A note on G -normal distributions

Accepted Manuscript A note on G-normal distributions Yongsheng Song PII: DOI: Reference:

S0167-7152(15)00247-3 http://dx.doi.org/10.1016/j.spl.2015.07.014 STAPRO 7354

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Statistics and Probability Letters

Received date: 16 January 2015 Accepted date: 8 July 2015 Please cite this article as: Song, Y., A note on G-normal distributions. Statistics and Probability Letters (2015), http://dx.doi.org/10.1016/j.spl.2015.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A Note On G-normal Distributions Yongsheng Song∗ January 16, 2015

Abstract As is known, the convolution µ ∗ ν of two G-normal distributions µ, ν with different intervals of variances may not be G-normal. We shows that µ ∗ ν is a G-normal distribution σ if and only if σµ = σσν . µ

ν

Key words: G-normal distribution, Cramer’s decomposition MSC-classification: 60E10

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Introduction

Peng (2007) introduced the notion of G-normal distribution via the viscosity solutions of the G-heat equation below ∂t u − G(∂x2 u) = 0, (t, x) ∈ (0, ∞) × R, u(0, x) = ϕ(x),

where G(a) = 21 (σ 2 x+ − σ 2 x− ), a ∈ R with 0 ≤ σ ≤ σ < ∞, and ϕ ∈ Cb,Lip (R), the collection of bounded Lipstchiz functions on R. Then the one-dimensional G-normal distribution is defined by NG [ϕ] = uϕ (1, 0), where uϕ is the viscosity solution to the G-heat equation with the initial value ϕ. We denote by G 0 the collection of functions G defined above. For G1 , G2 ∈ G 0 , the convolution between NG1 , NG2 is defined as NG1 ∗ NG2 [ϕ] := NG1 [φ] with φ(x) := NG2 [ϕ(x + ·)]. As is well-known, the convolution of two normal distributions is also normal. How about G-normal distributions? We state the question in the PDE language: For G1 , G2 ∈ G 0 , set G(t, a) = G2 (a), t ∈ (0, 21 ] and G(t, a) = G1 (a), t ∈ ( 21 , 1]. Let v ϕ be the viscosity solution to the following PDE ∂t v − G(t, ∂x2 v) = 0,

v(0, x) = ϕ(x).

Can we find a function G ∈ G 0 such that v ϕ (1, 0) = uϕ (1, 0), for all ϕ ∈ Cb,Lip (R)? ∗ Academy of Mathematics and Systems Science, CAS, Beijing, China, [email protected]. Research supported by by NCMIS; Youth Grant of NSF (No. 11101406); Key Project of NSF (No. 11231005); Key Lab of Random Complex Structures and Data Science, CAS (No. 2008DP173182).

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Here uϕ is the viscosity solution to the G-heat equation. However, [2] gave a counterexample to show that generally NG1 ∗ NG2 is not G-normal any more. We denote by G the subset of G 0 consisting of the elements with σ > 0. NG is called non-degenerate if G belongs to G. For G ∈ G, set βG :=

σ σ+σ and σG = . σ 2

For abbreviation, we write β, σ instead of βG , σG when no confusion can arise. We shall prove that NG1 ∗ NG2 is a G-normal distribution if and only if βG1 = βG2 . Besides, the convolution between G-normal distributions is not commutative. We also prove that NG1 ∗ NG2 = NG2 ∗ NG1 if and only if βG1 = βG2 . Therefore, in order to emphasize the importance of the ratio β, we denote the G-normal distribution NG by Nβ (0, σ 2 ).

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Characteristic Functions

First we shall consider the solutions of special forms to the G-heat equation ∂t u − G(∂x2 u) = 0.

(2.1)

Assume that u(t, x) = a(t)φ(x) with a(t) ≥ 0 is a solution to the G-heat equation (2.1). Then we conclude that a0 (t) G(φ00 (x)) = a(t) φ(x) 2

is a constant. Assuming that they are equal to − ρ2 , we have a(t) = e− G(φ00 (x)) = −

ρ2 t 2

and

ρ2 φ(x). 2

(2.2)

If φ00 (x) is positive, the equation (2.2) reduces to 00

φ (x) = −

ρ2 φ(x). σ2

(2.3)

and the solution is

ρ φ(x) = λ cos( x + θ). σ 00 On the other hand, if φ (x) is negative, the equation (2.2) reduces to φ00 (x) = − and the solution is

ρ2 φ(x). σ2

(2.4)

ρ φ(x) = λ cos( x + θ). σ

Motivated by the arguments above, we shall construct the solutions to the equation (2.2) in the following way. Denoting by β = σσ , set φβ (x) =

(

2 1+β 2β 1+β

cos( 1+β 2 x) cos( 1+β 2β x +

β−1 2β π)

2

π π for x ∈ [− 1+β , 1+β );

π for x ∈ [ 1+β , (2β+1)π 1+β ).

(2.5)

Figure 1: φβ (x) This is a variant of the trigonometric function cos x (see Figure 1). Then extend the definition of φβ to the whole real line by the property φ(x + 2kπ) = φ(x), k ∈ Z. Clearly, φ1 (x) = cos x and φβ belongs to C 2,1 (R), the space of bounded functions on R with uniformly Lipschitz continuous second-order derivatives. Proposition 2.1 φβ is a solution to equation (2.2) with ρ =

σ+σ 2

=: σ.

π π Proof. The proof follows from simple calculations. For x ∈ [− 1+β , 1+β ),

φ00β (x) = −

1+β 1+β (1 + β)2 cos( x) = − φβ (x) ≤ 0. 2 2 4

So G(φ00β (x)) = −

(1 + β)2 σ 2 σ2 φβ (x) = − φβ (x). 8 2

π For x ∈ [ 1+β , (2β+1)π 1+β ),

φ00β (x) = −

1+β 1+β β−1 (1 + β)2 cos( x+ π) = − φβ (x) ≥ 0. 2β 2β 2β 4β 2

So G(φ00β (x)) = −

(1 + β)2 σ 2 σ2 φ (x) = − φβ (x). β 8β 2 2

 σ+σ 2

and β := σσ . e−

σ2 t 2

φβ (x) is the solution to equation (2.1) with the √ σ2 initial value φβ (x). In other words, NG [φβ (x + t·)] = e− 2 t φβ (x). Corollary 2.2 Let σ :=

For λ > 0 and c, θ ∈ R, set φλ,c,θ (x) := λφβ (cx + θ). It’s easy to check that β G((φλ,c,θ )00 ) = − β

c2 σ 2 λ,c,θ φ . 2 β

c2 σ 2

So e− 2 t φλ,c,θ (x) is the solution to equation (2.1) with the initial value φλ,c,θ (x). For any β β β > 1, we call (φλ,c,θ (x))λ,c,θ β the characteristic functions of G-normal distributions Nβ (0, σ 2 ).

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3

Cramer’s Type Decomposition for G-normal Distributions

Let’s introduce more properties on the characteristic functions {φβ (x)}β≥1 . For 1 ≤ α < β < ∞, we have 2(β − α) φα (x) − φβ (x) ≥ =: e(α, β) > 0. (1 + α)(1 + β)

Figure 2: φα (x) and φβ (x) Theorem 3.1 For G1 , G2 ∈ G, the convolution NG1 ∗ NG2 is a G-normal distribution if and σ σ only if βG1 := σG1 = σG2 =: βG2 . G1

G2

Proof. The sufficiency is obvious, and we only prove the necessity. Assume N := NG1 ∗ NG2 is a G-normal distribution. Then σ 2N = σ 2G1 + σ 2G2 and σ 2N = σ 2G1 + σ 2G2 . If βG1 6= βG2 , we may assume that βG1 < βG2 , and the other case can be proved similarly. Then we have βG1 < βN < βG2 . On one hand we have 2 σN √ N [φβN ( t·)] = e− 2 t φβN (0) =

2 σN 2 e− 2 t . 1 + βN

On the other hand we have √ NG2 [φβN ( t(x + ·))] √ ≥ NG2 [φβG2 ( t(x + ·))] + e(βN , βG2 ) = e−

2 σG 2 2

t

√ φβG2 ( tx)] + e(βN , βG2 ) 2

≥−

2βG2 − σG2 t e 2 + e(βN , βG2 ). 1 + βG2

Consequently, 2 √ √ 2βG2 − σG2 t N [φβN ( t·)] = NG1 ∗ NG2 [φβN ( t·)] ≥ − e 2 + e(βN , βG2 ). 1 + βG2

Combining the above arguments we have 2

2 σN 2 2βG2 − σG2 t e− 2 t ≥ − e 2 + e(βN , βG2 ), for any t > 0, 1 + βN 1 + β G2

which is a contradiction noting that e(βN , βG2 ) > 0.  4

Theorem 3.2 For G1 , G2 ∈ G, NG1 ∗ NG2 = NG2 ∗ NG1 if and only if βG1 = βG2 . Proof. We shall only prove the necessity. Assume that βG1 < βG2 . Then √

NG1 [φβG1 ( tx + So NG2

√

t·)] = e

−

2 σG 1 2

t

√ φβG1 ( tx) ≤

√ ∗ NG1 [φβG1 ( t·)] ≤

2

σG 2 1 e− 2 t . 1 + βG1

2

σG 2 1 e− 2 t . 1 + βG1

On the other hand √ √ NG2 [φβG1 ( tx + t·)] √ √ ≥ NG2 [φβG2 ( tx + t·)] + e(βG1 , βG2 ) =e

−

2 σG 2 2

t

√ φβG2 ( tx) + e(βG1 , βG2 ) 2

≥−

2βG2 − σG2 t e 2 + e(βG1 , βG2 ), 1 + βG2

which implies that 2 √ 2βG2 − σG2 t NG1 ∗ NG2 [φβG1 ( t·)] ≥ − e 2 + e(βG1 , βG2 ). 1 + β G2

Noting that e(βG1 , βG2 ) > 0, we have √ √ NG1 ∗ NG2 [φβG1 ( t·)] > NG2 ∗ NG1 [φβG1 ( t·)] for t large enough. 

References [1] Peng, S.(2007) G-expectation, G-Brownian Motion and Related Stochastic Calculus of Itˆ o type, Stochastic analysis and applications, 541-567, Abel Symp., 2, Springer, Berlin. [2] Hu, M. (2010) Nonlinear Expectations and Related Topics, Doctoral thesis.

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