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A New Central Limit Theorem & Law of LargeNumbers under Mean and Variance Uncertainty
and Applications to Finance
Shige Peng
Institute of Mathematics, Qilu Institute of FinanceShandong University
Workshop on
“Finance and Related Mathematical and Statistical Issues”September 3-6, 2008, Kyoto Research Park, Kyoto, Japan
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Example:A typical financial product and related risk
Φ(X1 + X2 + · · ·+ XN ), Xi small, N large.
where Φ(x) = max0, x− k. An important problem is how tocalculate the price and the related risk. A popular method is to usethe central limit theorem (CLT) and/or the law of large number(LLN). For this some basic assumption for Xi such as iid is needed.
For many situations this assumption is too strong.But banks need to calculate their risk regardless the rigorous
conditions: sometimes it works, some times it fails.
1
N
N∑i=1
Xi∼= 0,
1√N
N∑i=1
X2i∼= σ2
=⇒ Φ(X1 + X2 + · · ·+ XN ) ∼=1√
2πσ2
∫ ∞
−∞Φ(x) exp[
−x2
σ2]dx
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
A sublinear expectation is also called:
* upper expectation (Robust Statistics, [P. Huber1987, P. Huber& Strassen 1973]);
* coherent expectation, coherent prevision [P. Walley,1991, Statistical Reasoning with ImpreciseProbabilities];
* coherent risk measure [ADEH1991];
* Choquet expectation in potential theory [Choquet1953] is also a type of sublinear expectation
A more generalized form is convex risk measure, or convexexpectation.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation, from [Knight 1921], [Keynes 1936]
Knight, F.H. (1921), Risk, Uncertainty, and Profit
”Mathematical, or a priori, type of probability is practically nevermet with in business ... business decisions, for example, deal withsituations which are far too unique, generally speaking,, for anysort of statistical tabulation to have any value for guidance ... (sothat) the concept of an objectively measurable probability orchance is simply inapplicable .”
The framework of sublinear expectation can take the uncertaintyinto consideration, in a systematic, beautiful and robust way.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation, from [Knight 1921], [Keynes 1936]
Knight, F.H. (1921), Risk, Uncertainty, and Profit
”Mathematical, or a priori, type of probability is practically nevermet with in business ... business decisions, for example, deal withsituations which are far too unique, generally speaking,, for anysort of statistical tabulation to have any value for guidance ... (sothat) the concept of an objectively measurable probability orchance is simply inapplicable .”
The framework of sublinear expectation can take the uncertaintyinto consideration, in a systematic, beautiful and robust way.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Space of random variables (Risk losses) (See [ADEH1999],[FS2004]
(Ω,F): a measurable space;
H a linear space of real functions (random variables, risklosses) defined on Ω) s.t.
X1, · · · , Xn ∈ H ⇒ ϕ(X) ∈ H, ∀ϕ ∈ Cb(Rn)
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Space of random variables (Risk losses) (See [ADEH1999],[FS2004]
(Ω,F): a measurable space;
H a linear space of real functions (random variables, risklosses) defined on Ω) s.t.
X1, · · · , Xn ∈ H ⇒ ϕ(X) ∈ H, ∀ϕ ∈ Cb(Rn)
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation
Definition
A nonlinear expectation: a functional E : H 7→ R
(a) Monotonicity: if X ≥ Y then E[X] ≥ E[Y ].
(b) Cash invariant: E[X + c] = E[X] + c.
A sublinear expectation: (a)+(b)+
(c) Convexity:
E[αX + (1− α)Y ] ≤ αE[X] + (1− α)E[Y ], α ∈ [0, 1].
(d) Positive homogeneity: E[λX] = λE[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation
Definition
A nonlinear expectation: a functional E : H 7→ R
(a) Monotonicity: if X ≥ Y then E[X] ≥ E[Y ].
(b) Cash invariant: E[X + c] = E[X] + c.
A sublinear expectation: (a)+(b)+
(c) Convexity:
E[αX + (1− α)Y ] ≤ αE[X] + (1− α)E[Y ], α ∈ [0, 1].
(d) Positive homogeneity: E[λX] = λE[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation
Definition
A nonlinear expectation: a functional E : H 7→ R
(a) Monotonicity: if X ≥ Y then E[X] ≥ E[Y ].
(b) Cash invariant: E[X + c] = E[X] + c.
A sublinear expectation: (a)+(b)+
(c) Convexity:
E[αX + (1− α)Y ] ≤ αE[X] + (1− α)E[Y ], α ∈ [0, 1].
(d) Positive homogeneity: E[λX] = λE[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation
Definition
A nonlinear expectation: a functional E : H 7→ R
(a) Monotonicity: if X ≥ Y then E[X] ≥ E[Y ].
(b) Cash invariant: E[X + c] = E[X] + c.
A sublinear expectation: (a)+(b)+
(c) Convexity:
E[αX + (1− α)Y ] ≤ αE[X] + (1− α)E[Y ], α ∈ [0, 1].
(d) Positive homogeneity: E[λX] = λE[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sublinear Expectation
Definition
A nonlinear expectation: a functional E : H 7→ R
(a) Monotonicity: if X ≥ Y then E[X] ≥ E[Y ].
(b) Cash invariant: E[X + c] = E[X] + c.
A sublinear expectation: (a)+(b)+
(c) Convexity:
E[αX + (1− α)Y ] ≤ αE[X] + (1− α)E[Y ], α ∈ [0, 1].
(d) Positive homogeneity: E[λX] = λE[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Coherent Risk Measures and Sunlinear Expectations
When X is risk position, −X is the loss.
ρ(X) := E[−X]
Definition–Coherent risk measure
ρ(X) : H 7→ R is a coherent risk measure if it satisfies:
(a) Monotonicity: if X≥Y then ρ[X]≤ρ[Y ].
(b) Constant translatability: ρ[X+c] = ρ[X]−c.
(c) Convexity: (or self–dominated property):
ρ[αX + (1− α)Y ] ≤ αρ[X] + (1− α)ρ[Y ].
(d) Positive homogeneity: ρ[λX] = λρ[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Coherent Risk Measures and Sunlinear Expectations
When X is risk position, −X is the loss.
ρ(X) := E[−X]
Definition–Coherent risk measure
ρ(X) : H 7→ R is a coherent risk measure if it satisfies:
(a) Monotonicity: if X≥Y then ρ[X]≤ρ[Y ].
(b) Constant translatability: ρ[X+c] = ρ[X]−c.
(c) Convexity: (or self–dominated property):
ρ[αX + (1− α)Y ] ≤ αρ[X] + (1− α)ρ[Y ].
(d) Positive homogeneity: ρ[λX] = λρ[X], ∀λ ≥ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Proposition
ρ is a coherent risk measureif and only ifE[·] = ρ(−·) is a sublinear expectation.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Meaning of the robust representation:Statistic model uncertainty
E[X] = supP∈P
EP [X], ∀X ∈ H.
The size of the uncertainty 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Why introduce (Ω,H, E)?
Sublinear expectation in a Proba. space
Let (Ω,F ,P ) be a prob. space with BM (Wt)t≥0 andFt := σWs : s ≤ t.
g-expectation [P. 1997]
For each X ∈ L2(FT ) we consider the following BSDE:
yt = X +
∫ T
tg(zs)ds−
∫ T
tzsdWs, t ∈ [0, T ].
here g : Rd 7→ R is a sublinear function. We define
Eg[X] := y0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Why introduce (Ω,H, E)?
Sublinear expectation in a Proba. space
Let (Ω,F ,P ) be a prob. space with BM (Wt)t≥0 andFt := σWs : s ≤ t.
g-expectation [P. 1997]
For each X ∈ L2(FT ) we consider the following BSDE:
yt = X +
∫ T
tg(zs)ds−
∫ T
tzsdWs, t ∈ [0, T ].
here g : Rd 7→ R is a sublinear function. We define
Eg[X] := y0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Why introduce (Ω,H, E)?
Sublinear expectation in a Proba. space
Let (Ω,F ,P ) be a prob. space with BM (Wt)t≥0 andFt := σWs : s ≤ t.
g-expectation [P. 1997]
For each X ∈ L2(FT ) we consider the following BSDE:
yt = X +
∫ T
tg(zs)ds−
∫ T
tzsdWs, t ∈ [0, T ].
here g : Rd 7→ R is a sublinear function. We define
Eg[X] := y0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
There exists a convex subset Λ ⊂ Rd s.t.
g(z) := supλ∈Λ
〈λ, z〉
It is proved that
Eg[X] = supη∈L∞F (0,T ;Λ)
Eη[X],dPη
dP= exp
∫ T
0ηtdWt −
1
2
∫ T
0|ηt|2dt
Meaning
We take
P := Pη P, Bt = Wt +
∫ t
0ηsds is BM under Pη
as the uncertainty subset.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
There exists a convex subset Λ ⊂ Rd s.t.
g(z) := supλ∈Λ
〈λ, z〉
It is proved that
Eg[X] = supη∈L∞F (0,T ;Λ)
Eη[X],dPη
dP= exp
∫ T
0ηtdWt −
1
2
∫ T
0|ηt|2dt
Meaning
We take
P := Pη P, Bt = Wt +
∫ t
0ηsds is BM under Pη
as the uncertainty subset.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
There exists a convex subset Λ ⊂ Rd s.t.
g(z) := supλ∈Λ
〈λ, z〉
It is proved that
Eg[X] = supη∈L∞F (0,T ;Λ)
Eη[X],dPη
dP= exp
∫ T
0ηtdWt −
1
2
∫ T
0|ηt|2dt
Meaning
We take
P := Pη P, Bt = Wt +
∫ t
0ηsds is BM under Pη
as the uncertainty subset.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The advantages of using g-expectation:
For state–dependent risk positions (e.g. X = Φ(ST )) we caneven apply nonlinear Feynman-Kac formula ([P. 1991],[Pardoux & P. 1992], [El Karoui, P. & Quenez 1997]) to usePDE methods.
Even if X is a path-dependent risk position (e.g.X = Ψ(St0≤t≤T )), we still can use many existing backwardalgorithms to calculate E[X]
In general directly calculating supPη∈P Eη[·] is impossible.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The advantages of using g-expectation:
For state–dependent risk positions (e.g. X = Φ(ST )) we caneven apply nonlinear Feynman-Kac formula ([P. 1991],[Pardoux & P. 1992], [El Karoui, P. & Quenez 1997]) to usePDE methods.
Even if X is a path-dependent risk position (e.g.X = Ψ(St0≤t≤T )), we still can use many existing backwardalgorithms to calculate E[X]
In general directly calculating supPη∈P Eη[·] is impossible.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The advantages of using g-expectation:
For state–dependent risk positions (e.g. X = Φ(ST )) we caneven apply nonlinear Feynman-Kac formula ([P. 1991],[Pardoux & P. 1992], [El Karoui, P. & Quenez 1997]) to usePDE methods.
Even if X is a path-dependent risk position (e.g.X = Ψ(St0≤t≤T )), we still can use many existing backwardalgorithms to calculate E[X]
In general directly calculating supPη∈P Eη[·] is impossible.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
[Coquet-Hu-Memin-P. 2002, PTRF], [P. 2004, CIME]:
If E is a dynamic nonlinear expectation in (Ω,F , Ftt≥0, P ) andis dominated by Eg, i.e.
E[X]− E[Y ] ≤ Eg[X − Y ], ∀X, Y ∈ L∞(FT )
then there exists a unique function g dominated by g, i.e.,g(x)− g(y) ≤ g(x− y), such that E[X] = Eg[X], for allX ∈ L∞(FT ).
[Delbaen-P. -Rosassa Gianin,2008,arxiv]
If E is a convex dynamic nonlinear expectation in(Ω,F , Ftt≥0, P ), then there exists a unique convex function gsuch that E = Eg.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
[Coquet-Hu-Memin-P. 2002, PTRF], [P. 2004, CIME]:
If E is a dynamic nonlinear expectation in (Ω,F , Ftt≥0, P ) andis dominated by Eg, i.e.
E[X]− E[Y ] ≤ Eg[X − Y ], ∀X, Y ∈ L∞(FT )
then there exists a unique function g dominated by g, i.e.,g(x)− g(y) ≤ g(x− y), such that E[X] = Eg[X], for allX ∈ L∞(FT ).
[Delbaen-P. -Rosassa Gianin,2008,arxiv]
If E is a convex dynamic nonlinear expectation in(Ω,F , Ftt≥0, P ), then there exists a unique convex function gsuch that E = Eg.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
BUT for an important situation: volatility uncertainty
Risky asset model in Black-Scholes-Merton:
dSt = µStdt + σStdWt
Thusd ln(St) = µdt + σdWt
But data analysis only supports
1
n
n∑i=1
(lnSti+1 − lnSti)2 ∈ [σ2, σ2]
meaning:d 〈ln(S)〉t
dt∈ [σ2, σ2]
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Meaning: the uncertainty subset P is:
P := Pv : Bvt =
∫ t
0vsdWs is BM under Pv, vt ∈ L2
F (0, T ; [σ, σ])
P,Q ∈ P cannot be absolutely continuous with each other!
Example. LetBv, vt ≡ σ is a BM under P ,Bv, vt ≡ σ is a BM under PIf P P then 〈σW 〉t ≡ 〈σW 〉t, P -a.s =⇒ σ = σ.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Meaning: the uncertainty subset P is:
P := Pv : Bvt =
∫ t
0vsdWs is BM under Pv, vt ∈ L2
F (0, T ; [σ, σ])
P,Q ∈ P cannot be absolutely continuous with each other!
Example. LetBv, vt ≡ σ is a BM under P ,Bv, vt ≡ σ is a BM under PIf P P then 〈σW 〉t ≡ 〈σW 〉t, P -a.s =⇒ σ = σ.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
We propose a natural solution
a
s if it is an expectation.
¡2-¿ In dynamical situation we have very simple method tocalculate E[X] instead of supP∈P EP [X].
¡3-¿ P ∈ P is absolutely continuous w.r.t. E:
‖X‖ := E[|X|] = 0, =⇒ EP [|X|] = 0.
‖ · ‖ is a very natural Banach norm.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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.
¡2-¿ FX [·] forms a sublinear expectation on Cb(Rn), thus
FX [ϕ] = supθ∈Θ
∫Rn
ϕ(x)Fθ(dy).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Definition
The distribution of X is said to be stronger than Y :
E[ϕ(X)] ≥ E[ϕ(Y )], ∀ϕ ∈ Cb(Rn).
Definition
X,Y are said to be identically distributed under a nonlinearexpectation E[ϕ(X)], (X ∼ Y , or X is a copy of Y ), if they havesame distributions:
E[ϕ(X)] = E[ϕ(Y )], ∀ϕ ∈ Cb(Rn).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Definition
The distribution of X is said to be stronger than Y :
E[ϕ(X)] ≥ E[ϕ(Y )], ∀ϕ ∈ Cb(Rn).
Definition
X,Y are said to be identically distributed under a nonlinearexpectation E[ϕ(X)], (X ∼ Y , or X is a copy of Y ), if they havesame distributions:
E[ϕ(X)] = E[ϕ(Y )], ∀ϕ ∈ Cb(Rn).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Remark.
X is stronger than Y in distribution means that the uncertainty ofX is bigger than that of Y . X ∼ Y means they have the samedegree of uncertainty.
Remark.
Whether X is distributionally stronger than Y can be verysubjective. In many cases, for the sake of simplification in riskmanagement, one can raise the degree of uncertainty of Y in orderto make X ∼ Y .
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Remark.
X is stronger than Y in distribution means that the uncertainty ofX is bigger than that of Y . X ∼ Y means they have the samedegree of uncertainty.
Remark.
Whether X is distributionally stronger than Y can be verysubjective. In many cases, for the sake of simplification in riskmanagement, one can raise the degree of uncertainty of Y in orderto make X ∼ Y .
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Remark.
X is stronger than Y in distribution means that the uncertainty ofX is bigger than that of Y . X ∼ Y means they have the samedegree of uncertainty.
Remark.
Whether X is distributionally stronger than Y can be verysubjective. In many cases, for the sake of simplification in riskmanagement, one can raise the degree of uncertainty of Y in orderto make X ∼ Y .
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Independence in a nonlinear expectation space (Ω,H, E)
Definition
An m-dim. r.v. Y is said to be independent to a n-dim. r.v. X if:
E[ϕ(X, Y )] = E[E[ϕ(x, Y )]x=Y ], ∀ϕ ∈ Cb(Rn × Rm).
Remark.
Meaning: a realization of X (X = x) does not change (improve)the distributional uncertainty of Y .
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Lemma
(Existence of independent copies) Fix the distributions of X andY , we can find (X, Y ) such that X is a copy of X (i.e. X ∼ X),Y is a copy of Y (Y ∼ Y ) and such that Y is independent to X.The distribution of and (X, Y ) is uniquely determined.
Remark.
If Xi+1 is independent to Xi for each i Then the computationalcomplexity of E[ϕ(X1, · · · , Xk)] will enormously be reduced:From order mk to k ×m.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Independence under (Ω,H, E) can be subjective
Example
X, Z: two r.v. in (Ω,F , P ) a classical prob. space ,Y = h(η(X), Z), Z is independent to X under P .If about the function η(x): we only know η(x) ∈ Θ.The robust expectation of ϕ(X, Y ):
E[ϕ(X, Y )] := EP
[supθ∈Θ
EP [ϕ(x, h(θ, Z))]
x=X
].
Y is not independent to X w.r.t. Pbut is independent under E.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The notion of independence
Example (An extreme example)
In reality, Y = X(Thus Y is not independent to Xbut our information is still so poorand we know nothing about the relation of X and Y , the onlyinformation we know is: X(ω), Y (ω) ∈ 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The notion of independence
Remark.
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The notion of independence
Remark.
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The notion of independence
Remark.
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Classical Normal distribution by P. Levy’s observation
If X ∼ N (0, σ2) and Y is an independent copy of X, thenaX + bY ∼ N (0, (a2 + b2)σ2). ConsequentlyaX + bY ∼
√a2 + b2X, for all a, b ∈ R.
The converse is true:
aX + bY ∼√
a2 + b2X, ∀a, b ∈ R =⇒ X ∼ N (0, σ2),
where Y is an independent copy of X, σ2 = E[X2].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Classical Normal distribution by P. Levy’s observation
If X ∼ N (0, σ2) and Y is an independent copy of X, thenaX + bY ∼ N (0, (a2 + b2)σ2). ConsequentlyaX + bY ∼
√a2 + b2X, for all a, b ∈ R.
The converse is true:
aX + bY ∼√
a2 + b2X, ∀a, b ∈ R =⇒ X ∼ N (0, σ2),
where Y is an independent copy of X, σ2 = E[X2].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Normal distribution under (Ω,H, E)
An fundamentally important distribution
Definition (Normal distribution under (Ω,H, E))
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 can check: E[X] = E[−X] = 0.
We also denote X ∼ N (0, [σ2, σ2]), where
σ2 := E[X2], σ2 := −E[−X2].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Normal distribution under (Ω,H, E)
An fundamentally important distribution
Definition (Normal distribution under (Ω,H, E))
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 can check: E[X] = E[−X] = 0.
We also denote X ∼ N (0, [σ2, σ2]), where
σ2 := E[X2], σ2 := −E[−X2].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Normal distribution under (Ω,H, E)
An fundamentally important distribution
Definition (Normal distribution under (Ω,H, E))
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 can check: E[X] = E[−X] = 0.
We also denote X ∼ N (0, [σ2, σ2]), where
σ2 := E[X2], σ2 := −E[−X2].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
If X ∼ X ∼ N (0, [σ2, σ2]), then
(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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
If X ∼ X ∼ N (0, [σ2, σ2]), then
(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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Remark.
If σ2 = σ2, then N (0; [σ2, σ2]) = N (0, σ2).
Remark.
The larger to [σ2, σ2] the stronger the uncertainty.
Remark.
But X ∼ N (0; [σ2, σ2]) does not simply implies
E[ϕ(X)] = supσ∈[σ2,σ2]
1√2πσ
∫ ∞
−∞ϕ(x) exp−x2
2σdx
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Cases with mean-uncertainty: U([µ, µ])-distribution
What happens if E[X1] > −E[−X1]?
Definition
A random variable Y in (Ω,H, E) is U([µ, µ])-distributed(Y ∼ U([µ, µ])) if
aY + bY ∼ (a + b)Y, ∀a, b ≥ 0.
where Y is an independent copy of Y , whereµ := E[Y ] > µ := −E[−Y ]
We can prove that
E[ϕ(Y )] = supy∈[µ,µ]
ϕ(y).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Cases with mean-uncertainty: U([µ, µ])-distribution
What happens if E[X1] > −E[−X1]?
Definition
A random variable Y in (Ω,H, E) is U([µ, µ])-distributed(Y ∼ U([µ, µ])) if
aY + bY ∼ (a + b)Y, ∀a, b ≥ 0.
where Y is an independent copy of Y , whereµ := E[Y ] > µ := −E[−Y ]
We can prove that
E[ϕ(Y )] = supy∈[µ,µ]
ϕ(y).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Mean-uncertainty: How to calculate U([µ, µ])-distribution
u(t, x) = E[ϕ(x + tY )]
∂tu = g(∂xu), u|t=0 = ϕ
g(x) = µx+ + µx−
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Theorem (Generally G-normal distribution is calculated by)
X ∼ N (0, [σ2, σ2]) in (Ω,H, E) iff for each ϕ ∈ Cb(R) thefunction
u(t, x) := E[ϕ(x +√
tX)], x ∈ R, t ≥ 0
is the solution of the PDE
ut = G(uxx), t > 0, x ∈ Ru|t=0 = ϕ,
where G(a) = 12(σ2a+−σ2a−)(= E[a
2X2]). G-normal distribution.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The generator G is a function defined by:
G(α) = E[αX2], ∀α ∈ R
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Law of Large Numbers (LLN), Central Limit Theorem(CLT)
Striking consequence of LLN & CLT
Accumulated independent and identically distributed randomvariables tends to a normal distributed random variable, whateverthe original distribution.
M
arinacci, M. Limit laws for non-additive probabilities and theirfrequentist interpretation, Journal of Economic Theory 84, 145-1951999. Nothing found for nonlinear CLT.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Law of Large Numbers (LLN), Central Limit Theorem(CLT)
Striking consequence of LLN & CLT
Accumulated independent and identically distributed randomvariables tends to a normal distributed random variable, whateverthe original distribution.
M
arinacci, M. Limit laws for non-additive probabilities and theirfrequentist interpretation, Journal of Economic Theory 84, 145-1951999. Nothing found for nonlinear CLT.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Definition
A sequence of random variables ηi∞i=1 in H is said to converge inlaw under E if the limit
limi→∞
E[ϕ(ηi)], for each ϕ ∈ Cb(R).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Law of large number under sunlinear expectation
Theorem
Let Xi∞i=1 in (Ω,H, E) be identically distributed: Xi ∼ X1, andeach Xi+1 is independent to (X1, · · · , Xi).We assume furthermore that E[|X1|1+α] < ∞.Sn := X1 + · · ·+ Xn.Then Sn/n converges in law to U(0; [µ, µ]):
limn→∞
E[ϕ(Sn
n)] = sup
µ∈[µ,µ]ϕ(µ), ∀ϕ ∈ Cb(R).
whereµ = E[X1], µ = −E[−X1].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Law of large number: Particularly
limn→∞
E[d[[µ, µ](Sn
n)] = sup
µ∈[µ,µ]d[[µ, µ](µ) = 0.
where dK(x) := inf|x− y| : y ∈ K.If µ = µ
limn→∞
E[|Sn
n− µ|] = 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Central Limit Theorem under Sublinear Expectation
Theorem
Let Xi∞i=1 in (Ω,H, E) be identically distributed: Xi ∼ X1, andeach Xi+1 is independent to (X1, · · · , Xi). We assumefurthermore that
E[|X1|2+α] < ∞ and E[X1] = E[−X1] = 0.
Sn := X1 + · · ·+ Xn. Then Sn/√
n converges in law toN (0; [σ2, σ2]):
limn→∞
E[ϕ(Sn√
n)] = E[ϕ(X)], ∀ϕ ∈ Cb(R),
where
X ∼ N (0, [σ2, σ2]), σ2 = E[X21 ], σ2 = −E[−X2
1 ].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Sketch of Proof: A new method
For a function ϕ ∈ CLip(R) and a small but fixed h > 0, let Vbe the solution of the PDE on (t, x) ∈ [0, 1]× R,
∂tV + G(∂2xxV ) = 0,
V |t=1 = ϕ.
We have, since X ∼ N (0, [σ2, σ2]),
V (t, x) = E[ϕ(x +√
1− tX)].
Particularly,
V (0, 0) = E[ϕ(X)], V (1, x) = ϕ(x).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
· · ·Sketch of Proof δ = 1/n. Thus E[ϕ(√
δSn)]− E[ϕ(X)]equals
E[V (1,√
δSn)− V (0, 0)] = En−1∑i=0
V ((i + 1)δ,√
δSi+1)− V (iδ,√
δSi)
by Taylor’s expansion = En−1∑i=0
(Iiδ + J i
δ), E[|J iδ|] ≤ Cδ1+α
Iiδ = ∂tV (iδ,
√δSi)δ + 1
2∂2xxV (iδ,
√δSi)X
2i+1δ
+∂xV (iδ,√
δSi)Xi+1
√δ.
We have
E[Iiδ] = E[∂tV (iδ,
√δSi) +
1
2∂2
xxV (iδ,√
δSi)X2i+1]δ
= E[∂tV + G(∂2xxV )(iδ,
√δSi)]δ = 0
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Case with mean and variance uncertainties
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[X2], σ2 := −E[−X], (E[X] = E[−X] = 0).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Theorem
(X, Y ) ∼ N ([µ, µ], [σ2, σ2]) in (Ω,H, E) iff for each ϕ ∈ Cb(R)the function
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 ∈ Ru|t=0 = ϕ,
whereG(p, a) := E[
a
2X2 + pY ].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
More general situation
Theorem
Let Xi + Yi∞i=1 be an independent and identically distributedsequence. 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/√
nconverges in law 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
G–Brownian
Definition
Under (Ω,F , E), a process Bt(ω) = ωt, t ≥ 0, is called aG–Brownian motion 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
G–Brownian
Definition
Under (Ω,F , E), a process Bt(ω) = ωt, t ≥ 0, is called aG–Brownian motion 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
G–Brownian
Definition
Under (Ω,F , E), a process Bt(ω) = ωt, t ≥ 0, is called aG–Brownian motion 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
G–Brownian
Remark.
Like N (0, [σ2, σ2])-distribution, the G-Brownian motionBt(ω) = ωt, t ≥ 0, can strongly correlated under the unknown‘objective probability’, it can even be have very long memory. Butit is i.i.d under the robust expectation E. By which we can havemany advantages in analysis, calculus and computation, comparewith, e.g. fractal B.M.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 ofB.
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 ofB.
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
We have the following isometry
E[(
∫ T
0η(s)dBs)
2] = E[
∫ T
0η2(s)d 〈B〉s],
η ∈ M2G(0, T )
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Prospectives
Risk measures and pricing under dynamic volatilityuncertainties ([A-L-P1995], [Lyons1995]) —for pathdependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinearMarkov process.
New Feynman-Kac formula for fully nonlinear PDE:path-interpretation.
u(t, x) = Ex[ϕ(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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Prospectives
Risk measures and pricing under dynamic volatilityuncertainties ([A-L-P1995], [Lyons1995]) —for pathdependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinearMarkov process.
New Feynman-Kac formula for fully nonlinear PDE:path-interpretation.
u(t, x) = Ex[ϕ(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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Prospectives
Risk measures and pricing under dynamic volatilityuncertainties ([A-L-P1995], [Lyons1995]) —for pathdependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinearMarkov process.
New Feynman-Kac formula for fully nonlinear PDE:path-interpretation.
u(t, x) = Ex[ϕ(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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Prospectives
Risk measures and pricing under dynamic volatilityuncertainties ([A-L-P1995], [Lyons1995]) —for pathdependent options;
Stochastic (trajectory) analysis of sublinear and/or nonlinearMarkov process.
New Feynman-Kac formula for fully nonlinear PDE:path-interpretation.
u(t, x) = Ex[ϕ(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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The sketch of proof: the existence of G-BM
Ω := (Rd)[0,∞), B(Ω) the σ-alg. generated by cylinder sets.Ω = Cd
0 (R+), ω0 = 0, equipped with the distance
ρ(ω1, ω2) :=∞∑i=1
2−i[( maxt∈[0,i]
|ω1t − ω2
t |) ∧ 1].
B(Ω) denotes the σ-algebra generated by all open sets.
Lip(Ω) := ϕ(Bt1 , Bt2 , · · · , Btn) : ∀n ≥ 1, t1, · · · , tn ∈ [0,∞),∀ϕ ∈ lip(Rd×n),
Lip(Ω) := ϕ(Bt1 , Bt2 , · · · , Btn) : ∀n ≥ 1, t1, · · · , tn ∈ [0,∞),∀ϕ ∈ lip(Rd×n),
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Lemma
For each fixed 0 ≤ t1 < t2 < · · · < tm < ∞ and for each sequenceϕn : n ≥ 1 ⊂ lip(Rd×m), ϕn ↓ 0, we haveE[ϕn(Bt1 , Bt2 , · · · , Btm)] ↓ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
We denoteT := t = (t1, . . . , tm) : ∀m ∈ N, 0 ≤ t1 < t2 < · · · < tm < ∞.
Lemma
Suppose E is a linear expectation dominated by E on Lip(Ω), thenthere exists a unique probability measure Q on (Ω,B(Ω)) such thatE[X] = EQ[X], ∀X ∈ Lip(Ω).
∀ t = (t1, . . . , tm) ∈ T and ϕn : n ≥ 1 ⊂ lip(Rd×m),ϕn ↓ 0,
Proof.
E[ϕn(Bt1 , Bt2 , · · · , Btm)] ≤ E[ϕn(Bt1 , Bt2 , · · · , Btm)] ↓ 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
By Daniell-Stone thm, ∃! extension prob. Qt on(Rd×m,B(Rd×m)) such that EQt [ϕ] = E[ϕ(Bt1 , Bt2 , · · · , Btm)],
∀ϕ ∈ lip(Rd×m).Qt : t ∈ T is consistent, then by Kolmogorov’s we extend Q
on (Ω,B(Ω)).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Lemma
∃ Pe on (Ω,B(Ω)) such thatE[X] = maxQ∈Pe EQ[X], ∀X ∈ Lip(Ω).
We set
c(A) := supQ∈Pe
Q(A), A ∈ B(Ω),
E[X] := supQ∈Pe
EQ[X].
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Lemma
For B = Bt : t ∈ [0,∞) , there exists a continuous modificationB = Bt : t ∈ [0,∞) of B (i.e. c(Bt 6= Bt) = 0, for eacht ≥ 0) such that B0 = 0.
Proof.
We know that E = E on Lip(Ω), thus
E[|Bt−Bs|4] = supQ∈Pe
EQ[|Bt−Bs|4] ≤ E[|Bt−Bs|4] = d|t−s|2,∀s, t,
By Kolmogorov’s criterion there exists a continuous modification Bof B. Since c(B0 6= 0) = 0, we can demand B0 = 0.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Now, we give the representation of G-expectation.
Theorem
We haveE[X] = max
P∈PEP [X], ∀X ∈ Lip(Ω).
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The completion of Lip(Ω)
With the Lp-distance, we denote:
Lpc the completion of Cb(Ω).
LpG(Ω) the completion of Lip(Ω).
We haveLp
c = LpG(Ω) ⊂ Lp
where
Lp := X ∈ L0(Ω) : E[|X|p] = supP∈P
EP [|X|p] < ∞
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The talk is based on:
Peng, S. (2006) G–Expectation, G–Brownian Motion andRelated Stochastic Calculus of Ito’s type, inarXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005Abel Symbosium, Springer.Peng, S. (2006) Multi-Dimensional G-Brownian Motion andRelated Stochastic Calculus under G-Expectation, inarXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA.Peng, S. Law of large numbers and central limit theoremunder nonlinear expectations, in arXiv:math.PR/0702358v1 13Feb 2007Peng, S. A New Central Limit Theorem under SublinearExpectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008Peng, S.L. Denis, M. Hu and S. Peng, Function spaces andcapacity related to a Sublinear Expectation: application toG-Brownian Motion Pathes, 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The talk is based on:
Peng, S. (2006) G–Expectation, G–Brownian Motion andRelated Stochastic Calculus of Ito’s type, inarXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005Abel Symbosium, Springer.Peng, S. (2006) Multi-Dimensional G-Brownian Motion andRelated Stochastic Calculus under G-Expectation, inarXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA.Peng, S. Law of large numbers and central limit theoremunder nonlinear expectations, in arXiv:math.PR/0702358v1 13Feb 2007Peng, S. A New Central Limit Theorem under SublinearExpectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008Peng, S.L. Denis, M. Hu and S. Peng, Function spaces andcapacity related to a Sublinear Expectation: application toG-Brownian Motion Pathes, 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The talk is based on:
Peng, S. (2006) G–Expectation, G–Brownian Motion andRelated Stochastic Calculus of Ito’s type, inarXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005Abel Symbosium, Springer.Peng, S. (2006) Multi-Dimensional G-Brownian Motion andRelated Stochastic Calculus under G-Expectation, inarXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA.Peng, S. Law of large numbers and central limit theoremunder nonlinear expectations, in arXiv:math.PR/0702358v1 13Feb 2007Peng, S. A New Central Limit Theorem under SublinearExpectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008Peng, S.L. Denis, M. Hu and S. Peng, Function spaces andcapacity related to a Sublinear Expectation: application toG-Brownian Motion Pathes, 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The talk is based on:
Peng, S. (2006) G–Expectation, G–Brownian Motion andRelated Stochastic Calculus of Ito’s type, inarXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005Abel Symbosium, Springer.Peng, S. (2006) Multi-Dimensional G-Brownian Motion andRelated Stochastic Calculus under G-Expectation, inarXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA.Peng, S. Law of large numbers and central limit theoremunder nonlinear expectations, in arXiv:math.PR/0702358v1 13Feb 2007Peng, S. A New Central Limit Theorem under SublinearExpectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008Peng, S.L. Denis, M. Hu and S. Peng, Function spaces andcapacity related to a Sublinear Expectation: application toG-Brownian Motion Pathes, 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
The talk is based on:
Peng, S. (2006) G–Expectation, G–Brownian Motion andRelated Stochastic Calculus of Ito’s type, inarXiv:math.PR/0601035v2 3Jan 2006, in Proceedings of 2005Abel Symbosium, Springer.Peng, S. (2006) Multi-Dimensional G-Brownian Motion andRelated Stochastic Calculus under G-Expectation, inarXiv:math.PR/0601699v2 28Jan 2006, to appear in SPA.Peng, S. Law of large numbers and central limit theoremunder nonlinear expectations, in arXiv:math.PR/0702358v1 13Feb 2007Peng, S. A New Central Limit Theorem under SublinearExpectations, arXiv:0803.2656v1 [math.PR] 18 Mar 2008Peng, S.L. Denis, M. Hu and S. Peng, Function spaces andcapacity related to a Sublinear Expectation: application toG-Brownian Motion Pathes, 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 A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
Thank you
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance
In the classic period of Newton’s mechanics, including A.Einstein, people believe that everything can be deterministicallycalculated. The last century’s research affirmatively claimed theprobabilistic behavior of our universe: God does plays dice!
Nowadays people believe that everything has its ownprobability distribution. But a deep research of human behaviorsshows that for everything involved human or life such, as finance,this may not be true: a person or a community may prepare manydifferent p.d. for your selection. She change them, also purposelyor randomly, time by time.
Shige Peng A New Central Limit Theorem & Law of Large Numbers under Mean and Variance Uncertaintyand Applications to Finance