我们考虑了一种Monge-Kantorovich类型的微分方程系统，该系统描述了颗粒物质由恒定源倒入网络时的平衡配置。基于P.-L. Lions和P. E. Souganidis最近引入的网络上Hamilton-Jacobi方程粘性解定义，我们证明了该系统的解的存在性和唯一性，并讨论了其数值近似。一些数值实验也进行了。
英文摘要：
We consider a system of differential equations of Monge-Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton-Jacobi equations on networks, recently introduced by P.-L. Lions and P. E. Souganidis, we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.
Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
ETH Zurich（苏黎世联邦理工学院）；University of Duisburg-Essen（杜伊斯堡-埃森大学）；University of Kassel（卡塞尔大学）
1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients
Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel technique, this paper focuses on the mean-square convergence and stability of the backward Euler method (BEM) for SDDEs whose drift and diffusion coefficients can both grow polynomially. The upper mean-square error bounds of BEM are obtained. Then the convergence rate, which is one-half, is revealed without using the moment boundedness of numerical solutions. F…
Tamed-adaptive Euler-Maruyama approximation for SDEs with superlinearly growing and piecewise continuous drift, superlinearly growing and locally Hölder continuous diffusion
Strong order one-half convergence of a coupled tamed Euler--Peano scheme for reflected stochastic differential equations with super-linearly growing coefficients
Yunnan University（云南大学）；Central University of Finance and Economics（中央财经大学）
Convergence of the tamed Euler scheme for stochastic differential equations with Piecewise Continuous Arguments under non-Lipschitz continuous coefficients
Department of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学系）
High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given space-time point. This gave rise to a growing literature on deep learning based methods for solving partial differential equations; results for integro-differential equations on the other hand are scarce. In this paper we consider an extension of the deep splitting scheme due to Beck, Becker, Cheridito, Jentzen, Neufeld (2021) and Germain, Pham, Warin (2022) to PID…
In this paper, we consider scalar stochastic differential equations (SDEs) with a superlinearly growing and piecewise continuous drift coefficient. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate 1/2 for all $p\in [1, +\infty)$ is recovered for the tamed Euler scheme. A numerical example is provided to support our conclusion.
Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with the sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ODEs. One possibility to construct this system is to approximate the Laplace spectrum of the …
This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. However, we require that the Lévy measure is finite. New arguments are develop…
