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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¿÷XØ(½S§??¿÷X‘Åy–quot;
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5. ‘Åy–Ä:êÆnصVÇÓ†‘Å©Û
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6. ‘Åy–Ä:êÆnصVÇÓ†‘Å©Û
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y“VÇnØ'ú
nÄ:§ú@´y“VÇnØ'gÄŠquot;
‘Åv§'êÆnØ
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7. ‘Åy–Ä:êÆnصVÇÓ†‘Å©Û
VÇÓ'nØÄ:
Kolmogorov (1933c¤5VÇnØ'Ä:6µïá
y“VÇnØ'ú
nÄ:§ú@´y“VÇnØ'gÄŠquot;
‘Åv§'êÆnØ
Brown6Ä£1900cµL.Bachelier, 1905cµA. Einstein, 1926cWiener¤;
Markovv§nØ£{s!€é!{s9Ù¦Æ 'ÏRý'íc¤
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o
Q¼êÆWalføžWolf™”‘Å(s¥'Úî½Æ”quot;
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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
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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
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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
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11. ‘Ň©§nØ­‡quot;õŸþ´•
Itˆ
o
•~‡©§£ ¤
t0 ≤ t ≤ T
dx (t) = b(x (t))dt, x (t0 ) = x0 (Щ^‡)
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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^‡)
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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
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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)
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15. •‘Ň©§Ä½n:
3˜5½n[Pardoux,Peng1990],'½n[Peng1991]
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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
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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 ), · · ·
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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;
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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
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20. û)7K¬½d¯K[El Karoui, Peng, Quenez1997]
g[Bachelier1900]±5§UKÆ[Ï—åuAûϽdù˜‡
•'‘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;
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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
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22. ˜‡š~Ä:¢´ϱ5cÐ$¢'êƯK´µ
Feynman-Kac úªUØUíP š‚Sº
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23. ˜‡š~Ä:¢´ϱ5cÐ$¢'êƯK´µ
Feynman-Kac úªUØUíP š‚Sº
˜‡Ñ¿GuyÚy²:
˜Œa0Œ‚S‚S ‡©§'AŒ±Ïv•‘Ň©§'A
5vǤ
Ù‚Sœ¹Ò´Feynman-Kac úªquot;
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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),
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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
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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«
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27. ù‡Ñ¿GuyÚy²k '¹Âµ
˜‡•‘Ň©§¢Sþ´˜«´»6' ‡©§:
´»˜m´[0,T]þ'뉴»'NµΩ = C [0, T ]
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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;
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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
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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
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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
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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
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() 37

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¥'­‡'ïĐ•§UKêÆ¥­‡'ïÄ!OŽó
äquot;
gQ{sލ'IÊQ•‘Ň©§Æâï?¬§qÑy
Ĭ
õ­‡ïĤtquot;
$¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
14 /
() 37

34. êÆŒ“…#xâÅ1933cïáy“VÇØ
®PA^ ØÓ'+
ù‡nØ')Ÿ´µêÆÏ4´‚S'quot;
ŽÑ‚SÏ4QAº²L
y–ž'Øv§kxõ²LÆ[†êÆ[—åuïÄš‚SêÆÏ
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. êÆŒ“…#xâÅ1933cïáy“VÇØ
®PA^ ØÓ'+
ù‡nØ')Ÿ´µêÆÏ4´‚S'quot;
ŽÑ‚SÏ4QAº²L
y–ž'Øv§kxõ²LÆ[†êÆ[—åuïÄš‚SêÆÏ
4§XͶêÆ[!³Ø;[Choquet ²½Âvš‚SVÇ
£=Choquet NݤÚChoquet Ï4§¢Q½Ât ž®8Ee'^
‡Ï4ž‘ )Ÿ(Jquot;
g¨Ï49^‡g-Ï4[Peng1997]–Äš‚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. 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. 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. g‚5êÆÏquot;Eµ rÑ…#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. 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. 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. 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. 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'­
ί
$¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
19 /
() 37

43. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
19 /
() 37

44. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
19 /
() 37

45. $¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
20 /
() 37

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. 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. 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. ¥%4½n
$¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
21 /
() 37

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. ¥%4½n
Y¥74½n
´VÇÚO¥'˜‡ P
q›©9…'@t§de Moivre, Laplace,
Kolmogorov, · · · ,
Y¥74½n'¿Â
ÕáÓ©Ù‘Ågþƒ''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. ¥%4½n
Y¥74½n
´VÇÚO¥'˜‡ P
q›©9…'@t§de Moivre, Laplace,
Kolmogorov, · · · ,
Y¥74½n'¿Â
ÕáÓ©Ù‘Ågþƒ''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. 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. 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. 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. 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. 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. 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. ¡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. 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. 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. 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. 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. 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. 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. 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. 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. 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¢
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29 /
() 37

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. ‘ÅÈ©
G–ÙK$ÄItˆ
o
9å©FÙK6Ä5§G–ÙK6ÄŒ±éN´G½ÂItˆF‘ÅÈ©
o
$¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
30 /
() 37

71. ‘ÅÈ©
G–ÙK$ÄItˆ
o
9å©FÙK6Ä5§G–ÙK6ÄŒ±éN´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. 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. 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. 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. 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
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33 /
() 37

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. 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. ·‚ÒŒ±|^Qù‡5'g‚SÏ4˜m'µee'5'Ú‘Å©
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. ·‚ÒŒ±|^Qù‡5'g‚SÏ4˜m'µee'5'Ú‘Å©
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. œ
$¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!­è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
c Ä ¯§w
ú G-Ù K $
36 /
() 37

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. 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. 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. 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. 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