❇é ●✐➳♦ ❉ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱Ü♥❤ ❍♦➭♥❣ ▼➠ ❤×♥❤ t❤ó ♠å✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥✲ ❉❡❛♥❣❡❧✐s ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ❇é ●✐➳♦ ❉ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱Ü♥❤ ❍♦➭♥❣ ▼➠ ❤×♥❤ t❤ó ♠å✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥✲ ❉❡❛♥❣❡❧✐s ❈❤✉②➟♥ ♥❣➭♥❤✿ ▲ý t❤✉②Õt ①➳❝ s✉✃t ✈➭ ❚❤è♥❣ ❦➟ t♦➳♥ ❤ä❝ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✻ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ❚❙✳ ◆❣✉②Ơ♥ ❚❤❛♥❤ ❉✐Ư✉ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ▼ô❝ ❧ô❝ ▼ô❝ ❧ô❝ ✐ ▼ë ➤➬✉ ✶ ✶ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✹ ✶✳✶ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➠♥❣ t❤ø❝ ■t➠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ✾ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼➠ ❤×♥❤ t❤ó ♠å✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥✲❉❡❆♥❣❡❧✐s ✶✸ ✷✳✶ ❚å♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ❞➢➡♥❣ ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✭✵✳✷✮ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ➛í❝ ❧➢ỵ♥❣ ♠♦♠❡♥t ❝đ❛ ♥❣❤✐Ư♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ➜✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ề ữ tệt ủ ủ ì ✳ ✳ ✳ ✳ ✷✷ ❑Õt ❧✉❐♥ ✷✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✷✾ ▼ë ➤➬✉ ❚r♦♥❣ t❤ù❝ tÕ ❤Ö s✐♥❤ t❤➳✐ ❝ã r✃t ♥❤✐Ị✉ t➢➡♥❣ t➳❝ ❣✐÷❛ ❝➳❝ ❧♦➭✐ ❝❤➻♥❣ ❤➵♥ ♥❤➢ q✉❛♥ ❤Ư ❝➵♥❤ tr❛♥❤ ✭✈Ị ❝❤ỉ ë✱ tứ q ệ tú q ệ t ỗ ỗ q ệ ú t ó tể ự ợ ì t ọ t ứ tr♦♥❣ t➭✐ ❧✐Ư✉ ❬✶✹❪ ❱♦❧t❡rr❛ ➤➲ ①Ðt ♠➠ ❤×♥❤ t❤ó ♠å✐ s❛✉ x˙ t = (a1 − b1 x(t) − c1 y(t))x(t) ✭✵✳✶✮ y˙ t = (−a2 − b2 y(t) + c2 x(t))y(t), tr♦♥❣ ➤ã a1 , a2 , b1 , b2 , c1 , c2 ❧➭ ❝➳❝ ❤➺♥❣ sè ❞➢➡♥❣✳ ❚r♦♥❣ ♠➠ ❤×♥❤ x(t) ✈➭ y(t) ❧➬♥ ❧➢ỵt ❧➭ ♠❐t ➤é ♠å✐ ✈➭ t❤ó t➵✐ t❤ê✐ ➤✐Ó♠ t✱ a1 ❧➭ tè❝ ➤é t➝♥❣ tr➢ë♥❣ ♥é✐ t➵✐ ❝đ❛ ♠å✐ ❦❤✐ ❦❤➠♥❣ ❝ã t❤ó✱ a2 ❧➭ tè❝ ➤é ❝❤Õt ❝đ❛ t❤ó ❦❤✐ ❦❤➠♥❣ ❝ã ♠å✐✱ b1 , b2 ❧➭ ❤Ö sè ❝➵♥❤ tr❛♥❤ tr♦♥❣ tõ♥❣ ❧♦➭✐✱ c1 , c2 ❤Ư sè ➯♥❤ ❤➢ë♥❣ ❝đ❛ ❧♦➭✐ ♥➭② ➤Õ♥ ❧♦➭✐ r ì t ộ ữ t➝♥❣ t❤❡♦ tè❝ ➤é t✉②Õ♥ tÝ♥❤✳ ◆❣❤Ü❛ ❧➭ t❤ó sÏ s➝♥ ♠å✐ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❞➵♥❣ c1 x(t)y(t) ✈➭ t➝♥❣ tr➢ë♥❣ ❝đ❛ ♥ã ✈í✐ tè❝ ➤é ❧➭ c2 x(t)y(t)✳ ◆❣➢ê✐ t❛ ➤➲ ❝❤Ø r❛ r➺♥❣ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✵✳✶✮ ỉ♥ ➤Þ♥❤ t✐Ư♠ ❝❐♥✳ ❙❛✉ ❦❤✐ ①✉✃t ❤✐Ư♥✱ ♠➠ ì ợ ề t q t ứ ❚r♦♥❣ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ♠➠ ❤×♥❤ ♥➭② ❝❤đ ②Õ✉ t tr ị ợ tr ì ị s➝♥ ❜ë✐ ➤➡♥ ✈Þ ❝♦♥ t❤ó tr♦♥❣ ➤➡♥ ✈Þ t❤ê✐ ❣✐❛♥ ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ➤➳♣ ø♥❣✳ ❍➭♠ ➤➳♣ ø♥❣ ❝ã t❤Ĩ ♣❤ơ t❤✉é❝ ✈➭♦ ♠❐t ➤é ❝đ❛ ♠å✐ ❤♦➷❝ ❝➯ t❤ó ✈➭ ♠å✐ ✈Ý ❞ơ ♥❤➢ ❤➭♠ ➤➳♣ ø♥❣ ❍♦❧❧✐♥❣ II x(t)y(t) x(t)y(t) α+βx(t) ✭①❡♠ ❬✶✷❪✮✱ tû ❧Ö ♣❤ô t❤✉é❝ β+γy(t) ✭①❡♠ ❬✶❪✮✳ ❚✉② ♥❤✐➟♥ ♠ét sè ♥❤➭ s✐♥❤ ❤ä❝ ➤➲ ❝❤Ø r❛ r➺♥❣ tr♦♥❣ ♠➠ ❤×♥❤ t❤ó ♠å✐ t❤× ❤➭♠ ➤➳♣ ø♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ❝➯ ♠❐t ➤é ❝đ❛ t❤ó ❧➱♥ ♠å✐ ✭①❡♠ ❬✶✱ ✽❪✮✳ ▼ét tr♦♥❣ ❝➳❝ ❤➭♠ ➤➳♣ ø♥❣ ❤❛② ➤➢ỵ❝ ①Ðt ➤Õ♥ ❧➭ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥ x(t)y(t) ❉❡❆♥❣❡❧✐s ❝ã ❞➵♥❣ α+βx(t)+γy(t) ❍➭♠ ➤➳♣ ø♥❣ ♥➭② ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ❜ë✐ ❇❡❞✲ ❞✐♥❣t♦♥ tr♦♥❣ ❬✷❪✱ ❉❡❆♥❣❡❧✐s ✈➭ ❝é♥❣ sù tr♦♥❣ ❬✸❪ ♠ét ❝➳❝❤ ➤é❝ ❧❐♣✳ ❙❛✉ ❦❤✐ ①✉✃t ❤✐Ö♥ ❤❛✐ ♥❣❤✐➟♥ ❝ø✉ tr➟♥ ❝ã ♥❤✐Ị✉ ♥❣❤✐➟♥ ❝ø✉ t➢➡♥❣ tù ❝❤♦ ♠➠ ❤×♥❤ t✃t ị ì tt ➤Þ♥❤ t❤➢ê♥❣ ❤✐Õ♠ ❣➷♣ tr♦♥❣ t❤ù❝ tÕ✳ ❚❤❛② ✈➭♦ ➤ã ❧➭ ♠➠ ❤×♥❤ ❝ã t➳❝ ➤é♥❣ ❜ë✐ ❝➳❝ ②Õ✉ tè ♥❣➱✉ ♥❤✐➟♥✳ ▼➠ ❤×♥❤ t❤ó ♠å✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥✲❉❡❆♥❣❡❧✐s ➤➢ỵ❝ ❚✳ ❱✳ ❚♦♥ ✈➭ ❝é♥❣ sù ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❬✶✸❪ ❝ã ❞➵♥❣ ♥❤➢ s❛✉ dx(t) = x(t) a1 − b1 x(t) − c1 y(t) dt + σ1 x(t)dW (t), α + βx(t) + γy(t) c2 x(t) dt + σ2 y(t)dW (t) dy(t) = y(t) − a2 − b2 y(t) + α + βx(t) + γy(t) ✭✵✳✷✮ tr♦♥❣ ➤ã , bi , ci , α, β, γ ❧➭ ❝➳❝ ❤➺♥❣ sè ❞➢➡♥❣✱ σi ❧➭ ❝➳❝ ❤➺♥❣ sè✱ W (t) ❧➭ ❝❤✉②Ó♥ ➤é♥❣ ❇r♦✇♥ ①➳❝ ➤Þ♥❤ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ➤➬② ➤đ (Ω, F, {Ft }t , P)✳ ▼➠ ❤×♥❤ ♥➭② ➤➢ỵ❝ ◆✳ ❍✳ ❉✉ ✈➭ ❝é♥❣ sù ♥❣❤✐➟♥ ❝ø✉ ✈➭ ♠ë ré♥❣ tr♦♥❣ ❬✹❪✳ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ❝đ❛ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ trì ự tr ế tứ ợ trì ❜➭② tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶✸❪ ✈➭ ❬✹❪✳ ❉ù❛ tr➟♥ ý t➢ë♥❣ ➤ã✱ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐ ✧▼➠ ❤×♥❤ t❤ó ♠å✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥✲❉❡❆♥❣❡❧✐s✧✳ ◆é✐ ❞✉♥❣ ❝đ❛ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶✳ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❧✉❐♥ ✈➝♥ t❐♣ tr✉♥❣ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ❝➡ ❜➯♥ ❝đ❛ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❝❤➢➡♥❣ s❛✉✳ ❈❤➢➡♥❣ ✷✳ ▼➠ ❤×♥❤ t❤ó ♠å✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ❤➭♠ ➤➳♣ ø♥❣ ❇❡❞❞✐♥❣t♦♥✲ ❉❡❆♥❣❡❧✐s✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ✈Ị ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ♥❣❤✐Ư♠ ❞➢➡♥❣ ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ trì ợ t ủ tr×♥❤✳ ◆❣♦➭✐ r❛✱ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ sù tå♥ t➵✐ ❜Ị♥ ữ tệt ủ ủ ì ũ ợ tr×♥❤ ❜➭② tr♦♥❣ ♠ơ❝ ✷✳✸ ❝đ❛ ❝❤➢➡♥❣ ♥➭②✳ ▲✉❐♥ ✈➝♥ ợ t t tì ❝❤✉ ➤➳♦ ❝đ❛ ❚❙✳ ◆❣✉②Ơ♥ ❚❤❛♥❤ ❉✐Ư✉✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ ❝❤➞♥ t❤➭♥❤ ♥❤✃t ➤Õ♥ t❤➬②✳ ◗✉❛ ➤➞② ❝ò♥❣ ①✐♥ ❝➯♠ ➡♥ ❝➳❝ t❤➬② ❝➠ tr♦♥❣ ♥❣➭♥❤ ❚♦➳♥ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ ✷ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤✱ ➤➷❝ ❜✐Öt ❧➭ ❝➳❝ t❤➬② ❝➠ tr♦♥❣ ❜é ♠➠♥ ▲ý t❤✉②Õt ❳➳❝ s✉✃t ❚❤è♥❣ ❦➟ ✈➭ ❚♦➳♥ ø♥❣ ❞ơ♥❣✳ ▼➷❝ ❞ï ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣✱ ♥❤➢♥❣ ❞♦ ❤➵♥ ❝❤Õ ✈Ị t❤ê✐ ❣✐❛♥ ❝ị♥❣ ♥❤➢ tr×♥❤ ➤é ♥➟♥ ❧✉❐♥ tr ỏ ữ tế sót rt ợ sù ❣ã♣ ý ❝ñ❛ q✉ý t❤➬② ❝➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ◆❣❤Ư ❆♥✱ t❤➳♥❣ ✵✽ ♥➝♠ ✷✵✶✼ ❚➳❝ ❣✐➯ ✸ ❈❤➢➡♥❣ ✶ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✳✶ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➠♥❣ t❤ø❝ ■t➠ ▲✃② ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈í✐ ❧ä❝ ❧➭ (Ω, F, P) ❦✐Ö♥ t❤➠♥❣ t❤➢ê♥❣ ✭tø❝ ❦✐Ö♥✿ s>t Fs ✈í✐ ♠ä✐ Ft = W = {W (t)}t t ✈➭ F0 t❤á❛ ♠➲♥ ➤✐Ò✉ σ ✲➤➵✐ sè ❝♦♥ ❝đ❛ F t❤á❛ ♠➲♥ ➤✐Ị✉ ❝❤ø❛ ♠ä✐ t❐♣ ❝ã ①➳❝ s✉✃t ✵✮✳ ▲✃② ❧➭ q✉➳ tr×♥❤ ❝❤✉②Ĩ♥ ➤é♥❣ ❇r♦✇♥ ①➳❝ ➤Þ♥❤ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ♣❤ï ợ ọ {Ft }t ị ĩ ▲✃② ❝➳❝ q✉➳ tr×♥❤ ❧➭ ❤ä t➝♥❣ ❝➳❝ {Ft }t {Ft }t b < ∞ a f = {f (t)}t ❑ý ❤✐Ư✉ ♥❤❐♥ ❣✐➳ trÞ t❤ù❝✱ M2 ([a, b]; R) ❧➭ ❦❤➠♥❣ ❣✐❛♥ (Ft )✲♣❤ï ❤ỵ♣ s❛♦ ❝❤♦ b f a,b |f (t)|2 dt < ∞ =E ✭✶✳✶✮ a ❈❤ó♥❣ t❛ ➤å♥❣ ♥❤✃t f ✈➭ f ❤ỵ♣ ♥➭② ❝❤ó♥❣ t❛ ♥ã✐ r➺♥❣ ❘â r➭♥❣✱ · tr➟♥ f M2 ([a, b]; R) ♥Õ✉ f − f ✈➭ f t➢➡♥❣ ➤➢➡♥❣ ✈➭ ✈✐Õt ❧➭ a,b ①➳❝ ➤Þ♥❤ ♠ét ♠❡tr✐❝ tr➟♥ ♠❡tr✐❝ ♥➭② ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ➤➬② ➤ñ✳ ỗ a,b = r trờ f = f M2 ([a, b]; R) ✈➭ ❦❤➠♥❣ ❣✐❛♥ f ∈ M2 ([a, b]; R) tå♥ t➵✐ ♠ét q✉➳ tr×♥❤ ➤♦ ➤➢ỵ❝ ❧ị② t✐Õ♥ f ∈ M2 ([a, b]; R) ❧➭ ❜➯♥ s❛♦ ❝ñ❛ f ✳ ➜➷t f (t) = lim sup h h↓0 ❑❤✐ ➤ã✱ f t f (s)ds t−h ❧➭ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ ➤♦➳♥ ✈➭ tỉ♥❣ q✉➳t ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❣✐➯ t❤✐Õt f = f ❉♦ ➤ã✱ ❦❤➠♥❣ ♠✃t tÝ♥❤ f ∈ M2 ([a, b]; R) q trì ị ♥❣❤Ü❛ ✶✳✶✳✷✳ ▼ét q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ g = {g(t)}a t b ợ ọ q trì ế tå♥ t➵✐ ♠ét ♣❤➞♥ ❤♦➵❝❤ ❝ñ❛ ➤♦➵♥ [a, b]✿ a = t0 < t1 < < tk = b k−1 ξi , ✈➭ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❜Þ ❝❤➷♥ i s❛♦ ❝❤♦ ξi ❧➭ Fti ✲➤♦ ➤➢ỵ❝ ✈➭ k−1 g(t) = ξ0 I[t0 ,t1 ] (t) + ξi I(ti ,ti+1 ] (t) ✭✶✳✷✮ i=1 ❑ý ❤✐Ö✉ t❐♣ ❝➳❝ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ➤➡♥ ❣✐➯♥ ❧➭ M0 ([a, b]; R) ❘â r➭♥❣ M0 ([a, b]; R) ⊂ M2 ([a, b]; R)✳ b ❘â r➭♥❣✱ a g(t)dWt ❧➭ Fb ✲➤♦ ➤➢ỵ❝ ❝❤ó♥❣ t❛ sÏ ❝❤Ø r❛ r➺♥❣ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ t❤✉é❝ ❧í♣ ❇ỉ ➤Ò ✶✳✶✳✸✳ ◆Õ✉ L2 (Ω, R) g ∈ M0 ([a, b]; R)✱ t❤× b g(t)dWt = E ✭✶✳✸✮ a b g(t)dWt E b |g(t)|2 dt =E a ✭✶✳✹✮ a ❚r♦♥❣ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✾❪ ➤➲ tr×♥❤ ❜➭② ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❇ỉ ➤Ị s❛✉✿ ❇ỉ ➤Ị ✶✳✶✳✹✳ ●✐➯ sö f ∈ M2 ([a, b]; R)✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞➲② q✉➳ tr×♥❤ ➤➡♥ ❣✐➯♥ {gn } s❛♦ ❝❤♦ b |f (t) − gn (t)|2 dt = lim E n→+∞ a ✺ ✭✶✳✺✮ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳ ●✐➯ sư ✭✶✳✷✮ tr♦♥❣ g ❧➭ ♠ét q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ➤➡♥ ❣✐➯♥ ①➳❝ ➤Þ♥❤ ❜ë✐ M0 ([a, b]; R) ❑❤✐ ➤ã tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ❝đ❛ g ❝❤✉②Ĩ♥ ➤é♥❣ ❇r♦✇♥ {W (t)} ợ ị s k1 b i (Wti+1 Wti ) g(t)dW (t) = a ỗ ❣✐➯♥ ➤è✐ ✈í✐ q✉➳ tr×♥❤ ✭✶✳✻✮ i=0 f ∈ M2 ([a, b]; R)✱ t❤❡♦ ❇ỉ ➤Ị ✶✳✶✳✹✱ tå♥ t➵✐ ❞➲② q✉➳ tr×♥❤ ➤➡♥ {gn } s❛♦ ❝❤♦ b |g(t) − gn (t)|2 dt = lim E n→∞ a ❚❛ ❝ã✱ b gn (t)dW (t) − E b [gn (t) − gm (t)]dW (t) gm (t)dW (t) = E a b a a b =E |gn (t) − gm (t)|2 dt → ❦❤✐ m, n → ∞ a ❚ø❝ ❧➭✱ b a gn (t)dW (t) ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ L2 (Ω; R)✳ ❉♦ ➤ã✱ tå♥ t➵✐ ❣✐í✐ ❤➵♥ ❝đ❛ ♥ã ✈➭ ❝❤ó♥❣ t❛ ①❡♠ ❣✐í✐ ❤➵♥ ➤ã ♥❤➢ ❧➭ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ■t➠✳ ❈❤ó♥❣ t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✻✳ ●✐➯ sư f f ∈ M2 ([a, b]; R)✱ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ❝ñ❛ q✉➳ tr×♥❤ t❤❡♦ q✉➳ tr×♥❤ ❝❤✉②Ĩ♥ ➤é♥❣ ❇r♦✇♥ {W (t)} tr➟♥ [a, b]✱ ❦ý ❤✐Ö✉ ❧➭ b a f (t)dW (t) ợ ị b b f (t)dW (t) = L2 − lim n→∞ a tr♦♥❣ ➤ã gn (t)dW (t), ✭✶✳✼✮ a {gn } ❧➭ ❞➲② ❝➳❝ q✉➳ tr×♥❤ t❤✉é❝ M0 ([a, b]; R) s❛♦ ❝❤♦ b |f (t) − gn (t)|2 dt = lim E n→∞ a ✻ ✭✶✳✽✮ ➜Þ♥❤ ♥❣❤Ü❛ tr➟♥ ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ❝➳❝❤ ❝❤ä♥ ❞➲② {gn }✳ ❈➳❝ tÝ♥❤ ❝❤✃t s❛✉ ➤➞② ủ tí ợ trì ứ ❝❤✐ t✐Õt tr♦♥❣ ❬✾❪✳ ➜Þ♥❤ ❧ý ✶✳✶✳✼✳ ●✐➯ sư ✐✮ b a f (t)dW (t) ❧➭ ✐✐✮ E ✐✐✐✮ E ✐✈✮ b a f (t)dW (t) f, g ∈ M2 ([a, b]; R) ✈➭ α, β Fb ✲➤♦ ➤➢ỵ❝❀ = 0; b f (t)dW (t) a b a [αf (t) ▲✃② =E b a |f (t)| dt; + βg(t)]dW (t) = α a ✈➭ f ∈ M2 ([0, T ]; R) ❱í✐ M2 ([a, b]; R) ❞♦ ➤ã ❧➭ ❤❛✐ sè t❤ù❝ ❑❤✐ ➤ã c f (t)dW (t) = b a