SDEs with constraints driven by semimartingales and processes with bounded p-variation

We study the existence, uniqueness and stability of solutions of general stochastic differential equations with constraints driven by semimartingales and processes with bounded $p$-variation. Applications to SDEs with constraints driven by fractional Brownian motion and standard Brownian motion are given.     

Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

Super-Brownian motion with reflecting historical paths. II. Convergence of approximations

We prove that the sequence of finite reflecting branching Brownian motion forests defined by Burdzy and Le Gall ([1]) converges in probability to the “super-Brownian motion with reflecting historical paths.” This solves an open problem posed in [1], where only tightness was proved for the sequence of approximations. Several results on path behavior were proved in [1] for all subsequential limits–they obviously hold for the unique limit found in the present paper.Mathematics Subject Classification (2000): Primary 60H15, Secondary 35R60Supported in part by NSF Grant DMS-0071486, Israel Science Foundation Grants 12/98 and 116/01 - 10.0, and the U.S.-Israel Binational Science Foundation (grant No. 2000065).     

Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous N-particle systems with singular interactions

Assuming that Ω⊂Rn, n?2, is an open, relatively compact set with boundary ∂Ω of Lebesgue measure zero we prove strong Feller properties for a class of distorted Brownian motions in with reflecting boundary condition. Dirichlet form techniques give the existence of a weak solution to the corresponding stochastic differential equation for quasi all starting points in the sense of the associated martingale problem. Combining this result with the strong Feller properties we can construct a weak solution for specified starting points. If Ω has C²-boundary the construction works for all starting points, where the drift term is not singular, even on the boundary. But also for a certain class of sets with less smooth boundary our approach works for all points in Ω, where the drift term is not singular, and at least some points from ∂Ω. Our techniques allow very singular drift terms. This enables us to construct continuous N-particle gradient stochastic dynamics in cuboids Λ⊂Rd, d∈N, with reflecting boundary condition and singular interactions for dN?2. We can start the stochastic dynamics in all initial configurations having at most one particle in ∂Λ, provided ∂Λ is locally smooth there.     

Method of all-optical frequency-encoded binary adder system using nonlinear waveguide and reflecting semiconductor optical amplifiers

Arithmetic and algebraic operations are the most important part of optical computation and data processing. To implement the optical logic operations, different data encoding/decoding techniques have been reported. Frequency variant encoding/decoding technique is now performing a very faithful role in this regard. Frequency is the fundamental character of any signal and it remains unaltered in reflection, refraction, absorption, etc. during propagation and transmission of the signal. This is the most potential advantage of the frequency encoding technique over any other conventional encoding techniques. Here, in our proposed scheme of addition of binary bits made of two encoded frequencies in C-band (1535-1560 nm), the conjugate beam is generated in LiNbO₃ waveguide using cascaded sum and difference frequency generation by the nonlinear interaction with a third frequency, exploiting the nonlinear response character of periodically poled LiNbO₃ waveguide. The cross-gain modulation property of reflecting semiconductor optical amplifier (RSOA) has also been exploited here for frequency conversion purposes.     

Equivalence of Differential System

Using reflecting function of Mironenko we construct some differential systems which are equivalentto the given differential system.This gives us an opportunity to find out the monodromic matrix of these periodicsystems which are not integrable in finite terms.     

Milne's problem for anisotropic scattering medium with specular reflecting boundary

The Milne problem is investigated subject to reflecting boundary conditions. The original version of the problem with vacuum boundary condition is generalized assigning, to the surface x=0, a specular reflection coefficient . Linearly anisotropic case is studied. The integral version of the transport equation solved using trial functions based on Case's eigenvalues and exponential integral function. Solution of the Milne problem is formulated in terms of characteristic parameters such as extrapolated end point, emergent angular distribution and total neutron density. Numerical results for the analytically evaluated parameters are then present. Some of our numerical results are compared with the available published results.     

Research on the properties of some planar polynomial differential equations

In this article, we use the new method of reflecting function to study the behavior of solutions of nonlinear time-vary differential equations, and give the sufficient conditions for these equations which have the reflecting function in the form of linear and fractional. We applied the obtained results to discuss the qualitative behavior of solutions of the higher degree polynomial differential systems and derive the sufficient conditions for a critical point to be a center.     

Backward Stochastic Differential Equation with Two Reflecting Barriers and Jumps

Abstract

In this paper, by using a penalization as well as a fixed point methods, we prove existence and uniqueness of the solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process.