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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