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Yingchao Xie

Publications and source records attributed to Yingchao Xie.

At least 19 recordsLinked to original sources
arXiv2026

Hua-Chen New Theory of Economic Optimization

Between 1957-1985, Chinese mathematician Loo-Keng Hua pioneered economic optimization theory through three key contributions: establishing economic stability's fundamental theorem, proving the uniqueness of equilibrium solutions in economic systems, and developing a consumption-integrated model 50 days before his death. Since 1988, Mu-Fa Chen has been working on Hua's theory. He introduced stochastics, namely Markov chains, to economic optimization theory. He updated and developed Hua's model and came up with a new model (Chen's model) which has become the starting point of a new economic optimization theory. Chen's theory can be applied to economic stability test, bankruptcy prediction, product ranking and classification, economic prediction and adjustment, economic structure optimization. Chen's theory can also provide efficient algorithms that are programmable and intelligent. {Stochastics} is the cornerstone of Chen's theory. There is no overlap between Chen's theory, and the existing mathematical economy theory and the economics developments that were awarded Nobel Prizes in Economics between 1969 and 2024. The distinguished features of Chen's theory from the existing theories are quantitative, calculable, predictable, optimizable, programmable and can be intelligent. This survey provides a theoretical overview of the newly published monograph \cite{5rw24}. Specifically, the invariant of the economic structure matrix, also known as the Chen's invariant, was first published in this survey.

econ.TH↗
arXiv2025

Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients

In this paper, we consider a class of nonautonomous multi-scale stochastic partial differential equations with fully local monotone coefficients. By introducing the evolution system of measures for time-inhomogeneous Markov semigroups, we study the averaging principle for such kind of system. Specifically, we first prove the slow component in the multi-scale stochastic system converges strongly to the solution of an averaged equation, whose coefficients retain the dependence of the scaling parameter. Furthermore, if the coefficients satisfy uniformly almost periodic conditions, we establish that the slow component converges strongly to the solution of another averaged equation, whose coefficients are independent of the scaling parameter. The main contribution of this paper extends the basic nonautonomous framework investigated by Cheng and Liu in [11] to a fully coupled framework, as well as the autonomous framework explored by Liu et al. in [27] to the more general nonautonomous framework. Additionally, we improve the locally monotone coefficients discussed in [11,27] to the fully local monotone coefficients, thus our results can be applied to a wide range of cases in nonlinear nonautonomous stochastic partial differential equations, such as multi-scale stochastic Cahn-Hilliard-heat equation and multi-scale stochastic liquid-crystal-porous-media equation.

math.PR↗
arXiv2025

Strong averaging principle for nonautonomous slow-fast SPDEs driven by $α$-stable processes

This paper considers a class of nonautonomous slow-fast stochastic partial differential equations driven by $α$-stable processes for $α\in (1,2)$. By introducing the evolution system of measures, we establish an averaging principle for this stochastic system. Specifically, we first prove the strong convergence (in the $L^p$ sense for $p\in (1,α)$) of the slow component to the solution of a simplified averaged equation with coefficients depend on the scaling parameter. Furthermore, under conditions that coefficients are time-periodic or satisfy certain asymptotic convergence, we prove that the slow component converges strongly to the solution of an averaged equation, whose coefficients are independent of the scaling parameter. Finally, a concrete example is provided to illustrate the applicability of our assumptions. Notably, the absence of finite second moments in the solution caused by the $α$-stable processes requires new technical treatments, thereby solving a problem mentioned in [1,Remark 3.3].

math.PR↗
arXiv2025

Averaging principles for time-inhomogeneous multi-scale SDEs with partially dissipative coefficients

In this paper, we study averaging principles for a class of time-inhomogeneous stochastic differential equations (SDEs) with slow and fast time-scales, where the drift term in the fast component is time-dependent and only partially dissipative. Under asymptotic assumptions on the coefficients, we prove that the slow component $(X^{\varepsilon}_t)_{t\geq 0}$ converges strongly to the unique solution $(\bar{X}_t)_{t\geq 0}$ to an averaged SDE, when the diffusion coefficient in the slow component is independent of the fast component; on the other hand, we establish the weak convergence of $(X_t^{\varepsilon})_{t\ge0}$ in the space $C([0,T];\mathbb{R}^n)$ and identify the limiting process by the martingale problem approach, when the diffusion coefficient of the slow component depends on the fast component. The proofs of strong and weak averaging principles are partly based on the study of the existence and uniqueness of an evolution system of measures for time-inhomogeneous SDEs with partially dissipative drift.

math.PR↗
arXiv2025

Diffusion Approximation for Slow-Fast SDEs with State-Dependent Switching

In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component $X^{\varepsilon}$ is the solution of a stochastic differential equation with additional homogenization term, while the fast component $α^{\varepsilon}$ is a switching process. We first prove the weak convergence of $\{X^\varepsilon\}_{0<\varepsilon\leq 1}$ to $\bar{X}$ in the space of continuous functions, as $\varepsilon\rightarrow 0$. Using the martingale problem approach and Poisson equation associated with a Markov chain, we identify this weak limiting process as the unique solution $\bar{X}$ of a new stochastic differential equation, which has new drift and diffusion terms that differ from those in the original equation. Next, we prove the order $1/2$ of weak convergence of $X^{\varepsilon}_t$ to $\bar{X}_t$ by applying suitable test functions $ϕ$, for any $t\in [0, T]$. Additionally, we provide an example to illustrate that the order we achieve is optimal.

math.PR↗
arXiv2024

Averaging principles for time-inhomogeneous multi-scale SDEs via nonautonomous Poisson equations

The purpose of this paper is to establish asymptotic behaviors of time-inhomogeneous multi-scale stochastic differential equations (SDEs). To achieve them, we analyze the evolution system of measures for time-inhomogeneous Markov semigroups, and investigate regular properties of nonautonomous Poisson equations. The strong and the weak averaging principle for time-inhomogeneous multi-scale SDEs, as well as explicit convergence rates, are provided. Specifically, we show the slow component in the multi-scale stochastic system converges strongly or weakly to the solution of an averaged equation, whose coefficients retain the dependence of the scaling parameter. When the coefficients of the fast component exhibit additional asymptotic or time-periodic behaviors, we prove the slow component converges strongly or weakly to the solution of an averaged equation, whose coefficients are independent of the scaling parameter. Finally, two examples are given to indicate the effectiveness of all the averaged equations mentioned above.

math.PR↗
arXiv2023

Poisson Equation and Application to Multi-Scale SDEs with State-Dependent Switching

In this paper, we study the averaging principle and central limit theorem for multi-scale stochastic differential equations with state-dependent switching. To accomplish this, we first study the Poisson equation associated with a Markov chain and the regularity of its solutions. As applications of the results on the Poisson equations, we prove three averaging principle results and two central limit theorems results. The first averaging principle result is a strong convergence of order $1/2$ of the slow component $X^{\varepsilon}$ in the space $C([0,T],\mathbb{R}^n)$. The second averaging principle result is a weak convergence of $X^{\varepsilon}$ in $C([0,T],\mathbb{R}^n)$. The third averaging principle result is a weak convergence of order $1$ of $X^{\varepsilon}_t$ in $\mathbb{R}^n$ for any fixed $t\ge 0$. The first central limit theorem type result is a weak convergence of $(X^{\varepsilon}-\bar{X})/\sqrt{\varepsilon}$ in $C([0,T],\mathbb{R}^n)$, where $\bar{X}$ is the solution of the averaged equation. The second central limit theorem type result is a weak convergence of order $1/2$ of $(X^{\varepsilon}_t-\bar{X}_t)/\sqrt{\varepsilon}$ in $\mathbb{R}^n$ for fixed $t\ge 0$. Several examples are given to show that all the achieved orders are optimal.

math.PR↗
arXiv2023

Asymptotic behavior for multi-scale SDEs with monotonicity coefficients driven by Lévy processes

In this paper, we study the asymptotic behavior for multi-scale stochastic differential equations driven by Lévy processes. The optimal strong convergence order 1/2 is obtained by studying the regularity estimates for the solution of Poisson equation with polynomial growth coefficients, and the optimal weak convergence order 1 is got by using the technique of Kolmogorov equation. The main contribution is that the obtained results can be applied to a class of multi-scale stochastic differential equations with monotonicity coefficients, as well as the driven processes can be the general Lévy processes, which seems new in the existing literature.

math.PR↗
arXiv2023

Stochastic Generalized Porous Media Equations over $σ$-finite Measure Spaces with Non-Continuous Diffusivity Function

In this paper, we prove that stochastic porous media equations over $σ$-finite measure spaces $(E,\mathcal{B},μ)$, driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator $L$ and the diffusivity function given by a maximal monotone multi-valued function $Ψ$ of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions $Ψ$, for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an $L^p(μ)$-Itô formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular SOC models) and cases where $E$ is a manifold or a fractal, and to non-local operators $L$, as e.g. $L=-f(-Δ)$, where $f$ is Bernstein function.

math.PR↗
arXiv2023

Stochastic Porous Media Equation on General Measure Spaces with Increasing Lipschitz Nonlinearties

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E, \mathscr{B}(E), μ)$, and the Laplacian replaced by a self-adjoint operator $L$. In the case of Lipschitz nonlinearities $Ψ$, we in particular generalize previous results for open $E\subset \mathbb{R}^d$ and $L\!\!=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $Ψ$ at infinity for $L^2(μ)$-initial data.

math.PR↗
arXiv2021

Orders of strong and weak averaging principle for multiscale SPDEs driven by $α$-stable process

In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $α$-stable process, where $α\in(1,2)$. Using the technique of Poisson equation, the orders of strong and weak convergence are given $1-1/α$ and $1-r$ for any $r\in (0,1)$ respectively. The main results extend Wiener noise considered by Bréhier in [6] and Ge et al. in [17] to $α$-stable process, and the finite dimensional case considered by Sun et al. in [39] to the infinite dimensional case.

math.PR↗
arXiv2020

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

This paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time-scales, where the drift coefficients satisfy local Lipschitz conditions with respect to the slow and fast variables, and the coefficients in the slow equation depend on time $t$ and $ω$. Making use of the techniques of time discretization and truncation, we prove that the slow component strongly converges to the solution of the corresponding averaged equation.

math.PR↗
arXiv2019

Small Time Asymptotics for SPDEs with Locally Monotone Coefficients

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

math.PR↗
arXiv2019

Strong convergence order for slow-fast McKean-Vlasov stochastic differential equations

In this paper, we consider the averaging principle for a class of McKean-Vlasov stochastic differential equations with slow and fast time-scales. Under some proper assumptions on the coefficients, we first prove that the slow component strongly converges to the solution of the corresponding averaged equation with convergence order $1/3$ using the approach of time discretization. Furthermore, under stronger regularity conditions on the coefficients, we use the technique of Poisson equation to improve the order to $1/2$, which is the optimal order of strong convergence in general.

math.PR↗