动态金融风险度量,稳型中心极限定理和G-Brown运动

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2009年4月23日16:20,彭实戈院士在厦门大学克立楼三楼报告厅做演讲,题目是动态金融风险度量,稳型中心极限定理和G-Brown运动。

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动态金融风险度量,稳型中心极限定理和G-Brown运动

  1. 1. Ä7KºxÝþ! ­è.¥%4½nÚG-ÙK$Ä 6¢{ ìÀŒÆêÆÆ ¥sêƬ2009c¬,ú¯§w 2009c4'23Fuf€ŒÆŽá¢ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 1/ () 37
  2. 2. Ä7KºxÝþ! ­è.¥%4½nÚG-ÙK$Ä 6¢{ ìÀŒÆêÆÆ ¥sêƬ2009c¬,ú¯§w 2009c4'23Fuf€ŒÆŽá¢ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 2/ () 37
  3. 3. ²Y'‡È©Ú‡©§nØ ”û½Ø”'BXQ‰ÆFÏÓÚ£G § ƒA'êÆNXÒ´© uÚî-4ÙZ]'‡È©Ú‡©§nØquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 3/ () 37
  4. 4. ²Y'‡È©Ú‡©§nØ ”û½Ø”'BXQ‰ÆFÏÓÚ£G § ƒA'êÆNXÒ´© uÚî-4ÙZ]'‡È©Ú‡©§nØquot; ‘Åy–µéu5'êÆóä'‡¦ ²vÏ'¢§‚@£ §·‚'­F)Ÿþ´‘Å'§Ù¥?? ¿÷XØ(½S§??¿÷X‘Åy–quot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 3/ () 37
  5. 5. ‘Åy–Ä:êÆnصVÇӆ‘Å©Û VÇÓ'nØÄ: Kolmogorov (1933c¤5VÇnØ'Ä:6µïá y“VÇnØ'ú nÄ:§ú@´y“VÇnØ'gĊquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 4/ () 37
  6. 6. ‘Åy–Ä:êÆnصVÇӆ‘Å©Û VÇÓ'nØÄ: Kolmogorov (1933c¤5VÇnØ'Ä:6µïá y“VÇnØ'ú nÄ:§ú@´y“VÇnØ'gĊquot; ‘Åv§'êÆnØ Brown6Ä£1900cµL.Bachelier, 1905cµA. Einstein, 1926cWiener¤; Markovv§nØ£{s!€é!{s9٦Ơ'ÏRý'íc¤ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 4/ () 37
  7. 7. ‘Åy–Ä:êÆnصVÇӆ‘Å©Û VÇÓ'nØÄ: Kolmogorov (1933c¤5VÇnØ'Ä:6µïá y“VÇnØ'ú nÄ:§ú@´y“VÇnØ'gĊquot; ‘Åv§'êÆnØ Brown6Ä£1900cµL.Bachelier, 1905cµA. Einstein, 1926cWiener¤; Markovv§nØ£{s!€é!{s9٦Ơ'ÏRý'íc¤ ‘Å©Û—-‘Å(s¥'Úî½Æ Itˆ o “OÚî-4ÙZ]'NX§é‘Åy–cI½þ©ÛÚïÄ'­ ‡'êÆóäÒ´ItˆmM'‘ŇȩڑŇ©§nاù‡nØ o Q¼êÆWalføžWolf™”‘Å(s¥'Úî½Æ”quot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 4/ () 37
  8. 8. ‘Ň©§ÚItˆúª£1942¤ Itˆ o o ‘Ň©§ Itˆ o dx (t) = b(x (t))dt + σ(x (t))dB(t), t ≥ t0 x (t0 ) = x0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 5/ () 37
  9. 9. ‘Ň©§ÚItˆúª£1942¤ Itˆ o o ‘Ň©§ Itˆ o dx (t) = b(x (t))dt + σ(x (t))dB(t), t ≥ t0 x (t0 ) = x0 úª Itˆ o σ2 du(x (t), t) = (ut + uxx + bux )(x (t), t)dt + σux (x (t), t)dBt 2 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 5/ () 37
  10. 10. ‘Ň©§ÚItˆúª£1942¤ Itˆ o o ‘Ň©§ Itˆ o dx (t) = b(x (t))dt + σ(x (t))dB(t), t ≥ t0 x (t0 ) = x0 úª Itˆ o σ2 du(x (t), t) = (ut + uxx + bux )(x (t), t)dt + σux (x (t), t)dBt 2 ‘Å©ÛmV y“‘Å©Û'5V£µ´»‘Å©Ûž“ Itˆ o ùTЊ‘ ÔnÆ'Ø(½S'€·S'g€£ÿØO¦n¤ ÚͶÔnÆ[¤ùJÑ'þfåÆ'´»©ÛnØ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 5/ () 37
  11. 11. ‘Ň©§nØ­‡quot;õŸþ´• Itˆ o •~‡©§£ ¤ t0 ≤ t ≤ T dx (t) = b(x (t))dt, x (t0 ) = x0 (Щ^‡) $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 6/ () 37
  12. 12. ‘Ň©§nØ­‡quot;õŸþ´• Itˆ o •~‡©§£ ¤ t0 ≤ t ≤ T dx (t) = b(x (t))dt, x (t0 ) = x0 (Щ^‡) •~‡©§ dx (t) = b(x (t))dt, x (T ) = xT (ªà^‡) $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 6/ () 37
  13. 13. ‘Ň©§nØ­‡quot;õŸþ´• Itˆ o •~‡©§£ ¤ t0 ≤ t ≤ T dx (t) = b(x (t))dt, x (t0 ) = x0 (Щ^‡) •~‡©§ dx (t) = b(x (t))dt, x (T ) = xT (ªà^‡) éì: Itˆ‘Ň©§£U½ÂЩ^‡¯K¤µ o dx (t) = b(x (t))dt + σ(x (t))dB(t), t ≥ t0 x (t0 ) = x0 T t $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 6/ () 37
  14. 14. ‘Ň©§nØ­‡quot;õŸþ´• Itˆ o •~‡©§£ ¤ t0 ≤ t ≤ T dx (t) = b(x (t))dt, x (t0 ) = x0 (Щ^‡) •~‡©§ dx (t) = b(x (t))dt, x (T ) = xT (ªà^‡) éì: Itˆ‘Ň©§£U½ÂЩ^‡¯K¤µ o dx (t) = b(x (t))dt + σ(x (t))dB(t), t ≥ t0 x (t0 ) = x0 ¢Sþ,QUK'û)y ½d, UK#¬'ºx'ŒþOŽ¥¯K´ •'‘Å,ïÄJb 1900c(L. Bachelier) $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 6/ () 37
  15. 15. •‘Ň©§Ä½n: 3˜5½n[Pardoux,Peng1990],'½n[Peng1991] $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 7/ () 37
  16. 16. •‘Ň©§Ä½n: 3˜5½n[Pardoux,Peng1990],'½n[Peng1991] •‘Ň©§— kü‡™v§y(t)Úz(t) dy (t) = −g (t, y (t), z(t))dt − z(t)dB(t), t ≤ T, y (T ) = ξ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 7/ () 37
  17. 17. •‘Ň©§Ä½n: 3˜5½n[Pardoux,Peng1990],'½n[Peng1991] •‘Ň©§— kü‡™v§y(t)Úz(t) dy (t) = −g (t, y (t), z(t))dt − z(t)dB(t), t ≤ T, y (T ) = ξ éì: •‘Ň©§([Itˆ,1942])µ o dx (t) = b(x (t))dt + σ(x (t))dB(t), t ≥ t0 x (t0 ) = x0 ¡Óƒ?µy(t),z(t)Úx(t)Ñ´ÙK´»BQt±c'¼ê: y (t) = y (t, (B(s))0≤s≤t ), · · · $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 7/ () 37
  18. 18. Ôn¿Âµ 1. du·‚QtžBÿ 'B´»(B(s)) §ØU#T žŸo Šy(T) = ξ((B(s)) )¬u)ŸoŠ,k žT·‚âU 0≤s≤t #quot;¢´Œ±OŽÑ˜é‘Åv§(y(t),z(t))§AyGUQz˜‡tž 0≤s≤T OŽÑz˜‡(y(t),z(t))'Šquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 8/ () 37
  19. 19. Ôn¿Âµ 1. du·‚QtžBÿ 'B´»(B(s)) §ØU#T žŸo Šy(T) = ξ((B(s)) )¬u)ŸoŠ,k žT·‚âU 0≤s≤t #quot;¢´Œ±OŽÑ˜é‘Åv§(y(t),z(t))§AyGUQz˜‡tž 0≤s≤T OŽÑz˜‡(y(t),z(t))'Šquot; 2. ²LÆ[¡ù'v§/E›0µQ˜‡A½'ÄýŽ!åe(g K])Œ¨i˜‡Ý]üѧE›Ñ'y(T) = ξQ¨cžt ž„´‘ Ågþ§† S½žTžâg®þquot; 0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 8/ () 37
  20. 20. û)7K¬½d¯K[El Karoui, Peng, Quenez1997] g[Bachelier1900]±5§UKÆ[ϗåuAûϽdù˜‡ •'‘ůKquot; Black-Scholes-Mertonu1973c¼'Ïd‚§§ ¢Sþ´˜‡AÏ'‚S•‘Ň©§µ dy (t) = −[ry (t) + σ−1 (µ − r )z(t)]dt + z(t)dB(t), y (T ) = ξ §'AҴͶ'Black-Scholesúª£Scholes-MertonÏ ¼1997cì ²LÆø¤,¦‚?n'´š~nŽ'‚¸e'Ïd‚'E›¯ K§˜„'^‡e'Ïd‚us‡^š‚S'•‘Ň©§5 £ã§X dy (t) = −[ry (t) + σ−1 (µ − r )z(t) − (R − r )σ−1 z(t)]dt + z(t)dB(t), y (T ) = ξ ÙE›Úš@|'Šó§éAu•‘Ň©§A'QS!˜ S!Ú9quot;½nquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚG-ÙK$23Fuf€ŒÆŽá¢ 2009c 4 F Ùc Ä c Ä ¯§w ú 9/ () 37
  21. 21. uyÚy²š‚5¤ù-k]£Feynman-Kac¤úª ÄuéþfåƥͶ'Feynman´»nØ'ïħêÆ[Kac Q1951 c¼ VÇ؆‚S0 ‡©§9X'Ͷ'Feynman-Kac ú ª§§¤y“VÇØ'˜‡­‡'Ä:S¤t§´´Í¶ 'Monte-Carlo {OŽ ‡©§A'nØÄ:quot; Feynman-Kacúª Xe'‚SÔG ‡©§'A 1 ∂t u + ∆u + c(x )u + h(x ) = 0, u|t=T = Φ 2 '‘Åv«´µ T s T c(B x (r ))dr c(B x (r ))dr Φ(B(T )x )] h(B x (s))e u(x, t) = E [ ds + e t t t $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 10 / () 37
  22. 22. ˜‡š~Ä:¢´ϱ5cÐ$¢'êƯK´µ Feynman-Kac úªUØUíP š‚Sº $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 11 / () 37
  23. 23. ˜‡š~Ä:¢´ϱ5cÐ$¢'êƯK´µ Feynman-Kac úªUØUíP š‚Sº ˜‡Ñ¿GuyÚy²: ˜Œa0Œ‚S‚S ‡©§'AŒ±Ïv•‘Ň©§'A 5v«§ قSœ¹Ò´Feynman-Kac úªquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 11 / () 37
  24. 24. š‚5Feynman-Kacúª[Peng1991] Xe'š‚S‚SÔG ‡©§'A 1 ∂t u + ∆u + g (x, u, u) = 0, u|t=T = Φ 2 'VÇv«´u(B x (s), s) = y (s), u(B x (s), s) = z(s), $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 12 / () 37
  25. 25. š‚5Feynman-Kacúª[Peng1991] Xe'š‚S‚SÔG ‡©§'A 1 ∂t u + ∆u + g (x, u, u) = 0, u|t=T = Φ 2 'VÇv«´u(B x (s), s) = y (s), u(B x (s), s) = z(s), Ù¥(y(s),z(s))´±e•‘Ň©§'Aµ dy (s) = −g (B x (s), y (s), z(s))ds + z(s)dB x (s), y (T ) = Φ(B x (T )). AyGkµu(x,t) = y (t). ‚S¤ù¨k]úªg(s,y,z) = c(B (s))y + h(B (s)). x x $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 12 / () 37
  26. 26. š‚5Feynman-Kacúª[Peng1991] Xe'š‚S‚SÔG ‡©§'A 1 ∂t u + ∆u + g (x, u, u) = 0, u|t=T = Φ 2 'VÇv«´u(B x (s), s) = y (s), u(B x (s), s) = z(s), Ù¥(y(s),z(s))´±e•‘Ň©§'Aµ dy (s) = −g (B x (s), y (s), z(s))ds + z(s)dB x (s), y (T ) = Φ(B x (T )). AyGkµu(x,t) = y (t). ‚S¤ù¨k]úªg(s,y,z) = c(B (s))y + h(B (s)). x x ·‚¼ ƒA'ýG§'v« $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 12 / () 37
  27. 27. ù‡Ñ¿GuyÚy²k '¹Âµ ˜‡•‘Ň©§¢Sþ´˜«´»6' ‡©§: ´»˜m´[0,T]þ'뉴»'NµΩ = C [0, T ] $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 13 / () 37
  28. 28. ù‡Ñ¿GuyÚy²k '¹Âµ ˜‡•‘Ň©§¢Sþ´˜«´»6' ‡©§: ´»˜m´[0,T]þ'뉴»'NµΩ = C [0, T ] ‰½ Tž'ªà^‡y(T,ω) = ξ(ω)£´»'¼ê¤ ÙAéuz˜‡'½'t,Ñ´´»'¼êy(t,ω) : Ω → Rquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 13 / () 37
  29. 29. ù‡Ñ¿GuyÚy²k '¹Âµ ˜‡•‘Ň©§¢Sþ´˜«´»6' ‡©§: ´»˜m´[0,T]þ'뉴»'NµΩ = C [0, T ] ‰½ Tž'ªà^‡y(T,ω) = ξ(ω)£´»'¼ê¤ ÙAéuz˜‡'½'t,Ñ´´»'¼êy(t,ω) : Ω → Rquot; AyG ¨ªà^‡ÚÙ¦XêÑ´G'¼êž§•‘Ň©§Òg¤  ‡©§ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 13 / () 37
  30. 30. UK¥kŒþ'´»6'©ÛÚOŽ duQy“UK½|¥kˆaGGÚÚ'´»6'ϧƒA'Ï ½d¯K¤éA'•‘Ň©§Ò´ùa´»6' ‡©§ , •N‡,v—PTŒ‘g:_®Re¹zf/“ï_—Oš“Vv—PO_®Re¹z ! $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 14 / () 37
  31. 31. UK¥kŒþ'´»6'©ÛÚOŽ duQy“UK½|¥kˆaGGÚÚ'´»6'ϧƒA'Ï ½d¯K¤éA'•‘Ň©§Ò´ùa´»6' ‡©§ ˜‡k'¯K: ÔnÆ!åÆ­F´ÄQù«y–º $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 14 / () 37
  32. 32. UK¥kŒþ'´»6'©ÛÚOŽ duQy“UK½|¥kˆaGGÚÚ'´»6'ϧƒA'Ï ½d¯K¤éA'•‘Ň©§Ò´ùa´»6' ‡©§ ˜‡k'¯K: ÔnÆ!åÆ­F´ÄQù«y–º Úu '9uPDEÚBSDE'5'êŠOŽ{'ïÄ (gA‡ïÄ¢|¼ pu10‘'PDE'Monte-CarloêŠOŽ{ Ú§ƒ) $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 14 / () 37
  33. 33. UK¥kŒþ'´»6'©ÛÚOŽ duQy“UK½|¥kˆaGGÚÚ'´»6'ϧƒA'Ï ½d¯K¤éA'•‘Ň©§Ò´ùa´»6' ‡©§ ˜‡k'¯K: ÔnÆ!åÆ­F´ÄQù«y–º Úu '9uPDEÚBSDE'5'êŠOŽ{'ïÄ (gA‡ïÄ¢|¼ pu10‘'PDE'Monte-CarloêŠOŽ{ Ú§ƒ) •‘Ň©§nØ ®²¤VÇØ¥'­‡'ïЕ§UKêÆ¥­‡'ïÄ!OŽó äquot; gQ{sލ'IÊQ•‘Ň©§Æâï?¬§qÑy ƒ¨ õ­‡ïĤtquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 14 / () 37
  34. 34. êƌ“…#xâÅ1933cïáy“VÇØ ®PA^ ØÓ'+ ù‡nØ')Ÿ´µêÆÏ4´‚S'quot; ŽÑ‚SÏ4QAº²L y–ž'Øv§kxõ²LÆ[†êÆ[—åuï̂SêÆÏ 4§XͶêÆ[!³Ø;[Choquet ²½Âvš‚SVÇ £=Choquet NݤÚChoquet Ï4§¢Q½Ât ž®8Ee'^ ‡Ï4ž‘ )Ÿ(Jquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 15 / () 37
  35. 35. êƌ“…#xâÅ1933cïáy“VÇØ ®PA^ ØÓ'+ ù‡nØ')Ÿ´µêÆÏ4´‚S'quot; ŽÑ‚SÏ4QAº²L y–ž'Øv§kxõ²LÆ[†êÆ[—åuï̂SêÆÏ 4§XͶêÆ[!³Ø;[Choquet ²½Âvš‚SVÇ £=Choquet NݤÚChoquet Ï4§¢Q½Ât ž®8Ee'^ ‡Ï4ž‘ )Ÿ(Jquot; g¨Ï49^‡g-Ï4[Peng1997]–̂SêÆÏ4nØÄ: Äk¦A dy (t) = −g (t, y (t), z(t))dt + z(t)dB(t), t ≤ T, y (T ) = ξ Eg [ξ] := y (0) $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 15 / () 37
  36. 36. I˜‡Äš‚SêÆÏ4 gs©Æöuy§g-Ï4´OŽ”ºxÿÝ”ÚcIš‚SÚO©Û' ˜‡­‡óäquot; ·‚y² ˜‡š~k'@ t:£[Delbaen,Peng,Rosazza2007](Ω,F,P)¥'˜‡ë‰'ăN' ºxÝþ§˜½´g¨Ï4§ùv²g¨Ï4´˜‡Ä:S'­‡Vgquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 16 / () 37
  37. 37. I˜‡Äš‚SêÆÏ4 gs©Æöuy§g-Ï4´OŽ”ºxÿÝ”ÚcIš‚SÚO©Û' ˜‡­‡óäquot; ·‚y² ˜‡š~k'@ t:£[Delbaen,Peng,Rosazza2007](Ω,F,P)¥'˜‡ë‰'ăN' ºxÝþ§˜½´g¨Ï4§ùv²g¨Ï4´˜‡Ä:S'­‡Vgquot; ¢´volatility uncertainty¦·‚Ã{QDÚ'Vǘm(Ω, F , P)¥½ ÂÚ?nÄUKºxÝþ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 16 / () 37
  38. 38. g‚5êÆÏquot;Eµ rх#xâÅ¿Âe²; Vǘm üxS(Monotonicity): (a) E[X ] ≥ E[Y ] ˆ ˆ X ≥Y =⇒ ¢~êS(Constant preserving): (b) E[c] = c. ˆ gŒS: ∀X , Y ∈ H, (c) E[X + Y ] ≤ E[X ] + E[Y ]. ˆ ˆ ˆ àgS(Positive homogeneity): (d) E[λX ] = λE[X ], ˆ ˆ ∀λ ≥ 0. ¨Xv«¢´'›”Þ€£‘Ågþ¤ž§E[X ]Ò¤˜«ƒNº ˆ $¢xÝþ Æ ¥IêƬ c¬ ú¯§w c Fu f € Œ Æ Ž á ¢ { ìÀŒÆêÆ Ä,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä G-Ù K $ Ù 23F 17 / 2009c () 37
  39. 39. Sublinear Expectation, from [Knight 1921], [Keynes 1936] Knight, F.H. (1921), Risk, Uncertainty, and Profit ”Mathematical, or a priori, type of probability is practically never met with in business ... business decisions, for example, deal with situations which are far too unique, generally speaking,, for any sort of statistical tabulation to have any value for guidance ... (so that) the concept of an objectively measurable probability or chance is simply inapplicable .” $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 18 / () 37
  40. 40. Sublinear Expectation, from [Knight 1921], [Keynes 1936] Knight, F.H. (1921), Risk, Uncertainty, and Profit ”Mathematical, or a priori, type of probability is practically never met with in business ... business decisions, for example, deal with situations which are far too unique, generally speaking,, for any sort of statistical tabulation to have any value for guidance ... (so that) the concept of an objectively measurable probability or chance is simply inapplicable .” The framework of sublinear expectation can take the uncertainty into consideration, in a systematic, beautiful and robust way. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 18 / () 37
  41. 41. g‚5êÆÏquot;§¡µ Upper expectation [P. Huber 1987, P. Huber Strassen 1973]); Coherent expectation, coherent prevision [P. Walley, 1991]; Choquet expectation in potential theory [Choquet 1953] is also a type of sublinear expectation; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 19 / () 37
  42. 42. g‚5êÆÏquot;§¡µ Upper expectation [P. Huber 1987, P. Huber Strassen 1973]); Coherent expectation, coherent prevision [P. Walley, 1991]; Choquet expectation in potential theory [Choquet 1953] is also a type of sublinear expectation; d©ÙCoherent risk measure [ADEH1997,1999]Úu 9uºxÝþ'­ ŒØ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 19 / () 37
  43. 43. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 19 / () 37
  44. 44. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 19 / () 37
  45. 45. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 20 / () 37
  46. 46. g‚5Ïquot;­èL«½n [Huber, 1973, 1981], [Artzner, Delbean, Eber Heath, 1997,1999], [F¨llmer Schied, 2004] o Theorem ´½Â3(Ω,H)þ'g‚5Ïquot; E[·] ˆ ¨…=¨ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 20 / () 37
  47. 47. g‚5Ïquot;­èL«½n [Huber, 1973, 1981], [Artzner, Delbean, Eber Heath, 1997,1999], [F¨llmer Schied, 2004] o Theorem ´½Â3(Ω,H)þ'g‚5Ïquot; E[·] ˆ ¨…=¨ 3˜‡(kŒ')VÇ'8Ü{P , θ ∈ Θ} ¦eãL«¤áµ θ E[X ] = sup EPθ [X ] = sup ˆ ∀X ∈ H. X (ω)dPθ (ω), θ∈Θ Ω θ∈Θ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 20 / () 37
  48. 48. g‚5Ïquot;­èL«½n [Huber, 1973, 1981], [Artzner, Delbean, Eber Heath, 1997,1999], [F¨llmer Schied, 2004] o Theorem ´½Â3(Ω,H)þ'g‚5Ïquot; E[·] ˆ ¨…=¨ 3˜‡(kŒ')VÇ'8Ü{P , θ ∈ Θ} ¦eãL«¤áµ θ E[X ] = sup EPθ [X ] = sup ˆ ∀X ∈ H. X (ω)dPθ (ω), θ∈Θ Ω θ∈Θ ‡NQ @£Ø¯Kþ û½Ø'BX =⇒ VÇØ'BX£X´‘Å'§¢ÙVÇ©Ù´(½'¤ =⇒ g‚SÏ4'BX£X´‘Å'§ÙVǩُäkØ(½S¤ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 20 / () 37
  49. 49. ¥%4½n $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 21 / () 37
  50. 50. ¥%4½n Y¥74½n ´VÇÚO¥'˜‡ P q›©9…'@t§de Moivre, Laplace, Kolmogorov, · · · , $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 21 / () 37
  51. 51. ¥%4½n Y¥74½n ´VÇÚO¥'˜‡ P q›©9…'@t§de Moivre, Laplace, Kolmogorov, · · · , Y¥74½n'¿Â ÕáөّÅgþƒ''ÚU©ÙÂñu©Ù§ØC‘Å gþ¦5´'x'©Ù: ù˜X´y“UKnØÚ¢‚¥Œþ¦^“ F”'̇â $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 21 / () 37
  52. 52. ¥%4½n Y¥74½n ´VÇÚO¥'˜‡ P q›©9…'@t§de Moivre, Laplace, Kolmogorov, · · · , Y¥74½n'¿Â ÕáөّÅgþƒ''ÚU©ÙÂñu©Ù§ØC‘Å gþ¦5´'x'©Ù: ù˜X´y“UKnØÚ¢‚¥Œþ¦^“ F”'̇â ¥74½n X1 + · · · + Xn E[ϕ( )] → E[ϕ(ξ)], ξ ∼ N (0, σ2 ), √ n E[X1 ] = 0, σ2 = E[X1 ] 2 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 21 / () 37
  53. 53. g‚5Ïquot;e¥%4½n[Peng2007-2008] Theorem (Ω,H, E)‘ÅCþS{X } Ñ´Ó©Ù': X ∼ X . ∞ ˆ 1 i i =1 i $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 22 / () 37
  54. 54. g‚5Ïquot;e¥%4½n[Peng2007-2008] Theorem (Ω,H, E)‘ÅCþS{X } Ñ´Ó©Ù': X ∼ X . ∞ ˆ …z˜‡X ÑÕáu (X , · · · , X ) i = 1, 2, · · · . qX þeþ 1 i i =1 i ŠÑquot;: i +1 1 1 i E[X1 ] = E[−X1 ] = 0. ˆ ˆ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 22 / () 37
  55. 55. g‚5Ïquot;e¥%4½n[Peng2007-2008] Theorem (Ω,H, E)‘ÅCþS{X } Ñ´Ó©Ù': X ∼ X .∞ ˆ …z˜‡X ÑÕáu (X , · · · , X ) i = 1, 2, · · · . qX þeþ 1 i i =1 i ŠÑquot;: i +1 1 1 i E[X1 ] = E[−X1 ] = 0. ˆ ˆ KS U©ÙÂñuN(0,[σ , σ ])µ∀ϕ ∈ C 2 2 = X1 + · · · + Xn (R), n unif Sn lim E[ϕ( √ )] = E[ϕ(X )], X ∼ N (0, [σ2 , σ2 ]). ˆ ˜ (CLT) n n→∞ Ù¥σ = E[X1 ], σ2 = −E[−X1 ] 0. ˆ2 ˆ 2 2 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 22 / () 37
  56. 56. g‚5Ïquot;e©Ù(½pd©Ù): ½ N (0, [σ2 , σ2 ]) Definition g‚SÏ4˜m(Ω,H, E)¥§¡˜‡‘ÅgþX '© ˆ (ðŽd‰, L´vy) e ُN(0,[σ PX ∼ N (0, [σ , σ ])) Xtéuz˜‡†XÕá 2 , σ2 ])', 2 2 Ó©Ù'YÑk: a2 + b 2 X . aX + bY ∼ Ù¥: σ2 := E[X 2 ], σ2 = −E[−X 2 ]. ˆ ˆ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 23 / () 37
  57. 57. G-©Ù éz˜‡à¼ê ϕ, à ∞ y2 1 E[ϕ(X )] = √ ˆ ϕ(y ) exp(− )dy 2σ2 2πσ2 −∞ éz˜‡]¼ê ϕ, we have, ] ∞ y2 1 E[ϕ(X )] = ˆ ϕ(y ) exp(− )dy 2σ2 2πσ2 −∞ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 24 / () 37
  58. 58. G-©Ù éz˜‡à¼ê ϕ, à Black-Scholes Ql_v—nåu( ∞ y2 1 E[ϕ(X )] = √ ˆ ϕ(y ) exp(− )dy 2σ2 2πσ2 −∞ éz˜‡]¼ê ϕ, we have, ] ∞ y2 1 E[ϕ(X )] = ˆ ϕ(y ) exp(− )dy 2σ2 2πσ2 −∞ ù‡¥74½nAºµŸoUKŨ Œ±QFš~Ø(½'œ¹e ?^¥74½ncIºxÝþÚ½ d $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 24 / () 37
  59. 59. ¡N(0,[σ , σ ])G ©Ù§Ù¥G£)¤¼ê¤d 2 2 eª½Âµ G (α) = E[αX 2 ], ∀α ∈ R ˆ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 25 / () 37
  60. 60. Œ±y²§deª½Â'¼êµ ´d±e √ u(t, x ) := E[ϕ(x + ˆ tX )] š‚S ‡©§'A ∂t u = G (∂2 u), u|t=0 = ϕ xx Ù¥ G-píe¹z 1 G (α) := E[αX 2 ] = [σ2 α+ − σ2 α− ] ˆ 2 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 26 / () 37
  61. 61. If σ2 = σ2 , then N (0, [σ2 , σ2 ]) = N (0, σ2 ) Definition. If ξ ∼ N (0, [σ2 , σ2 ]), then ξ + x ∼ N (x, [σ2 , σ2 ]) $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 27 / () 37
  62. 62. G–ÙK$Ä[Peng2005] Definition Qg‚SÏ4˜m(Ω,F, E)¥§¡˜‡‘Åv§B (ω) = ω , t ≥ 0 ˆ G–ÙK6Ä motion Xt§÷v±e^‡: Ù t t (i) Bt+s − Bs is N (0, [σ2 t, σ2 t]) distributed ∀ s, t ≥ 0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 28 / () 37
  63. 63. G–ÙK$Ä[Peng2005] Definition Qg‚SÏ4˜m(Ω,F, E)¥§¡˜‡‘Åv§B (ω) = ω , t ≥ 0 ˆ G–ÙK6Ä motion Xt§÷v±e^‡: Ù t t (i) Bt+s − Bs is N (0, [σ2 t, σ2 t]) distributed ∀ s, t ≥ 0 (ii) For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to (Bt1 , · · · , Btn−1 ). $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 28 / () 37
  64. 64. G–ÙK$Ä[Peng2005] Definition Qg‚SÏ4˜m(Ω,F, E)¥§¡˜‡‘Åv§B (ω) = ω , t ≥ 0 ˆ G–ÙK6Ä motion Xt§÷v±e^‡: Ù t t (i) Bt+s − Bs is N (0, [σ2 t, σ2 t]) distributed ∀ s, t ≥ 0 (ii) For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to (Bt1 , · · · , Btn−1 ). For simplification, we set σ2 = 1. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 28 / () 37
  65. 65. ½n XtQ˜‡g‚SÏ4˜m(Ω,F, E)¥, ˜‡‘Åv§B (ω), t ≥ 0 ÷ ˆ vµ t $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 29 / () 37
  66. 66. ½n XtQ˜‡g‚SÏ4˜m(Ω,F, E)¥, ˜‡‘Åv§B (ω), t ≥ 0 ÷ ˆ vµ t é?‰'ž t ≤ · · · ≤ t , B − B ÑÕá u(B , · · · , B ). 1 n tn tn−1 t1 tn−1 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 29 / () 37
  67. 67. ½n XtQ˜‡g‚SÏ4˜m(Ω,F, E)¥, ˜‡‘Åv§B (ω), t ≥ 0 ÷ ˆ vµ t é?‰'ž t ≤ · · · ≤ t , B − B ÑÕá u(B , · · · , B ). 1 n tn tn−1 t1 tn−1 Bt ∼ Bs+t − Bs , ∀s, t ≥ 0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 29 / () 37
  68. 68. ½n XtQ˜‡g‚SÏ4˜m(Ω,F, E)¥, ˜‡‘Åv§B (ω), t ≥ 0 ÷ ˆ vµ t é?‰'ž t ≤ · · · ≤ t , B − B ÑÕá u(B , · · · , B ). 1 n tn tn−1 t1 tn−1 Bt ∼ Bs+t − Bs , ∀s, t ≥ 0 limt→0 t −1 E[|Bt |3 ] = 0. ˆ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 29 / () 37
  69. 69. ½n XtQ˜‡g‚SÏ4˜m(Ω,F, E)¥, ˜‡‘Åv§B (ω), t ≥ 0 ÷ ˆ vµ t é?‰'ž t ≤ · · · ≤ t , B − B ÑÕá u(B , · · · , B ). 1 n tn tn−1 t1 tn−1 Bt ∼ Bs+t − Bs , ∀s, t ≥ 0 limt→0 t −1 E[|Bt |3 ] = 0. ˆ uB ˜½´˜‡G-ÙK6Ä. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 29 / () 37
  70. 70. ‘ÅÈ© G–ÙK$ÄItˆ o 9å©FÙK6Ä5§G–ÙK6ȱéN´G½ÂItˆF‘ÅÈ© o $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 30 / () 37
  71. 71. ‘ÅÈ© G–ÙK$ÄItˆ o 9å©FÙK6Ä5§G–ÙK6ȱéN´G½ÂItˆF‘ÅÈ© o Lemma We have T E[ ˆ η(s)dBs ] = 0 0 and T T E[( η(s)dBs )2 ] = E ˆ ˆ (η(t))2 d B . t 0 0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 30 / () 37
  72. 72. G-ÙK$IJCL§ B t ²gv§Pµ N−1 t ∑ (Bt = Bt2 − 2 − Btk )2 Bs dBs = B lim N t k+1 max(tk+1 −tk )→0 k=0 0 ´˜‡yv§§¡B'²gv§quadratic variation process B . E[ B t ] = t, but E[− B t ] = −σ2 t ˆ ˆ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 31 / () 37
  73. 73. G-ÙK$IJCL§ B t ²gv§Pµ N−1 t ∑ (Bt = Bt2 − 2 − Btk )2 Bs dBs = B lim N t k+1 max(tk+1 −tk )→0 k=0 0 ´˜‡yv§§¡B'²gv§quadratic variation process B . E[ B t ] = t, but E[− B t ] = −σ2 t ˆ ˆ Lemma Bts := Bt+s − Bs , t ≥ 0 is still a G -ÙK$Ä. We also have ≡ Bs −B B t+s s t $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 31 / () 37
  74. 74. úª G–ÙK$ÄItˆ o t t t Xt = X0 + αs ds + s+ ηs d B β s dBs 0 0 0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 33 / () 37
  75. 75. úª G–ÙK$ÄItˆ o t t t Xt = X0 + αs ds + s+ ηs d B β s dBs 0 0 0 ½n. α, β and η L ué?¿'t ≥ 0Ñkµ 2 (0, ∞)¥'‘Åv§. G t t Φ(Xt ) = Φ(X0 ) + Φx (Xu )β u dBu + Φx (Xu )αu du 0 0 t 1 [Φx (Xu )ηu + Φxx (Xu )β2 ]d B + u u 2 0 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 33 / () 37
  76. 76. g‚5Ïquot;˜m¥Itˆ.‘Ň©§ o Problem ÄXe‘Ň©§(SDE: Stochastic differential equations: ) t t t Xt = X0 + b(Xs )ds + h(Xs )d B + σ(Xs )dBs , t 0. s 0 0 0 Ù¥X ∈ R ´‰½'Щ^‡¶b,h,σ : R → R ´‰½''¼ê§ n n n ÷vLipschitz^‡quot; 0 ¡÷v±þSDE'‘ÅL§X ∈ M (0, T ; R )SDE') 2 n G $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 34 / () 37
  77. 77. g‚5Ïquot;˜m¥Itˆ.‘Ň©§ o Problem ÄXe‘Ň©§(SDE: Stochastic differential equations: ) t t t Xt = X0 + b(Xs )ds + h(Xs )d B + σ(Xs )dBs , t 0. s 0 0 0 Ù¥X ∈ R ´‰½'Щ^‡¶b,h,σ : R → R ´‰½''¼ê§ n n n ÷vLipschitz^‡quot; 0 ¡÷v±þSDE'‘ÅL§X ∈ M (0, T ; R )SDE') 2 n G Theorem 3˜'L§X ∈ M G (0, T ; R )÷vSDE. 2 n $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 34 / () 37
  78. 78. ·‚ÒŒ±|^Qù‡5'g‚SÏ4˜m'µee'5'ڑũ Û!‘ÅOŽcIû)y '½d§±9ºxÝþ'OŽÚ©Ûquot;ŠÑ |ºxSr'­è'ûüquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 35 / () 37
  79. 79. ·‚ÒŒ±|^Qù‡5'g‚SÏ4˜m'µee'5'ڑũ Û!‘ÅOŽcIû)y '½d§±9ºxÝþ'OŽÚ©Ûquot;ŠÑ |ºxSr'­è'ûüquot; ,˜¡§Q·‚'g‚SÏ4˜m§Ñy ¿¬U‰uyéõš~k 'êÆSŸquot;Aû´´»˜m§´»˜m¼ê¥Ñy'éõ­‡ Ä :'¯K§G¤5'–G¨ÙK6ÄnØquot; $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 35 / () 37
  80. 80. œ $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 36 / () 37
  81. 81. The second part of the talk is based on: Peng, S. (2006) G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Ito’s type, in arXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005 Abel Symbosium, Springer. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 37 / () 37
  82. 82. The second part of the talk is based on: Peng, S. (2006) G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Ito’s type, in arXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 37 / () 37
  83. 83. The second part of the talk is based on: Peng, S. (2006) G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Ito’s type, in arXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA. Peng, S. Law of large numbers and central limit theorem under nonlinear expectations, in arXiv:math.PR/0702358v1 13 Feb 2007 Peng, S. A New Central Limit Theorem under Sublinear Expectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008 $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 37 / () 37
  84. 84. The second part of the talk is based on: Peng, S. (2006) G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Ito’s type, in arXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA. Peng, S. Law of large numbers and central limit theorem under nonlinear expectations, in arXiv:math.PR/0702358v1 13 Feb 2007 Peng, S. A New Central Limit Theorem under Sublinear Expectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008 Peng, S.L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes, see arXiv:0802.1240v1 [math.PR] 9 Feb 2008. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 37 / () 37
  85. 85. The second part of the talk is based on: Peng, S. (2006) G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Ito’s type, in arXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005 Abel Symbosium, Springer. Peng, S. (2006) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, in arXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA. Peng, S. Law of large numbers and central limit theorem under nonlinear expectations, in arXiv:math.PR/0702358v1 13 Feb 2007 Peng, S. A New Central Limit Theorem under Sublinear Expectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008 Peng, S.L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes, see arXiv:0802.1240v1 [math.PR] 9 Feb 2008. Song, Y. (2007) A general central limit theorem under Peng’s G-normal distribution, Preprint. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢ Ùc 23F c Ä ¯§w ú G-Ù K $ 37 / () 37

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