Law of Large Number and Central Limit Theorem under …prima2009.maths.unsw.edu.au/plenarylectures/PRIMA2009... · 2010-03-04 · Law of Large Number and Central Limit Theorem under

Law of Large Number and Central Limit Theoremunder Uncertainty, the Related New Ito’s Calculus

and Applications to Risk Measures

Shige Peng

Shandong University, Jinan, China

Presented at

1st PRIMA Congress, July 06-10, 2009UNSW, Sydney, Australia

Shige Peng () Law of Large Number and Central Limit Theorem under Uncertainty, the Related New Ito’s Calculus and Applications to Risk MeasuresShandong University, Jinan, China Presented at 1st PRIMA Congress, July 06-10, 2009 UNSW, Sydney, Australia 1

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A Chinese Understanding of Beijing Opera

A popular point of view of Beijing opera by Chinese artists

U/Z,

ZU/

Translation (with certain degree of uncertainty)

The universe is a big opera stage,

An opera stage is a small universe

Real world ⇐⇒ Modeling world


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U/Z,

ZU/






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U/Z,

ZU/






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Scientists point of view of the real world after Newton:

People strongly believe that our universe is deterministic: it can beprecisely calculated by (ordinary) differential equations.

After longtime explore

our scientists begin to realize that our universe is full of uncertainty


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Scientists point of view of the real world after Newton:

People strongly believe that our universe is deterministic: it can beprecisely calculated by (ordinary) differential equations.

After longtime explore

our scientists begin to realize that our universe is full of uncertainty


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A fundamental tool to understand and to treatuncertainty: Probability Theory

Pascal & Fermat, Huygens, de Moivre, Laplace, · · ·Ω: a set of events: only one event ω ∈ Ω happens but we don’t knowwhich ω (at least before it happens).

The probability of A ⊂ Ω:

P(A) ' number times A happens

total number of trials


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A fundamental tool to understand and to treatuncertainty: Probability Theory

Pascal & Fermat, Huygens, de Moivre, Laplace, · · ·Ω: a set of events: only one event ω ∈ Ω happens but we don’t knowwhich ω (at least before it happens).

The probability of A ⊂ Ω:

P(A) ' number times A happens

total number of trials


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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(∞

∑i=1

Ai ) =∞

∑i=1

P(Ai )

The basic idea of Kolmogorov:

Using (possibly infinite dimensional) Lebesgue integral to calculate theexpectation of a random variable X (ω)

E [X ] =∫

ΩX (ω)dP(ω)


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P(Ω) = 1, P(∅) = 0, P(A + B) = P(A) + P(B)

P(∞

∑i=1

Ai ) =∞

∑i=1

P(Ai )



E [X ] =∫

ΩX (ω)dP(ω)


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P(Ω) = 1, P(∅) = 0, P(A + B) = P(A) + P(B)

P(∞

∑i=1

Ai ) =∞

∑i=1

P(Ai )



E [X ] =∫

ΩX (ω)dP(ω)


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vNM Expected Utility Theory: The foundation of moderneconomic theory

E [u(X )] ≥ E [u(Y )] ⇐⇒ The utility of X is bigger than that of Y


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Probability Theory

Markov Processes¶Ito’s calculus: and pathwise stochastic analysisStatistics: life science and medical industry; insurance, politics;Stochastic controls;Statistic Physics;Economics, Finance;Civil engineering;Communications, internet;Forecasting: Whether, pollution, · · ·


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Probability measure P(·) vs mathematical expectationE [·]: who is more fundamental?

We can also first define a linear functional E : H 7−→ R, s.t.

E [αX + c ] = αE [X ] + c , E [X ] ≥ E [Y ], if X ≥ Y

E [Xi ] → 0, if Xi ↓ 0.

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!!


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E [Xi ] → 0, if Xi ↓ 0.






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E [Xi ] → 0, if Xi ↓ 0.






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Why nonlinear expectation?

M. Allais Paradox §Ellesberg Paradox in economic theoryµThe linearityof expectation E cannot be a linear operator[1997§1999Artzner-Delbean-Eden-Heath] Coherent risk measure


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A well-known puzzle: why normal distributions are sowidely used?

If someone faces a random variable X with uncertain distribution, he willfirst try to use normal distribution N (µ, σ2).


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The density of a normal distribution

p(x) = 1√2π

exp(−x2

2

)


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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

limi→∞

E[ϕ(1√n

n

∑i=1

(Xi − µ))] = E [ϕ(X )], X ∼ N (0, σ2).

The beauty and power

of this result comes from: the above sum tends to N (0, σ2) regardless theoriginal distribution of Xi , provided that Xi ∼ X1, for all i = 2, 3, · · · andthat X1, X2,· · · are mutually independent.


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limi→∞

E[ϕ(1√n

n

∑i=1

(Xi − µ))] = E [ϕ(X )], X ∼ N (0, σ2).




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limi→∞

E[ϕ(1√n

n

∑i=1

(Xi − µ))] = E [ϕ(X )], X ∼ N (0, σ2).




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History of CLT

Abraham de Moivre 1733,

Pierre-Simon Laplace, 1812: Theorie Analytique des Probabilite

Aleksandr Lyapunov, 1901

Cauchy’s, Bessel’s and Poisson’s contributions, von Mises, Polya,Lindeberg, Levy, Cramer


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‘dirty work’ through “contaminated data”?

But in, real world in finance, as well as in many other human sciencesituations, 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 deeplyabuse this beautiful mathematical result to do ‘dirty work’ through“contaminated data”


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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 unknowndistributions. We only assume that the distribution of Xi , i = 1, 2, · · · arewithin some subset of distribution functions

L(Xi ) ∈ Fθ(x) : θ ∈ Θ.


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In the situation of distributional uncertainty and/or probabilisticuncertainty (or model uncertainty) the problem of decision become morecomplicated. A well-accepted method is the following robust calculation:

supθ∈Θ

Eθ [ϕ(ξ)], infθ∈Θ

Eθ [ϕ(η)]

where Eθ represent the expectation of a possible probability in ouruncertainty model.


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But this turns out to be a new and basic mathematical tool:

E[X ] = supθ∈Θ

Eθ [X ]

The operator E has the properties:

a) X ≥ Y then E[X ] ≥ E[Y ]b) E[c ] = cc) E[X + Y ] ≤ E[X ] + E[Y ]d) E[λX ] = λE[X ], λ ≥ 0.


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Sublinear Expectation

Definition

A nonlinear expectation: a functional E : H 7→ R

(a) Monotonicity: if X ≥ Y then E[X ] ≥ E[Y ].(b) Constant preserving: E[c ] = c .

A sublinear expectation:

(c) Sub-additivity (or self–dominated property):

E[X ]− E[Y ] ≤ E[X − Y ].

(d) Positive homogeneity: E[λX ] = λE[X ], ∀λ ≥ 0.


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Definition


(a) Monotonicity: if X ≥ Y then E[X ] ≥ E[Y ].

(b) Constant preserving: E[c ] = c .



E[X ]− E[Y ] ≤ E[X − Y ].



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Definition





E[X ]− E[Y ] ≤ E[X − Y ].



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Definition





E[X ]− E[Y ] ≤ E[X − Y ].



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Definition





E[X ]− E[Y ] ≤ E[X − Y ].



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Coherent Risk Measures and Sunlinear Expectations

H is the set of bounded and measurable random variables on (Ω,F ).

ρ(X ) := E[−X ]


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Robust representation of sublinear expectations

Huber Robust Statistics (1981).

Artzner, Delbean, Eber & Heath (1999)

Follmer & Schied (2004)

Theorem

E[·] is a sublinear expectation on H if and only if there exists a subsetP ∈ M1,f (the collection of all finitely additive probability measures) suchthat

E[X ] = supP∈P

EP [X ], ∀X ∈ H.

(For interest rate uncertainty, see Barrieu & El Karoui (2005)).


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Robust representation of sublinear expectations

Huber Robust Statistics (1981).

Artzner, Delbean, Eber & Heath (1999)

Follmer & Schied (2004)

Theorem

E[·] is a sublinear expectation on H if and only if there exists a subsetP ∈ M1,f (the collection of all finitely additive probability measures) suchthat

E[X ] = supP∈P

EP [X ], ∀X ∈ H.

(For interest rate uncertainty, see Barrieu & El Karoui (2005)).


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Meaning of the robust representation:Statistic model uncertainty

E[X ] = supP∈P

EP [X ], ∀X ∈ H.

The size of the subset P represents the degree ofmodel uncertainty: The stronger the E the more the uncertainty

E1[X ] ≥ E2[X ], ∀X ∈ H ⇐⇒ P1 ⊃ P2


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The distribution of a random vector in (Ω,H, E)

Definition (Distribution of X )

Given X = (X1, · · · , Xn) ∈ Hn. We define:

FX [ϕ] := E[ϕ(X )] : ϕ ∈ Cb(Rn) 7−→ R.

We call FX [·] the distribution of X under E.

Sublinear Distribution

FX [·] forms a sublinear expectation on (Rn, Cb(Rn)). Briefly:

FX [ϕ] = supθ∈Θ

∫Rn

ϕ(x)Fθ(dy).


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FX [ϕ] := E[ϕ(X )] : ϕ ∈ Cb(Rn) 7−→ R.





∫Rn

ϕ(x)Fθ(dy).


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FX [ϕ] := E[ϕ(X )] : ϕ ∈ Cb(Rn) 7−→ R.





∫Rn

ϕ(x)Fθ(dy).


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Definition

Definition

X ,Y are said to be identically distributed, (X ∼ Y , or X is a copy of Y ),if they have same distributions:

E[ϕ(X )] = E[ϕ(Y )], ∀ϕ ∈ Cb(Rn).

The meaning of X ∼ Y :

X and Y has the same subset of uncertain distributions.


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Definition

Definition

X ,Y are said to be identically distributed, (X ∼ Y , or X is a copy of Y ),if they have same distributions:

E[ϕ(X )] = E[ϕ(Y )], ∀ϕ ∈ Cb(Rn).

The meaning of X ∼ Y :

X and Y has the same subset of uncertain distributions.


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Independence under E

Definition

Y (ω) is said to be independent from X (ω) if we have:

E[ϕ(X , Y )] = E[E[ϕ(x , Y )]x=X ], ∀ϕ(·).

Fact

Meaning: the realization of X does not change (improve) the distributionuncertainty of Y .


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Independence under E

Definition

Y (ω) is said to be independent from X (ω) if we have:

E[ϕ(X , Y )] = E[E[ϕ(x , Y )]x=X ], ∀ϕ(·).

Fact

Meaning: the realization of X does not change (improve) the distributionuncertainty of Y .


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The notion of independence: it can be subjective

Example (An extreme example)

In reality, Y = Xbut we are in the very beginningand 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:

E[ϕ(X , Y )] = supx ,y∈Θ

ϕ(x , y).

Y is independent to X , X is also independent to Y .


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The notion of independence

Fact

Y is independent to X DOES NOT IMPLIESX is independent to Y

Example

σ2 := E[Y 2] > σ2 := −E[−Y 2] > 0, E[X ] = E[−X ] = 0.Then

If Y is independent to X :

E[XY 2] = E[E[xY 2]x=X ] = E[X+σ2 − X−σ2]

= E[X+](σ2 − σ2) > 0.

But if X is independent to Y :

E[XY 2] = E[E[X ]Y 2] = 0.


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Fact


Example

σ2 := E[Y 2] > σ2 := −E[−Y 2] > 0, E[X ] = E[−X ] = 0.Then


E[XY 2] = E[E[xY 2]x=X ] = E[X+σ2 − X−σ2]

= E[X+](σ2 − σ2) > 0.


E[XY 2] = E[E[X ]Y 2] = 0.


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Fact


Example

σ2 := E[Y 2] > σ2 := −E[−Y 2] > 0, E[X ] = E[−X ] = 0.Then


E[XY 2] = E[E[xY 2]x=X ] = E[X+σ2 − X−σ2]

= E[X+](σ2 − σ2) > 0.


E[XY 2] = E[E[X ]Y 2] = 0.


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Central Limit Theorem (CLT)

Let Xi∞i=1 be a i.i.d. in a sublinear expectation space (Ω,H, E) in the

following sense:(i) identically distributed:

E[ϕ(Xi )] = E[ϕ(X1)], ∀i = 1, 2, · · ·

(ii) independent: for each i , Xi+1 is independent to (X1, X2, · · · , Xi )under E.We also assume that E[X1] = −E[−X1]. We denote

σ2 = E[X 21 ], σ2 = −E[−X 2

1 ]

Then for each convex function ϕ we have

E[ϕ(Sn√

n)] → 1√

2πσ2

∫ ∞

−∞ϕ(x) exp(

−x2

2σ2)dx

and for each concave function ψ we have

E[ψ(Sn√

n)] → 1√

2πσ2

∫ ∞

−∞ψ(x) exp(

−x2

2σ2)dx .


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Theorem ( CLT under distribution uncertainty)

Let Xi∞i=1 be a i.i.d. in a sublinear expectation space (Ω,H, E) in the

following sense:(i) identically distributed:

E[ϕ(Xi )] = E[ϕ(X1)], ∀i = 1, 2, · · ·

(ii) independent: for each i , Xi+1 is independent to (X1, X2, · · · , Xi )under E.We also assume that E[X1] = −E[−X1]. Then for each convex functionϕ we have

E[ϕ(Sn√

n)] → E[ϕ(X )], X ∼ N (0, [σ2, σ2])


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Normal distribution under Sublinear expectation

An fundamentally important sublinear distribution

Definition

A random variable X in (Ω,H, E) is called normally distributed if

aX + bX ∼√

a2 + b2X , ∀a, b ≥ 0.

where X is an independent copy of X .

We have E[X ] = E[−X ] = 0.

We denote X ∼ N (0, [σ2, σ2]), where

σ2 := E[X 2], σ2 := −E[−X 2].


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Definition


aX + bX ∼√

a2 + b2X , ∀a, b ≥ 0.


We have E[X ] = E[−X ] = 0.


σ2 := E[X 2], σ2 := −E[−X 2].


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Definition


aX + bX ∼√

a2 + b2X , ∀a, b ≥ 0.


We have E[X ] = E[−X ] = 0.


σ2 := E[X 2], σ2 := −E[−X 2].


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G-normal distribution: a under sublinear expectation E[·]

(1) For each convex ϕ, we have

E[ϕ(X )] =1√

2πσ2

∫ ∞

−∞ϕ(y) exp(− y2

2σ2)dy

(2) For each concave ϕ, we have,

E[ϕ(X )] =1√

2πσ2

∫ ∞


2σ2)dy


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G-normal distribution: a under sublinear expectation E[·]

(1) For each convex ϕ, we have

E[ϕ(X )] =1√

2πσ2

∫ ∞


2σ2)dy

(2) For each concave ϕ, we have,

E[ϕ(X )] =1√

2πσ2

∫ ∞


2σ2)dy


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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]!!


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Theorem

X ∼ N (0, [σ2, σ2]) in (Ω,H, E) iff for each ϕ ∈ Cb(R) the function

u(t, x) := E[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 (σ2a+ − σ2a−)(= E[ a

2X 2]). G-normal distribution.


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Law of Large Numbers (LLN), Central Limit Theorem(CLT)

Striking consequence of LLN & CLT

Accumulated independent and identically distributed random variablestends to a normal distributed random variable, whatever the originaldistribution.

LLN under Choquet capacities:

Marinacci, M. Limit laws for non-additive probabilities and theirfrequentist interpretation, Journal of Economic Theory 84, 145-195 1999.Nothing found for nonlinear CLT.


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Law of Large Numbers (LLN), Central Limit Theorem(CLT)

Striking consequence of LLN & CLT

Accumulated independent and identically distributed random variablestends to a normal distributed random variable, whatever the originaldistribution.

LLN under Choquet capacities:

Marinacci, M. Limit laws for non-additive probabilities and theirfrequentist interpretation, Journal of Economic Theory 84, 145-195 1999.Nothing found for nonlinear CLT.


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Definition

A sequence of random variables ηi∞i=1 in H is said to converge in law

under E if the limit

limi→∞

E[ϕ(ηi )], for each ϕ ∈ Cb(R).


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Central Limit Theorem under Sublinear Expectation

Theorem

Let Xi∞i=1 in (Ω,H, E) be identically distributed: Xi ∼ X1, and each

Xi+1 is independent to (X1, · · · , Xi ). We assume furthermore that

E[|X1|2+α] < ∞ and E[X1] = E[−X1] = 0.

Sn := X1 + · · ·+ Xn. Then Sn/√

n converges in law to N (0; [σ2, σ2]):

limn→∞

E[ϕ(Sn√

n)] = E[ϕ(X )], ∀ϕ ∈ Cb(R),

where

where X ∼ N (0, [σ2, σ2]), σ2 = E[X 21 ], σ2 = −E[−X 2

1 ].


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Cases with mean-uncertainty

What happens if E[X1] > −E[−X1]?

Definition

A random variable Y in (Ω,H, E) is N ([µ, µ]× 0)-distributed

(Y ∼ N ([µ, µ]× 0)) if

aY + bY ∼ (a + b)Y , ∀a, b ≥ 0.

where Y is an independent copy of Y , where µ := E[Y ] > µ := −E[−Y ]

We can prove that

E[ϕ(Y )] = supy∈[µ,µ]

ϕ(y).


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Cases with mean-uncertainty

What happens if E[X1] > −E[−X1]?

Definition

A random variable Y in (Ω,H, E) is N ([µ, µ]× 0)-distributed

(Y ∼ N ([µ, µ]× 0)) if

aY + bY ∼ (a + b)Y , ∀a, b ≥ 0.

where Y is an independent copy of Y , where µ := E[Y ] > µ := −E[−Y ]

We can prove that

E[ϕ(Y )] = supy∈[µ,µ]

ϕ(y).


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Case with mean-uncertainty E[·]

Definition

A pair of random variables (X , Y ) in (Ω,H, E) isN ([µ, µ]× [σ2, σ2])-distributed ((X , Y ) ∼ N ([µ, µ]× [σ2, σ2])) if

(aX + bX , a2Y + b2Y ) ∼ (√

a2 + b2X , (a2 + b2)Y ), ∀a, b ≥ 0.

where (X , Y ) is an independent copy of (X , Y ),

µ := E[Y ], µ := −E[−Y ]

σ2 := E[X 2], σ2 := −E[−X ], (E[X ] = E[− X ] = 0).


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Theorem

(X , Y ) ∼ N ([µ, µ], [σ2, σ2]) in (Ω,H, E) iff for each ϕ ∈ Cb(R) thefunction

u(t, x , y) := E[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,

whereG (p, a) := E[

a

2X 2 + pY ].


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Central Limit Theorem under Sublinear Expectation

Theorem

Let Xi + Yi∞i=1 be an independent and identically distributed sequence.

We assume furthermore that

E[|X1|2+α] + E[|Y1|1+α] < ∞ and E[X1] = E[−X1] = 0.

SXn := X1 + · · ·+ Xn, SY

n := Y1 + · · ·+ Yn. Then Sn/√

n converges inlaw to N (0; [σ2, σ2]):

limn→∞

E[ϕ(SX

n√n

+SY

n

n)] = E[ϕ(X + Y )], ∀ϕ ∈ Cb(R),

where (X , Y ) is N ([µ, µ], [σ2, σ2])-distributed.


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Remarks.

1.

Like classical case, these new LLN and CLT plays fundamental roles forthe 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 calculateE[X ] is to use our theorem, instead of calculating E[X ] = supθ∈Θ Eθ [X ];

3.

Many statistic methods must be revisited to clarify their meaning ofuncertainty: it’s very risky to treat this new notion in a classical way. Thiswill cause serious confusions.

4.

The third revolution: Deterministic point of view =⇒ Probabilistic pointof view =⇒ uncertainty of probabilities.


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Remarks.

1.


2.


3.


4.



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Remarks.

1.


2.


3.


4.



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Remarks.

1.


2.


3.


4.



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G–Brownian Motion

Definition

Under (Ω,F , E), a process Bt(ω) = ωt , t ≥ 0, is called a G–Brownianmotion if:

(i) Bt+s − Bs is N (0, [σ2t, σ2t]) distributed ∀ s , t ≥ 0

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


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G–Brownian Motion

Definition



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

For simplification, we set σ2 = 1.


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G–Brownian Motion

Definition



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


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Theorem

If, under some (Ω,F , E), 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

E[|Bt |3] = o(t).Then B is a G-Brownian motion.


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Theorem


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



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Theorem





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Theorem



E[|Bt |3] = o(t).

Then B is a G-Brownian motion.


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Theorem





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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’, itcan even be have very long memory. But it is i.i.d under the robustexpectation E. By which we can have many advantages in analysis,calculus and computation, compare with, e.g. fractal B.M.


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Ito’s integral of G–Brownian motion

For each process (ηt)t≥0 ∈ L2,0F (0, T ) of the form

ηt(ω) =N−1

∑j=0

ξj (ω)I[tj ,tj+1)(t), ξj ∈L2(Ftj ) (Ftj -meas. & E[|ξj |2] < ∞)

we define

I (η) =∫ T

0η(s)dBs :=

N−1

∑j=0

ξj (Btj+1 − Btj ).

Lemma

We have

E[∫ T

0η(s)dBs ] = 0

and

E[(∫ T

0η(s)dBs)2] ≤

∫ T

0E[(η(t))2]dt.


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Ito’s integral of G–Brownian motion

For each process (ηt)t≥0 ∈ L2,0F (0, T ) of the form

ηt(ω) =N−1

∑j=0

ξj (ω)I[tj ,tj+1)(t), ξj ∈L2(Ftj ) (Ftj -meas. & E[|ξj |2] < ∞)

we define

I (η) =∫ T

0η(s)dBs :=

N−1

∑j=0

ξj (Btj+1 − Btj ).

Lemma

We have

E[∫ T

0η(s)dBs ] = 0

and

E[(∫ T

0η(s)dBs)2] ≤

∫ T

0E[(η(t))2]dt.


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Definition

Under the Banach norm ‖η‖2 :=∫ T0 E[(η(t))2]dt,

I (η) : L2,0(0, T ) 7→ L2(FT ) is a contract mapping

We then extend I (η) to L2(0, T ) and define, the stochastic integral∫ T

0η(s)dBs := I (η), η ∈ L2(0, T ).


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Lemma

We have(i)

∫ ts ηudBu =

∫ rs ηudBu +

∫ tr ηudBu .

(ii)∫ ts (αηu + θu)dBu = α

∫ ts ηudBu +

∫ ts θudBu, α ∈ L1(Fs)

(iii) E[X +∫ Tr ηudBu |Hs ] = E[X ], ∀X ∈ L1(Fs).


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Quadratic variation process of G–BM

We denote:

〈B〉t = B2t − 2

∫ t

0BsdBs = lim

max(tk+1−tk )→0

N−1

∑k=0

(BtNk+1− Btk )

2

〈B〉 is an increasing process called quadratic variation process of B.

E[ 〈B〉t ] = σ2t but E[− 〈B〉t ] = −σ2t

Lemma

Bst := Bt+s − Bs , t ≥ 0 is still a G-Brownian motion. We also have

〈B〉t+s − 〈B〉s ≡ 〈Bs〉t ∼ U ([σ2t, σ2t]).


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Quadratic variation process of G–BM

We denote:

〈B〉t = B2t − 2

∫ t

0BsdBs = lim

max(tk+1−tk )→0

N−1

∑k=0

(BtNk+1− Btk )

2

〈B〉 is an increasing process called quadratic variation process of B.

E[ 〈B〉t ] = σ2t but E[− 〈B〉t ] = −σ2t

Lemma

Bst := Bt+s − Bs , t ≥ 0 is still a G-Brownian motion. We also have

〈B〉t+s − 〈B〉s ≡ 〈Bs〉t ∼ U ([σ2t, σ2t]).


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We have the following isometry

E[(∫ T

0η(s)dBs)2] = E[

∫ T

0η2(s)d 〈B〉s ],

η ∈ M2G (0, T )


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Ito’s formula for G–Brownian motion

Xt = X0 +∫ t

0αsds +

∫ t

0ηsd 〈B〉s +

∫ t

0βsdBs

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) +∫ t

0Φx (Xu)βudBu +

∫ t

0Φx (Xu)αudu

+∫ t

0[Φx (Xu)ηu +

1

2Φxx (Xu)β2

u ]d 〈B〉u


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Ito’s formula for G–Brownian motion

Xt = X0 +∫ t

0αsds +

∫ t

0ηsd 〈B〉s +

∫ t

0βsdBs

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) +∫ t

0Φx (Xu)βudBu +

∫ t

0Φx (Xu)αudu

+∫ t

0[Φx (Xu)ηu +

1

2Φxx (Xu)β2

u ]d 〈B〉u


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Stochastic differential equations

Problem

We consider the following SDE:

Xt = X0 +∫ t

0b(Xs)ds +

∫ t

0h(Xs)d 〈B〉s +

∫ t

0σ(Xs)dBs , t > 0.

where X0 ∈ Rn is given

b, h, σ : Rn 7→ Rn are given Lip. functions.

The solution: a process X ∈ M2G (0, T ; Rn) satisfying the above SDE.

Theorem

There exists a unique solution X ∈ M2G (0, T ; Rn) of the stochastic

differential equation.


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Stochastic differential equations

Problem

We consider the following SDE:

Xt = X0 +∫ t

0b(Xs)ds +

∫ t

0h(Xs)d 〈B〉s +

∫ t

0σ(Xs)dBs , t > 0.

where X0 ∈ Rn is given

b, h, σ : Rn 7→ Rn are given Lip. functions.

The solution: a process X ∈ M2G (0, T ; Rn) satisfying the above SDE.

Theorem

There exists a unique solution X ∈ M2G (0, T ; Rn) of the stochastic

differential equation.


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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 Markovprocess.

New Feynman-Kac formula for fully nonlinear PDE:path-interpretation.

u(t, x) = Ex [(Bt) exp(∫ t

0c(Bs)ds)]

∂tu = G (D2u) + c(x)u, u|t=0 = ϕ(x).

Fully nonlinear Monte-Carlo simulation.

BSDE driven by G -Brownian motion: a challenge.


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Prospectives





0c(Bs)ds)]

∂tu = G (D2u) + c(x)u, u|t=0 = ϕ(x).




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Prospectives





0c(Bs)ds)]

∂tu = G (D2u) + c(x)u, u|t=0 = ϕ(x).




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Prospectives





0c(Bs)ds)]

∂tu = G (D2u) + c(x)u, u|t=0 = ϕ(x).




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The talk is based on:

Peng, S. (2006) G–Expectation, G–Brownian Motion and RelatedStochastic Calculus of Ito’s type, in arXiv:math.PR/0601035v2 3Jan2006, to appear in Proceedings of 2005 Abel Symbosium, Springer.

Peng, S. (2006) Multi-Dimensional G-Brownian Motion and RelatedStochastic Calculus under G-Expectation, inarXiv:math.PR/0601699v2 28Jan 2006.

Peng, S. Law of large numbers and central limit theorem undernonlinear 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 capacityrelated to a Sublinear Expectation: application to G-Brownian MotionPathes, see arXiv:0802.1240v1 [math.PR] 9 Feb 2008.

Song, Y. (2007) A general central limit theorem under Peng’sG-normal distribution, Preprint.


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Thank you,


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In the classic period of Newton’s mechanics, including A. Einstein, peoplebelieve that everything can be deterministically calculated. The lastcentury’s research affirmatively claimed the probabilistic behavior of ouruniverse: God does plays dice!Nowadays people believe that everything has its own probabilitydistribution. But a deep research of human behaviors shows that foreverything involved human or life such, as finance, this may not be true: aperson or a community may prepare many different p.d. for your selection.She change them, also purposely or randomly, time by time.


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