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
$¢xÝþ Æ ¥IêƬ c¬ ú¯§w c Fu f € Œ Æ Ž á ¢
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39. Sublinear Expectation, from [Knight 1921], [Keynes 1936]
Knight, F.H. (1921), Risk, Uncertainty, and Profit
”Mathematical, or a priori, type of probability is practically never met with
in business ... business decisions, for example, deal with situations which
are far too unique, generally speaking,, for any sort of statistical tabulation
to have any value for guidance ... (so that) the concept of an objectively
measurable probability or chance is simply inapplicable .”
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40. Sublinear Expectation, from [Knight 1921], [Keynes 1936]
Knight, F.H. (1921), Risk, Uncertainty, and Profit
”Mathematical, or a priori, type of probability is practically never met with
in business ... business decisions, for example, deal with situations which
are far too unique, generally speaking,, for any sort of statistical tabulation
to have any value for guidance ... (so that) the concept of an objectively
measurable probability or chance is simply inapplicable .”
The framework of sublinear expectation can take the uncertainty into
consideration, in a systematic, beautiful and robust way.
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41. g‚5êÆÏquot;§¡µ
Upper expectation [P. Huber 1987, P. Huber Strassen 1973]); Coherent
expectation, coherent prevision [P. Walley, 1991];
Choquet expectation in potential theory [Choquet 1953] is also a type of
sublinear expectation;
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72. G-ÙK$IJCL§ B t
²gv§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'²gv§quadratic variation process
B
.
E[ B t ] = t, but E[− B t ] = −σ2 t
ˆ ˆ
$¢{ ìÀŒÆêÆÆ ¥IêƬ2009c¬,ú7KºxÝþ!è.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
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73. G-ÙK$IJCL§ B t
²gv§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'²gv§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$Ä. We also have
≡ Bs
−B
B t+s s t
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() 37
74. úª
G–ÙK$ÄItˆ
o
t t t
Xt = X0 + αs ds + s+
ηs d B β s dBs
0 0 0
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() 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.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
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() 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.¥%4½nÚ2009c4Ä 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.¥%4½nÚ2009c4Ä 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.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
Ùc 23F
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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.¥%4½nÚ2009c4Ä Fuf€ŒÆŽá¢
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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.
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