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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•‘Å‡©§Ä½n:
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o

x (t0 ) = x0

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‚S¤ù¨k]úªg(s,y,z) = c(B (s))y + h(B (s)).
x x

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∂t u + ∆u + g (x, u, u) = 0, u|t=T = Φ
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‚S¤ù¨k]úªg(s,y,z) = c(B (s))y + h(B (s)).
x x

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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)

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g‚5êÆÏquot;Eµ rÑ…#xâÅ¿Âe²;
VÇ˜m
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.

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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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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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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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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

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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

¨…=¨

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o
Theorem
E[·]
ˆ

¨…=¨
3˜‡(kŒ')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

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o
Theorem
E[·]
ˆ

¨…=¨
3˜‡(kŒ')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

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¥%4½n
Y¥74½n
´VÇÚO¥'˜‡ P
q›©9…'@t§de Moivre, Laplace,
Kolmogorov, · · · ,

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F”'Ì‡â

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¥%4½n
Y¥74½n
´VÇÚO¥'˜‡ P

Y¥74½n'¿Â
ÕáÓ©Ù‘Ågþƒ''ÚU©ÙÂñu©Ù§ØC‘Å
gþ¦5´'x'©Ù: ù˜X´y“UKnØÚ¢‚¥Œþ¦^“
F”'Ì‡â
¥74½n
X1 + · · · + Xn
E[ϕ( )] → E[ϕ(ξ)], ξ ∼ N (0, σ2 ),
√
n

E[X1 ] = 0, σ2 = E[X1 ]
2

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g‚5Ïquot;e¥%4½n[Peng2007-2008]
Theorem
(Ω,H, E)‘ÅCþS{X } Ñ´Ó©Ù': X ∼ X .
∞
ˆ 1
i i =1 i

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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

ŠÑquot;:
i +1 1 1
i

E[X1 ] = E[−X1 ] = 0.
ˆ ˆ

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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

ŠÑquot;:
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

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g‚5Ïquot;e©Ù(½pd©Ù):
½
N (0, [σ2 , σ2 ])

Deﬁnition
g‚SÏ4˜m(Ω,H, E)¥§¡˜‡‘ÅgþX '©
ˆ
(ðŽd‰, L´vy)
e
ÙN(0,[σ PX ∼ 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

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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 )] =
2σ2
2πσ2 −∞

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G-©Ù

éz˜‡à¼ê ϕ,
à Black-Scholes
Ql_v—nåu(
∞ y2
1
E[ϕ(X )] = √
2σ2
2πσ2 −∞

éz˜‡]¼ê ϕ, we have,
]
∞ y2
1
E[ϕ(X )] =
2σ2
2πσ2 −∞

ù‡¥74½nAºµŸoUKÅ¨
Œ±QFš~Ø(½'œ¹e ?^¥74½ncIºxÝþÚ½
d
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¡N(0,[σ , σ ])G ©Ù§Ù¥G£)¤¼ê¤d
2
2

eª½Âµ
G (α) = E[αX 2 ], ∀α ∈ R
ˆ

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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

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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 ])

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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

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Deﬁnition
Qg‚SÏ4˜m(Ω,F, E)¥§¡˜‡‘Åv§B (ω) = ω , t ≥ 0
ˆ
Ù
t t

(ii) For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to
(Bt1 , · · · , Btn−1 ).

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Deﬁnition
Qg‚SÏ4˜m(Ω,F, E)¥§¡˜‡‘Åv§B (ω) = ω , t ≥ 0
ˆ
Ù
t t

(ii) For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to
(Bt1 , · · · , Btn−1 ).
For simpliﬁcation, we set σ2 = 1.

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½n
XtQ˜‡g‚SÏ4˜m(Ω,F, E)¥, ˜‡‘Åv§B (ω), t ≥ 0 ÷
ˆ
vµ
t

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½n
ˆ
vµ
t

é?‰'ž t ≤ · · · ≤ t , B − B ÑÕá
u(B , · · · , B ).
1 n tn tn−1
t1 tn−1

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½n
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

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½n
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

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½n
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.

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‘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

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G-ÙK$Ä²CL§ 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

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G-ÙK$Ä²CL§ 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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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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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

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g‚5Ïquot;˜m¥Itˆ.‘Å‡©§
o

Problem
ÄXe‘Å‡©§(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'Ð©^‡¶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

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g‚5Ïquot;˜m¥Itˆ.‘Å‡©§
o

Problem
ÄXe‘Å‡©§(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'Ð©^‡¶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

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·‚ÒŒ±|^Qù‡5'g‚SÏ4˜m'µee'5'Ú‘Å©
Û!‘ÅOŽcIû)y '½d§±9ºxÝþ'OŽÚ©Ûquot;ŠÑ
|ºxSr'è'ûüquot;
,˜¡§Q·‚'g‚SÏ4˜m§Ñy
¿¬U‰uyéõš~k
'êÆSŸquot;Aû´´»˜m§´»˜m¼ê¥Ñy'éõ‡
Ä
:'¯K§G¤5'–G¨ÙK6ÄnØquot;

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œ

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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.

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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.

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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

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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.

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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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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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