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Yongsheng Song

Publications and source records attributed to Yongsheng Song.

At least 19 recordsLinked to original sources
arXiv2026

Limit Laws of the Iterated Logarithm Under Sub-linear Expectations

Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k\le n}\frac{S_k}{\sqrt{2k \log\log n}}. \end{align*} Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of $Y_1,Y_2,\ldots$, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.

math.PR↗
arXiv2026

Infinite-Time Mean Field FBSDEs and the Associated Elliptic Master Equations

This paper presents a further investigation of the properties of infinite-time mean field forward-backward stochastic differential equations (FBSDEs) and the associated elliptic master equations, which were introduced in [18] as mathematical tools for solving discounted infinite-time mean field games. By establishing the continuous dependence of the FBSDE solutions on their initial values, we prove the flow property of the mean field FBSDEs. And then, we prove that, at the Nash equilibrium, the value function of the representative player constitutes a viscosity solution to the corresponding elliptic master equation. In particular, when the coefficients of the equations are distribution-independent, we construct a classical solution to the elliptic partial differential equation (PDE) via fully coupled infinite-time FBSDEs. Furthermore, for classical solutions possessing displacement monotonicity and certain growth conditions, we establish their uniqueness for the elliptic master equation.

math.PR↗
arXiv2026

On Discounted Infinite-Time Mean Field Games

In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player has the same feedback form as the strategy in the social equilibrium, we say that the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations (FBSDEs). By employing elliptic master equations, a class of distribution-dependent elliptic partial differential equations (PDEs), we provide a representation for the Nash equilibrium strategies. We prove the Yamada-Watanabe type theorem and show weak uniqueness for infinite-time FBSDEs. Furthermore, we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.

math.OC↗
arXiv2026

Regularity of the Value Function in Discounted Infinite-Time Mean Field Games

In [17], we introduced the discounted infinite-time mean field games. Subsequently, in [18], we studied the connection between infinite-time mean field FBSDEs and elliptic master equations. In this paper, we further investigate the regularity of the representative player's value function. Specifically, we first prove the strong existence and uniqueness, as well as the uniqueness in law, for an extended class of infinite-time FBSDEs. We then establish the Lions-differentiability for the derivative of the representative player's value function with respect to the measure argument, and provide an explicit characterization for it using solutions to FBSDEs.

math.PR↗
arXiv2024

Invariant Sublinear Expectations

We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}=\sup_{P\inΘ}\mathrm{E}_P$, and show that each component $\mathbb{E}^{(d)}=\sup_{P\inΘ^{(d)}}\mathrm{E}_P$ of the decomposition has a finite period $p_d\in\mathbb{N}$, i.e., \[\mathbb{E}^{(d)}\left[f-f\circ T^{p_d}\right]=0, \quad f\in\mathcal{H}.\] Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component $Θ^{(d)}$ of its periodic decomposition is the convex hull of a finite set of $T^{p_d}$-ergodic probabilities. As an application of the characterization, we prove an ergodicity result which shows that the limit of the $p_{\mathbb{E}}$-step time means achieves the upper expectation.

math.PR↗
arXiv2023

Continuous Ergodic Capacities

The objective of this paper is to characterize the structure of the set $Θ$ for a continuous ergodic upper probability $\mathbb{V}=\sup_{P\inΘ}P$ (Theorem \ref {main result}): . $Θ$ contains a finite number of ergodic probabilities; . Any invariant probability in $Θ$ is a convex combination of those ergodic ones in $Θ$; . Any probability in $Θ$ coincides with an invariant one in $Θ$ on the invariant $σ$-algebra. The last property has already been obtained in \textsl{Cerreia-Vioglio, Maccheroni, and Marinacci} \cite{ergodictheorem}, which firstly studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result (Theorem \ref {improve}), which improves the result in \cite{ergodictheorem} in the sense that the limit of the time mean of $ξ$ is bounded by the upper expectation $\sup_{P\inΘ}E_P[ξ]$, instead of the Choquet integral. Generally, the former is strictly smaller.

math.PR↗
arXiv2022

A Strong Law of Large Numbers under Sublinear Expectations

We consider a sequence of i.i.d. random variables $\{ξ_k\}$under a sublinear expectation $\mathbb{E}=\sup_{P\inΘ}E_P$. We first give a new proof to the fact that, under each $P\inΘ$, any cluster point of the empirical averages $\barξ_n=(ξ_1+\cdots+ξ_n)/n$ lies in $[\underlineμ, \barμ]$ with $\underlineμ=-\mathbb{E}[-ξ_1], \barμ=\mathbb{E}[ξ_1]$. Then, we consider sublinear expectations on a Polish space $Ω$, and show that for each constant $μ\in [\underlineμ,\barμ]$, there exists a probability $P_μ\inΘ$ such that \begin {eqnarray}\label {intro-a.s.} \lim_{n\rightarrow\infty}\barξ_n=μ, \ P_μ\textmd{-a.s.}, \end {eqnarray} supposing that $Θ$ is weakly compact and $\{ξ_n\}\in L^1_{\mathbb{E}}(Ω)$. Under the same conditions, we can get a generalization of (\ref {intro-a.s.}) in the product space $Ω=\mathbb{R}^{\mathbb{N}}$ with $μ\in [\underlineμ,\barμ]$ replaced by $Π=π(ξ_1, \cdots,ξ_d)\in [\underlineμ,\barμ]$, where $π$ is a Borel measurable function on $\mathbb{R}^d$, $d\in\mathbb{R}$. Finally, we characterize the triviality of the tail $σ$-algebra of i.i.d. random variables under a sublinear expectation.

math.PR↗
arXiv2021

Forward-backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the existence and uniqueness of solutions to the fully coupled nonlinear forward-backward stochastic differential equations driven by G-Brownian motion. Assuming that the diffusion coefficient $σ$ is uniformly elliptic and all coefficients are differentiable, combining the results of fully nonlinear PDEs, we prove the existence and uniqueness of solutions to these equations.

math.PR↗
arXiv2019

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections

In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method.

math.PR↗
arXiv2018

Limit theorems with rate of convergence under sublinear expectations

Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{θ\in Θ} E_θ[\cdot]$ for a given set of linear expectations $\{E_θ: θ\in Θ\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng (2007)'s central limit theorem, in a probability space.

math.PR↗
arXiv2017

Normal Approximation by Stein's Method under Sublinear Expectations

Peng (2008)(\cite{P08b}) proved the Central Limit Theorem under a sublinear expectation: \textit{Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d random variables under a sublinear expectation $\hat{\mathbf{E}}$ with $\hat{\mathbf{E}}[X_1]=\hat{\mathbf{E}}[-X_1]=0$ and $\hat{\mathbf{E}}[|X_1|^3]<\infty$. Setting $W_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$, we have, for each bounded and Lipschitz function $φ$, \[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[φ(W_n)]-\mathcal{N}_G(φ)\bigg|=0,\] where $\mathcal{N}_G$ is the $G$-normal distribution with $G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]$, $a\in \mathbb{R}$.} In this paper, we shall give an estimate of the rate of convergence of this CLT by Stein's method under sublinear expectations: \textit{Under the same conditions as above, there exists $α\in(0,1)$ depending on $\underlineσ$ and $\overlineσ$, and a positive constant $C_{α, G}$ depending on $α, \underlineσ$ and $\overlineσ$ such that \[\sup_{|φ|_{Lip}\le1}\bigg|\hat{\mathbf{E}}[φ(W_n)]-\mathcal{N}_G(φ)\bigg|\leq C_{α,G}\frac{\hat{\mathbf{E}}[|X_1|^{2+α}]}{n^{\fracα{2}}},\] where $\overlineσ^2=\hat{\mathbf{E}}[X_1^2]$, $\underlineσ^2=-\hat{\mathbf{E}}[-X_1^2]>0$ and $\mathcal{N}_G$ is the $G$-normal distribution with \[G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overlineσ^2a^+-\underlineσ^2a^-), \ a\in \mathbb{R}.\]}

math.PR↗
arXiv2017

Supermartingale Decomposition Theorem under G-expectation

The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales have the decomposition similar to the classical case. The main ideas are to apply the uniformly continuous property of $S_G^β(0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.

math.PR↗
arXiv2016

Properties of $G$-martingales with finite variation and the application to $G$-Sobolev spaces

As is known, a process of form $\int_0^tη_sd\langle B\rangle_s-\int_0^t2G(η_s)ds$, $η\in M^1_G(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale could not be form of $\int_0^tη_sds$ or $\int_0^tγ_sd\langle B\rangle_s$, $η, γ\in M^1_G(0,T)$, which implies that the decomposition for generalized $G$-Itô processes is unique: For $ζ\in H^1_G(0,T)$, $η\in M^1_G(0,T)$ and non-increasing $G$-martingales $K, L$, if \[\int_0^tζ_s dB_s+\int_0^tη_sds+K_t=L_t,\ t\in[0,T],\] then we have $η\equiv0$, $ζ\equiv0$ and $K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).

math.PR↗
arXiv2016

Stein Type Characterization for $G$-normal Distributions

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

math.PR↗
arXiv2014

A Note On $G$-normal Distributions

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 if and only if $\frac{\overlineσ_μ}{\underlineσ_μ}=\frac{\overlineσ_ν}{\underlineσ_ν}$.

math.PR↗
arXiv2014

Gradient Estimates for Nonlinear Diffusion Semigroups by Coupling Methods

Our purpose is to obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First we derive uniform gradient estimates for a certain semi-linear PDEs based on the coupling method introduced in Wang (2011) and the theory of backward SDEs. Then we generalize Wang's coupling to the $G$-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of a certain fully nonlinear PDEs.

math.PR↗
arXiv2014

G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE

We introduce a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian motion are in fact path dependent PDEs in the corresponding Sobolev spaces under G-norms. For the linear case of G corresponding the classical Wiener probability space with Wiener measure P, we have established a 1-1 correspondence between BSDE and such new type of quasilinear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness of such type of fully nonlinear path-dependence PDE in G-Sobolev space have been obtained via a recent results of BSDE driven by G-Brownian motion.

math.PR↗