Download Law of Large Number and Central Limit Theorem under

Law of Large Number and Central Limit Theorem
under Uncertainty, the Related New Itô’s Calculus
and Applications to Risk Measures
Shige Peng
Shandong University, Jinan, China
Presented at
1st PRIMA Congress, July 06-10, 2009
UNSW, Sydney, Australia
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A Chinese Understanding of Beijing Opera
A popular point of view of Beijing opera by Chinese artists
,
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A Chinese Understanding of Beijing Opera
A popular point of view of Beijing opera by Chinese artists
,
Translation (with certain degree of uncertainty)
The universe is a big opera stage,
An opera stage is a small universe
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A Chinese Understanding of Beijing Opera
A popular point of view of Beijing opera by Chinese artists
,
Translation (with certain degree of uncertainty)
The universe is a big opera stage,
An opera stage is a small universe
Real world ⇐⇒ Modeling world
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Scientists point of view of the real world after Newton:
People strongly believe that our universe is deterministic: it can be
precisely calculated by (ordinary) differential equations.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Scientists point of view of the real world after Newton:
People strongly believe that our universe is deterministic: it can be
precisely calculated by (ordinary) differential equations.
After longtime explore
our scientists begin to realize that our universe is full of uncertainty
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A fundamental tool to understand and to treat
uncertainty: Probability Theory
Pascal & Fermat, Huygens, de Moivre, Laplace, · · ·
Ω: a set of events: only one event ω ∈ Ω happens but we don’t know
which ω (at least before it happens).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A fundamental tool to understand and to treat
uncertainty: Probability Theory
Pascal & Fermat, Huygens, de Moivre, Laplace, · · ·
Ω: a set of events: only one event ω ∈ Ω happens but we don’t know
which ω (at least before it happens).
The probability of A ⊂ Ω:
number times A happens
P (A) �
total number of trials
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A fundamental revolution of probability theory (1933
Kolmogorov’s Foundation of Probability Theory (Ω, F , P )
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A fundamental revolution of probability theory (1933
Kolmogorov’s Foundation of Probability Theory (Ω, F , P )
The basic rule of probability theory
P (Ω) = 1, P (∅) = 0, P (A + B ) = P (A) + P (B )
∞
P ( ∑ Ai ) =
i =1
Shige Peng ()
∞
∑ P (Ai )
i =1
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A fundamental revolution of probability theory (1933
Kolmogorov’s Foundation of Probability Theory (Ω, F , P )
The basic rule of probability theory
P (Ω) = 1, P (∅) = 0, P (A + B ) = P (A) + P (B )
∞
P ( ∑ Ai ) =
i =1
∞
∑ P (Ai )
i =1
The basic idea of Kolmogorov:
Using (possibly infinite dimensional) Lebesgue integral to calculate the
expectation of a random variable X (ω )
E [X ] =
Shige Peng ()
�
Ω
X (ω )dP (ω )
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
vNM Expected Utility Theory: The foundation of modern
economic theory
E [u (X )] ≥ E [u (Y )]
Shige Peng ()
⇐⇒ The utility of X is bigger than that of Y
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Probability Theory
Markov Processes
Itô’s calculus: and pathwise stochastic analysis
Statistics: life science and medical industry; insurance, politics;
Stochastic controls;
Statistic Physics;
Economics, Finance;
Civil engineering;
Communications, internet;
Forecasting: Whether, pollution, · · ·
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Probability measure P (·) vs mathematical expectation
E [·]: who is more fundamental?
We can also first define a linear functional E : H �−→ R, s.t.
E [αX + c ] = αE [X ] + c,
E [Xi ] → 0, if Xi ↓ 0.
Shige Peng ()
E [X ] ≥ E [Y ], if X ≥ Y
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Probability measure P (·) vs mathematical expectation
E [·]: who is more fundamental?
We can also first define a linear functional E : H �−→ R, s.t.
E [αX + c ] = αE [X ] + c,
E [Xi ] → 0, if Xi ↓ 0.
E [X ] ≥ E [Y ], if X ≥ Y
By Daniell-Stone Theorem
There is a unique probability measure (Ω, F , P ) such that P (A) = E [1A ]
for each A ∈ F
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Probability measure P (·) vs mathematical expectation
E [·]: who is more fundamental?
We can also first define a linear functional E : H �−→ R, s.t.
E [αX + c ] = αE [X ] + c,
E [Xi ] → 0, if Xi ↓ 0.
E [X ] ≥ E [Y ], if X ≥ Y
By Daniell-Stone Theorem
There is a unique probability measure (Ω, F , P ) such that P (A) = E [1A ]
for each A ∈ F
For nonlinear expectation
No more such equivalence!!
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Why nonlinear expectation?
M. Allais Paradox Ellesberg Paradox in economic theory The linearity
of expectation E cannot be a linear operator
[1997 1999Artzner-Delbean-Eden-Heath] Coherent risk measure
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A well-known puzzle: why normal distributions are so
widely used?
If someone faces a random variable X with uncertain distribution, he will
first try to use normal distribution N (µ, σ2 ).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The density of a �normal
distribution
�
2
1
x
p (x ) = √2π exp − 2
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Explanation for this phenomenon is the well-known central limit theorem:
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Explanation for this phenomenon is the well-known central limit theorem:
Theorem (Central Limit Theorem (CLT))
{Xi }i∞=1 is assumed to be i.i.d. with µ = E [X1 ] and σ2 = E [(X1 − µ)2 ].
Then for each bounded and continuous function ϕ ∈ C (R ), we have
1 n
lim E [ ϕ( √ ∑ (Xi − µ))] = E [ ϕ(X )], X ∼ N (0, σ2 ).
i →∞
n i =1
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Explanation for this phenomenon is the well-known central limit theorem:
Theorem (Central Limit Theorem (CLT))
{Xi }i∞=1 is assumed to be i.i.d. with µ = E [X1 ] and σ2 = E [(X1 − µ)2 ].
Then for each bounded and continuous function ϕ ∈ C (R ), we have
1 n
lim E [ ϕ( √ ∑ (Xi − µ))] = E [ ϕ(X )], X ∼ N (0, σ2 ).
i →∞
n i =1
The beauty and power
of this result comes from: the above sum tends to N (0, σ2 ) regardless the
original distribution of Xi , provided that Xi ∼ X1 , for all i = 2, 3, · · · and
that X1 , X2 ,· · · are mutually independent.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
History of CLT
Abraham de Moivre 1733,
Pierre-Simon Laplace, 1812: Théorie Analytique des Probabilité
Aleksandr Lyapunov, 1901
Cauchy’s, Bessel’s and Poisson’s contributions, von Mises, Polya,
Lindeberg, Lévy, Cramer
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
‘dirty work’ through “contaminated data”?
But in, real world in finance, as well as in many other human science
situations, it is not true that the above {Xi }i∞=1 is really i.i.d.
Many academic people think that people in finance just widely and deeply
abuse this beautiful mathematical result to do ‘dirty work’ through
“contaminated data”
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
A new CLT under Distribution Uncertainty
1. We do not know the distribution of Xi ;
2. We don’t assume Xi ∼ Xj , they may have difference unknown
distributions. We only assume that the distribution of Xi , i = 1, 2, · · · are
within some subset of distribution functions
L(Xi ) ∈ {Fθ (x ) : θ ∈ Θ}.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
In the situation of distributional uncertainty and/or probabilistic
uncertainty (or model uncertainty) the problem of decision become more
complicated. A well-accepted method is the following robust calculation:
sup Eθ [ ϕ(ξ )], inf Eθ [ ϕ(η )]
θ ∈Θ
θ ∈Θ
where Eθ represent the expectation of a possible probability in our
uncertainty model.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
But this turns out to be a new and basic mathematical tool:
Ê [X ] = sup Eθ [X ]
θ ∈Θ
The operator Ê has the properties:
a) X ≥ Y then Ê [X ] ≥ Ê [Y ]
b) Ê [c ] = c
c) Ê [X + Y ] ≤ Ê [X ] + Ê [Y ]
d) Ê [λX ] = λÊ [X ], λ ≥ 0.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Sublinear Expectation
Definition
A nonlinear expectation: a functional Ê : H �→ R
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Sublinear Expectation
Definition
A nonlinear expectation: a functional Ê : H �→ R
(a) Monotonicity: if X ≥ Y then Ê [X ] ≥ Ê [Y ].
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Sublinear Expectation
Definition
A nonlinear expectation: a functional Ê : H �→ R
(a) Monotonicity: if X ≥ Y then Ê [X ] ≥ Ê [Y ].
(b) Constant preserving: Ê [c ] = c.
A sublinear expectation:
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Sublinear Expectation
Definition
A nonlinear expectation: a functional Ê : H �→ R
(a) Monotonicity: if X ≥ Y then Ê [X ] ≥ Ê [Y ].
(b) Constant preserving: Ê [c ] = c.
A sublinear expectation:
(c) Sub-additivity (or self–dominated property):
Ê [X ] − Ê [Y ] ≤ Ê [X − Y ].
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Sublinear Expectation
Definition
A nonlinear expectation: a functional Ê : H �→ R
(a) Monotonicity: if X ≥ Y then Ê [X ] ≥ Ê [Y ].
(b) Constant preserving: Ê [c ] = c.
A sublinear expectation:
(c) Sub-additivity (or self–dominated property):
Ê [X ] − Ê [Y ] ≤ Ê [X − Y ].
(d) Positive homogeneity: Ê [λX ] = λÊ [X ],
Shige Peng ()
∀λ ≥ 0.
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Coherent Risk Measures and Sunlinear Expectations
H is the set of bounded and measurable random variables on (Ω, F ).
ρ(X ) := Ê [−X ]
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Robust representation of sublinear expectations
Huber Robust Statistics (1981).
Artzner, Delbean, Eber & Heath (1999)
Föllmer & Schied (2004)
(For interest rate uncertainty, see Barrieu & El Karoui (2005)).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Robust representation of sublinear expectations
Huber Robust Statistics (1981).
Artzner, Delbean, Eber & Heath (1999)
Föllmer & Schied (2004)
Theorem
Ê [·] is a sublinear expectation on H if and only if there exists a subset
P ∈ M1,f (the collection of all finitely additive probability measures) such
that
Ê [X ] = sup EP [X ], ∀X ∈ H.
P ∈P
(For interest rate uncertainty, see Barrieu & El Karoui (2005)).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Meaning of the robust representation:
Statistic model uncertainty
Ê [X ] = sup EP [X ],
P ∈P
∀X ∈ H .
The size of the subset P represents the degree of
model uncertainty: The stronger the Ê the more the uncertainty
Ê1 [X ] ≥ Ê2 [X ],
Shige Peng ()
∀X ∈ H ⇐⇒
P1 ⊃ P2
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The distribution of a random vector in (Ω, H, Ê )
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The distribution of a random vector in (Ω, H, Ê )
Definition (Distribution of X )
Given X = (X1 , · · · , Xn ) ∈ Hn . We define:
F̂X [ ϕ] := Ê [ ϕ(X )] : ϕ ∈ Cb (Rn ) �−→ R.
We call F̂X [·] the distribution of X under Ê.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The distribution of a random vector in (Ω, H, Ê )
Definition (Distribution of X )
Given X = (X1 , · · · , Xn ) ∈ Hn . We define:
F̂X [ ϕ] := Ê [ ϕ(X )] : ϕ ∈ Cb (Rn ) �−→ R.
We call F̂X [·] the distribution of X under Ê.
Sublinear Distribution
FX [·] forms a sublinear expectation on (Rn , Cb (Rn )). Briefly:
FX [ ϕ] = sup
θ ∈Θ
Shige Peng ()
�
Rn
ϕ(x )Fθ (dy ).
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Definition
Definition
X ,Y are said to be identically distributed, (X ∼ Y , or X is a copy of Y ),
if they have same distributions:
Ê [ ϕ(X )] = Ê [ ϕ(Y )],
Shige Peng ()
∀ ϕ ∈ Cb (Rn ).
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Definition
Definition
X ,Y are said to be identically distributed, (X ∼ Y , or X is a copy of Y ),
if they have same distributions:
Ê [ ϕ(X )] = Ê [ ϕ(Y )],
∀ ϕ ∈ Cb (Rn ).
The meaning of X ∼ Y :
X and Y has the same subset of uncertain distributions.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Independence under Ê
Definition
Y (ω ) is said to be independent from X (ω ) if we have:
Ê [ ϕ(X , Y )] = Ê [Ê [ ϕ(x, Y )]x =X ], ∀ ϕ(·).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Independence under Ê
Definition
Y (ω ) is said to be independent from X (ω ) if we have:
Ê [ ϕ(X , Y )] = Ê [Ê [ ϕ(x, Y )]x =X ], ∀ ϕ(·).
Fact
Meaning: the realization of X does not change (improve) the distribution
uncertainty of Y .
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The notion of independence: it can be subjective
Example (An extreme example)
In reality, Y = X
but we are in the very beginning
and we know noting about the relation of X and Y ,
the only information we know is X , Y ∈ Θ.
The robust expectation of ϕ(X , Y ) is:
Ê [ ϕ(X , Y )] = sup ϕ(x, y ).
x,y ∈Θ
Y is independent to X , X is also independent to Y .
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The notion of independence
Fact
Y is independent to X DOES NOT IMPLIES
X is independent to Y
Example
σ2 := Ê [Y 2 ] > σ2 := −Ê [−Y 2 ] > 0,
Then
Shige Peng ()
Ê [X ] = Ê [−X ] = 0.
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The notion of independence
Fact
Y is independent to X DOES NOT IMPLIES
X is independent to Y
Example
σ2 := Ê [Y 2 ] > σ2 := −Ê [−Y 2 ] > 0,
Then
Ê [X ] = Ê [−X ] = 0.
If Y is independent to X :
Ê [XY 2 ] = Ê [Ê [xY 2 ]x =X ] = Ê [X + σ2 − X − σ2 ]
= Ê [X + ](σ2 − σ2 ) > 0.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
The notion of independence
Fact
Y is independent to X DOES NOT IMPLIES
X is independent to Y
Example
σ2 := Ê [Y 2 ] > σ2 := −Ê [−Y 2 ] > 0,
Then
Ê [X ] = Ê [−X ] = 0.
If Y is independent to X :
Ê [XY 2 ] = Ê [Ê [xY 2 ]x =X ] = Ê [X + σ2 − X − σ2 ]
= Ê [X + ](σ2 − σ2 ) > 0.
But if X is independent to Y :
Ê [XY 2 ] = Ê [Ê [X ]Y 2 ] = 0.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Central Limit Theorem (CLT)
Let {Xi }i∞=1 be a i.i.d. in a sublinear expectation space (Ω, H, Ê ) in the
following sense:
(i) identically distributed:
Ê [ ϕ(Xi )] = Ê [ ϕ(X1 )], ∀i = 1, 2, · · ·
(ii) independent: for each i, Xi +1 is independent to (X1 , X2 , · · · , Xi )
under Ê.
We also assume that Ê [X1 ] = −Ê [−X1 ]. We denote
σ2 = Ê [X12 ], σ2 = −Ê [−X12 ]
Then for each convex function ϕ we have
Sn
1
Ê [ ϕ( √ )] → √
n
2πσ2
� ∞
−x 2
ϕ(x ) exp( 2 )dx
2σ
−∞
and for each concave function ψ we have
Shige Peng ()
Sn
Prese
Shandong University, Jinan, China
Law of Large Number and∞
Central Limit Theorem under
53 I
2 Uncertainty, the Related/ New
1
�
−x
Theorem ( CLT under distribution uncertainty)
Let {Xi }i∞=1 be a i.i.d. in a sublinear expectation space (Ω, H, Ê ) in the
following sense:
(i) identically distributed:
Ê [ ϕ(Xi )] = Ê [ ϕ(X1 )], ∀i = 1, 2, · · ·
(ii) independent: for each i, Xi +1 is independent to (X1 , X2 , · · · , Xi )
under Ê.
We also assume that Ê [X1 ] = −Ê [−X1 ]. Then for each convex function
ϕ we have
Sn
√
Ê [ ϕ(
)] → Ê [ ϕ(X )], X ∼ N (0, [σ2 , σ2 ])
n
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Normal distribution under Sublinear expectation
An fundamentally important sublinear distribution
Definition
A random variable X in (Ω, H, Ê ) is called normally distributed if
�
aX + b X̄ ∼ a2 + b 2 X , ∀a, b ≥ 0.
where X̄ is an independent copy of X .
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Normal distribution under Sublinear expectation
An fundamentally important sublinear distribution
Definition
A random variable X in (Ω, H, Ê ) is called normally distributed if
�
aX + b X̄ ∼ a2 + b 2 X , ∀a, b ≥ 0.
where X̄ is an independent copy of X .
We have Ê [X ] = Ê [−X ] = 0.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Normal distribution under Sublinear expectation
An fundamentally important sublinear distribution
Definition
A random variable X in (Ω, H, Ê ) is called normally distributed if
�
aX + b X̄ ∼ a2 + b 2 X , ∀a, b ≥ 0.
where X̄ is an independent copy of X .
We have Ê [X ] = Ê [−X ] = 0.
We denote X ∼ N (0, [σ2 , σ2 ]), where
σ2 := Ê [X 2 ], σ2 := −Ê [−X 2 ].
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
G-normal distribution: a under sublinear expectation E [·]
(1) For each convex ϕ, we have
Ê [ ϕ(X )] = √
Shige Peng ()
1
2πσ2
� ∞
y2
ϕ(y ) exp(− 2 )dy
2σ
−∞
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
G-normal distribution: a under sublinear expectation E [·]
(1) For each convex ϕ, we have
Ê [ ϕ(X )] = √
1
2πσ2
� ∞
y2
ϕ(y ) exp(− 2 )dy
2σ
−∞
(2) For each concave ϕ, we have,
Ê [ ϕ(X )] = �
Shige Peng ()
1
2πσ2
� ∞
y2
ϕ(y ) exp(− 2 )dy
2σ
−∞
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Fact
If σ2 = σ2 , then N (0; [σ2 , σ2 ]) = N (0, σ2 ).
Fact
The larger to [σ2 , σ2 ] the stronger the uncertainty.
Fact
But the uncertainty subset of N (0; [σ2 , σ2 ]) is not just consisting of
N (0; σ), σ ∈ [σ2 , σ2 ]!!
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
X ∼ N (0, [σ2 , σ2 ]) in (Ω, H, Ê ) iff for each ϕ ∈ Cb (R ) the function
√
u (t, x ) := Ê [ ϕ(x + tX )], x ∈ R, t ≥ 0
is the solution of the PDE
ut = G (uxx ), t > 0, x ∈ R
u |t =0 = ϕ,
where G (a) = 12 (σ2 a+ − σ2 a− )(= Ê [ 2a X 2 ]). G -normal distribution.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Law of Large Numbers (LLN), Central Limit Theorem
(CLT)
Striking consequence of LLN & CLT
Accumulated independent and identically distributed random variables
tends to a normal distributed random variable, whatever the original
distribution.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Law of Large Numbers (LLN), Central Limit Theorem
(CLT)
Striking consequence of LLN & CLT
Accumulated independent and identically distributed random variables
tends to a normal distributed random variable, whatever the original
distribution.
LLN under Choquet capacities:
Marinacci, M. Limit laws for non-additive probabilities and their
frequentist interpretation, Journal of Economic Theory 84, 145-195 1999.
Nothing found for nonlinear CLT.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Definition
A sequence of random variables {ηi }i∞=1 in H is said to converge in law
under Ê if the limit
lim Ê [ ϕ(ηi )],
i →∞
Shige Peng ()
for each ϕ ∈ Cb (R ).
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Central Limit Theorem under Sublinear Expectation
Theorem
Let {Xi }i∞=1 in (Ω, H, Ê ) be identically distributed: Xi ∼ X1 , and each
Xi +1 is independent to (X1 , · · · , Xi ). We assume furthermore that
Ê [|X1 |2+α ] < ∞ and Ê [X1 ] = Ê [−X1 ] = 0.
√
Sn := X1 + · · · + Xn . Then Sn / n converges in law to N (0; [σ2 , σ2 ]):
Sn
lim Ê [ ϕ( √ )] = Ê [ ϕ(X )], ∀ ϕ ∈ Cb (R ),
n→∞
n
where
where X ∼ N (0, [σ2 , σ2 ]),
Shige Peng ()
σ2 = Ê [X12 ], σ2 = −Ê [−X12 ].
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Cases with mean-uncertainty
What happens if Ê [X1 ] > −Ê [−X1 ]?
Definition
A random variable Y in (Ω, H, Ê ) is N ([µ, µ] × {0})-distributed
(Y ∼ N ([µ, µ] × {0})) if
aY + b Ȳ ∼ (a + b )Y ,
∀a, b ≥ 0.
where Ȳ is an independent copy of Y , where µ := Ê [Y ] > µ := −Ê [−Y ]
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Cases with mean-uncertainty
What happens if Ê [X1 ] > −Ê [−X1 ]?
Definition
A random variable Y in (Ω, H, Ê ) is N ([µ, µ] × {0})-distributed
(Y ∼ N ([µ, µ] × {0})) if
aY + b Ȳ ∼ (a + b )Y ,
∀a, b ≥ 0.
where Ȳ is an independent copy of Y , where µ := Ê [Y ] > µ := −Ê [−Y ]
We can prove that
Ê [ ϕ(Y )] = sup ϕ(y ).
y ∈[µ,µ]
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Case with mean-uncertainty E [·]
Definition
A pair of random variables (X , Y ) in (Ω, H, Ê ) is
N ([µ, µ] × [σ2 , σ2 ])-distributed ((X , Y ) ∼ N ([µ, µ] × [σ2 , σ2 ])) if
�
(aX + bX̄ , a2 Y + b2 Ȳ ) ∼ ( a2 + b2 X , (a2 + b2 )Y ),
∀a, b ≥ 0.
where (X̄ , Ȳ ) is an independent copy of (X , Y ),
µ := Ê [Y ], µ := −Ê [−Y ]
σ2 := Ê [X 2 ], σ2 := −Ê [−X ], (Ê [X ] = Ê [ − X ] = 0).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
(X , Y ) ∼ N ([µ, µ], [σ2 , σ2 ]) in (Ω, H, Ê ) iff for each ϕ ∈ Cb (R ) the
function
√
u (t, x, y ) := Ê [ ϕ(x + tX , y + tY )], x ∈ R, t ≥ 0
is the solution of the PDE
ut = G (uy , uxx ), t > 0, x ∈ R
u |t =0 = ϕ,
where
a 2
G (p, a) := Ê [ X + pY ].
2
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Central Limit Theorem under Sublinear Expectation
Theorem
Let {Xi + Yi }i∞=1 be an independent and identically distributed sequence.
We assume furthermore that
Ê [|X1 |2+α ] + Ê [|Y1 |1+α ] < ∞ and Ê [X1 ] = Ê [−X1 ] = 0.
√
X
Y
Sn := X1 + · · · + Xn , Sn := Y1 + · · · + Yn . Then Sn / n converges in
law to N (0; [σ2 , σ2 ]):
SnX
SnY
)] = Ê [ ϕ(X + Y )], ∀ ϕ ∈ Cb (R ),
lim Ê [ ϕ( √ +
n→∞
n
n
where (X , Y ) is N ([µ, µ], [σ2 , σ2 ])-distributed.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Remarks.
1.
Like classical case, these new LLN and CLT plays fundamental roles for
the case with probability and distribution uncertainty;
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Remarks.
1.
Like classical case, these new LLN and CLT plays fundamental roles for
the case with probability and distribution uncertainty;
2.
According to our LLN and CLT, the uncertainty accumulates, in general,
there are infinitely many probability measures. The best way to calculate
Ê [X ] is to use our theorem, instead of calculating Ê [X ] = supθ ∈Θ Eθ [X ];
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Remarks.
1.
Like classical case, these new LLN and CLT plays fundamental roles for
the case with probability and distribution uncertainty;
2.
According to our LLN and CLT, the uncertainty accumulates, in general,
there are infinitely many probability measures. The best way to calculate
Ê [X ] is to use our theorem, instead of calculating Ê [X ] = supθ ∈Θ Eθ [X ];
3.
Many statistic methods must be revisited to clarify their meaning of
uncertainty: it’s very risky to treat this new notion in a classical way. This
will cause serious confusions.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Remarks.
1.
Like classical case, these new LLN and CLT plays fundamental roles for
the case with probability and distribution uncertainty;
2.
According to our LLN and CLT, the uncertainty accumulates, in general,
there are infinitely many probability measures. The best way to calculate
Ê [X ] is to use our theorem, instead of calculating Ê [X ] = supθ ∈Θ Eθ [X ];
3.
Many statistic methods must be revisited to clarify their meaning of
uncertainty: it’s very risky to treat this new notion in a classical way. This
will cause serious confusions.
4.
The third revolution: Deterministic point of view =⇒ Probabilistic point
of view =⇒ uncertainty of probabilities.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
G–Brownian Motion
Definition
Under (Ω, F , Ê ), a process Bt (ω ) = ωt , t ≥ 0, is called a G –Brownian
motion if:
(i) Bt +s − Bs is N (0, [σ2 t, σ2 t ]) distributed ∀ s, t ≥ 0
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
G–Brownian Motion
Definition
Under (Ω, F , Ê ), a process Bt (ω ) = ωt , t ≥ 0, is called a G –Brownian
motion if:
(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 ).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
G–Brownian Motion
Definition
Under (Ω, F , Ê ), a process Bt (ω ) = ωt , t ≥ 0, is called a G –Brownian
motion if:
(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.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
If, under some (Ω, F , Ê ), a stochastic process Bt (ω ), t ≥ 0 satisfies
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
If, under some (Ω, F , Ê ), a stochastic process Bt (ω ), t ≥ 0 satisfies
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to
(Bt1 , · · · , Btn−1 ).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
If, under some (Ω, F , Ê ), a stochastic process Bt (ω ), t ≥ 0 satisfies
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to
(Bt1 , · · · , Btn−1 ).
Bt is identically distributed as Bs +t − Bs , for all s, t ≥ 0
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
If, under some (Ω, F , Ê ), a stochastic process Bt (ω ), t ≥ 0 satisfies
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to
(Bt1 , · · · , Btn−1 ).
Bt is identically distributed as Bs +t − Bs , for all s, t ≥ 0
Ê [|Bt |3 ] = o (t ).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Theorem
If, under some (Ω, F , Ê ), a stochastic process Bt (ω ), t ≥ 0 satisfies
For each t1 ≤ · · · ≤ tn , Btn − Btn−1 is independent to
(Bt1 , · · · , Btn−1 ).
Bt is identically distributed as Bs +t − Bs , for all s, t ≥ 0
Ê [|Bt |3 ] = o (t ).
Then B is a G-Brownian motion.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
G–Brownian
Fact
Like N (0, [σ2 , σ2 ])-distribution, the G -Brownian motion Bt (ω ) = ωt ,
t ≥ 0, can strongly correlated under the unknown ‘objective probability’, it
can even be have very long memory. But it is i.i.d under the robust
expectation Ê. By which we can have many advantages in analysis,
calculus and computation, compare with, e.g. fractal B.M.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Itô’s integral of G–Brownian motion
For each process (ηt )t ≥0 ∈ L2,0
F (0, T ) of the form
ηt ( ω ) =
N −1
∑
j =0
ξ j (ω )I[tj ,tj +1 ) (t ), ξ j ∈L2 (Ftj ) (Ftj -meas. & Ê [|ξ j |2 ] < ∞)
we define
I (η ) =
Shige Peng ()
� T
0
η (s )dBs :=
N −1
∑
j =0
ξ j (Btj +1 − Btj ).
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Itô’s integral of G–Brownian motion
For each process (ηt )t ≥0 ∈ L2,0
F (0, T ) of the form
ηt ( ω ) =
N −1
∑
j =0
ξ j (ω )I[tj ,tj +1 ) (t ), ξ j ∈L2 (Ftj ) (Ftj -meas. & Ê [|ξ j |2 ] < ∞)
we define
I (η ) =
� T
0
η (s )dBs :=
Lemma
We have
Ê [
and
Ê [(
Shige Peng ()
� T
0
� T
0
N −1
∑
j =0
ξ j (Btj +1 − Btj ).
η (s )dBs ] = 0
η (s )dBs )2 ] ≤
� T
0
Ê [(η (t ))2 ]dt.
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Definition
2
Under the Banach norm �η � :=
�T
0
Ê [(η (t ))2 ]dt,
I (η ) : L2,0 (0, T ) �→ L2 (FT ) is a contract mapping
We then extend I (η ) to L2 (0, T ) and define, the stochastic integral
� T
0
Shige Peng ()
η (s )dBs := I (η ), η ∈ L2 (0, T ).
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Lemma
We�have
�r
�t
t
(i) s ηu dBu = s ηu dBu + r ηu dBu .
�t
�t
�t
(ii) s (αηu + θu )dBu = α s ηu dBu + s θu dBu , α ∈ L1 (Fs )
�T
(iii) Ê [X + r ηu dBu |Hs ] = Ê [X ], ∀X ∈ L1 (Fs ).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Quadratic variation process of G–BM
We denote:
�B �t = Bt2 − 2
� t
0
Bs dBs =
lim
N −1
2
(
B
−
B
)
N
t
∑ t
k
max(tk +1 −tk )→0 k =0
k +1
�B � is an increasing process called quadratic variation process of B.
Ê [ �B �t ] = σ2 t but Ê [− �B �t ] = −σ2 t
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Quadratic variation process of G–BM
We denote:
�B �t = Bt2 − 2
� t
0
Bs dBs =
lim
N −1
2
(
B
−
B
)
N
t
∑ t
k
max(tk +1 −tk )→0 k =0
k +1
�B � is an increasing process called quadratic variation process of B.
Ê [ �B �t ] = σ2 t but Ê [− �B �t ] = −σ2 t
Lemma
Bts := Bt +s − Bs , t ≥ 0 is still a G -Brownian motion. We also have
�B �t +s − �B �s ≡ �B s �t ∼ U ([σ2 t, σ2 t ]).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
We have the following isometry
Ê [(
Shige Peng ()
� T
0
η (s )dBs )2 ] = Ê [
� T
0
η 2 (s )d �B �s ],
η ∈ MG2 (0, T )
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Itô’s formula for G–Brownian motion
Xt = X0 +
Shige Peng ()
� t
0
αs ds +
� t
0
ηs d � B � s +
� t
0
β s dBs
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Itô’s formula for G–Brownian motion
Xt = X0 +
� t
0
αs ds +
� t
0
ηs d � B � s +
� t
0
β s dBs
Theorem.
Let α, β and η be process in L2G (0, T ). Then for each t ≥ 0 and in
L2G (Ht ) we have
Φ(Xt ) = Φ(X0 ) +
+
Shige Peng ()
� t
0
� t
0
Φx (Xu ) β u dBu +
� t
0
Φx (Xu )αu du
1
[Φx (Xu )ηu + Φxx (Xu ) β2u ]d �B �u
2
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Stochastic differential equations
Problem
We consider the following SDE:
Xt = X0 +
� t
0
b (Xs )ds +
� t
0
h(Xs )d �B �s +
� t
0
σ(Xs )dBs , t > 0.
where X0 ∈ Rn is given
b, h, σ : Rn �→ Rn are given Lip. functions.
The solution: a process X ∈ MG2 (0, T ; Rn ) satisfying the above SDE.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Stochastic differential equations
Problem
We consider the following SDE:
Xt = X0 +
� t
0
b (Xs )ds +
� t
0
h(Xs )d �B �s +
� t
0
σ(Xs )dBs , t > 0.
where X0 ∈ Rn is given
b, h, σ : Rn �→ Rn are given Lip. functions.
The solution: a process X ∈ MG2 (0, T ; Rn ) satisfying the above SDE.
Theorem
There exists a unique solution X ∈ MG2 (0, T ; Rn ) of the stochastic
differential equation.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Prospectives
Risk measures and pricing under dynamic volatility uncertainties
([A-L-P1995], [Lyons1995]) —for path dependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov
process.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Prospectives
Risk measures and pricing under dynamic volatility uncertainties
([A-L-P1995], [Lyons1995]) —for path dependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov
process.
New Feynman-Kac formula for fully nonlinear PDE:
path-interpretation.
u (t, x ) = Êx [(Bt ) exp(
� t
0
c (Bs )ds )]
∂t u = G (D 2 u ) + c (x )u, u |t =0 = ϕ(x ).
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Prospectives
Risk measures and pricing under dynamic volatility uncertainties
([A-L-P1995], [Lyons1995]) —for path dependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov
process.
New Feynman-Kac formula for fully nonlinear PDE:
path-interpretation.
u (t, x ) = Êx [(Bt ) exp(
� t
0
c (Bs )ds )]
∂t u = G (D 2 u ) + c (x )u, u |t =0 = ϕ(x ).
Fully nonlinear Monte-Carlo simulation.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Prospectives
Risk measures and pricing under dynamic volatility uncertainties
([A-L-P1995], [Lyons1995]) —for path dependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinear Markov
process.
New Feynman-Kac formula for fully nonlinear PDE:
path-interpretation.
u (t, x ) = Êx [(Bt ) exp(
� t
0
c (Bs )ds )]
∂t u = G (D 2 u ) + c (x )u, u |t =0 = ϕ(x ).
Fully nonlinear Monte-Carlo simulation.
BSDE driven by G -Brownian motion: a challenge.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
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, to appear in Proceedings of 2005 Abel Symbosium, Springer.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
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, to appear 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.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
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, to appear 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.
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
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
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, to appear 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.
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.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
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, to appear 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.
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.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
Thank you,
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I
In the classic period of Newton’s mechanics, including A. Einstein, people
believe that everything can be deterministically calculated. The last
century’s research affirmatively claimed the probabilistic behavior of our
universe: God does plays dice!
Nowadays people believe that everything has its own probability
distribution. But a deep research of human behaviors shows that for
everything involved human or life such, as finance, this may not be true: a
person or a community may prepare many different p.d. for your selection.
She change them, also purposely or randomly, time by time.
Shige Peng ()
Prese
Shandong University, Jinan, China
Law of Large Number and Central Limit Theorem under Uncertainty, the Related/ New
53 I