Stein Type Characterization for $G$-normal Distributions

In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N}[\varphi]=\max_{\mu\in\Theta}\mu[\varphi],\ \varphi\in C_{b,Lip}(\mathbb{R}),$ be a sublinear expectation. $\mathcal{N}$ is $G$-normal if and only if for any $\varphi\in C_b^2(\mathbb{R})$, we have \[\int_\mathbb{R}[\frac{x}{2}\varphi'(x)-G(\varphi"(x))]\mu^\varphi(dx)=0,\] where $\mu^\varphi$ is a realization of $\varphi$ associated with $\mathcal{N}$, i.e., $\mu^\varphi\in \Theta$ and $\mu^\varphi[\varphi]=\mathcal{N}[\varphi]$.

Then the one-dimensional G-normal distribution is defined by where u ϕ is the viscosity solution to the G-heat equation with the initial value ϕ. The above G-heat equation has a unique viscosity solution. We refer to [2] for the definition, existence, uniqueness and comparison theorem of this type of parabolic PDEs (see also [10] for this specific situation). In this article, we consider only the non-degenerate G, i.e., σ > 0. Then the above G-heat equation has a unique C 1,2 -solution (see, e.g., [6]). More precisely, there exists α ∈ (0, 1) such that for any 0 < a < b < ∞, By the comparison theorem of the G-heat equation, it can be easily checked that N G is a sublinear expectation on C b,Lip (R), i.e., a functional on C b,Lip (R) satisfies Moreover, N G is continuous from above: for ϕ n ∈ C b,Lip (R), ϕ n ↓ 0, we have N G [ϕ n ] ↓ 0.
A sublinear expectation with this property is called regular, which happens if and only if it can be represented as the supremum expectation of a tight family of probability measures Θ on (R, B(R)) (see [3]).
Throughout this article, we shall only consider sublinear expectations which are regular.
As a regular sublinear expectation, the G-normal distribution can be represented as where Θ G is a tight family of probability measures on (R, B(R)). For ϕ ∈ C b,Lip (R), we call µ ∈ Θ G a realization of ϕ associated with N G if N G [ϕ] = µ[ϕ]. To ensure that each ϕ ∈ C b,Lip (R) has a realization, Θ G will always be chosen as weakly compact.
As is well known, the fact that µ = N (0, σ 2 ) if and only if This is the characterization of the normal distribution presented in Stein (1972), which is the basis of Stein's method for normal approximation (see Chen, Goldstein and Shao (2011) and the references therein for more details).
What is the proper counterpart of (1.1) for G-normal distributions? An immediate conjecture should be ). However, the above equality does not hold generally as was pointed out in Hu et al. (2015) by a counterexample.
By calculating some examples, we try to find the proper generalization of (1.1) for G-normal distributions. Example 1.1. Set β = σ σ and σ = σ+σ 2 . Song (2015) defined a periodic function φ β as a variant of the trigonometric function cos x (see Figure 1).
It was proved that and that u(t, x) := e − 1 2 σ 2 t φ β (x) is a solution to the G-heat equation. Therefore considered as a function of s attains its maximum at s = t, we get by taking the formal derivation on the equality (1.3) with respect to t. On the other hand, Combining the above arguments, we get the following equality Inspired by this example, we predict the following result generally holds.
To convince ourselves, let us calculate another simple example.
It is easy to check that u(t, where µ t,x is a realization of φ(x + √ t·). By taking the formal derivation on (1.4) with respect to t, we get (1.6) Similarly, by taking the formal derivation on (1.4) with respect to x, we get (1.7) Note that (1.
More precisely, we have which is exactly the conclusion of Proposition 1.2.
Returning to the linear case, the closed linear span of the family of functions considered in either of the previous two examples is the space of continuous functions, which increases our confidence that the conclusion of Proposition 1.2 is correct.
Just like Stein's characterization of (classical) normal distributions, we are also concerned about the converse problem: Throughout this article, we suppose the following additional properties: (H1) Θ is weakly compact; Clearly, Θ and Θ w generate the same sublinear expectation on C b,Lip (R). Here, we emphasize by (H1) that Θ is weakly compact, which ensures that there exists a realization µ ϕ for any ϕ ∈ C b,Lip (R). (H2) is a condition (strictly) stronger than N [|x|] < ∞, but weaker than N [|x| α ] < ∞ for some α > 1, which is employed to ensure that the functions generated by N have better analytic properties.
Actually, we also find evidence for the converse statement from some simple examples.
So, from Hypothesis (SH), we get .
Our purpose is to prove the Stein type formula for G-normal distributions (Proposition 1.2) and its converse problem (Q). In order to do so, we first prove a weaker version of the Stein type characterization below.
Since (SH) implies (SHw), the necessity part of Theorem 1.5 follows from Proposition 1.2. At the same time, the converse argument (Q) follows from the sufficiency part of Theorem 1.5.
In Section 2, we provide several lemmas to show how the differentiation penetrates the sublinear expectations, which makes sense the "formal derivation" in the above examples. In Section 3, we give a proof to Theorem 1.5. We shall prove Proposition 1.2 in Section 5 based on the G-expectation theory, and as a preparation we list some basic definitions and notations concerning G-expectation in Section 4.

Some useful lemmas
We denote by C b,loc ([a, b] × R) the totality of such functions. For ψ ∈ C b,loc ([a, b] × R), we sometimes employ the following assumption: there exists a continuous functionψ t0 (x) such that at point t 0 ∈ [a, b] the properties below hold.
For a function α : if the corresponding limits exist.
] and µ n ∈ Θ tn , n ≥ 1, such that t n → t 0 and µ n weakly −−−−→ µ as n goes to infinity, we have µ ∈ Θ t0 . Denote by Θ t0 the totality of µ ∈ Θ t0 defined above corresponding to t n ↓ t 0 and by Θ t0 corresponding to t n ↑ t 0 .
Since ψ belongs to C b,loc ([a, b] × R), it is easy to prove that {α(t)} t∈[a,b] is continuous based on the assumption (H2) on N . By similar arguments, we can show that, as n goes to infinity, By the definition of the function α we have, for any µ δ ∈ Θ t0+δ , The last equality follows from Assumption (A1). Let δ n ↓ 0 be a sequence such that lim sup Since Θ is weakly compact, there exists a subsequence, also denoted by δ n , such that µ δn weakly −−−−→ µ ∈ Θ.
Applying Lemma 2.2, we shall present the derivative formulas for two types of functions in the remainder of this section.
For a sublinear expectation N on C b,Lip (R) and ϕ ∈ C b,Lip (R), set For a sublinear expectation N on C b,Lip (R) and ϕ ∈ C 2 b (R), we have, for t > 0, (2.14)

Similar relations hold for
Proof. We shall only give proof to (2.11) and (2.13). The other conclusions can be proved similarly.

Proof to Theorem 1.5
We shall prove Theorem 1.5 based mainly on the lemmas introduced in Section 2.
tξ)] is the solution to G-heat equation with initial value ϕ.
So, by Lemma 2.2, we have Sufficiency. Assume N is a sublinear expectation on C b,Lip (R) satisfying Hypothesis (SHw). For any ϕ ∈ C b,Lip (R), let v(t, x) be the solution to the G-heat equation with initial value ϕ. For s ∈ [0, 1], set w(s) To prove the theorem, it suffices to show that w(0) = w(1).

Some definitions and notations about G-expectation
We review some basic notions and definitions of the related spaces under G-expectation. The readers may refer to [9], [10], [11], [12] and [14] for more details.
Let us recall the definitions of G-Brownian motion and its corresponding G-expecta- We are given a function G : S d → R satisfying the following monotonicity, sublinearity and positive homogeneity: A3. G(λa) = λG(a) for a ∈ S d and λ ≥ 0.
. From this construction we obtain a natural norm ξ L p G := E[|ξ| p ] 1/p , p ≥ 1. The completion of L ip (Ω T ) under · L p G is a Banach space, denoted by L p G (Ω T ). The canonical process B t (ω) := ω(t), t ≥ 0, is called a G-Brownian motion in this sublinear expectation space (Ω, L 1 G (Ω), E).

Proof to Proposition 1.2
Let P be a weakly compact set that represents E. Then, the corresponding G-normal distribution can be represented as Clearly, N G satisfies condition (H2) and Θ := {P • B −1 1 | P ∈ P} is weakly compact. Also, Proposition 1.2 can be restated in the following form.

Remark 5.2.
In [4], the authors used a similar idea to obtain the variation equation for the cost functional associated with the stochastic recursive optimal control problem.