Strong Law of Large Numbers under an Upper Probability

Strong law of large numbers is a fundamental theory in probability and statistics. When the measure tool is nonadditive, this law is very different from additive case. In 2010 Chen investigated the strong law of large numbers under upper probabilityVby assumingVis continuous. This assumption is very strong. Upper probabilities may not be continuous. In this paper we prove the strong law of large numbers for an upper probability without the continuity assumption whereby random variables are quasi-continuous and the upper probability is generated by a weakly compact family of probabilities on a complete and separable metric sample space.

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).

A Property of g-probabilities

Most work on capacities has focused on the 2-alternating and sub 2-alternating.In this paper we consider the capacities-g-probabilities derived from peng＇s g-expectations.When g satisfies some conditions,we show that the g-probabilities fail to be 2-alternating but are sub 2-alternating.

Representation Theorems for Generators of BSDEs in L^p Spaces

In this paper, we prove that the generator g of a class of backward stochastic differential equations （BSDEs） can be represented by the solutions of the corresponding BSDEs at point （t, y, z）, when the terminal data is in Lp spaces, for 1 〈 p ≤ 2.

Control and Nash Games with Mean Field Effect

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007,of Gueant-Lasry-Lions in 2011,of HuangCaines-Malham in 2007 and many others).There are a lot of applications.In general,the applications concern approximating an infinite number of players with common behavior by a representative agent.This agent has to solve a control problem perturbed by a field equation,representing in some way the behavior of the average infinite number of agents.This approach does not lead easily to the problems of Nash equilibrium for a finite number of players,perturbed by field equations,unless one considers averaging within different groups,which has not been done in the literature,and seems quite challenging.In this paper,the authors approach similar problems with a different motivation which makes sense for control and also for differential games.Thus the systems of nonlinear partial differential equations with mean field terms,which have not been addressed in the literature so far,are considered here.

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.