General Solutions of Reflection Equations for Bariev Model

The integrable general open-boundary conditions for the one-dimensional Bariev chain are considered. All kinds of solutions to the reflection equation （RE） and its dual are obtained.

NADARAYA-WATSON ESTIMATORS FOR REFLECTED STOCHASTIC PROCESSES

We study the Nadaraya-Watson estimators for the drift function of two-sided reflected stochastic differential equations.The estimates,based on either the continuously observed process or the discretely observed process,are considered.Under certain conditions,we prove the strong consistency and the asymptotic normality of the two estimators.Our method is also suitable for one-sided reflected stochastic differential equations.Simulation results demonstrate that the performance of our estimator is superior to that of the estimator proposed by Cholaquidis et al.(Stat Sin,2021,31:29-51).Several real data sets of the currency exchange rate are used to illustrate our proposed methodology.

Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.

An Alternative Method to Study Wave Scattering by Semi-infinite Inertial Surfaces

A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green＇s second identity to the potential functions and appropriate Green＇s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.

Distribution dependent reflecting stochastic differential equations

To characterize the Neumann problem for nonlinear Fokker-Planck equations,we investigate distribution dependent reflecting stochastic differential equations(DDRSDEs)in a domain.We first prove the well-posedness and establish functional inequalities for reflecting stochastic differential equations with singular drifts,and then extend these results to DDRSDEs with singular or monotone coefficients,for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting stochastic differential equations is established.

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.

Reflected stochastic differential equations driven by G-Brownian motion with nonlinear resistance

We study the uniqueness and existence of solutions of reflected G-stochastic differential equations （RGSDEs） with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the comparison theorem for RGSDEs with nonlinear resistance.

Reflected Backward Doubly Stochastic Differential Equations with Discontinuous Coefficients

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations （RBDSDEs） with one continuous barrier and discontinuous （left or right continuous） genera- tor. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.

Hölder Continuity of Solutions of SPDEs with Reflection

In this paper,we obtain the Hölder continuity of the solutions of SPDEs with reflection,which have singular drifts(random measures).

Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations

We consider in this paper random batch interacting particle methods forsolving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann(PB) equation as the equilibrium, in the external unbounded domain. To justify thesimulation in a truncated domain, an error estimate of the truncation is proved inthe symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are O(N) per time step. The particle methods cannot only be considered as a numerical method for solving the PNP and PB equations,but also can be used as a direct simulation approach for the dynamics of the chargedparticles in solution. The particle methods are preferable due to their simplicity andadaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effectsand interactions in the particle methods and to describe phenomena beyond the scopeof the mean-field equations.

CONTINUOUS DEPENDENCE ON THE TERMINAL CONDITION OF SOLUTIONS TO NONLINEAR REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper, we derive the continuous dependence on the terminal condition of solutions to nonlinear reflected backward stochastic differential equations involving the subdifferential operator convex function under non-Lipschitz of a lower semi-continuous, proper and condition by means of the corollary of Bihari inequality.

Convergence of Solutions of Discrete Reflected Backward SDE's and Simulations

The objective of this paper is to introduce elementary discrete reflected backward equations and to give a simple method to discretize in time a （continuous） reflected backward equation. A presentation of numerical simulations is also described.

Existence, Uniqueness and Approximation for Lp Solutions of Reflected BSDEs with Generators of One-sided Osgood Type

We prove several existence and uniqueness results for Lp （p 〉 1） solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.

Exact solution of an anisotropic J_(1)–J_(2)model with the Dzyloshinsky–Moriya interactions at boundaries

We propose a method to construct new quantum integrable models.As an example,we construct an integrable anisotropic quantum spin chain which includes the nearest-neighbor,next-nearestneighbor and chiral three-spin couplings.It is shown that the boundary fields can enhance the anisotropy of the first and last bonds,and can induce the Dzyloshinsky–Moriya interactions along the z-direction at the boundaries.By using the algebraic Bethe ansatz,we obtain the exact solution of the system.The energy spectrum of the system and the associated Bethe ansatz equations are given explicitly.The method provided in this paper is universal and can be applied to constructing other exactly solvable models with certain interesting interactions.

RBSDEs with optional barriers:monotone approximation

In this short note we consider reflected backward stochastic differential equations(RBSDEs)with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous.In this case,the barrier is represented as a nondecreasing limit of right continuous with left limit(RCLL)barriers.We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions.Finally,we highlight the connection of these RBSDEs with standard RCLL BSDEs.