law of large number and central limit theorem under...
Post on 02-Aug-2020
2 Views
Preview:
TRANSCRIPT
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
/ 53
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
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 2
/ 53
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
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 2
/ 53
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
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 2
/ 53
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
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 3
/ 53
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
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 3
/ 53
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
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 4
/ 53
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
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 4
/ 53
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)
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 5
/ 53
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)
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 5
/ 53
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)
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 5
/ 53
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
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 6
/ 53
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, · · ·
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 7
/ 53
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!!
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 8
/ 53
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!!
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 8
/ 53
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!!
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 8
/ 53
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
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 9
/ 53
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).
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 10
/ 53
The density of a normal distribution
p(x) = 1√2π
exp(−x2
2
)
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 11
/ 53
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.
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 12
/ 53
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.
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 12
/ 53
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.
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 12
/ 53
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
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 13
/ 53
‘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”
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 14
/ 53
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.
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 15
/ 53
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.
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 16
/ 53
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.
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 17
/ 53
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.
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 18
/ 53
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.
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 18
/ 53
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.
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 18
/ 53
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.
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 18
/ 53
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.
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 18
/ 53
Coherent Risk Measures and Sunlinear Expectations
H is the set of bounded and measurable random variables on (Ω,F ).
ρ(X ) := E[−X ]
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 19
/ 53
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)).
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 20
/ 53
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)).
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 20
/ 53
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
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 21
/ 53
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).
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 22
/ 53
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).
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 22
/ 53
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).
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 22
/ 53
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.
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 23
/ 53
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.
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 23
/ 53
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 .
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 24
/ 53
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 .
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 24
/ 53
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 .
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 25
/ 53
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.
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 26
/ 53
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.
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 26
/ 53
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.
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 26
/ 53
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 .
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 27
/ 53
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])
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 28
/ 53
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].
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 29
/ 53
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].
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 29
/ 53
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].
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 29
/ 53
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
∫ ∞
−∞ϕ(y) exp(− y2
2σ2)dy
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 30
/ 53
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
∫ ∞
−∞ϕ(y) exp(− y2
2σ2)dy
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 30
/ 53
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 () 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 31
/ 53
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.
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 32
/ 53
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 33
/ 53
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.
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 33
/ 53
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.
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 33
/ 53
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).
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 34
/ 53
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 ].
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 35
/ 53
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).
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 36
/ 53
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).
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 36
/ 53
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).
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 37
/ 53
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 ].
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 38
/ 53
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 39
/ 53
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.
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 39
/ 53
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.
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 40
/ 53
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.
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 40
/ 53
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.
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 40
/ 53
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.
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 40
/ 53
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.
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 41
/ 53
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.
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 41
/ 53
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.
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 41
/ 53
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.
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 42
/ 53
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.
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 42
/ 53
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.
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 42
/ 53
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.
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 42
/ 53
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.
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 42
/ 53
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.
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 43
/ 53
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.
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 44
/ 53
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.
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 44
/ 53
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 ).
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 45
/ 53
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).
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 46
/ 53
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]).
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 47
/ 53
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]).
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 47
/ 53
We have the following isometry
E[(∫ T
0η(s)dBs)2] = E[
∫ T
0η2(s)d 〈B〉s ],
η ∈ M2G (0, T )
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 48
/ 53
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
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 49
/ 53
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
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 49
/ 53
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.
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 50
/ 53
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.
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 50
/ 53
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.
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 51
/ 53
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.
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 51
/ 53
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.
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 51
/ 53
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.
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 51
/ 53
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.
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 52
/ 53
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.
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 52
/ 53
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.
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 52
/ 53
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.
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 52
/ 53
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.
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 52
/ 53
Thank you,
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 53
/ 53
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.
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 53
/ 53
top related