➤➢ỵ❝ ①➞② ❞ù♥❣ t❤❡♦ ❝➳❝❤ t❤➠♥❣ t❤➢ê♥❣✳ ✭✷✮ ◆Õ✉ T = hZ✱ h > t❤× t h Et (M ) = + (Mhτ − Mh(τ −1) ) τ = +1 ✭✸✮ ◆Õ✉ T = q N0 = {q n : n ∈ N0 } ✈í✐ q > t❤× ln t ln q Et (M ) = [1 + (Mqτ − Mqτ −1 )] a τ = ln ln q +1 ✹✷ ❈❤➢➡♥❣ ✸ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❝❤✐ t✐Õt✱ ❝ã ❤Ư t❤è♥❣ ✈Ị ♣❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ♥❤✐Ơ✉ ❧➭ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✱ ❝❤Ø r❛ ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ▼❛r❦♦✈ ♥❣❤✐Ư♠✳ ❑Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ❞ù❛ ✈➭♦ t➭✐ ❧✐Ö✉ ❬✹❪ ✸✳✶ P❤➢➡♥❣ tr×♥❤ ➤é♥❣ ❧ù❝ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ▲✃② ❝❤♦ M ∈ M2 ✈➭ xa ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ Ex2a < ∞ f : [a, T ] × R R Fa ợ trị t❤ù❝ s❛♦ g : [a, T ] × R → R ❧➭ ✷ ❤➭♠ ❇♦r❡❧✳ ❳Ðt ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ■t➠ ❞➵♥❣   d∇ X(t) = f (t, X(t− ))d∇ t + g(t, X(t− ))d∇ M (t) ∀ t ∈ [a, T ]  X(a) = xa ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ ❣✐➯ t❤✐Õt ✹✸ ✭✸✳✶✮ t M Nτ ∇τ, = t ✭✸✳✷✮ a Nt tr♦♥❣ ➤ã (Ft ) q trì ị ù ợ tứ tå♥ t➵✐ ❤➺♥❣ sè N s❛♦ ❝❤♦ P{ sup |Nt | N } = ✭✸✳✸✮ a t T ➜Þ♥❤ ĩ ột q trì trị tự {X(t)}t[a,T ] ợ ọ ệ ủ trì ✭✸✳✶✮ ♥Õ✉ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ✭✐✮ ✭✐✐✮ {X(t)} q trì (Ft )ù ợ f (Ã, X(Ã )) ∈ L1 ([a, T ]; R) ✈➭ g(·, X(·− )) ∈ L2 ([a, T ]; M ), ✭✐✐✐✮ P❤➢➡♥❣ tr×♥❤ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ t X(t) = xa + t g(τ, X(τ− ))∇Mτ , ∀ t ∈ [a, T ], f (τ, X(τ− ))∇τ + a a ✭✸✳✹✮ ✈í✐ ①➳❝ st P X(t) X(t) trì ợ ọ ❝ã ❞✉② ♥❤✃t ♥❣❤✐Ö♠ tr➟♥ ❧➭ [a, T ] ♥Õ✉ ❦❤✐ ❜➯♥ s❛♦ ❧➭ ✷ ♥❣❤✐➟♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ t❤× P {X(t) = X(t) ∀ t ∈ [a, T ]} = ❈❤ó♥❣ t❛ ❝ã ❝❛❞❧❛❣✳ ❉♦ ➤ã✱ Mt ❧➭ t a g(τ, X(τ− ))∇Mτ X(t) rd− t❤á❛ ♠➲♥ ❧✐➟♥ tơ❝ t❤× X(t) ✭✸✳✹✮ ❝ị♥❣ (Ft )− ❧➭ t❤× ♠❛rt✐♥❣❛❧❡✱ X(t) s✉② r❛ ♥ã ❝ã ❝ã tÝ♥❤ ❝❤✃t ❝❛❞❧❛❣✳ ❍➡♥ ữ ế rd tụ ị ý ự tồ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ư♠✮✳ ●✐➯ sư tå♥ t➵✐ ❤❛✐ ❤➺♥❣ sè ❞➢➡♥❣ K ✈➭ K s❛♦ ❝❤♦ (i) ✭➜✐Ò✉ ❦✐Ư♥ ▲✐♣s❝❤✐t③✮ ❱í✐ ♠ä✐ x, y ∈ R ✈➭ t ∈ [a, T ] t❤× (f (t, x) − f (t, y))2 ∨ (g(t, x) − g(t, y))2 ✹✹ K(x − y)2 ; ✭✸✳✺✮ (ii) ✭➜✐Ị✉ ❦✐Ư♥ t➝♥❣ t✉②Õ♥ tÝ♥❤✮ ❱í✐ ♠ä✐ (t, x) ∈ [a, T ] × R t❤× f (t, x) ∨ g (t, x) K(1 + x2 ) ❑❤✐ ➤ã✱ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✶✮ tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ X(t) ✭✸✳✻✮ ✈➭ ♥❣❤✐Ö♠ ❧➭ s❡♠✐✲ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt✱ ❝❤ó♥❣ t❛ ❝❤Ø r❛ r➺♥❣ ♥Õ✉ ➤✐Ị✉ ❦✐Ư♥ t➝♥❣ t✉②Õ♥ tÝ♥❤ X(t) ệ ủ trì tì ợ t❤á❛ ♠➲♥ ✈➭ sup X (t) E (1 + 3Ex2a )e3K(T −a+4N ) (T, a), a t T tr♦♥❣ ó N ợ ị t ♠ä✐ sè ♥❣✉②➟♥ n 1✱ ①➳❝ ➤Þ♥❤ t❤ê✐ ➤✐Ĩ♠ ❞õ♥❣ υn = T ∧ inf{t ∈ [a, T ] : |X(t)| ❘â r➭♥❣✱ ✭✸✳✼✮ υn ↑ T ❤✳❝✳❝✱ ❦❤✐ n → ∞✳ ➜➷t n} un (t) := X(t ∧ υn ) ✈í✐ t ∈ [a, T ] ❈❤ó♥❣ t❛ t❤✃② t un (t) = xa + t f (τ, un (τ− ))1[a,υn ] (τ )∇τ + a ✈í✐ g(τ, un (τ− ))1[a,υn ] (τ )∇Mτ , a t ∈ [a, T ] ❜✃t ❦ú✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❝➡ ❜➯♥ (a+b+c)2 3(a2 +b2 +c2 ) ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ o ăr s r t u2n (t) 3x2a f (τ, un (τ− ))1[a,υn ] (τ )∇τ + 3(t − a) a t +3 g(τ, un (τ− ))1[a,υn ] (τ )∇Mτ a ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✶✳✸✳✶✷ ✈➭ ➤✐Ị✉ ❦✐Ö♥ ✭✸✳✻✮✱ t❛ ❝ã t E sup a s t u2n (s) 3Ex2a f (τ, un (τ− ))1[a,υn ] (τ ))∇τ + 3(T − a) a t g (τ, un (τ− ))1[a,υn ] (τ )∇ M + 12E τ a t 3Ex2a + 3K(T − a + 4N ) (1 + Eu2n (τ− ))∇τ a ✹✺ ❙✉② r❛ t 1+ E( sup u2n (s)) a s t 1+ 3Ex2a (1 + Eu2n (τ− ))∇τ + 3K(T − a + 4N ) a t + 3Ex2a + 3K(T − a + 4N ) 1+E a sup u2n (s) ∇τ a s τ− ❚õ ➤✐Ị✉ ♥➭② ✈➭ ❇ỉ ➤Ò ✶✳✶✳✶✻ t❛ ❝ã sup u2n (s) 1+E ❈❤♦ (1 + 3Ex2a )e3K(T −a+4N ) (T, a) a s t n → ∞ t❛ ❝ã ❜✃t ➤➻♥❣ t❤ø❝ ✭✸✳✼✮ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❚Ý♥❤ ❞✉② ♥❤✃t✿ X(t) ●✐➯ sö ✈➭ X(t) ❧➭ ❤❛✐ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✶✮✳ ❚õ ❤Ư t❤ø❝ t f (τ, X(τ− )) − f (τ, X(τ− )) ∇τ X(t) − X(t) = a t g(τ, X(τ− )) − g(τ, X(τ− )) ∇Mτ , + a ❜✃t ➤➻♥❣ tứ o ăr ị ý ề ệ st ✈➭ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ❜✃t ➤➻♥❣ t❤ø❝ ✭✸✳✼✮ t❛ ❝ã t E sup (X(s) − X(s)) ((X(τ− ) − X(τ− ))2 )∇τ 2K(T − a) a s t a t (X(τ− ) − X(τ− ))2 ∇ M + 8KE τ a t 2K T − a + 4N E a sup (X(s) − X(s))2 ∇τ a s τ− ❑Õt ❤ỵ♣ ➤✐Ị✉ ♥➭② ✈í✐ ❇ỉ ➤Ị ✶✳✶✳✶✻ s✉② r❛ E sup (X(s) − X(s))2 = a s T ❱❐②✱ X(t) = X(t) ❤✳❝✳❝✱ ✈í✐ ♠ä✐ a t ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ✹✻ T✳ ❙✉② r❛ tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ❙ù tå♥ t➵✐✿ ➜➷t X0 (t) := xa ✈➭ ✈í✐ n = 1, 2, · · · t Xn (t) = xa + t f (τ, Xn−1 (τ− ))∇τ + g(τ, Xn−1 (τ− ))∇Mτ , a ✈í✐ t ∈ [a; T ]✳ ①➳❝ ➤Þ♥❤ ①✃♣ ①Ø P✐❝❛r❞ ✭✸✳✽✮ a ❈❤ó♥❣ t❛ t❤✃② E sup X02 (s) = Ex2a < ∞ ✈í✐ ♠ä✐ t > a✳ E sup Xn2 (s) < ∞ ✈í✐ ♠ä✐ n ∈ N∗ ❉♦ a s t ➤ã✱ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ s✉② r❛ a s t t ✈➭ a✳ ❍➡♥ ♥÷❛✱ (X1 (t) − X0 (t))2 = (X1 (t) − xa )2 t f (τ, xa )∇τ t +2 a g(τ, xa )∇Mτ a ❚õ ➤ã t❛ ❝ã E sup (X1 (s) − X0 (s))2 C, ✭✸✳✾✮ a s t ✈í✐ C = 2K (T − a)2 + 4N (T − a) (1 + Ex2a )✳ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✱ t❛ ❝ã t E sup (Xn+2 (s) − Xn+1 (s)) E sup (Xn+1 (s− ) − Xn (s− ))2 ∇τ P a s t a s τ a t E sup (Xn+1 (s) − Xn (s))2 ∇τ, P a ✈í✐ a s τ− P = 2K(T − a + 4N )✳ ❱× ✈❐②✱ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ ❝❤ó♥❣ t❛ ❝❤Ø r❛ ➤➢ỵ❝ r➺♥❣ E sup (Xn+2 (s) − Xn+1 (s))2 CP n hn (t, a), a s t hn (t, s) ợ ị ✭✶✳✶✮✳ ❚õ ✭✸✳✶✵✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ❈❤❡❜②❝❤❡✈ s✉② r❛ r➺♥❣ P sup (Xn+1 (t) − Xn (t)| a t T 2n C(4P )n hn (t, a) ❚❤❡♦ ❝➠♥❣ t❤ø❝ ❦❤❛✐ tr✐Ó♥ ❚❛②❧♦r s✉② r❛ ∞ (4P )n hn (T, a) = e2P (T, a) n=0 ✹✼ ✭✸✳✶✵✮ ➳♣ ❞ơ♥❣ ❇ỉ ➤Ị ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✈➭ ➤Þ♥❤ ❧ý ❲❡✐❡rstr❛ss✱ s✉② r❛ ỗ (Xn+1 (s) Xn (s)) xa + n=0 ❤é✐ tơ ➤Ị✉ ❤➬✉ ❝❤➽❝ ❝❤➽♥ ➤Õ♥ ♠ét q✉➳ trì X(t) ữ lim E sup Xn (t) − X(t) n→∞ ❚❛ ❝ã = a t T (X(t)) ❧➭ q✉➳ tr×♥❤ ❝❛❞❧❛❣✱ (Ft )− ♣❤ï ❤ỵ♣✳ ◆❣♦➭✐ r❛ f (·, X(·− )) ∈ L1 ([a, T ]; R) ❇➞② ❣✐ê✱ ❝❤ó♥❣ t❛ ❝❤Ø r❛ r➺♥❣ X(t) t❤á❛ ♠➲♥ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✹✮✳ ❚❛ ❝ã t t f (τ, Xn (τ− ))∇τ − E g(·, X(·− )) ∈ L2 ([a, T ]; M ) ✈➭ f (τ, X(τ− ))∇τ a a t g(τ, Xn (τ− ))∇Mτ − +E t a g(τ, X(τ− ))∇Mτ a T E(Xn (τ− ) − X(τ− ))2 ∇τ K(T − a) a T E(Xn (τ− ) − X(τ− ))2 ∇τ → + KN ❦❤✐ n → ∞, a ❝❤♦ n → ∞ tõ ✭✸✳✽✮✱ s✉② r❛ t X(t) = xa + t f (τ, X(τ− ))∇τ + a g(τ, X(τ− ))∇Mτ tr➟♥ a t T a ❙✉② r❛ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✸✳✶✳✷ ❝❤ó♥❣ t❛ ➤➲ ❝❤Ø r❛ r➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❧➷♣ P✐❝❛r❞ ❝❤♦ t❛ ❞➲② (Xn (t)) ❤é✐ tơ ✈Ị ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ị ý s ợ tố ộ ộ tụ ❝đ❛ ❞➲② ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✶✮✳ ✹✽ X(t) (Xn (t)) ❝đ❛ trì ề ệ X(t) ủ ị ý sư r➺♥❣ ❝➳❝ ❣✐➯ t❤✐Õt tr♦♥❣ ➜Þ♥❤ ❧ý ✸✳✶✳✷ ➤ó♥❣✳ ▲✃② X(t) ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✶✮ ✈➭ (Xn (t)) ❧➭ ❞➲② ❝ã ➤➢ỵ❝ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❧➷♣ P✐❝❛r❞ ①➳❝ ➤Þ♥❤ ❜ë✐ ✭✸✳✽✮✳ ❑❤✐ ➤ã✱ sup (Xn (t) − X(t))2 E CP n hn (T, a), ✭✸✳✶✶✮ a t T ✈í✐ ♠ä✐ n 1, tr♦♥❣ ➤ã C ✈➭ P ợ ị tr ứ ủ ị ý ✸✳✶✳✷✱ ♥❣❤Ü❛ ❧➭✱ C = 2K (T − a)2 + 4N (T − a) (1 + Ex2a ); P = 2K(T − a + 4N ) ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã t [f (τ, Xn−1 (τ− )) − f (τ, X(τ− ))]∇τ Xn (t) − X(t) = a t [g(τ, Xn−1 (τ− )) − g(τ, X(τ− ))]∇Mτ , + a s✉② r❛ t E sup (Xn (s) − X(s)) E(Xn−1 (τ− ) − X(τ− ))2 ∇τ 2K(T − a) a s t a t E(xn−1 (τ− ) − x(τ− ))2 ∇τ + 8KN a t E(Xn−1 (τ− ) − X(τ− ))2 ∇τ = 2K(T − a + 4N ) a t E(Xn−1 (τ− ) − X(τ− ))2 ∇τ =P a t E[ sup (Xn−1 (s) − X(s))2 ]∇τ P a a s τ− ❇➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ tr✉② ❤å✐✱ t❛ ❝ã ➤➢ỵ❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✸✳✶✶✮✳ ❱Ý ❞ơ ✸✳✶✳✹✳ ❳Ðt ♣❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤   d∇ X(t) = rX(t− )d∇ t + pX(t− )d∇ Mt ∀ t ∈ [a, T ]  X(a) = 1, ✹✾ ✭✸✳✶✷✮ tr♦♥❣ ➤ã r, p ❧➭ ❤❛✐ ❤➺♥❣ sè ✈➭ M ❧➭ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤ t❤á❛ ♠➲♥ ✭✸✳✷✮ ✈➭ ✭✸✳✸✮✳ ❘â r➭♥❣ ♣❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✸✳✶✷✮✳ ✭✸✳✶✷✮ ❝ã ❞✉② ♥❤✃t ♥❣❤✐Ư♠✳ ●✐➯ sư ◆Õ✉ tå♥ t➵✐ s ∈ Ta X(t) s❛♦ ❝❤♦ rν(s) + p∇∗ Ms = −1, t❤× ∇∗ X(s) = rX(s− )ν(s) + pX(s− )∇∗ Ms = −X(s− ), ➤✐Ò✉ ♥➭② s✉② r❛ X(t) = ∀ t ✈í✐ ❜✃t ❦ú X(s) = s s ∈ Ta ✳ ❉♦ tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤✱ ❉♦ ➤ã✱ ❝❤ó♥❣ t❛ ❝❤Ø ①Ðt tr➢ê♥❣ ❤ỵ♣ s✉② r❛ + rν(s) + p∇∗ Ms = ✳ ➜➷t 1 Zt := r(t − a) + pMt − p2 [M ]t + p2 2 (∇∗ Ms )2 , s∈(a,t] (1 + rν(s) + p∇∗ Ms )e−(rν(s)+p∇ Yt := ∗ Ms ) s∈(a,t] ❈❤ó♥❣ t❛ t❤✃② r➺♥❣ ➜➷t Zt ❧➭ s❡♠✐♠❛rt✐♥❣❛❧❡ ✈➭ Λs := {|rν(s) + p∇∗ Ms | ➤➻♥❣ t❤ø❝ 2} x2 | ln(1 + x) − x| Us2 ✈➭ Yt ❧➭ q trì (Ft ) ù ợ Us := (r(s) + p∇∗ Ms )1Λs ✈í✐ ♠ä✐ |x| ❙ư ❞ơ♥❣ ❜✃t ✈➭ ν(s) + p2 [M ]t , r s t s t s r ỗ (ln(1 + Us ) − Us ) , ζt := s∈(a,t] ❤é✐ tơ t✉②Ưt ➤è✐ ➤Õ♥ q✉➳ tr×♥❤ ❝ã ❜✐Õ♥ ♣❤➞♥ ❣✐í✐ ♥é✐✳ ❉♦ ➤ã✱ (1 + (rν(s) + p∇∗ Ms )1Λs ) e−(rν(s)+p∇ exp{ζt } = s∈(a,t] ✺✵ ∗ Ms )1Λs , ❝ị♥❣ ❝ã ❜✐Õ♥ ♣❤➞♥ ❣✐í✐ ♥é✐✳ ❱× ❤➵♥ ➤✐Ĩ♠ Yt ❙✉② r❛ s s❛♦ ❝❤♦ rt + pMt ❧➭ q trì ỉ tồ t ữ |r(s) + p Ms ()| > tr ỗ t t ỗ q trì ó ế ❣✐í✐ ♥é✐✳ ❈❤ó♥❣ t❛ ❝❤Ø r❛ r➺♥❣ ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ủ trì ị X(t) = e(r(t−a)+pMt − p [M ]t )+ 12 p2 a s t (∇ ∗ Ms )2 (1 + rν(s) + p∇∗ Ms )e−rν(s)−p∇ × ∗ Ms = Yt eZt ✭✸✳✶✸✮ a