Mean-field backward stochastic differential equations on Markov chains

In this paper, we deal with a class of mean-field backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We obtain the existence and uniqueness theorem and a comparison theorem for solutions of one-dimensional mean-field BSDEs under Lipschitz condition

Deep Competitive Pathway Networks

In the design of deep neural architectures, recent studies have demonstrated the benefits of grouping subnetworks into a larger network. For examples, the Inception architecture integrates multi-scale subnetworks and the residual network can be regarded that a residual unit combines a residual subnetwork with an identity shortcut. In this work, we embrace this observation and propose the Competitive Pathway Network (CoPaNet). The CoPaNet comprises a stack of competitive pathway units and each unit contains multiple parallel residual-type subnetworks followed by a max operation for feature competition. This mechanism enhances the model capability by learning a variety of features in subnetworks. The proposed strategy explicitly shows that the features propagate through pathways in various routing patterns, which is referred to as pathway encoding of category information. Moreover, the cross-block shortcut can be added to the CoPaNet to encourage feature reuse. We evaluated the proposed CoPaNet on four object recognition benchmarks: CIFAR-10, CIFAR-100, SVHN, and ImageNet. CoPaNet obtained the state-of-the-art or comparable results using similar amounts of parameters. The code of CoPaNet is available at: https://github.com/JiaRenChang/CoPaNet.Comment: To appear in ACML1

Reflected backward stochastic differential equations with jumps in time-dependent random convex domains

In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domain ${\cal{D}}=\{D_t, t\in[0,T]\}$. We prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of $\cal{D}$, via a penalization method.Comment: 43 pages. arXiv admin note: text overlap with arXiv:1307.2124 by other author

Magnetostatics of Magnetic Skyrmion Crystals

Magnetic skyrmion crystals are topological magnetic textures arising in the chiral ferromagnetic materials with Dzyaloshinskii-Moriya interaction. The magnetostatic fields generated by magnetic skyrmion crystals are first studied by micromagnetic simulations. For N\'eel-type skyrmion crystals, the fields will vanish on one side of the crystal plane, which depend on the helicity; while for Bloch-type skyrmion crystals, the fields will distribute over both sides, and are identical for the two helicities. These features and the symmetry relations of the magetostatic fields are understood from the magnetic scalar potential and magnetic vector potential of the hybridized triple-Q state. The possibility to construct magnetostatic field at nanoscale by stacking chiral ferromagnetic layers with magnetic skyrmion crystals is also discussed, which may have potential applications to trap and manipulate neutral atoms with magnetic moments.Comment: 5 pages, 2 figure

Kernel Bayesian Inference with Posterior Regularization

We propose a vector-valued regression problem whose solution is equivalent to the reproducing kernel Hilbert space (RKHS) embedding of the Bayesian posterior distribution. This equivalence provides a new understanding of kernel Bayesian inference. Moreover, the optimization problem induces a new regularization for the posterior embedding estimator, which is faster and has comparable performance to the squared regularization in kernel Bayes' rule. This regularization coincides with a former thresholding approach used in kernel POMDPs whose consistency remains to be established. Our theoretical work solves this open problem and provides consistency analysis in regression settings. Based on our optimizational formulation, we propose a flexible Bayesian posterior regularization framework which for the first time enables us to put regularization at the distribution level. We apply this method to nonparametric state-space filtering tasks with extremely nonlinear dynamics and show performance gains over all other baselines.Comment: NIPS 201

Multivalued backward doubly stochastic differential equations with time delayed coefficients

In this paper, we deal with a class of multivalued backward doubly stochastic differential equations with time delayed coefficients. Based on a slight extension of the existence and uniqueness of solutions for backward doubly stochastic differential equations with time delayed coefficients, we establish the existence and uniqueness of solutions for these equations by means of Yosida approximation.Comment: 12 page

Non-smooth analysis method in optimal investment- a BSDE approach

In this paper, our aim is to investigate necessary conditions for optimal investment. We model the wealth process by Backward differential stochastic equations (shortly for BSDE) with or without constraints on wealth and portfolio process. The constraints can be very general thanks the non-smooth analysis method we adopted

Stochastic viscosity solution for stochastic PDIEs with nonlinear Neumann boundary condition

This paper is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential integral equations with nonlinear Neumann boundary condition. Using the recently developed theory on generalized backward doubly stochastic differential equations driven by a L\'evy process, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula.Comment: 16 pag

Control of Ultracold Atoms with a Chiral Ferromagnetic Film

We show that the magnetic field produced by a chiral ferromagnetic film can be applied to control ultracold atoms. The film will act as a magnetic mirror or a reflection grating for ultracold atoms when it is in the helical phase or the skyrmion crystal phase respectively. By applying a bias magnetic field and a time-dependent magnetic field, one-dimensional or two-dimensional magnetic lattices including honeycomb, Kagome, triangular types can be created to trap the ultracold atoms. We have also discussed the trapping height, potential barrier, trapping frequency, and Majorana loss rate for each lattice. Our results suggest that the chiral ferromagnetic film can be a platform to develop artificial quantum systems with ultracold atoms based on modern spintronics technologies.Comment: 9 pages, 6 figure

Superconducting state properties of a d-wave superconductor with mass anisotropy

YBa$_2$Cu$_3$O$_7$ (YBCO) exhibits a large anisotropy between the $a$ and $b$ axes in the CuO$_2$ planes because of the presence of CuO chains. In order to account for such an anisotropy we develop a Ginzburg-Landau (GL) theory for an anisotropic d-wave superconductor in an external magnetic field, based on an anisotropic effective mass approximation within CuO$_2$ planes. The anisotropic parameter $\lambda=m_x/m_y$, where $m_x$ ($m_y$) is the effective mass in the $x$ ($y$) direction, is found to have significant physical consequences: In the bulk case, there exist both the $s$- and $d$-wave order parameters with the same transition temperature, as long as $\lambda\ne 1$. The GL equations are also solved both analytically and numerically for the vortex state, and it is shown that both the $s$- and $d$-wave components show a two-fold symmetry, in contrast to the four-fold symmetry around the vortex, as expected for the purely $d$-wave vortex. With the deviation of $\lambda$ from unity, the opposite winding between the $s$- and $d$-wave components observed in the purely $d$-wave case is gradually taken over by the same winding number. The vortex lattice is found to have oblique structure in a wide temperature range with the precise shape depending on the anisotropy.Comment: 16 pages, latex, 11 figures availabe on reques