We consider the Cauchy problem for a semilinear stochastic differential inclusion in a Hilbert space. The linear operator generates a strongly continuous semigroup and the nonlinear term is multivalued and satisfies a condition which is more heneral than the Lipschitz condition. We prove the existence of a mild solution to this problem. This solution is not "strong" in the probabilistic sense, that is, it is not defined on the underlying probability space, but on a larger one, which provides a "very good extension" in the sense of Jacod and Memin. Actually, we construct this solution as a Young measure, limit of approximated solutions provided by the Euler scheme. The compactness in the space of Young measures of this sequence of approximated solutions is obtained by proving that some measure of noncompactness equals zero.
We consider a Markov chain $(x_n)$ whose kernel is indexed by a scaling parameter $γ>0$, refered to as the step size. The aim is to analyze the behavior of the Markov chain in the doubly asymptotic regime where $n\to\infty$ then $γ\to 0$. First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putti…
We present a new Aumann-like integral for a Hölder multifunction with respect to a Hölder signal, based on the Young integral of a particular set of Hölder selections. This restricted Aumann integral has continuity properties that allow for numerical approximation as well as an existence theorem for an abstract stochastic differential inclusion. This is applied to concrete examples of first order and second order stochastic differential inclusions directed by fractional Brownian motion.
This work develops new results for stochastic approximation algorithms. The emphases are on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, and non-smooth analysis, and stochastic differential inclusions. Under broad conditions, it is shown that a suitably scaled sequence of the iterates has a differential inclusion limit. In addition, it is shown for the first time that a centered and scaled sequence of the iterates converges weakly to a stochastic differential inclusion limit. The results are then used…
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equation…
Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme
CNRS - Université Lille 1（法国国家科学研究中心-里尔第一大学）；Laboratoire de mathématiques Paul Painlevé（保罗·潘勒韦数学实验室）；LAMAV Université de Valenciennes et du Hainaut-Cambrésis（瓦朗谢纳与康布雷-汉诺威大学拉马夫研究所）
The two dimensional stochastic Euler equations (EE) perturbed by a linear multiplicative noise of Itô type on the bounded domain $\mathcal{O}$ have been considered in this work. Our first aim is to prove the existence of \textsl{global weak (analytic) solutions} for stochastic EE when the divergence free initial data $\boldsymbol{u}^*\in\mathbb{H}^1(\mathcal{O})$, and the external forcing $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H}^1(\mathcal{O}))$. In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if $\mat…
This work is concerned about the asymptotic behavior of the solutions of the two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations driven by white noise with nonlinear diffusion terms. We prove the existence and uniqueness of weak pullback mean random attractors for the 2D SCBF equations (for $r\geq1$) as well as 3D SCBF equations (for $r>3$, any $μ,β>0$ and for $r=3$, $2μβ\geq1$) in Bochner spaces, when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we establish the existence of weak pullback mean random attractors for a class of locally mo…
In this paper, we study backward doubly stochastic recursive optimal control problem where the cost function is described by the solution of a backward doubly stochastic differential equation. We give the dynamical programming principle for this kind of optimal control problem and show that the value function is the unique Sobolev weak solution for the corresponding stochastic Hamilton-Jacobi-Bellman equation.
Aggregation of weakly dependent doubly stochastic processes
本文的目的是将Dacunha-Castelle and Fermin 2005，Dacunha-Castelle and Fermin 2008中的聚合收敛结果推广到具有弱相关新息的双随机线性和非线性过程.首先，基于Doukhan和Louhichi（1999）给出的弱相依定义，我们引入了双随机过程的弱相依概念，并给出了满足这一概念的几个模型，如：双随机Volterra过程和新息弱相依的双随机Bernoulli方案.在弱相依双随机过程的情况下，得到了部分集结序列的中心极限定理。最后，在双随机Volterra过程具有交互新息的情况下，给出了部分集结过程的协方差函数的一个新的SLLN。关键词：聚集，弱相关，双随机过程，Volterra过程，伯努利位移，TCL，SLLN。
