Forward–backward stochastic differential equations with monotone functionals and mean field games with common noise

We consider a system of forward–backward stochastic differential equations (FBSDEs) with monotone functionals. We show that such a system is well-posed by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. Lastly, we show that mean field games with common noise are uniquely solvable under a linear-convex setting and weak-monotone cost functions and prove that the optimal control is in a feedback form depending only on the current state and conditional law.     

具有时滞的Volterra反应扩散差分方程的稳定性

本文考虑一类时滞Volterra反应扩散差分方程的初边值问题的正稳态解的稳定性。利用上下解方法和单调迭代方法得到了每一个解趋于方程的正稳态解的充分条件。     

单调型条件下随机微分方程θ-方法的均方指数稳定性

本文研究高度非线性随机微分方程(SDEs)的数值解稳定性性质.给出θ-方法均方指数稳定性的充分条件.与现有文献不同,本文无需单边线性增长条件和充分小的步长.本文在单调型的条件下,并且至于要步长满足一个很弱的条件即可.因此本文是对现有文献的很大改进.     

Convex hull properties in location theory

We present new characterizations of inner product spaces which bring into play a property of a family of optimization problems related to the norm of the space. This property concerns the existence of a solution .to some optimization problems which belongs to the convex hull of some set. We thus obtain a generalization of results of V. Klee and A. Garkavi about the Chebychev centers and also of more recent results of the author about Fermat points. Intermediate propositions concerning unicity in some optimization problems, a geometric characterization of finite dimensional inner product spaces and monotone norms seem to have their own interest.     

Bayesian Free-Knot Monotone Cubic Spline Regression

This article proposes a new Bayesian approach for monotone curve fitting based on the isotonic regression model. The unknown monotone regression function is approximated by a cubic spline and the constraints are represented by the intersection of quadratic cones. We treat the number and locations of knots as free parameters and use reversible jump Markov chain Monte Carlo to obtain posterior samples of knot configurations. Given the number and locations of the knots, second-order cone programming is used to estimate the remaining parameters. Simulation results suggest the method performs well and we illustrate the approach using the ASA car data.     

External Approximation of Nonlinear Operator Equations

Based on an external approximation scheme for the underlying Banach space, a nonlinear operator equation is approximated by a sequence of coercive problems. The equation is supposed to be governed by the sum of two nonlinear operators acting between a reflexive Banach space and its dual. Under suitable stability assumptions and if the underlying operators can be approximated consistently, weak convergence of a subsequence of approximate solutions is shown. This also proves existence of solutions to the original equation.     

A Regularized Hybrid Steepest Descent Method for Variational Inclusions

This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems (see, e.g., [9 I. Yamada , M. Yukawa , and M. Yamagishi ( 2011 ). Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings . In Fixed Point Algorithms for Inverse Problems in Science and Engineering ( H.H. Bauschke , R. Burachik , P.L. Combettes , V. Elser , D.R. Luke , and H. Wolkowicz , eds.), Springer-Verlag , New York , Chapter 17 , pp. 343 – 388 . [Google Scholar], 10 I. Yamada , Ogura , and N. Shirakawa ( 2002 ). A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems . In Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics ( Z. Nashed and O. Scherzer , eds.), American Mathematical Society , Providence , RI , Vol. 313 , pp. 269 – 305 . [Google Scholar]]).     

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.      

Convergence of Monotone Iterations for Differential Problem

We use the method of lower and upper solutions combined with monotone iterations to differential problems with a parameter. Existence of extremal solutions to such problems is proved. Received April 15, 1999, Revised December 6, 1999, Accepted July 18, 2000     

Combined Relaxation Method for Monotone Equilibrium Problems

A scalar equilibrium problem which involves a monotone differentiable cost bifunction is considered. For such bifunction, a skew-symmetric type property with respect to the partial gradients is established. This property enables us to construct a new combined relaxation method and essentially simplify its line search procedure. An application to an inverse equilibrium problem is also presented.