Law of large numbers and central limit theorem under nonlinear expectations

The main achievement of this paper is the finding and proof of Central Limit Theorem(CLT,see Theorem 12)under the framework of sublinear expectation.Roughly speaking under some reasonable assumption,the random sequence{1/√n(X1+···+Xn)}i∞=1 converges in law to a nonlinear normal distribution,called G-normal distribution,where{Xi}i∞=1 is an i.i.d.sequence under the sublinear expectation.It’s known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties.Under such situation,this new CLT plays a similar role as the one of classical CLT.The classical CLT can be also directly obtained from this new CLT,since a linear expectation is a special case of sublinear expectations.A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT.This paper is originally exhibited in arXiv.(math.PR/0702358v1).

G-Lévy processes under sublinear expectations

We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations.We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes.We also introduce G-Poisson processes.

ON CONTROLLABILITY FOR STOCHASTIC CONTROL SYSTEMS WHEN THE COEFFICIENT IS TIME-VARIANT

This paper investigates the controllability problem of time-variant linear stochastic controlsystems.A sufficient and necessary condition is established for stochastic exact controllability,whichprovides a useful algebraic criterion for stochastic control systems.Furthermore,when the stochasticsystems degenerate to deterministic systems,the algebraic criterion becomes the counterpart for thecomplete controllability of deterministic control systems.

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.

Survey on Path-Dependent PDEs

In this paper,the authors provide a brief introduction of the path-dependent partial differential equations(PDEs for short)in the space of continuous paths,where the path derivatives are in the Dupire(rather than Fréchet)sense.They present the connections between Wiener expectation,backward stochastic differential equations(BSDEs for short)and path-dependent PDEs.They also consider the well-posedness of path-dependent PDEs,including classical solutions,Sobolev solutions and viscosity solutions.

Spatial and temporal white noises under sublinear G-expectation

Under the framework of sublinear expectation,we introduce a new type of G-Gaussian random fields,which contains a type of spatial white noise as a special case.Based on this result,we also introduce a spatial-temporal G-white noise.Different from the case of linear expectation,in which the probability measure needs to be known,under the uncertainty of probability measures,spatial white noises are intrinsically different from temporal cases.

A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation

This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

Extended conditional G-expectations and related stopping times

In this paper,we extend the definition of conditional G-expectation to a larger space on which the dynamical consistency still holds.We can consistently define,by taking the limit,the conditional G-expectation for each random variable X,which is the downward limit(respectively,upward limit)of a monotone sequence (Xi) in L_(G)^(1)(Ω).To accomplish this procedure,some careful analysis is needed.Moreover,we present a suitable definition of stopping times and obtain the optional stopping theorem.We also provide some basic and interesting properties for the extended conditional G-expectation.

Optimal unbiased estimation for maximal distribution

Unbiased estimation for parameters of maximal distribution is a fundamental problem in the statistical theory of sublinear expectations.In this paper,we proved that the maximum estimator is the largest unbiased estimator for the upper mean and the minimum estimator is the smallest unbiased estimator for the lower mean.

Special issue dedicated to Alain Bensoussan on the occasion of his 80th birthday:Preface

We are delighted to present this special issue of PUQR in honor of Professor Alain Bensoussan on the occasion of his 80th birthday.While this birthday provides a good opportunity to celebrate the life and the successes of an outstanding researcher,the COVID-19 epidemic has made it hard for a normal meeting.We hope that this special issue of collected papers will nevertheless provide a lasting mark for his birthday and express the appreciation and best wishes to Alain Bensoussan,from his colleagues and former students,from his co-authors and co-co-authors around the world,for a long life in good health and creative power.

Editorial

Dear All,It is with great pleasure that we welcome you to the first issue of our journal,PUQR–Probability,Uncertainty and Quantitative Risk,a peer-reviewed openaccess journal.Considering its recent and very dynamic development,the theory of backward stochastic differential equations has attracted many researchers,with its vast field of applications in stochastic control,games,finance,and deterministic and stochastic partial differential equations.This has spurred the development of new areas for research such as nonlinear dynamic expectation theory,e.g.,g and G-expectation,and path-dependent partial differential equations,while also finding new applications for problems of ambiguity,uncertainty,quantitative risk,and recursive utility in finance and economics.As we further this field,it is important to provide a forum to stimulate future development with a journal that focuses on these topics.More precisely。