Oracle inequalities for the stochastic differential equations

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi-martingale noises with jumps. The noises are modeled by the Lévy processes, the Ornstein–Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.     

Large deviation for stochastic Cahn-Hilliard partial differential equations

In this paper, we prove a large deviation principle for a class of stochastic Cahn-Hilliard partial differential equations driven by space-time white noises.     

Transportation inequalities for stochastic differential equations with jumps

For stochastic differential equations with jumps, we prove that _W1H$W_{1}H$ transportation inequalities hold for their invariant probability measures and for their process-level laws on the right-continuous path space w.r.t. the L¹$L^1$-metric and uniform metric, under dissipative conditions, via Malliavin calculus. Several applications to concentration inequalities are given.     

Diffusion processes on graphs: stochastic differential equations,large deviation principle

Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices. Received: 12 February 1997 / Revised version: 3 March 1999     

Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations

Some inequalities concerning the Itô stochastic integral and solutions of stochastic different equations are obtained.     

Large deviation for mean-field stochastic differential equations with subdifferential operator

In this article, we prove the existence and uniqueness of a solution for a class of mean-field stochastic differential equations with subdifferential operator (i.e., mean-field MSDEs) by means of the Moreau–Yosida type penalization method. Moreover, we prove a large deviation principle of its path solution via the weak convergence method.     

Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales

The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles for a sequence of Markov processes.     

Concentration inequalities for stochastic differential equations of pure non-Poissonian jumps

We provide concentration inequalities for solutions to stochastic differential equations of pure not-necessarily Poissonian jumps. Our proofs are based on transportation cost inequalities for square integrable functionals of point processes with stochastic intensity and elements of stochastic calculus with respect to semi-martingales. We apply the general results to solutions of stochastic differential equations driven by renewal and non-linear Hawkes point processes.     

Lyapunov type inequalities for second-order half-linear differential equations

In this paper, we establish some new Lyapunov type inequalities for second-order half-linear differential equations, which almost generalize and extend all related existing results in the literature.     

Harmonic analysis of stochastic equations and backward stochastic differential equations

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in ${\mathcal{R}^p}$ ( ${p\in [1,\infty)}$ ) and backward stochastic differential equations (BSDEs) in ${\mathcal{R}^p\times \mathcal{H}^p}$ ( ${p\in (1, \infty)}$ ) and in ${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$ , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Hölder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.     

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.     

Large deviation principle of stochastic differential equations with non-Lipschitzian coefficients

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang＇s results.     

Itô type measure-valued stochastic differential equations

A class of Itô type measure-valued stochastic differential equations is studied on a locally compact Polish space. The SDEs are driven by countably many Brownian motions with interactions caused by the diffusion and the drift coefficients through countably many continuous functions. Explicit conditions are presented for the existence, uniqueness and ergodicity of the solution.     

Skorohod stochastic differential equations of diffusion type

Leta, b beC ²(R ¹)-functions with bounded derivatives of first and second order. We study stochastic differential equations

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.     

Large deviation principle for reflected Poisson driven stochastic differential equations in epidemic models

Abstract

We establish a large deviation principle for a reflected Poisson driven stochastic differential equation. Our motivation is to study in a forthcoming paper the problem of exit of such a process from the basin of attraction of a locally stable equilibrium associated with its law of large numbers. Two examples are described in which we verify the assumptions that we make to establish the large deviation principle.     

Large deviation principle for stochastic evolution equations

Summary The large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail estimates for infinite-dimensional stochastic convolutions are used as main tools.     

A general non-convex large deviation result with applications to stochastic equations

We prove an abstract large deviation result for a sequence of random elements of a vector space satisfying an “abstract exponential martingale condition”. The framework naturally generates non-convex rate functions. We apply the result to solutions of It? stochastic equations in R ^d driven by Brownian motion and a Poisson random measure. Received: 23 June 1999 / Revised version: 17 February 2000 / Published online: 22 November 2000