RELATIVE ENTROPY AND LARGE DEVIATIONS UNDER SUBLINEAR EXPECTATIONS

We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.

A strong law of large numbers under sublinear expectations

We consider a sequence of independent and identically distributed(i.i.d.)random variables{ξ_(k)}under a sublinear expectation E=sup_(P∈Θ).We first give a new proof to the fact that,under each P∈Θ,any cluster point of the empirical averages.Next,we consider sublinear expectations on a Polish space,and show that for each constantμ∈[μ,μ^(-)],there exists a probability P_(μ)∈Θsuch thatlim_(n→∞)ξ_(n)=μ,P_(μ-a.s.,(0.1))supposing thatΘis weakly compact and.Under the same conditions,we obtain a generalization of(0.1)in the product space with replaced by.Here is a Borel measurable function on,.Finally,we characterize the triviality of the tail-algebra of the i.i.d.random variables under a sublinear expectation.

Moment bounds for IID sequences under sublinear expectations

With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality,we get the following result:For any continuous functionsatisfying the growth condition |(x) | C(1 + |x|p) for some C > 0,p 1 depending on ,the central limit theorem under sublinear expectations obtained by Peng still holds.

The minimal sublinear expectations and their related properties

In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined on L 2(Ω, $ \mathcal{F} $ )? is linearthe two-dimensional Jensen’s inequality for ? holds.Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.

Multi-dimensional Central Limit Theorems and Laws of Large Numbers under Sublinear Expectations

In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.

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.

Convergence rate of Peng’s law of large numbers under sublinear expectations

This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations,which improves the results presented by Song[15]and Fang et al.[3].

Some Inequalities and Limit Theorems Under Sublinear Expectations

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob＇s inequality for submartingale and Kolmogrov＇s inequality. By Kolmogrov＇s inequality, we obtain a special version of Kolmogrov＇s law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.

On the laws of the iterated logarithm with mean-uncertainty under sublinear expectations

A new Hartman-Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.

Local Lipschitz-α Mappings and Applications to Sublinear Expectations

The aim of this paper is to establish a series of important properties of local Lipschitz-α mappings from a subset of a normed space into a normed space. These mappings include Lipschitz operators, Lipschitz-α operators and local Lipschitz functions. Some applications to the theory of sublinear expectation spaces are given.

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.

Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables,the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.

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.

Non-linear affine processes with jumps

We present a probabilistic construction of R^(d)-valued non-linear affine processes with jumps.Given a setΘof affine parameters,we define a family of sublinear expectations on the Skorokhod space under which the canonical process X is a(sublinear)Markov process with a non-linear generator.This yields a tractable model for Knightian uncertainty for which the sublinear expectation of a Markovian functional can be calculated via a partial integro-differential equation.

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.

How big are the increments of G-Brownian motion?

In this paper,we investigate the problem:How big are the increments of G-Brownian motion.We obtain the Csrg and R′ev′esz’s type theorem for the increments of G-Brownian motion.As applications of this result,we get the law of iterated logarithm and the Erds and R′enyi law of large numbers for G-Brownian motion.Furthermore,it turns out that our theorems are natural extensions of the classical results obtained by Csrg and R′ev′esz(1979).

Optimal Contract for the Principal-Agent Under Knightian Uncertainty

Under the Knightian uncertainty,this paper constructs the optimal principal(he)-agent(she)contract model based on the principal’s expected profit and the agent’s expected utility function by using the sublinear expectation theory.The output process in the model is provided by the agent’s continuous efforts and the principal cannot directly observe the agent’s efforts.In the process of work,risk-averse agent will have the opportunity to make external choices.In order to promote the agent’s continuous efforts,the principal will continuously provide the agents with consumption according to the observable output process after the probation period.In this paper,the Hamilton–Jacobi–Bellman equation is deduced by using the optimality principle under sublinear expectation while the smoothness viscosity condition of the principal-agent optimal contract is given.Moreover,the continuation value of the agent is taken as the state variable to characterize the optimal expected profit of the principal,the agent’s effort and the consumption level under different degrees of Knightian uncertainty.Finally,the behavioral economics is used to analyze the simulation results.The research findings are that the increasing Knightian uncertainty incurs the decline of the principal’s maximum profit;within the probation period,the increasing Knightian uncertainty leads to the shortening of probation period and makes the agent give higher effort when she faces the outside option;what’s more,after the smooth completion of the probation period for the agent,the agent’s consumption level will rise and her effort level will drop as Knightian uncertainty increasing.

Multiple G-Ito integral in G-expectation space

In 2007, Peng introduced Ito integral with respect to G-Brownian motion and the related Ito＇s formula in G-expectation space. Motivated by the properties of multiple Wiener integral obtained by Ito in 1951, we introduce multiple G-Ito integral in G-expectation space, and investigate how to calculate it. Furthermore, We establish a relationship between Hermite polynomials and multiple G-Ito integrals.

The Risk Transfer of Non-tradable Risks under Model Uncertainty

In the context of model uncertainty, we study the optimal design and the pricing of financial instruments aiming to hedge some of non-tradable risks. For the existence of model uncertainty, the preference can be represented by the robust expected utility （also called maxmin expected utility） which can be put in the framework of sublinear expectation. The problem of maximizing the issuer＇s robust expected utility under the constraint imposed by the buyer can be transformed to the problem of minimizing the issuer＇s convex measure under the corresponding constraint. And here the convex measure measures not only the risks but also the model uncertainties.

Reduced-form setting under model uncertainty with non-linear affine intensities

In this paper we extend the reduced-form setting under model uncertainty introduced in[5]to include intensities following an affine process under parameter uncertainty,as defined in[15].This framework allows us to introduce a longevity bond under model uncertainty in a way consistent with the classical case under one prior and to compute its valuation numerically.Moreover,we price a contingent claim with the sublinear conditional operator such that the extended market is still arbitrage-free in the sense of“no arbitrage of the first kind”as in[6].