次线性数学期望及其在博弈论中的应用
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摘要
1933年,Andrey Kolmogorov发表了著作《概率论基础》(初稿为德语,Grundbegriffe der Wahrscheinlichkeitsrechnung),建立了现代概率论的公理体系.给定可测空间(Ω,F)上的概率测度P,则关于F可测的随机变量X的期望EO[X]定义为积分∫ΩXdP.显然,由于概率测度P是线性的,从而EP[·]为一个线性泛函.然而,许多不确定性现象并不能很好的用上述线性概率和线性期望来建模.
一个很有趣的问题是如何建立非线性期望以及相应的条件期望.容度和Choquet期望(或Choquet积分)的概念由Choquet[17]引入,它们被广泛的应用于位势论(参见,Choquet[17],Doob[34])和决策论(参见,Schmeidler [108], Gilboa和Schmeidler[48]).然而据我们所知,条件Choquet期望的概念并没有被人们很好的理解,从而它们很难用于处理经济中的动态问题.另一重要的非线性期望一g-期望在Peng[85]中通过倒向随机微分方程来引入.它是一个理想的框架来度量概率不确定模型的随机性和风险(参见,Chen和Epstein[12],Fritelli和Rossaza Gianin[38],Peng[88]).然而,g-期望的一个局限性是它仅用来处理所涉及的不确定的概率测度关于一个参考概率测度(如,Wiener测度)绝对连续的情形.但是对于金融中著名的波动率不确定性问题,存在不可数个未知的概率测度,它们之间相互奇异Avellaneda等[3]和Lyons[73]研究了带有波动率不确定性的状态依赖的期权定价问题,而路径依赖的情形更有挑战性,需要创建一个比经典概率论更一般的框架.
上述路径依赖下的非线性期望由Peng在[88,90]中建立,并且给出两种完全不同的方法来解决所涉及的动态相容性问题.前一种方法采用路径依赖下的更一般的动态规划原理；第二种方法使用Nisio形式的单调半群(参见Nisio[78,79]),称其为非线性Markov链,通过建立非线性形式的Kolmogorov相容性定理来构造非线性期望空间,这在经典概率论中同样重要.
在上述所提到的非线性期望中,一类非常典型的非线性期望-G-期望首先由Peng[92]于2006年提出.事实上,G-期望也是一种非常典型的次线性期望,它保留了除去线性性质外线性期望所具有的其他良好性质.分布和独立性的概念在整个理论中起到非常重要的作用Peng的一项开创性工作是直接使用次线性期望E[·]来定义分布和独立,而不是使用容度这一看上去更加自然的概念来推广它们的定义.基于这些新概念Peng引入了一种非常重要的分布G-正态分布,它可以由所谓的G-热方程来刻画.G-期望和G-Brown运动的概念可以视为Wiener测度和经典的Brown运动的非线性推广.这些概念和相应的极限定理(大数定律和中心极限定理)以及关于G-Brown运动的Ito随机积分由Peng[92-102]引入并且做了系统研究.最近,很多作者基于Peng的开创性工作做了许多相应的推广.关于大数定律,Chen[11],Chen和Wu[14],Chen等[15]研究了强大数定律,这些结果是对Peng[93,96]中“弱”大数定律的推广.在Peng首先在[93]中证明了次线性期望空间中独立同分布(简记为i.i.d.)假设下的中心极限定理之后,许多作者在不假设同分布但是仍假设独立的情形下推广了这一结果,参见Li和Shi[66],Hu和Zhang[51],Hu[50],Hu和Zhou[58]等.关于G-期望理论框架下的Ito随机计算,特别的,关于Ito公式,由Gao[39]和Zhang等[123]等做出相应的推广.关于次线性期望理论以及G-期望理论的进一步工作可以参见Bai和Buckdahn[6],Bai和Lin[7],Chen和Hu[13],Denis等[30]Dolinsky等[33],Epstein和Ji[35],Gao[42],Gao和Jiang[41,42],Gao和Xu[43,44],Hu[52,53],Hu等[54：55],Hu和Peng[56,57],Lin[71],Lin[72],Nutz[80],Nutz和van Handel[81],Nutz和Zhang[82],Peng等[103],Soner等[112]Song[113-116],Xu和Zhang[122],等.本论文的出发点与Peng的开创性工作有一点不同.我们将次线性期望EP[·]视为
一族概率测度集合P的上期望,其中P中的元素P为定义在可测空间(Ω,B(Ω))上的概率测度,这使得我们可以方便的利用线性期望EP[·]的现有性质来研究EP[·]的性质.并且我们从一个新的观点来研究独立的定义,即通过经典的条件期望来定义独立.这些新的构想使得我们可以将Peng[92-100]中相应的极限定理和Ito随机计算以及其他作者的工作推广到我们的框架中来.本论文中的第1章至第3章将重点研究这些问题.第4章是本论文的亮点.我们研究次线性期望和G-期望的性质,包括严格比较定理,可加性Wasserstein距离,对偶,控制和最优转移问题,尽管这些性质在经典概率论中是众所周知的,并且其中的一些性质甚至是显然的.这些结果是经典结果的非平凡推广,它们在第5章和第6章中多次用到.
作为G-期望理论的应用.我们在第5章中介绍了连续最大变差鞅(简记为CMMV)的概念以及最大鞅变差问题.粗略地讲,最大鞅变差问题是指,给定一个定义在△(Rd)上的实值泛函M和一个概率测度μ∈△(Rd),我们的目标是在由所有长度为n的其终端值在Blackwell意义下受控于μ的取值于Rd的鞅所组成的集合上最大化所谓的M-变差.这一问题推广了Mertens和Zamir[77]中所引入的最大L1-变差问题.一维情形下的一般问题在De Meyer[26]中所研究,随后由Gensbittel[46]推广到多维情形.我们基于第1章至第4章中的结果给出这一问题的一个新的简洁的证明.两篇文章[26]和[46]中仅研究中心化的情形,即,泛函M仅定义在零均值的概率测度集上,我们将其推广到非中心化的情形,并且我们将在随后一章研究带有交易费用的博弈模型时发现这一推广非常有用.在第6章,我们基于Aumman和Maschler[4]的模型研究一类更广的单边不完全信息下的重复博弈模型,这类模型首先由De Meyer[25]以金融交易博弈的形式提出,随后由Gensbittel[45,47]推广到多维情形.这些博弈模型与第5章中所介绍的连续最大变差鞅的概念以及最大鞅变差问题有着紧密的联系.我们系统地研究了两个具体的博弈模型并且得到它们的Nash均衡显式解.通过这些模型我们可以看出连续最大变差鞅在股票市场中是一类非常稳健的价格过程.我们指出第5章和第6章的内容仅仅是从G-期望观点来研究博弈论的一个初步尝试,还有许多有趣的问题有待进一步研究.
论文共分为六章,以下是本文的结构和得到的主要结论：
(Ⅰ)第1章主要研究不确定性下的随机游走和相应的极限定理.
我们将经典的Bernoulli随机游走和简单随机游走推广到不确定性的情形.所谓的“不确定性”是指参考的概率测度不是唯一的,而是一族概率测度组成的集合.设(Ω,B(Ω))为一个可测空间,P为定义在(Ω,B(Ω))上的概率测度组成的集合.给定随机变量X,参照Peng[93],X在P下的分布函数为Gb,Lip(R)到R上的泛函,其定义为
其中Cb,Lip(R)为R上的有界Lipschitz函数空间.为了简化记号,我们将SupP∈ρ EP[·]记为EP[·]Peng[93]中给出的在P下独立的定义如下定义1.4设{Xi}i=1∞为一列(Ω,B(Ω))上的随机变量.如果对任一φ∈Cb.Lip(Rn),那么我们称Xn在P下独立于(X1,…,Xn-1).如果对每个n∈N,Xn在P下独立于(X1.…,X-1),那么我们称{Xi}i=1∞在P下独立.
我们通过经典的条件期望给出一个独立的新定义：
定义1.5设{Xi}i=1∞为一列(Ω,B(Ω))上的随机变量.给定(Ω,B(Ω))上的一族概率测度P,如果下列条件成立.
(1)(?)P∈P,(?)φ∈Cb,Lip(R),EP[φ(Xn)|X1,…,Xn-1]≤EP[φ(Xn)],P-a.s.
(2)(?)φ∈Cb,Lip(R),存在依赖于φ的P∈P,使得EP[φ(Xn)|X1,…,Xn-1]=EP[φ(Xn)],P-a.s.那么我们称Xn在P下弱独立于(X1,…,Xn-1).如果对每个n∈N,Xn在P下弱独立于(X1,…,Xn-1),那么我们称{Xi}i=1∞在P下弱独立.
下面的定理给出了弱独立和Peng的独立定义之间的关系.
定理1.7设{Xi｝i=1∞为一列在P下弱独立的随机变量序列.我们定义P如下
P={P:(?)φ∈Cb,Lip(R),(?)n∈N,EP[φ(Xn)|X1,…,Xn-1]≤EP[φ(Xn)],P-a.s.｝.则{Xi｝i=1∞在定义1.4的意义下在P下独立,并且
我们可以证明在独立同分布假设下关于Bernoulli随机游走的大数定律,并且我们将它推广到不假设独立同分布的情形.下面的定理是更一般的形式.定理1.9设{Xk｝k=1∞为可测空间(Ω,B(Ω))上的一列随机变量,P为定义在(Ω,B(Ω))上所有满足下列条件的概率测度P组成的集合：(?)n∈N,μ≤EP[Xn|X1,…,Xn-1]≤μ且EP[|Xn|q|X1,…,Xn-1]≤Kq P-a.s.,其中μ,μ,K,q为常数且q>1.则我们有(i)对每一μ∈[μ,μ],存在Pμ∈P,使得(ii)对每一P∈P,(iii)对每一φ∈Cb,Lip(R),我们同样考虑关于简单随机游走的中心极限定理.G-正态分布的概念在中心极限定理中起到关键作用.在本章中,G-正态分布通过G-热方程的解来定义.定义1.10如果ξ的分布通过下式给出其中uφ(t,x)为如下G-热方程的解：
其中G(α)=1/2σ2α+-1/2σ2α-,0≤σ≤σ.那么我们称ξ为G-正态分布,记为ξN(0,[σ2,σ2]).
我们在此仅列出两个中心极限定理.事实上,它们在某种意义上是等价的.第二个多维情形下的中心极限定理会在第5章用到.我们将G-正态分布ξ的分布记为EG[φ(ξ)]：=Fξ(φ).
定理1.14设{Xi｝i=1∞为可测空间(Ω,B(Ω))上的一列随机变量.设P为定义在(Ω,B(Ω))上的所有满足下列条件的概率测度组成的集合：(?)P∈P,(?)i∈N,(1)EP[Xi|X1,…,Xi-1]=0,(2)σ2≤EP[X,2|X1,…,Xi-1]≤σ2,(3)EP[|Xi|q|X1,…,Xi-1]≤Kq.我们记Sn=∑i=1n Xi.
设Mn1(∑,K)为概率空间(Ω,B(Ω),P)上所有满足下列条件的长度为n的取值于Rd的鞅组成的集合：(i)EP[Sn]=0,
(ii)EP[(Sk+1-Sk)(Sk+1-Sk)T|S1,…,Sk]∈∑,0≤k≤n-1,其中∑为S+(d)中的有界凸闭子集.
(iii)EP[||Sk+1-Sk||q]≤K,0≤k≤n-1.设Vn[φ]:=supS∈mq(Σ,K)EP[φ[Sn/(?)n)].定理1.17我们假设q>2.设ξ为G-期望EG[·]下的G-正态分布N(0,∑),则(?)φ∈C(Rd)且满足增长条件|φ(x)|≤C(1+|x|p),其中1≤p＜q,我们有
在第1章的最后一节,我们给出G-Brown运动通过简单随机游走的逼近定理.设{Sn｝n=1∞为一个离散时间过程,则S的连续时间化过程定义为St=Sn+(t-n)(Sn+1-Sn), n≤t 设P为可测空间(Ω,B(Ω))上的概率测度组成的集合.次线性期望EP[·],上概率V(·)和下概率υ(·)分别定义为Ep[·]=supP∈ρEp[·];V(·)=supP∈ρ=P(·);v(·)=infP∈ρP(·).
我们给出乘积独立和加和独立的定义,这些定义比Peng[93]中给出的独立的定义要弱.
定义2.14设X1,X2,…,Xn为(Ω,B(Ω))上的一列可测随机变量.(i)如果对于每一个非负有界的Lipschitz函数φk,k=1,…,n,那么我们称Xn乘积独立于(X1,…,Xn-1).(ii)如果对于每一个φ∈Cb,Lip(R),那么我们称Xn加和独立于(X1,…,Xn-1).
下面给出的大数定律是Peng[93,96,98],Chen[11],Chen和Wu[14],以及Chen等[15]中的大数定律的推广.
定理2.17设{Xk｝k=1∞为一列随机变量满足：对某一q>1,supk≥1EP[|Xk|q]＜∞,且EP[Xk]三μ,-EP[-Xk]三μ,k=1,2,设Sn=∑k=1n Xk.
(i)如果{Xk｝k=1∞乘积独立,那么
(ii)如果{Xk｝k=1∞乘积独立且加和独立,那么
(iii)如果{Xk｝k=1∞加和独立且V(·)上连续,即,当An↓A时,V(An)↓V(A),其中,An,A∈B(Ω),
那么
关于次线性期望空间上的中心极限定理,Peng[93]首先证明了独立同分布的情形,随后由Li和Shi[66],Hu和Zhang[51],Hu[50],Hu和Zhou[58]推广到不假设同分布的情形.然而,所有的这些定理均要求随机变量的独立性.我们考虑一个比独立稍弱的条件,称之为m-相依,并且证明了相应的中心极限定理.这一结果已被Acta Mathematicae Applicatae Sinica, English Series所接受.
定义2.31如果存在一个整数m使得对每一个n和每一个j≥m+l,(Xn+m+1,…,Xn+j)均独立于(X1,…,Xn),那么我们称序列{Xi｝i=1∞为m-相依.特别的,如果m=0,那么称{Xi｝i=1∞为独立序列.
定理2.32设{Xi｝i=1∞为一列m-相依序列,满足
且对i=1,2,…,EP[|Xi|2+α]≤M,其中α>0且M为常数.设Sn=∑i=1n,则我们有
其中ξ～N(0；[σ2,σ2]).
(Ⅲ)第3章主要研究不假设拟连续条件的Ito随机计算,并且得到一般形式的Ito公式以及具有局部Lipschitz系数的随机微分方程的解的存在唯一性.
现有的关于G-Brown运动的Ito随机分析均建立在随机过程空间MGp(0,T)(p≥1)上(参见,Peng[92,95,97,98,100]以及Gao[39]和Zhang等[123]),其中空间MGp(0,T)由满足拟连续条件的随机变量所生成.然而停时这一在经典随机分析中非常重要的概念并不满足拟连续性,从而我们很难在空间MGp(0,T)中处理停时问题.这也导致现有的Ito公式均对C2-函数要求一定的增长条件.为了克服上述困难,我们在本章中引入一个更大的随机过程空间M*p(0,T).它是由未必满足拟连续条件的随机变量所生成.随后我们在这个更大的空间M*p(0,T)上定义Ito随机积分,并且我们考虑了定义在停时区间上的Ito随机积分.这使得我们可以在“局部可积”空间Mω2(0,T)上定义Ito随机积分.这一新的理论框架使得我们可以得到关于C1,2-函数的更一般的Ito公式,这一结果在本质上推广了Peng[92,95,97,98,100]以及Gao[39]和Zhang等[123]中的结果.该结果与导师彭实戈教授合作发表于Stochastic Process and Their Applications121(7)1492-1508.
定理3.41设Φ∈C1,2([0,T]×R)且则对任一t∈[0,T],我们有,
在本章的最后一节,我们考虑如下由d维G-Brown运动驱动的随机微分方程：
其中b(·,·),hij(·,·),σj(·,·)：[0,T]×R→R为连续函数,X0为常数.我们引入如下条件：
(H1)有界性条件：对任一s∈[0,T]
(H2)Lipschitz条件：对任何x,y∈R和s∈[0,T],max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤K|x-y|.
(H3)局部Lipschitz条件：对所有满足|x|,|y|≤R的x,y∈R以及s∈[0,T],
max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤KR|x-y|.
(H4)增长性条件：对任一x∈R和s∈[0,T],xb(s,x)≤K(1+x2), xhij(s,x)≤K(1+x2),|σj(s,x)|2≤K(1+x2).
第一个定理给出在空间M*2(0,T)上的解的存在唯一性.第二个定理研究具有局部Lipschitz系数的随机微分方程的可解性.
定理3.44假设条件(H1)和(H2)成立.则存在唯一的过程X∈M*2(0,T)满足(1).
定理3.45假设条件(H3)和(H4)成立.则随机微分方程(1)存在唯一的连续适应解X.
(Ⅳ)第4章研究次线性期望和G-期望的性质,包括严格比较定理,可加性,G-期望与Choquet期望之间的联系,Wasserstein距离,对偶,控制以及最优转移问题.
本章分为五节来研究上述性质.在第4.1节,我们研究严格比较定理.我们在此仅列出本节中两个重要的定理.定理4.4设X,Y∈Lc1(Ω)且X≤Y q.s.如果那么EP[X]＜EP[Y].定理4.9设旦＞0且X,Y∈Lip(Ω)具有如下形式X=φ(Bt1,Bt2-Bt1,…,Btn-Btn-1)和Y=ψ(Bt1,Bt2-Bt1,…,Btn-Btn-1),其中φ(x)≤ψ(x),(?)x∈Rn.则EG[X]＜EG[Y]当且仅当存在x0∈Rn使得φ(x0)＜ψ(xo).
在第4.2节,我们研究G-期望的可加性.设ξ为G-正态分布N(0,[σ2,σ2]),其中0<σ<σ.如下的两个定理是本节中的主要定理.定理4.13我们假设φ,ψ∈Cb,Lip(R).则EG[φ(ξ)+ψ(ξ)]=EG[φ(ξ)]+EG[ψ(ξ)]当且仅当(?)xxu(t,x)(?)xxσ(t,x)≥0,V(t,x)∈(0,1)×R.定理4.16如果存在x0θ>0使得φ,ψ∈C2((x0-θ,x0+θ))且φ″(x0)ψ″(x0)＜0,那么我们有
在第4.3节,我们比较了G-期望和Choquet期望并且给出了一些有趣的例子.我们证明了G-期望被相应的Choquet期望所控制.下面的定理是本节中的主要定理.定理4.26G-期望EG[·]可以表示成Choquet期望Ec[·]伊P,(?)X∈LG1(Ω),EG[X]=Ec[X])当且仅当EG[·]为线性(即,σ=σ.
在第4.4节,我们将经典的Wasserstein距离及其相应的性质推广到次线性空间.
设P1和P2为两个非空凸的弱紧的概率测度集合.我们定义P1和P2之间的Hausdorff-Wasserstein距离如下：其中Wp(P1,P2)是经典的P1和P2之间的Wasserstein距离.
我们首先给出次线性形式的Kantorovich-Rubinstein对偶公式.定理4.30||φ||Lip≤1表示Lipschitz函数φ的Lipschitz系数小于等于1.
众所周知,概率测度的弱收敛等价于Wasserstein距离下收敛,我们给出这一性质在次线性期望框架下的非平凡推广.
定义4.31设{EPn}n=1∞为一列次线性期望.如果对每一φ∈Cb,Lip(Ω),那么我们称{EPn｝n=1∞弱收敛于EP,或等价的,称{Pn｝n=1∞弱收敛于P.
定理4.35设{Pn｝n=1∞为一列凸的弱紧的概率测度集合满足下列条件：
设P为一凸的弱紧的概率测度集合.则下面的两个论述等价：(ⅰ)Pn弱收敛于P.(ⅱ)W1(Pn,P)→0.
在第4.5节,我们介绍了经典情形以及次线性情形下的对偶和控制的概念以及最优转移问题.首先介绍的Fenchel对偶及其性质在第6章中研究对偶博弈时会再次用到.
随后我们介绍了经典的Blackwell控制的概念,并且得到一个关于次线性期望的控制定理.
定理4.47如果P1和P2为(Ω,B(Ω))上的两个凸的弱紧的概率测度集合.那么下面的叙述等价：
(ⅰ)EP1[·]被EP2[·]所控制.
(ⅱ)P1(?)P2.
最后,我们研究次线性空间上的Kantorovich最优转移问题.设Q1和Ω2为两个完备可分的距离空间,Ⅱ(μ,υ)为定义在(Ω1(?)Q2,B(Q1(?)Q2))上的所有满足其边际分布在Q1和Ω2上分别为μ和υ的概率测度组成的集合.设P1和P2为分别定义在(Ω1,B(Q1))和(Ω2,B(Q2))上的弱紧的凸的概率测度集合.记Ⅱ(ρ1,ρ2)=∪μ∈ρ1,v∈ρ2Ⅱ(μ,v)对Ω1(?)Ω2上的连续函数c,Φc为所有满足下列条件的连续函数对(φ,ψ)组成的集合：φ(ω1)+φ(ω2)≥c(ω1,ω2),(?)ω1∈Ω1,ω2∈Ω2.我们得到如下形式的对偶公式：定理4.49设c：Q1(?)Ω2→R为连续函数,则我们有
我们同时考虑如下的最大协方差问题：给定一个概率测度μ和一个概率测度集合P.最大协方差函数C(μ,P)定义为
下面的定理在第5章中会用到.定理4.52设{Pn｝和P为凸的弱紧的概率测度集合,μ为任一概率测度.如果W2(Pn,P)→0,那么我们有(Ⅴ)第5章主要研究连续最大变差鞅和最大鞅变差问题.
我们首先介绍De Meyer[26]中所提出的连续最大变差鞅的概念,然后我们将其推广到G-期望框架.本章的主要目的是解决如下的最大鞅变差问题.
设Mn(μ)为(F,X)组成的集合,其中F：=(Fq)q=1,…,n为概率空间(Ω,B(Ω),P)上的信息族,X=(Xq)q=1,…,n为F鞅,其终端分布Xn在Blackwell意义下受控于μ.给定泛函M：△2(Rd)→R,我们可以定义M-变差VnM(F,X)如下则最大M变差VM(μ)定义为
对于一维的情形,我们得到如下的定理：
定理5.11如果M满足.
(ⅰ)正齐性(?)X∈L02(R),(?)α>0:M[αX]=α[X].
(ⅱ)Lipschitz连续性：存在p∈[1,2)和K∈R使得对所有的X,Y∈L02(R)：|M[X]-M[Y]|≤K||X-Y||Lp.
(ⅲ)常数平移不变性M[X+β]=M[X]+M[β],(?)β∈R.那么对所有的μ∈△2(R),我们有
(3)如果ρ>0并且对所有的n,(Fn,Xn)∈Mn(μ)满足VnM(Fn,Xn)=VnM(μ),那么Xn的连续时间表示Πtn：=X[nt]n依有限维分布收敛到连续最大变差鞅Πμ.对于多维情形,我们首先引入辅助函数r定义如下
其中cov(μ)为μ∈△02(Rd)的协方差矩阵.集合r的定义可以参见定义5.12.我们假设M：△2(Rd)→R满足如下假设：(H1)M≥0并且非退化：(?)x∈Rd,存在μ∈△02(Rd)使得μ(Rx)=1,且M(μ)≥0.
(H2)M在p阶Wasserstein距离下K-Lipschitz连续,其中p∈[1,2).
(H3)M满足正齐性：(?)X∈L2(Rd),λ>0,M[λX]=AM[X].
(H4)M在△02(Rd)上为凸泛函.
(H5)r为拟凸函数,即,(?)α∈R,{Y∈L2(Rd)|r(cov(Y))≤α}.在L2(Rd)中为凸集.
(H6)M[X+β]=M[X]+M[β],(?)β∈Rd.
我们有下面的定理：
定理5.14在假设(H1)-(H6)下,我们有
(Ⅵ)第6章主要研究一类广义的Aumann和Maschler[4]中所介绍的重复博弈模型.
这类模型首先在De Meyer[25]中作为金融交易模型引入,随后由Gensbittel[45,47]推广到多维情形.与Aumann-Maschler的模型所不同的是我们允许状态集和策略集为不可数集.本章包含四节.
在第6.1节,我们研究[45]中所提出的线性博弈模型并且将[45]中的Cav(u)定理从△(P)推广到△1(Rd),其中P为Rd中的凸紧子集.设Vn(μ)和瓦(μ)分别为单边信息不对称重复博弈模型Γn(μ)(参见第6.1节)中参与者1的最大支付和参与者2的最小支付,ui(μ)和u(μ)分别为完全信息下的博弈模型中相应的支付.我们有如下的Cav(u)定理：
定理6.5对所有的μ∈△1(Rd),我们有
如果我们进一步假设,(?)μ∈△∞(Rd),博弈r1(μ)存在值,即,V1(μ)=V1(μ)=V1(μ),那么我们有如下更精确的Cav(u)定理：
定理6.9如果V1满足如下假设：
(i)存在μ0∈△2(Rd)使得V1(μ0)＞0.
(ⅱ)V1([L+β])=V1([L])+V1([β]),(?)β∈Rd.
那么我们有,对所有的μ∈△2(Rd),其中ξ～N(0,Γ),Γ在第5章第5.3节中给出,我们需要用V1代替那里的M.
在第6.2节,我们研究一类特殊的线性博弈模型,称为金融交易模型.这一模型由De Meyer[25]提出.我们推广了[25]中的自然交易机制.
在博弈Γn(μ)中(参见第6.2节),自然交易机制的假设如下：
(H1)博弈值的存在性：(?)μ∈△∞(R),博弈Γ1(μ)存在值.
(H2)交易的有界性：(?)i,j:|Aij|≤K,其中K为常数.
(H3)正齐性：
(H4)关于风险资产中的无风险部分的平移不变性：
(H5)信息具有正价值：(?)L∈L2(R)：V1([L])>0.
基于第5章的结果,我们给出下面的值函数以及价格过程的逼近定理.定理6.15如果(H1)-(H5)成立,那么对所有的μ∈△2(R),(ⅲ)如果ρ>0且对所有的n,(Fn,Xn)∈Mn(μ)满足VnV1(Fn,Xn)=Vn(μ),那么Xn
的连续时间表示Ⅱtn：=X[nt]n依有限维分布收敛于Ⅱμ.
在第6.3节中,我们系统的研究了一个带有交易费用的博弈模型,它并不满足[25]中的自然交易机制但是满足我们在第6.2节中推广的自然交易机制.我们通过对偶方法得到这一模型的Nash均衡显式解.并且我们证明了由非内幕参与者所提出的价格过程在有限维分布下收敛到连续最大变差鞅.这一结果更好的表明连续最大变差鞅在股票市场中是一类非常稳健的价格过程.
在第6.4节,我们研究了一个博弈模型其中博弈双方均不是风险中性,这推广了DeMeyer[26]中的结果,原结果中非内幕参与者是风险厌恶而内幕参与者是风险中性.我们得到的结论非常有趣Nash均衡解不依赖于内幕参与者的风险态度,即,当内幕参与者是风险中性时的Nash均衡同时也是内幕参与者是风险厌恶或风险喜好时的均衡解.
In1933, Andrey Kolmogorov published his book Foundation of the Theory of Prob-ability (in German, Grundbegriffe der Wahrscheinlichkeitsrechnung), which established the modern axiomatic foundations of probability theory. Given a probability measure P on a measurable space (Ω,F), the expectation Ep[X] of a F-measurable random variable X is defined as the integral fΩXdP. Obviously, Ep[·] is a linear functional due to the linearity of probability measure P. However, a great number of uncertainty phenomena can not be well modeled by such linear probability or expectation.
A very interesting problem is to develop nonlinear expectation and related condition-al expectations. The notion of capacity and Choquet expectation (or Choquet integral) was introduced by Choquet [17], which has been widely used in the potential theory (e.g., Choquet [17], Doob [34]) and decision theory (e.g., Schmeidler [108], Gilboa and Schmeidler [48]). But to the best of our knowledge, the notion of conditional Choquet expectation has not been well understood and it was hardly used to deal with dynamic problems in economics. The other important nonlinear expectation called g-expectation was introduced via BSDE in Peng [85]. It is an ideal framework for the valuation of randomness and risk in the case of the uncertainty of probability models (e.g., Chen and Epstein [12], Fritelli and Rossaza Gianin [38], Peng [88]). However, one important limitation of g-expectation is that the involved uncertain probability measures have to be absolutely continuous with respect to a reference probability measure, e.g., a Wiener measure. But for the well-known problem of volatility model uncertainty in finance, there is an uncountable number of unknown probabilities which are essentially singular from each other. Avellaneda et al.[3] and Lyons [73] studied this volatility uncertainty problem for the situation of state-dependent options. The situation of path-dependence is more challenging and needed to create a new framework more general than the classical notion of probability.
Such types of fully nonlinear expectations for situations of path-dependence were constructed by Peng in [88,90] where two very different approaches were introduced to solve the involved problem of dynamic consistency. The first one is a generalized dynamic programming principle for the path-dependent situation. The second one is about to use the notion of nonlinear monotone semigroups of Nisio's type (see Nisio [78,79]), called nonlinear Markov chain, to develop a nonlinear version of Kolmogorov consistency theorem in order to construct nonlinear expectation spaces which plays the important role as in the classical probability theory.
The most typical example of the above-mentioned fully nonlinear expectation is the G-expectation which was first introduced by Peng [92] in2006. In fact, G-expectation is also a typical example of the sublinear expectation which keeps the well properties of linear expectations except linearity. The notion of distribution and independence plays an important role in the whole theory. One pioneering work of Peng is to define such distribution and independence directly by the sublinear expectation E[·] but not by capacity which seems as a natural way to generalize them. Based on these new notions, Peng introduced the most important distribution called G-normal distribution which can also be characterized by the so-called G-heat equation. The notion of G-expectation and G-Brownian motion can be regarded as a nonlinear generalization of Wiener measure and classical Brownian motion. These notions and the corresponding limit theorems (law of large numbers and central limit theorem) as well as stochastic calculus of Ito's type with respect to G-Brownian motion were introduced and systematically developed in Peng [92-102]. Recently, many authors gave a number of generalizations of Peng's initial works. As for law of larger numbers, Chen [11], Chen and Wu [14], Chen et al.[15] study the strong law of large numbers which generalizes the "weak" law of large numbers in Peng [93,96]. After Peng first established central limit theorem in sublinear expectation space under independent and identically distributed (i.i.d. for short) assumptions in [93], many authors generalized this result without the identically distributed assumption but still keeping the independent assumption, see Li and Shi [66], Hu and Zhang [51], Hu [50], Hu and Zhou [58] etc. About the Ito's calculus under the framework of G-expectation theory, especially, for Ito's formula, was generalized by Gao [39] and Zhang et al.[123], etc. The further development of sublinear expectation theory and G-expectation theory can be found in Bai and Buckdahn [6], Bai and Lin [7], Chen and Hu [13], Denis et al.[30], Dolinsky et al.[33], Epstein and Ji [35], Gao [42], Gao and Jiang [41,42], Gao and Xu [43,44], Hu [52,53], Hu et al.[54,55], Hu and Peng [56,57], Lin [71], Lin [72], Nutz [80], Nutz and van Handel [81], Nutz and Zhang [82], Peng et al.[103], Soner et al.[112], Song [113-116], Xu and Zhang [122], etc.
The starting point of this thesis is a little different from Peng's initial work. We will take the sublinear expectation Ev[·] as an upper expectation of a set V of probability measures P defined on some measurable space (Ω,B(Ω)), which allows us to study the properties of Ep[·] conveniently by the existing properties of linear expectation Ep[·]. And we study the notion of independence from a new view which defined via classical conditional expectations. These formulations allow us to generalize the corresponding limit theorems and Ito's calculus in Peng [92-100] and other authors'work to our setting. Chapter1to Chapter3of this thesis will focus on these topics. Chapter4is the high-light of this thesis. We study some properties of sublinear expectation and G-expectation, including the strict comparison theorem, additivity, Wasserstein distance, duality, dom-ination and optimal transportation, although these properties are well-known and some of them are even obvious in the classical probability theory. These results are non-trivial extensions of the classical ones and will be used in Chapter5and6.
As an application of the G-expectation theory, we introduce the notion of continuous martingale of maximal variation (CMMV for short) and the problem of maximal variation of martingales in Chapter5. Roughly speaking, the problem of maximal variation of martingales is that, given a real-valued function M defined on△(Rd) and a probability measure/μ∈△(Rd), we aim to maximize a functional called M-variation, over the set of all Rd-valued martingales of length n whose terminal distribution is Blackwell dominated by μ. This problem generalizes the problem of maximal L1-variation introduced in Mertens and Zamir [77]. The most general form has been studied in De Meyer [26] for one-dimensional case and then generalized by Gensbittel [46] for multi-dimensional case. We will give a new and simple proof for it based on the results in Chapter1to Chapter4. Both of the two papers [26] and [46] only studied the centered case, i.e., the functional M defined on the set of probability measures with zero-mean, we will generalize them to the non-centered case which turns out to be very useful when we study the games with transaction costs in the next chapter. In Chapter6, we study a general class of repeated games with incomplete information on one side a la Aumman and Maschler [4], which was first introduced by De Meyer [25] as financial exchange game and then generalized by Gensbittel [45,47] in multi-dimensional context. These games have the closed relation with the notion of CMMV and the maximal variation of martingales in Chapter5. We also systematically study two particular game models, and obtain the explicit solution of Nash equilibrium for them. These models show that CMMV is a very robust dynamic in the stock market. We point out that the contents in Chapter5and Chapter6is just a first attempt to study the game theory from the view of G-expectation. There are many interesting problems to study in the future.
This thesis consists of six chapters. In the following, we list the main results in this thesis:
(Ⅰ) In Chapter1, we study the random walks under uncertainty and the corresponding limit theorems.
We generalized the classical Bernoulli random walk and simple random walk to the uncertainty case."Uncertainty" means that the underlying probability measure is not unique, but correspond, is a set of probability measures. Let (Ω, B(Ω)) be a measurable space, V be the set of probability measures defined on (Ω,B(Ω)). Given a random variable X, following Peng [93], the distribution function of X under V is a functional from Cb,Lip(R) to R defined by where Cb,Lip(R) is the space of all bounded and Lipschitz functions on R. For simplicity of notation, we write Eρ[·] instead of sup P∈ρ[·]. The notion of independence under ρ denned in Peng [93] as following:
Definition1.4Let {Xi}i=1∞be a sequence of random variables on (Ω,B(Ω,)).{Xi}i=1∞is said to be independent under ρ, if for each n∈N, Xn is independent of (X1,…, Xn-1) under V, which is defined by (?)φ∈Cb,Lip(Rn), We give a new definition of independence via classical conditional expectations.
Definition1.5Let {Xi}i=1n be a sequence of random variables on (Ω,B(Ω)). Given a set V of probability measures on (Ω,B(Ω)),{Xi}i=1∞is said to be weak independent under ρ, if for each n∈N, Xn is weak independent of (X1,…, Xn-1) under ρ, which is defined by (1)(?)P∈ρ,(?)φ∈Cb,Lip(R), EP[φ(Xn)|X1,…,Xn-1]≤Eρ[φ(Xn)], P-a.s.,(2)(?)φ∈Cb,Lip(R), there exists P∈ρ depending on φ, such that EP[φ(Xn)[X1,…,Xn-1]=Eρ[φ(Xn)], P-a.s.
The following theorem give the relations of weak independence and Peng's indepen-dence.
Theorem1.7Let {Xi}i=1∞be a sequence of weak independent random variables under ρ. We define ρ by
ρ={P:(?)φ∈Cb,Lip(R),(?)n∈N, EP[φ{Xn)|X1,…,Xn-x]≤Eρ[φ(Xn)], P-a.s.}. Then {Xi}i=1∞is independent under ρ as in the Definition1.4, and Eρ[φ(Xi)]=Eρ[φ(Xi)],(?)i∈N,(?)φ∈Cb,Lip(R).
We can prove the law of larger numbers for Bernoulli random walk under i.i.d. assumption, and then we generalize this theorem without the i.i.d. assumption. The following theorem is the general version.
Theorem1.9Let {Xk}k=1∞be a sequence of random variables on measurable space (Ω, B(Ω)) and V a set of all probability measures P defined on (Ω,B(Ω)) such that (?)n∈N, μ≤EP[Xn|X1,…,Xn-1}≤μ, and EP[|Xn|q|X1,…,Xn-1]≤Kq P-a.s., where μ,μ, K, q are constants and q>1. Then we have (ⅰ) For each μ∈[μ,μ], there exists Pμ∈ρ, such that (ⅱ) For each P e ρ,(ⅲ) For each φ∈Cb,Lip(R), We also consider the central limit theorems for simple random walk. The notion of G-normal distribution play an important role in the central limit theorem. In this chapter, the G-normal distribution is defined as the solution of G-heat equation.
Definition1.10We call ξ is G-normal distributed, denoted by ξ~N(0,[σ2,σ2]), where0≤σ≤σ, if the distribution of ξ is given by Fξ(φ)=uφ(0,1),(?)φ∈Cb,Lip(R), where uφ{t,x) is the unique viscosity solution of the following G-heat equation:(?)tu-G((?)xx2u)=0u|t=1=φ where G(α)=1/2σ-2α+-1/2σ2α-
We only list two central limit theorems. In fact, they are equivalent in some sense. The second one in multi-dimensional case will be used in Chapter5. We denote the distribution function of G-normal distribution ξ by Eg[φ(ξ)]:=Fξ(φ).
Theorem1.14Let {Xi}i=1∞be a sequence of random variables on measurable space (Ω, B(Ω)). Let ρ be the set of all probability measures on (Ω, B(Ω)) such that, VP G V,(?)i∈N,
(1) EP[Xi|X1,…,Xi-1}=0,
(2)σ2≤EP[Xi2|X1,…,Xi-1}≤σ2,
(3)EP[|Xoi|q|X1…,Xi-1]≤Kq. We denote Sn=Σi=1n Xi.
If q>2and0<σ≤σ≤K<∞, then (?)φ∈Cb,Lip(R), where ξ~N(Q,[σ2,σ2]).
Let Mnq(Σ, K) be the set of n-stage Rd-valued martingales on some probability space (Ω,B(Ω), P) satisfying the following conditions:(ⅰ) EP[Sn]=0,(ⅱ) EP[(Sk+1-Sk)(Sk+1-Sk)T|S1,…,Sk]∈Σ,0≤k≤n-1, where Σ is a bounded,
convex and closed subset of S+(d).(ⅲ) EP[||Sk+1-Sk||q]≤K,0≤k≤n-1.
Let Vn[φ]:=supS∈Mnq(Σ,K) EP[φ(Sn/(?)n)].
Theorem1.17We assume that q>2. Let ξ be a G-normal distribution N (0,Σ) under G-expectation EG, then (?)φ∈C(Rd) with growth condition|φ(x)|≤C(1+|x|p), where1≤p≤q,
In the last section of Chapter1, we give the approximation of G-Brownian motion by simple random walk. Let {Sn}n=1∞be a discrete-time process,the continuous-time representation of S is given by St=Sn+(t-n)(Sn+1-Sn), n≤t Then Wt(n) weakly converges to G-Brownian motion, namely, for each k∈N and0≤t1 (Ⅱ) In Chapter2, we study the limit theorems on the sublinear expectation space.
Let ρ be a set of probability measures on measurable space (Ω,B(Ω)). The sublinear expectation Eρ[·], the upper probability V(·) and lower probability v(·) are respectively defined by
We introduce the notion of product independence and sum independence, which is weaker than Peng's independence in Peng [93]. Definition2.14Suppose that X1,X2,…, Xn is a sequence of real measurable random variables on (Ω,B(Ω)).(ⅰ) Xn is said to be product independent of(X1,…, Xn-1) if for each nonnegative bound-ed Lipschitz function φk,=1,…, n, (ⅱ) Xn is said to be sum independent of (X1,…, Xn-1) if for each φ∈Cb,Lip(R),
The following law of large numbers is an extension of which in Peng [93,96,98], Chen [11], Chen and Wu [14], and Chen et al.[15]. Theorem2.17Let {Xk}k=1∞be a sequence of random variables satisfying: supk≥1Eρ[|Xk|q]<∞, for some q>1, and Eρ[Xk]=μ,-Eρ[-Xk]≡μ,k=1,2,
Set Sn=Σk=1n Xk.(ⅰ) If {Xk}k=1∞is product independent, then (ⅱ) If{Xk}k=1∞is product and sum independent, then (ⅲ) If {Xk}k=1∞is sum independent, and V(·) is upper continuous, i.e., Then
About the central limit theorem on sublinear expectation space, Peng [93] first prove it under "i.i.d." assumptions, and then generalized by Li and Shi [66], Hu and Zhang [51], Hu [50], Hu and Zhou [58] without the assumption of identical distribution. However, all of these results require the independence of random variables. We consider one weaker condition called m-dependent, and prove the corresponding central limit theorem. This result was accepted by Acta Mathematicae Applicatae Sinica, English Series. Definition2.31The sequence {Xi}i=1∞is called m-dependent if there exits an integer m such that for every n and every j≥m+1,(Xn+m+1,…,Xn+j) is independent of (X1,…,Xn). In particular, if m=0, then {Xi}i=1∞is an independent sequence.
Theorem2.32Let {Xi}i=1∞be a sequence of m-dependent variables, such that
and Eρ[|Xi|2+α]≤M for i=1,2,…, where α>0and M is a constant. Let Sn=Σi=1n Xi, then we have where ξ~N(0;[σ2,σ2]).
(Ⅲ) In Chapter3, we study the Ito's calculus without quasi-continuity, and we obtain the general form of Ito's formula and the solvability of stochastic differential equation with local Lipschitz coefficients.
All the existing research on the Ito's calculus with respect to G-Brownian motion is based on the stochastic process space MGp(0,T),p≥1(see Peng [92,95,97,98,100], Gao [39], Zhang et al.[123]), which generalized by the random variables with quasi-continuity. But the notion of stopping time, which is the important notion in classical stochastic analysis, does not satisfy such quasi-continuity. It is difficult to consider the problem of stopping time on the space MGp(0,T). And the existing Ito's formula requires C2-function satisfying some growth condition. In order to overcome such difficulty, in this chapter, we introduce a larger space of stochastic process M*p(0, T), which is generalized by the random variables without quasi-continuity. Then we can define Ito's integral on this larger space M*p(0, T), and we can consider the Ito's integral on stopping time interval, which allows us to have a Ito's integral for a "locally integrable" space Mωp(0, T). This new formulation permits us to obtain Ito's formula for a general C1,2-function, which essentially generalizes the previous results of Peng [92,95,97,98,100] as well as those of Gao [39] and Zhang et al.[123]. This result was published in Stochastic Process and Their Applications121(7)1492-1508with Prof. Peng Shige.
Theorem3.41Let Φ∈C1.2([0,T]×R) and
where αv,ηvij∈Mω1(0,T), βvj∈Mω2(0,T). Then for each t∈[0,T], we have, quasi- surely, In the last section of this chapter, we consider the following stochastic differential equations driven by a d-dimensional G-Brownian motion: where b(·,·), hij(·,·), σj(·,·):[0, T]×R→R are continuous functions and X0is a con-stant. We introduce the following condition:(H1) Bounded condition: for any s∈[0, T](H2) Lipschitz condition: for any x,y∈R and s∈[0, T], max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤K|x-y|.(H3) Locally Lipschitz condition: for all x, y∈R satisfying|x|,|y|≤R and any s∈[0,T], max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤KR|x-y|.(H4) Growth condition: for any xeE and any5G [0,T], xb(s,x)≤K(1+x2), xhij(s,x)≤K(1+x2),|σj(s,x)|2≤K(1+x2). The first theorem consider the existence and uniqueness of the solution in the s-pace M*2(0, T). The second theorem study the solvability of SDE with locally Lipschitz coefficients.
Theorem3.44Let conditions (H1) and (H2) hold. Then there exists a unique contin-uous process X G M*2(0,T) satisfying (1).
Theorem3.45Let the condition (H3) and (H4) hold. Then there is a unique continuous adapted solution X of SDE (1).
(Ⅳ) In Chapter4, we study the properties of sublinear expectation and G-expectation, including the strict comparison theorem, additivity, Wasserstein distance, duality, domination and optimal transportation.
In order to study such properties, we divide this chapter into five sections. In Section4.1, we study the strict comparison theorem. We only list two important theorems in this section.
Theorem4.4Let X, Y∈Lc1(Ω) and X≤Y q.s. If thenEρ[X] Theorem4.9Let σ>0and X, Y∈Lip(Ω) with the forms X=φ(Bt1,Bt2-Bt1,…,Btn-Btn-1) and Y=φ(Bt1,Bt2-Bt1,…,Btn-Btn-1), where φ(x)≤φ(x),(?)x∈Rn. Then Eg[X] In Section4.2, we study the additivity of G-expectation. Let ξ be a G-normal distribution N(0,[σ2,σ2]), where0<σ<σ. The following two theorems is the main theorems in this section.
Theorem4.13We assume that φ,φ∈Cb,Lip(R). Then EG[φ(ξ)+φ(ξ)]=EG[φ(ξ)]+Eg[φ(ξ)] if and only if (?)xxu(t,x)(?)xxu(t,x)≥0,(?)(t,x)∈(0,1)×R.
Theorem4.16If there exists x0and θ>0such that φ,φ∈C2((x0-θ,x0+θ)) and φ"(x0)φ"(x0)<0,then we have
In Section4.3, we compare the G-expectation with Choquet expectation and provide some interesting examples. We show that the G-expectation is always dominated by the corresponding Choquet expectation. The following theorems are the main theorems in this section.
Theorem4.26G-expectation EG[·] can be represented by Choquet expectation EC[·](i.e.,(?)X∈LG1(Ω), EG[X]=EC[X]) if and only if EG is linear (i.e.,σ=σ).
In Section4.4, we generalize the classical Wasserstein distance and related properties to the sublinear expectation space.
Let ρ1and ρ2be two nonempty weakly compact and convex sets of probability measures. We define the Hausdorff-Wasserstein distance between ρ1and ρ2as follows: where Wp(P1, P2) is the classical Wasserstein distance between two probability measures P1and P2.
We first give the Kantorovich-Rubinstein duality formula for the sublinear case. Theorem4.30W1(ρ1,ρ2)=sup {|Eρ1[φ]-Eρ2[φ]|}.||φ||Lip<1||φ||Lip≤1means the Lipschitz constant of Lipschitz function φ is at most1.
As well known that weak convergence of probability measures is equivalent to con-vergence in Wasserstein distance, we give a non-trivial extension of this result to the setting of sublinear expectations.
Definition4.31A sequence {Eρn}n=1∞of sublinear expectations is said to be weakly con-vergent to Eρ, or equivalently,{ρn}n=1∞weakly converges to ρ, if for each φ∈Cb,Lip(Ω), limn→∞Eρn[φ]=Eρ[φ].
Theorem4.35Let {ρn}n=1∞be a sequence of convex and weakly compact sets of proba-bility measures satisfying supn Eρn[|X|1+α]<∞for some α>0and ρ be a convex and weakly compact set of probability measures. Then the following statements are equivalent.(ⅰ) ρn weakly converges to ρ.(ⅱ)W1(ρn,ρ)→0.
In Section4.5, we introduce the notion of duality, domination and optimal trans-portation from classical case to the sublinear expectation space. We first introduce the notion of Fenchel duality and related properties, which will be used in Chapter6when we study the dual game.
We also introduce the classical notion of Blackwell domination and we obtain a dominated theorem for sublinear expectations.
Theorem4.47If ρ1and ρ2are two convex and weakly compact sets of probability measures on (Ω,B(Ω)). Then the following statements are equivalent.(ⅰ) Eρ1[·] is dominated by Eρ2[·]. (ⅱ) ρ1(?)ρ2.
In the end of this section, we study the Kantorovich optimal transportation under sublinear expectation space. Let Ω1and Ω2be two complete separable metric spaces, Π(μ, v) be the set of probability measures defined on (Ω1(?)Ω2,B{Ω1(?)Ω2)) such that the marginal probability measures on Ω1and Ω2are μ and v respectively. Let ρ1and ρ2be two weakly compact and convex sets of probability measures defined on (Ω1, B(Ω1) and (Ω2,B(Ω2)) respectively. For a continuous function c on Ω1(?)Ω2,Φc denotes the set of all bounded continuous function pairs (φ,φ) satisfying φ(ω1)+φ(ω2)≥c(ω1,ω2),(?)ω1∈Ω1and (?)ω2∈Ω2.
Theorem4.49Let c: Ω1(?)Ω2→R be a continuous function, then we have
We also consider the following maximal covariance problem: Given a probability measure μ and ρ a set of probability measures. The maximal covariance function C(μ,ρ) is defined by
We give the main theorem in this section, which will be useful in Chapter5.
Theorem4.52Let {ρn} and ρ be the weakly compact and convex sets of probability measures, μ be a arbitrary probability measure. If W2(ρn,ρ)→0, then we have
(Ⅴ) In Chapter5, we introduce the notion of CMMV and the problem of maximal variation for martingales.
We first introduce the notion of CMMV which studied in De Meyer [26], and then generalize it to the framework of G-expectation. The main purpose of this chapter is to solve the following problem of maximal variation of martingales.
Let Mn{μ) be the set of pairs (F, X) where F:=(Fq)q=1,…,n is a filtration on a probability space (Ω, B(Ω), P), and X=(Xq)q=1,…,n is an F-martingale whose terminal value Xn is Blackwell dominated by μ. For a function M:△2(Rd)→R, we can define M-variation VnM(F,X) as Then the maximal M-variation VM(μ) is denned by
For one-dimensional case, we have the following theorem. Theorem5.11If M satisfies:(ⅰ) Positive homogeneity:(?)X∈L02(R),(?)α>0: M[αX]=αM[X].(ⅱ) Lipschitz continuity: There exists p∈[1,2) and K∈R such that for all X, Y∈
L02(R);|M[X]-M[Y]|≤K||X-Y||Lp.(ⅲ) Transportation invariance for constant: M [X+β]=M[X]+M[β],(?)β∈R. Then for all μ∈△2(R), we have
(3) If ρ>0and if, for all n,(Fn,Xn)∈Mn(μ) satisfies VnM(Fn,Xn)=VnM(μ), then the continuous time representation Πn of Xn defined as Πtn:=X[nt]n, converges in finite-dimensional distribution to the CMMV Πμ.
As for the multi-dimensional case, we first introduce a auxiliary function r defined by where cov(μ) denotes the covariance matrix of μ∈△02(Rd). Then we can define a set Γ as in Definition5.12.
We assume that function M:△2(Rd)→R satisfies the following hypotheses:
(H1) M≥0and has no degenerate directions:(?)x∈Rd, there exists μ∈△02(Rd) such that
(H2) M is K-Lipschitz for the Wasserstein distance of order p for some p∈[1,2).
(H3) M is positively homogenous:(?)X∈L2(Rd) and (?)λ>0, M[λX]=λM[X].
(H4) M is concave on△02(Rd).
(H5) r is quasiconvex, i.e.,(?)α∈R,{Y∈L2(Rd)|r(cov(Y))≤α} is convex in L2(Rd).
(H6) M[X+β}=M[X]+M[β],(?)β∈Rd.
Then we have the following theorem.
Theorem5.14Under hypotheses (H1)-(H6), we have
(Ⅵ) In Chapter6, we study a general class of repeated games with incomplete information on one side a la Aumann and Maschler [4].
This game was first introduced by De Meyer [25] as financial exchange game and then generalized by Gensbittel [45,47] in multi-dimensional context. The difference with Aumann-Maschler's model is that the state space and sets of actions of both players are allowed to be infinite. This chapter contains four sections.
In Section6.1, we study the linear game proposed in [45] and generalize the Cav(u) theorem in [45] from△(P) to△1(Rd), where P is a compact and convex subset of Rd. Let Vn(μ) and Vn(μ) denote the maximal and minimal payoffs for player1and player2respectively in the repeated game Γn(μ)(more details can be found in Section6.1). Let u(μ) and u(μ) denote the corresponding payoffs in the one-shot non-revealing game. Then we have the following Cav(u) theorem. Theorem6.5For all μ∈△1(Rd), we have
Furthermore, if we assume that (?)μ∈△2(Rd), the game Γ1(μ) always has the value V1(μ), i.e., V1(μ)=V1(μ)=V1(μ).Then we have the more precise Cav(u) theorem.
Theorem6.9If V1satisfies the following hypotheses:
(ⅰ) There exists μ0∈△2(Rd) such that V1(μ0)>0.
(ⅱ) V1([L+β)=V1([L])+V1([β]),(?)β∈Rd.
(ⅲ) For any α∈R,{X∈L2(Rd):supv∈△02(Rd):cov(v)=cov(X) V1(v)≤α} is convex in L2(Rd).
Then we have, for all μ∈△2(Rd), where ξ~N(0, Γ), Γ is given in Section5.3in Chapter5where V1is instead of M.
In Section6.2, we study a typical example of linear game called financial exchange game introduced by De Meyer [25]. We generalize the natural exchange mechanism in [25].
In the game Γn(μ)(see Section6.2), the hypotheses on the natural exchange mech-anism are the following:
(H1) Existence of the value:(?)μ∈△∞(R), the game Γ1(μ) has a value.
(H2) Bounded exchanges:(?)i,j:|Aij|≤K, where K is a constant.
(H3) Invariance with respect to the scale:(?)α>0,(?)L∈L2(R): V1([αL])=αV1([L]).
(H4) Transportation invariance with respect to the risk-less part of the risky asset:(?)β∈R:V1([L+β)=V1{[L])+V1{[β]).
(H5) Positive value of information:(?)L∈L2(R): V1([L])>0.
The following theorem gives the asymptotic characterizations of the value function and price process based on the results in Chapter5.
Theorem6.15If (H1)-(H5) is satisfied, then for all μ∈△2(R),(ⅰ)limn→∞1/nVn(μ)=V1([E(μ)]).
(ⅲ) If p>0and if, for all n,(Fn,Xn)∈Mn(μ) satisfies VnV1(Fn,Xn)=Vn(μ),then the continuous time representation Πn of Xn defined as Πtn:=X[nt]n converges in finite-dimensional distribution to the CMMV Πμ.
In Section6.3, we systematically study a game model with transaction costs, which does not satisfy the natural exchange mechanism in [25] but satisfies our general natural exchange mechanism in Section6.2. The explicit solution of Nash equilibrium of this game was obtained by the dual method, and we show that the price process posted by uninformed player converges to CMMV in finite distribution. This result confirms that CMMV is a very robust dynamic in stock market.
In Section6.4, we study a game model where both players are not risk-neutral, which generalize the results in De Meyer [26] while the uninformed player is risk-aversion but the informed player is risk-neutral. The results are very surprising:the Nash equilibrium in the game does not depend on the risk attitude of informed player, i.e., the equilibrium in the game where the informed player is risk-natural is also an equilibrium in the game when the informed player is risk-aversion or risk-seeking.
一个很有趣的问题是如何建立非线性期望以及相应的条件期望.容度和Choquet期望(或Choquet积分)的概念由Choquet[17]引入,它们被广泛的应用于位势论(参见,Choquet[17],Doob[34])和决策论(参见,Schmeidler [108], Gilboa和Schmeidler[48]).然而据我们所知,条件Choquet期望的概念并没有被人们很好的理解,从而它们很难用于处理经济中的动态问题.另一重要的非线性期望一g-期望在Peng[85]中通过倒向随机微分方程来引入.它是一个理想的框架来度量概率不确定模型的随机性和风险(参见,Chen和Epstein[12],Fritelli和Rossaza Gianin[38],Peng[88]).然而,g-期望的一个局限性是它仅用来处理所涉及的不确定的概率测度关于一个参考概率测度(如,Wiener测度)绝对连续的情形.但是对于金融中著名的波动率不确定性问题,存在不可数个未知的概率测度,它们之间相互奇异Avellaneda等[3]和Lyons[73]研究了带有波动率不确定性的状态依赖的期权定价问题,而路径依赖的情形更有挑战性,需要创建一个比经典概率论更一般的框架.
上述路径依赖下的非线性期望由Peng在[88,90]中建立,并且给出两种完全不同的方法来解决所涉及的动态相容性问题.前一种方法采用路径依赖下的更一般的动态规划原理；第二种方法使用Nisio形式的单调半群(参见Nisio[78,79]),称其为非线性Markov链,通过建立非线性形式的Kolmogorov相容性定理来构造非线性期望空间,这在经典概率论中同样重要.
在上述所提到的非线性期望中,一类非常典型的非线性期望-G-期望首先由Peng[92]于2006年提出.事实上,G-期望也是一种非常典型的次线性期望,它保留了除去线性性质外线性期望所具有的其他良好性质.分布和独立性的概念在整个理论中起到非常重要的作用Peng的一项开创性工作是直接使用次线性期望E[·]来定义分布和独立,而不是使用容度这一看上去更加自然的概念来推广它们的定义.基于这些新概念Peng引入了一种非常重要的分布G-正态分布,它可以由所谓的G-热方程来刻画.G-期望和G-Brown运动的概念可以视为Wiener测度和经典的Brown运动的非线性推广.这些概念和相应的极限定理(大数定律和中心极限定理)以及关于G-Brown运动的Ito随机积分由Peng[92-102]引入并且做了系统研究.最近,很多作者基于Peng的开创性工作做了许多相应的推广.关于大数定律,Chen[11],Chen和Wu[14],Chen等[15]研究了强大数定律,这些结果是对Peng[93,96]中“弱”大数定律的推广.在Peng首先在[93]中证明了次线性期望空间中独立同分布(简记为i.i.d.)假设下的中心极限定理之后,许多作者在不假设同分布但是仍假设独立的情形下推广了这一结果,参见Li和Shi[66],Hu和Zhang[51],Hu[50],Hu和Zhou[58]等.关于G-期望理论框架下的Ito随机计算,特别的,关于Ito公式,由Gao[39]和Zhang等[123]等做出相应的推广.关于次线性期望理论以及G-期望理论的进一步工作可以参见Bai和Buckdahn[6],Bai和Lin[7],Chen和Hu[13],Denis等[30]Dolinsky等[33],Epstein和Ji[35],Gao[42],Gao和Jiang[41,42],Gao和Xu[43,44],Hu[52,53],Hu等[54：55],Hu和Peng[56,57],Lin[71],Lin[72],Nutz[80],Nutz和van Handel[81],Nutz和Zhang[82],Peng等[103],Soner等[112]Song[113-116],Xu和Zhang[122],等.本论文的出发点与Peng的开创性工作有一点不同.我们将次线性期望EP[·]视为
一族概率测度集合P的上期望,其中P中的元素P为定义在可测空间(Ω,B(Ω))上的概率测度,这使得我们可以方便的利用线性期望EP[·]的现有性质来研究EP[·]的性质.并且我们从一个新的观点来研究独立的定义,即通过经典的条件期望来定义独立.这些新的构想使得我们可以将Peng[92-100]中相应的极限定理和Ito随机计算以及其他作者的工作推广到我们的框架中来.本论文中的第1章至第3章将重点研究这些问题.第4章是本论文的亮点.我们研究次线性期望和G-期望的性质,包括严格比较定理,可加性Wasserstein距离,对偶,控制和最优转移问题,尽管这些性质在经典概率论中是众所周知的,并且其中的一些性质甚至是显然的.这些结果是经典结果的非平凡推广,它们在第5章和第6章中多次用到.
作为G-期望理论的应用.我们在第5章中介绍了连续最大变差鞅(简记为CMMV)的概念以及最大鞅变差问题.粗略地讲,最大鞅变差问题是指,给定一个定义在△(Rd)上的实值泛函M和一个概率测度μ∈△(Rd),我们的目标是在由所有长度为n的其终端值在Blackwell意义下受控于μ的取值于Rd的鞅所组成的集合上最大化所谓的M-变差.这一问题推广了Mertens和Zamir[77]中所引入的最大L1-变差问题.一维情形下的一般问题在De Meyer[26]中所研究,随后由Gensbittel[46]推广到多维情形.我们基于第1章至第4章中的结果给出这一问题的一个新的简洁的证明.两篇文章[26]和[46]中仅研究中心化的情形,即,泛函M仅定义在零均值的概率测度集上,我们将其推广到非中心化的情形,并且我们将在随后一章研究带有交易费用的博弈模型时发现这一推广非常有用.在第6章,我们基于Aumman和Maschler[4]的模型研究一类更广的单边不完全信息下的重复博弈模型,这类模型首先由De Meyer[25]以金融交易博弈的形式提出,随后由Gensbittel[45,47]推广到多维情形.这些博弈模型与第5章中所介绍的连续最大变差鞅的概念以及最大鞅变差问题有着紧密的联系.我们系统地研究了两个具体的博弈模型并且得到它们的Nash均衡显式解.通过这些模型我们可以看出连续最大变差鞅在股票市场中是一类非常稳健的价格过程.我们指出第5章和第6章的内容仅仅是从G-期望观点来研究博弈论的一个初步尝试,还有许多有趣的问题有待进一步研究.
论文共分为六章,以下是本文的结构和得到的主要结论：
(Ⅰ)第1章主要研究不确定性下的随机游走和相应的极限定理.
我们将经典的Bernoulli随机游走和简单随机游走推广到不确定性的情形.所谓的“不确定性”是指参考的概率测度不是唯一的,而是一族概率测度组成的集合.设(Ω,B(Ω))为一个可测空间,P为定义在(Ω,B(Ω))上的概率测度组成的集合.给定随机变量X,参照Peng[93],X在P下的分布函数为Gb,Lip(R)到R上的泛函,其定义为
其中Cb,Lip(R)为R上的有界Lipschitz函数空间.为了简化记号,我们将SupP∈ρ EP[·]记为EP[·]Peng[93]中给出的在P下独立的定义如下定义1.4设{Xi}i=1∞为一列(Ω,B(Ω))上的随机变量.如果对任一φ∈Cb.Lip(Rn),那么我们称Xn在P下独立于(X1,…,Xn-1).如果对每个n∈N,Xn在P下独立于(X1.…,X-1),那么我们称{Xi}i=1∞在P下独立.
我们通过经典的条件期望给出一个独立的新定义：
定义1.5设{Xi}i=1∞为一列(Ω,B(Ω))上的随机变量.给定(Ω,B(Ω))上的一族概率测度P,如果下列条件成立.
(1)(?)P∈P,(?)φ∈Cb,Lip(R),EP[φ(Xn)|X1,…,Xn-1]≤EP[φ(Xn)],P-a.s.
(2)(?)φ∈Cb,Lip(R),存在依赖于φ的P∈P,使得EP[φ(Xn)|X1,…,Xn-1]=EP[φ(Xn)],P-a.s.那么我们称Xn在P下弱独立于(X1,…,Xn-1).如果对每个n∈N,Xn在P下弱独立于(X1,…,Xn-1),那么我们称{Xi}i=1∞在P下弱独立.
下面的定理给出了弱独立和Peng的独立定义之间的关系.
定理1.7设{Xi｝i=1∞为一列在P下弱独立的随机变量序列.我们定义P如下
P={P:(?)φ∈Cb,Lip(R),(?)n∈N,EP[φ(Xn)|X1,…,Xn-1]≤EP[φ(Xn)],P-a.s.｝.则{Xi｝i=1∞在定义1.4的意义下在P下独立,并且
我们可以证明在独立同分布假设下关于Bernoulli随机游走的大数定律,并且我们将它推广到不假设独立同分布的情形.下面的定理是更一般的形式.定理1.9设{Xk｝k=1∞为可测空间(Ω,B(Ω))上的一列随机变量,P为定义在(Ω,B(Ω))上所有满足下列条件的概率测度P组成的集合：(?)n∈N,μ≤EP[Xn|X1,…,Xn-1]≤μ且EP[|Xn|q|X1,…,Xn-1]≤Kq P-a.s.,其中μ,μ,K,q为常数且q>1.则我们有(i)对每一μ∈[μ,μ],存在Pμ∈P,使得(ii)对每一P∈P,(iii)对每一φ∈Cb,Lip(R),我们同样考虑关于简单随机游走的中心极限定理.G-正态分布的概念在中心极限定理中起到关键作用.在本章中,G-正态分布通过G-热方程的解来定义.定义1.10如果ξ的分布通过下式给出其中uφ(t,x)为如下G-热方程的解：
其中G(α)=1/2σ2α+-1/2σ2α-,0≤σ≤σ.那么我们称ξ为G-正态分布,记为ξN(0,[σ2,σ2]).
我们在此仅列出两个中心极限定理.事实上,它们在某种意义上是等价的.第二个多维情形下的中心极限定理会在第5章用到.我们将G-正态分布ξ的分布记为EG[φ(ξ)]：=Fξ(φ).
定理1.14设{Xi｝i=1∞为可测空间(Ω,B(Ω))上的一列随机变量.设P为定义在(Ω,B(Ω))上的所有满足下列条件的概率测度组成的集合：(?)P∈P,(?)i∈N,(1)EP[Xi|X1,…,Xi-1]=0,(2)σ2≤EP[X,2|X1,…,Xi-1]≤σ2,(3)EP[|Xi|q|X1,…,Xi-1]≤Kq.我们记Sn=∑i=1n Xi.
设Mn1(∑,K)为概率空间(Ω,B(Ω),P)上所有满足下列条件的长度为n的取值于Rd的鞅组成的集合：(i)EP[Sn]=0,
(ii)EP[(Sk+1-Sk)(Sk+1-Sk)T|S1,…,Sk]∈∑,0≤k≤n-1,其中∑为S+(d)中的有界凸闭子集.
(iii)EP[||Sk+1-Sk||q]≤K,0≤k≤n-1.设Vn[φ]:=supS∈mq(Σ,K)EP[φ[Sn/(?)n)].定理1.17我们假设q>2.设ξ为G-期望EG[·]下的G-正态分布N(0,∑),则(?)φ∈C(Rd)且满足增长条件|φ(x)|≤C(1+|x|p),其中1≤p＜q,我们有
在第1章的最后一节,我们给出G-Brown运动通过简单随机游走的逼近定理.设{Sn｝n=1∞为一个离散时间过程,则S的连续时间化过程定义为St=Sn+(t-n)(Sn+1-Sn), n≤t
我们给出乘积独立和加和独立的定义,这些定义比Peng[93]中给出的独立的定义要弱.
定义2.14设X1,X2,…,Xn为(Ω,B(Ω))上的一列可测随机变量.(i)如果对于每一个非负有界的Lipschitz函数φk,k=1,…,n,那么我们称Xn乘积独立于(X1,…,Xn-1).(ii)如果对于每一个φ∈Cb,Lip(R),那么我们称Xn加和独立于(X1,…,Xn-1).
下面给出的大数定律是Peng[93,96,98],Chen[11],Chen和Wu[14],以及Chen等[15]中的大数定律的推广.
定理2.17设{Xk｝k=1∞为一列随机变量满足：对某一q>1,supk≥1EP[|Xk|q]＜∞,且EP[Xk]三μ,-EP[-Xk]三μ,k=1,2,设Sn=∑k=1n Xk.
(i)如果{Xk｝k=1∞乘积独立,那么
(ii)如果{Xk｝k=1∞乘积独立且加和独立,那么
(iii)如果{Xk｝k=1∞加和独立且V(·)上连续,即,当An↓A时,V(An)↓V(A),其中,An,A∈B(Ω),
那么
关于次线性期望空间上的中心极限定理,Peng[93]首先证明了独立同分布的情形,随后由Li和Shi[66],Hu和Zhang[51],Hu[50],Hu和Zhou[58]推广到不假设同分布的情形.然而,所有的这些定理均要求随机变量的独立性.我们考虑一个比独立稍弱的条件,称之为m-相依,并且证明了相应的中心极限定理.这一结果已被Acta Mathematicae Applicatae Sinica, English Series所接受.
定义2.31如果存在一个整数m使得对每一个n和每一个j≥m+l,(Xn+m+1,…,Xn+j)均独立于(X1,…,Xn),那么我们称序列{Xi｝i=1∞为m-相依.特别的,如果m=0,那么称{Xi｝i=1∞为独立序列.
定理2.32设{Xi｝i=1∞为一列m-相依序列,满足
且对i=1,2,…,EP[|Xi|2+α]≤M,其中α>0且M为常数.设Sn=∑i=1n,则我们有
其中ξ～N(0；[σ2,σ2]).
(Ⅲ)第3章主要研究不假设拟连续条件的Ito随机计算,并且得到一般形式的Ito公式以及具有局部Lipschitz系数的随机微分方程的解的存在唯一性.
现有的关于G-Brown运动的Ito随机分析均建立在随机过程空间MGp(0,T)(p≥1)上(参见,Peng[92,95,97,98,100]以及Gao[39]和Zhang等[123]),其中空间MGp(0,T)由满足拟连续条件的随机变量所生成.然而停时这一在经典随机分析中非常重要的概念并不满足拟连续性,从而我们很难在空间MGp(0,T)中处理停时问题.这也导致现有的Ito公式均对C2-函数要求一定的增长条件.为了克服上述困难,我们在本章中引入一个更大的随机过程空间M*p(0,T).它是由未必满足拟连续条件的随机变量所生成.随后我们在这个更大的空间M*p(0,T)上定义Ito随机积分,并且我们考虑了定义在停时区间上的Ito随机积分.这使得我们可以在“局部可积”空间Mω2(0,T)上定义Ito随机积分.这一新的理论框架使得我们可以得到关于C1,2-函数的更一般的Ito公式,这一结果在本质上推广了Peng[92,95,97,98,100]以及Gao[39]和Zhang等[123]中的结果.该结果与导师彭实戈教授合作发表于Stochastic Process and Their Applications121(7)1492-1508.
定理3.41设Φ∈C1,2([0,T]×R)且则对任一t∈[0,T],我们有,
在本章的最后一节,我们考虑如下由d维G-Brown运动驱动的随机微分方程：
其中b(·,·),hij(·,·),σj(·,·)：[0,T]×R→R为连续函数,X0为常数.我们引入如下条件：
(H1)有界性条件：对任一s∈[0,T]
(H2)Lipschitz条件：对任何x,y∈R和s∈[0,T],max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤K|x-y|.
(H3)局部Lipschitz条件：对所有满足|x|,|y|≤R的x,y∈R以及s∈[0,T],
max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤KR|x-y|.
(H4)增长性条件：对任一x∈R和s∈[0,T],xb(s,x)≤K(1+x2), xhij(s,x)≤K(1+x2),|σj(s,x)|2≤K(1+x2).
第一个定理给出在空间M*2(0,T)上的解的存在唯一性.第二个定理研究具有局部Lipschitz系数的随机微分方程的可解性.
定理3.44假设条件(H1)和(H2)成立.则存在唯一的过程X∈M*2(0,T)满足(1).
定理3.45假设条件(H3)和(H4)成立.则随机微分方程(1)存在唯一的连续适应解X.
(Ⅳ)第4章研究次线性期望和G-期望的性质,包括严格比较定理,可加性,G-期望与Choquet期望之间的联系,Wasserstein距离,对偶,控制以及最优转移问题.
本章分为五节来研究上述性质.在第4.1节,我们研究严格比较定理.我们在此仅列出本节中两个重要的定理.定理4.4设X,Y∈Lc1(Ω)且X≤Y q.s.如果那么EP[X]＜EP[Y].定理4.9设旦＞0且X,Y∈Lip(Ω)具有如下形式X=φ(Bt1,Bt2-Bt1,…,Btn-Btn-1)和Y=ψ(Bt1,Bt2-Bt1,…,Btn-Btn-1),其中φ(x)≤ψ(x),(?)x∈Rn.则EG[X]＜EG[Y]当且仅当存在x0∈Rn使得φ(x0)＜ψ(xo).
在第4.2节,我们研究G-期望的可加性.设ξ为G-正态分布N(0,[σ2,σ2]),其中0<σ<σ.如下的两个定理是本节中的主要定理.定理4.13我们假设φ,ψ∈Cb,Lip(R).则EG[φ(ξ)+ψ(ξ)]=EG[φ(ξ)]+EG[ψ(ξ)]当且仅当(?)xxu(t,x)(?)xxσ(t,x)≥0,V(t,x)∈(0,1)×R.定理4.16如果存在x0θ>0使得φ,ψ∈C2((x0-θ,x0+θ))且φ″(x0)ψ″(x0)＜0,那么我们有
在第4.3节,我们比较了G-期望和Choquet期望并且给出了一些有趣的例子.我们证明了G-期望被相应的Choquet期望所控制.下面的定理是本节中的主要定理.定理4.26G-期望EG[·]可以表示成Choquet期望Ec[·]伊P,(?)X∈LG1(Ω),EG[X]=Ec[X])当且仅当EG[·]为线性(即,σ=σ.
在第4.4节,我们将经典的Wasserstein距离及其相应的性质推广到次线性空间.
设P1和P2为两个非空凸的弱紧的概率测度集合.我们定义P1和P2之间的Hausdorff-Wasserstein距离如下：其中Wp(P1,P2)是经典的P1和P2之间的Wasserstein距离.
我们首先给出次线性形式的Kantorovich-Rubinstein对偶公式.定理4.30||φ||Lip≤1表示Lipschitz函数φ的Lipschitz系数小于等于1.
众所周知,概率测度的弱收敛等价于Wasserstein距离下收敛,我们给出这一性质在次线性期望框架下的非平凡推广.
定义4.31设{EPn}n=1∞为一列次线性期望.如果对每一φ∈Cb,Lip(Ω),那么我们称{EPn｝n=1∞弱收敛于EP,或等价的,称{Pn｝n=1∞弱收敛于P.
定理4.35设{Pn｝n=1∞为一列凸的弱紧的概率测度集合满足下列条件：
设P为一凸的弱紧的概率测度集合.则下面的两个论述等价：(ⅰ)Pn弱收敛于P.(ⅱ)W1(Pn,P)→0.
在第4.5节,我们介绍了经典情形以及次线性情形下的对偶和控制的概念以及最优转移问题.首先介绍的Fenchel对偶及其性质在第6章中研究对偶博弈时会再次用到.
随后我们介绍了经典的Blackwell控制的概念,并且得到一个关于次线性期望的控制定理.
定理4.47如果P1和P2为(Ω,B(Ω))上的两个凸的弱紧的概率测度集合.那么下面的叙述等价：
(ⅰ)EP1[·]被EP2[·]所控制.
(ⅱ)P1(?)P2.
最后,我们研究次线性空间上的Kantorovich最优转移问题.设Q1和Ω2为两个完备可分的距离空间,Ⅱ(μ,υ)为定义在(Ω1(?)Q2,B(Q1(?)Q2))上的所有满足其边际分布在Q1和Ω2上分别为μ和υ的概率测度组成的集合.设P1和P2为分别定义在(Ω1,B(Q1))和(Ω2,B(Q2))上的弱紧的凸的概率测度集合.记Ⅱ(ρ1,ρ2)=∪μ∈ρ1,v∈ρ2Ⅱ(μ,v)对Ω1(?)Ω2上的连续函数c,Φc为所有满足下列条件的连续函数对(φ,ψ)组成的集合：φ(ω1)+φ(ω2)≥c(ω1,ω2),(?)ω1∈Ω1,ω2∈Ω2.我们得到如下形式的对偶公式：定理4.49设c：Q1(?)Ω2→R为连续函数,则我们有
我们同时考虑如下的最大协方差问题：给定一个概率测度μ和一个概率测度集合P.最大协方差函数C(μ,P)定义为
下面的定理在第5章中会用到.定理4.52设{Pn｝和P为凸的弱紧的概率测度集合,μ为任一概率测度.如果W2(Pn,P)→0,那么我们有(Ⅴ)第5章主要研究连续最大变差鞅和最大鞅变差问题.
我们首先介绍De Meyer[26]中所提出的连续最大变差鞅的概念,然后我们将其推广到G-期望框架.本章的主要目的是解决如下的最大鞅变差问题.
设Mn(μ)为(F,X)组成的集合,其中F：=(Fq)q=1,…,n为概率空间(Ω,B(Ω),P)上的信息族,X=(Xq)q=1,…,n为F鞅,其终端分布Xn在Blackwell意义下受控于μ.给定泛函M：△2(Rd)→R,我们可以定义M-变差VnM(F,X)如下则最大M变差VM(μ)定义为
对于一维的情形,我们得到如下的定理：
定理5.11如果M满足.
(ⅰ)正齐性(?)X∈L02(R),(?)α>0:M[αX]=α[X].
(ⅱ)Lipschitz连续性：存在p∈[1,2)和K∈R使得对所有的X,Y∈L02(R)：|M[X]-M[Y]|≤K||X-Y||Lp.
(ⅲ)常数平移不变性M[X+β]=M[X]+M[β],(?)β∈R.那么对所有的μ∈△2(R),我们有
(3)如果ρ>0并且对所有的n,(Fn,Xn)∈Mn(μ)满足VnM(Fn,Xn)=VnM(μ),那么Xn的连续时间表示Πtn：=X[nt]n依有限维分布收敛到连续最大变差鞅Πμ.对于多维情形,我们首先引入辅助函数r定义如下
其中cov(μ)为μ∈△02(Rd)的协方差矩阵.集合r的定义可以参见定义5.12.我们假设M：△2(Rd)→R满足如下假设：(H1)M≥0并且非退化：(?)x∈Rd,存在μ∈△02(Rd)使得μ(Rx)=1,且M(μ)≥0.
(H2)M在p阶Wasserstein距离下K-Lipschitz连续,其中p∈[1,2).
(H3)M满足正齐性：(?)X∈L2(Rd),λ>0,M[λX]=AM[X].
(H4)M在△02(Rd)上为凸泛函.
(H5)r为拟凸函数,即,(?)α∈R,{Y∈L2(Rd)|r(cov(Y))≤α}.在L2(Rd)中为凸集.
(H6)M[X+β]=M[X]+M[β],(?)β∈Rd.
我们有下面的定理：
定理5.14在假设(H1)-(H6)下,我们有
(Ⅵ)第6章主要研究一类广义的Aumann和Maschler[4]中所介绍的重复博弈模型.
这类模型首先在De Meyer[25]中作为金融交易模型引入,随后由Gensbittel[45,47]推广到多维情形.与Aumann-Maschler的模型所不同的是我们允许状态集和策略集为不可数集.本章包含四节.
在第6.1节,我们研究[45]中所提出的线性博弈模型并且将[45]中的Cav(u)定理从△(P)推广到△1(Rd),其中P为Rd中的凸紧子集.设Vn(μ)和瓦(μ)分别为单边信息不对称重复博弈模型Γn(μ)(参见第6.1节)中参与者1的最大支付和参与者2的最小支付,ui(μ)和u(μ)分别为完全信息下的博弈模型中相应的支付.我们有如下的Cav(u)定理：
定理6.5对所有的μ∈△1(Rd),我们有
如果我们进一步假设,(?)μ∈△∞(Rd),博弈r1(μ)存在值,即,V1(μ)=V1(μ)=V1(μ),那么我们有如下更精确的Cav(u)定理：
定理6.9如果V1满足如下假设：
(i)存在μ0∈△2(Rd)使得V1(μ0)＞0.
(ⅱ)V1([L+β])=V1([L])+V1([β]),(?)β∈Rd.
那么我们有,对所有的μ∈△2(Rd),其中ξ～N(0,Γ),Γ在第5章第5.3节中给出,我们需要用V1代替那里的M.
在第6.2节,我们研究一类特殊的线性博弈模型,称为金融交易模型.这一模型由De Meyer[25]提出.我们推广了[25]中的自然交易机制.
在博弈Γn(μ)中(参见第6.2节),自然交易机制的假设如下：
(H1)博弈值的存在性：(?)μ∈△∞(R),博弈Γ1(μ)存在值.
(H2)交易的有界性：(?)i,j:|Aij|≤K,其中K为常数.
(H3)正齐性：
(H4)关于风险资产中的无风险部分的平移不变性：
(H5)信息具有正价值：(?)L∈L2(R)：V1([L])>0.
基于第5章的结果,我们给出下面的值函数以及价格过程的逼近定理.定理6.15如果(H1)-(H5)成立,那么对所有的μ∈△2(R),(ⅲ)如果ρ>0且对所有的n,(Fn,Xn)∈Mn(μ)满足VnV1(Fn,Xn)=Vn(μ),那么Xn
的连续时间表示Ⅱtn：=X[nt]n依有限维分布收敛于Ⅱμ.
在第6.3节中,我们系统的研究了一个带有交易费用的博弈模型,它并不满足[25]中的自然交易机制但是满足我们在第6.2节中推广的自然交易机制.我们通过对偶方法得到这一模型的Nash均衡显式解.并且我们证明了由非内幕参与者所提出的价格过程在有限维分布下收敛到连续最大变差鞅.这一结果更好的表明连续最大变差鞅在股票市场中是一类非常稳健的价格过程.
在第6.4节,我们研究了一个博弈模型其中博弈双方均不是风险中性,这推广了DeMeyer[26]中的结果,原结果中非内幕参与者是风险厌恶而内幕参与者是风险中性.我们得到的结论非常有趣Nash均衡解不依赖于内幕参与者的风险态度,即,当内幕参与者是风险中性时的Nash均衡同时也是内幕参与者是风险厌恶或风险喜好时的均衡解.
In1933, Andrey Kolmogorov published his book Foundation of the Theory of Prob-ability (in German, Grundbegriffe der Wahrscheinlichkeitsrechnung), which established the modern axiomatic foundations of probability theory. Given a probability measure P on a measurable space (Ω,F), the expectation Ep[X] of a F-measurable random variable X is defined as the integral fΩXdP. Obviously, Ep[·] is a linear functional due to the linearity of probability measure P. However, a great number of uncertainty phenomena can not be well modeled by such linear probability or expectation.
A very interesting problem is to develop nonlinear expectation and related condition-al expectations. The notion of capacity and Choquet expectation (or Choquet integral) was introduced by Choquet [17], which has been widely used in the potential theory (e.g., Choquet [17], Doob [34]) and decision theory (e.g., Schmeidler [108], Gilboa and Schmeidler [48]). But to the best of our knowledge, the notion of conditional Choquet expectation has not been well understood and it was hardly used to deal with dynamic problems in economics. The other important nonlinear expectation called g-expectation was introduced via BSDE in Peng [85]. It is an ideal framework for the valuation of randomness and risk in the case of the uncertainty of probability models (e.g., Chen and Epstein [12], Fritelli and Rossaza Gianin [38], Peng [88]). However, one important limitation of g-expectation is that the involved uncertain probability measures have to be absolutely continuous with respect to a reference probability measure, e.g., a Wiener measure. But for the well-known problem of volatility model uncertainty in finance, there is an uncountable number of unknown probabilities which are essentially singular from each other. Avellaneda et al.[3] and Lyons [73] studied this volatility uncertainty problem for the situation of state-dependent options. The situation of path-dependence is more challenging and needed to create a new framework more general than the classical notion of probability.
Such types of fully nonlinear expectations for situations of path-dependence were constructed by Peng in [88,90] where two very different approaches were introduced to solve the involved problem of dynamic consistency. The first one is a generalized dynamic programming principle for the path-dependent situation. The second one is about to use the notion of nonlinear monotone semigroups of Nisio's type (see Nisio [78,79]), called nonlinear Markov chain, to develop a nonlinear version of Kolmogorov consistency theorem in order to construct nonlinear expectation spaces which plays the important role as in the classical probability theory.
The most typical example of the above-mentioned fully nonlinear expectation is the G-expectation which was first introduced by Peng [92] in2006. In fact, G-expectation is also a typical example of the sublinear expectation which keeps the well properties of linear expectations except linearity. The notion of distribution and independence plays an important role in the whole theory. One pioneering work of Peng is to define such distribution and independence directly by the sublinear expectation E[·] but not by capacity which seems as a natural way to generalize them. Based on these new notions, Peng introduced the most important distribution called G-normal distribution which can also be characterized by the so-called G-heat equation. The notion of G-expectation and G-Brownian motion can be regarded as a nonlinear generalization of Wiener measure and classical Brownian motion. These notions and the corresponding limit theorems (law of large numbers and central limit theorem) as well as stochastic calculus of Ito's type with respect to G-Brownian motion were introduced and systematically developed in Peng [92-102]. Recently, many authors gave a number of generalizations of Peng's initial works. As for law of larger numbers, Chen [11], Chen and Wu [14], Chen et al.[15] study the strong law of large numbers which generalizes the "weak" law of large numbers in Peng [93,96]. After Peng first established central limit theorem in sublinear expectation space under independent and identically distributed (i.i.d. for short) assumptions in [93], many authors generalized this result without the identically distributed assumption but still keeping the independent assumption, see Li and Shi [66], Hu and Zhang [51], Hu [50], Hu and Zhou [58] etc. About the Ito's calculus under the framework of G-expectation theory, especially, for Ito's formula, was generalized by Gao [39] and Zhang et al.[123], etc. The further development of sublinear expectation theory and G-expectation theory can be found in Bai and Buckdahn [6], Bai and Lin [7], Chen and Hu [13], Denis et al.[30], Dolinsky et al.[33], Epstein and Ji [35], Gao [42], Gao and Jiang [41,42], Gao and Xu [43,44], Hu [52,53], Hu et al.[54,55], Hu and Peng [56,57], Lin [71], Lin [72], Nutz [80], Nutz and van Handel [81], Nutz and Zhang [82], Peng et al.[103], Soner et al.[112], Song [113-116], Xu and Zhang [122], etc.
The starting point of this thesis is a little different from Peng's initial work. We will take the sublinear expectation Ev[·] as an upper expectation of a set V of probability measures P defined on some measurable space (Ω,B(Ω)), which allows us to study the properties of Ep[·] conveniently by the existing properties of linear expectation Ep[·]. And we study the notion of independence from a new view which defined via classical conditional expectations. These formulations allow us to generalize the corresponding limit theorems and Ito's calculus in Peng [92-100] and other authors'work to our setting. Chapter1to Chapter3of this thesis will focus on these topics. Chapter4is the high-light of this thesis. We study some properties of sublinear expectation and G-expectation, including the strict comparison theorem, additivity, Wasserstein distance, duality, dom-ination and optimal transportation, although these properties are well-known and some of them are even obvious in the classical probability theory. These results are non-trivial extensions of the classical ones and will be used in Chapter5and6.
As an application of the G-expectation theory, we introduce the notion of continuous martingale of maximal variation (CMMV for short) and the problem of maximal variation of martingales in Chapter5. Roughly speaking, the problem of maximal variation of martingales is that, given a real-valued function M defined on△(Rd) and a probability measure/μ∈△(Rd), we aim to maximize a functional called M-variation, over the set of all Rd-valued martingales of length n whose terminal distribution is Blackwell dominated by μ. This problem generalizes the problem of maximal L1-variation introduced in Mertens and Zamir [77]. The most general form has been studied in De Meyer [26] for one-dimensional case and then generalized by Gensbittel [46] for multi-dimensional case. We will give a new and simple proof for it based on the results in Chapter1to Chapter4. Both of the two papers [26] and [46] only studied the centered case, i.e., the functional M defined on the set of probability measures with zero-mean, we will generalize them to the non-centered case which turns out to be very useful when we study the games with transaction costs in the next chapter. In Chapter6, we study a general class of repeated games with incomplete information on one side a la Aumman and Maschler [4], which was first introduced by De Meyer [25] as financial exchange game and then generalized by Gensbittel [45,47] in multi-dimensional context. These games have the closed relation with the notion of CMMV and the maximal variation of martingales in Chapter5. We also systematically study two particular game models, and obtain the explicit solution of Nash equilibrium for them. These models show that CMMV is a very robust dynamic in the stock market. We point out that the contents in Chapter5and Chapter6is just a first attempt to study the game theory from the view of G-expectation. There are many interesting problems to study in the future.
This thesis consists of six chapters. In the following, we list the main results in this thesis:
(Ⅰ) In Chapter1, we study the random walks under uncertainty and the corresponding limit theorems.
We generalized the classical Bernoulli random walk and simple random walk to the uncertainty case."Uncertainty" means that the underlying probability measure is not unique, but correspond, is a set of probability measures. Let (Ω, B(Ω)) be a measurable space, V be the set of probability measures defined on (Ω,B(Ω)). Given a random variable X, following Peng [93], the distribution function of X under V is a functional from Cb,Lip(R) to R defined by where Cb,Lip(R) is the space of all bounded and Lipschitz functions on R. For simplicity of notation, we write Eρ[·] instead of sup P∈ρ[·]. The notion of independence under ρ denned in Peng [93] as following:
Definition1.4Let {Xi}i=1∞be a sequence of random variables on (Ω,B(Ω,)).{Xi}i=1∞is said to be independent under ρ, if for each n∈N, Xn is independent of (X1,…, Xn-1) under V, which is defined by (?)φ∈Cb,Lip(Rn), We give a new definition of independence via classical conditional expectations.
Definition1.5Let {Xi}i=1n be a sequence of random variables on (Ω,B(Ω)). Given a set V of probability measures on (Ω,B(Ω)),{Xi}i=1∞is said to be weak independent under ρ, if for each n∈N, Xn is weak independent of (X1,…, Xn-1) under ρ, which is defined by (1)(?)P∈ρ,(?)φ∈Cb,Lip(R), EP[φ(Xn)|X1,…,Xn-1]≤Eρ[φ(Xn)], P-a.s.,(2)(?)φ∈Cb,Lip(R), there exists P∈ρ depending on φ, such that EP[φ(Xn)[X1,…,Xn-1]=Eρ[φ(Xn)], P-a.s.
The following theorem give the relations of weak independence and Peng's indepen-dence.
Theorem1.7Let {Xi}i=1∞be a sequence of weak independent random variables under ρ. We define ρ by
ρ={P:(?)φ∈Cb,Lip(R),(?)n∈N, EP[φ{Xn)|X1,…,Xn-x]≤Eρ[φ(Xn)], P-a.s.}. Then {Xi}i=1∞is independent under ρ as in the Definition1.4, and Eρ[φ(Xi)]=Eρ[φ(Xi)],(?)i∈N,(?)φ∈Cb,Lip(R).
We can prove the law of larger numbers for Bernoulli random walk under i.i.d. assumption, and then we generalize this theorem without the i.i.d. assumption. The following theorem is the general version.
Theorem1.9Let {Xk}k=1∞be a sequence of random variables on measurable space (Ω, B(Ω)) and V a set of all probability measures P defined on (Ω,B(Ω)) such that (?)n∈N, μ≤EP[Xn|X1,…,Xn-1}≤μ, and EP[|Xn|q|X1,…,Xn-1]≤Kq P-a.s., where μ,μ, K, q are constants and q>1. Then we have (ⅰ) For each μ∈[μ,μ], there exists Pμ∈ρ, such that (ⅱ) For each P e ρ,(ⅲ) For each φ∈Cb,Lip(R), We also consider the central limit theorems for simple random walk. The notion of G-normal distribution play an important role in the central limit theorem. In this chapter, the G-normal distribution is defined as the solution of G-heat equation.
Definition1.10We call ξ is G-normal distributed, denoted by ξ~N(0,[σ2,σ2]), where0≤σ≤σ, if the distribution of ξ is given by Fξ(φ)=uφ(0,1),(?)φ∈Cb,Lip(R), where uφ{t,x) is the unique viscosity solution of the following G-heat equation:(?)tu-G((?)xx2u)=0u|t=1=φ where G(α)=1/2σ-2α+-1/2σ2α-
We only list two central limit theorems. In fact, they are equivalent in some sense. The second one in multi-dimensional case will be used in Chapter5. We denote the distribution function of G-normal distribution ξ by Eg[φ(ξ)]:=Fξ(φ).
Theorem1.14Let {Xi}i=1∞be a sequence of random variables on measurable space (Ω, B(Ω)). Let ρ be the set of all probability measures on (Ω, B(Ω)) such that, VP G V,(?)i∈N,
(1) EP[Xi|X1,…,Xi-1}=0,
(2)σ2≤EP[Xi2|X1,…,Xi-1}≤σ2,
(3)EP[|Xoi|q|X1…,Xi-1]≤Kq. We denote Sn=Σi=1n Xi.
If q>2and0<σ≤σ≤K<∞, then (?)φ∈Cb,Lip(R), where ξ~N(Q,[σ2,σ2]).
Let Mnq(Σ, K) be the set of n-stage Rd-valued martingales on some probability space (Ω,B(Ω), P) satisfying the following conditions:(ⅰ) EP[Sn]=0,(ⅱ) EP[(Sk+1-Sk)(Sk+1-Sk)T|S1,…,Sk]∈Σ,0≤k≤n-1, where Σ is a bounded,
convex and closed subset of S+(d).(ⅲ) EP[||Sk+1-Sk||q]≤K,0≤k≤n-1.
Let Vn[φ]:=supS∈Mnq(Σ,K) EP[φ(Sn/(?)n)].
Theorem1.17We assume that q>2. Let ξ be a G-normal distribution N (0,Σ) under G-expectation EG, then (?)φ∈C(Rd) with growth condition|φ(x)|≤C(1+|x|p), where1≤p≤q,
In the last section of Chapter1, we give the approximation of G-Brownian motion by simple random walk. Let {Sn}n=1∞be a discrete-time process,the continuous-time representation of S is given by St=Sn+(t-n)(Sn+1-Sn), n≤t
Let ρ be a set of probability measures on measurable space (Ω,B(Ω)). The sublinear expectation Eρ[·], the upper probability V(·) and lower probability v(·) are respectively defined by
We introduce the notion of product independence and sum independence, which is weaker than Peng's independence in Peng [93]. Definition2.14Suppose that X1,X2,…, Xn is a sequence of real measurable random variables on (Ω,B(Ω)).(ⅰ) Xn is said to be product independent of(X1,…, Xn-1) if for each nonnegative bound-ed Lipschitz function φk,=1,…, n, (ⅱ) Xn is said to be sum independent of (X1,…, Xn-1) if for each φ∈Cb,Lip(R),
The following law of large numbers is an extension of which in Peng [93,96,98], Chen [11], Chen and Wu [14], and Chen et al.[15]. Theorem2.17Let {Xk}k=1∞be a sequence of random variables satisfying: supk≥1Eρ[|Xk|q]<∞, for some q>1, and Eρ[Xk]=μ,-Eρ[-Xk]≡μ,k=1,2,
Set Sn=Σk=1n Xk.(ⅰ) If {Xk}k=1∞is product independent, then (ⅱ) If{Xk}k=1∞is product and sum independent, then (ⅲ) If {Xk}k=1∞is sum independent, and V(·) is upper continuous, i.e., Then
About the central limit theorem on sublinear expectation space, Peng [93] first prove it under "i.i.d." assumptions, and then generalized by Li and Shi [66], Hu and Zhang [51], Hu [50], Hu and Zhou [58] without the assumption of identical distribution. However, all of these results require the independence of random variables. We consider one weaker condition called m-dependent, and prove the corresponding central limit theorem. This result was accepted by Acta Mathematicae Applicatae Sinica, English Series. Definition2.31The sequence {Xi}i=1∞is called m-dependent if there exits an integer m such that for every n and every j≥m+1,(Xn+m+1,…,Xn+j) is independent of (X1,…,Xn). In particular, if m=0, then {Xi}i=1∞is an independent sequence.
Theorem2.32Let {Xi}i=1∞be a sequence of m-dependent variables, such that
and Eρ[|Xi|2+α]≤M for i=1,2,…, where α>0and M is a constant. Let Sn=Σi=1n Xi, then we have where ξ~N(0;[σ2,σ2]).
(Ⅲ) In Chapter3, we study the Ito's calculus without quasi-continuity, and we obtain the general form of Ito's formula and the solvability of stochastic differential equation with local Lipschitz coefficients.
All the existing research on the Ito's calculus with respect to G-Brownian motion is based on the stochastic process space MGp(0,T),p≥1(see Peng [92,95,97,98,100], Gao [39], Zhang et al.[123]), which generalized by the random variables with quasi-continuity. But the notion of stopping time, which is the important notion in classical stochastic analysis, does not satisfy such quasi-continuity. It is difficult to consider the problem of stopping time on the space MGp(0,T). And the existing Ito's formula requires C2-function satisfying some growth condition. In order to overcome such difficulty, in this chapter, we introduce a larger space of stochastic process M*p(0, T), which is generalized by the random variables without quasi-continuity. Then we can define Ito's integral on this larger space M*p(0, T), and we can consider the Ito's integral on stopping time interval, which allows us to have a Ito's integral for a "locally integrable" space Mωp(0, T). This new formulation permits us to obtain Ito's formula for a general C1,2-function, which essentially generalizes the previous results of Peng [92,95,97,98,100] as well as those of Gao [39] and Zhang et al.[123]. This result was published in Stochastic Process and Their Applications121(7)1492-1508with Prof. Peng Shige.
Theorem3.41Let Φ∈C1.2([0,T]×R) and
where αv,ηvij∈Mω1(0,T), βvj∈Mω2(0,T). Then for each t∈[0,T], we have, quasi- surely, In the last section of this chapter, we consider the following stochastic differential equations driven by a d-dimensional G-Brownian motion: where b(·,·), hij(·,·), σj(·,·):[0, T]×R→R are continuous functions and X0is a con-stant. We introduce the following condition:(H1) Bounded condition: for any s∈[0, T](H2) Lipschitz condition: for any x,y∈R and s∈[0, T], max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤K|x-y|.(H3) Locally Lipschitz condition: for all x, y∈R satisfying|x|,|y|≤R and any s∈[0,T], max{|b(s,x)-b(s,y)|,|hij(s,x)-hij(s,y)|,|σj(s,x)-σj(s,y)|}≤KR|x-y|.(H4) Growth condition: for any xeE and any5G [0,T], xb(s,x)≤K(1+x2), xhij(s,x)≤K(1+x2),|σj(s,x)|2≤K(1+x2). The first theorem consider the existence and uniqueness of the solution in the s-pace M*2(0, T). The second theorem study the solvability of SDE with locally Lipschitz coefficients.
Theorem3.44Let conditions (H1) and (H2) hold. Then there exists a unique contin-uous process X G M*2(0,T) satisfying (1).
Theorem3.45Let the condition (H3) and (H4) hold. Then there is a unique continuous adapted solution X of SDE (1).
(Ⅳ) In Chapter4, we study the properties of sublinear expectation and G-expectation, including the strict comparison theorem, additivity, Wasserstein distance, duality, domination and optimal transportation.
In order to study such properties, we divide this chapter into five sections. In Section4.1, we study the strict comparison theorem. We only list two important theorems in this section.
Theorem4.4Let X, Y∈Lc1(Ω) and X≤Y q.s. If thenEρ[X]
Theorem4.13We assume that φ,φ∈Cb,Lip(R). Then EG[φ(ξ)+φ(ξ)]=EG[φ(ξ)]+Eg[φ(ξ)] if and only if (?)xxu(t,x)(?)xxu(t,x)≥0,(?)(t,x)∈(0,1)×R.
Theorem4.16If there exists x0and θ>0such that φ,φ∈C2((x0-θ,x0+θ)) and φ"(x0)φ"(x0)<0,then we have
In Section4.3, we compare the G-expectation with Choquet expectation and provide some interesting examples. We show that the G-expectation is always dominated by the corresponding Choquet expectation. The following theorems are the main theorems in this section.
Theorem4.26G-expectation EG[·] can be represented by Choquet expectation EC[·](i.e.,(?)X∈LG1(Ω), EG[X]=EC[X]) if and only if EG is linear (i.e.,σ=σ).
In Section4.4, we generalize the classical Wasserstein distance and related properties to the sublinear expectation space.
Let ρ1and ρ2be two nonempty weakly compact and convex sets of probability measures. We define the Hausdorff-Wasserstein distance between ρ1and ρ2as follows: where Wp(P1, P2) is the classical Wasserstein distance between two probability measures P1and P2.
We first give the Kantorovich-Rubinstein duality formula for the sublinear case. Theorem4.30W1(ρ1,ρ2)=sup {|Eρ1[φ]-Eρ2[φ]|}.||φ||Lip<1||φ||Lip≤1means the Lipschitz constant of Lipschitz function φ is at most1.
As well known that weak convergence of probability measures is equivalent to con-vergence in Wasserstein distance, we give a non-trivial extension of this result to the setting of sublinear expectations.
Definition4.31A sequence {Eρn}n=1∞of sublinear expectations is said to be weakly con-vergent to Eρ, or equivalently,{ρn}n=1∞weakly converges to ρ, if for each φ∈Cb,Lip(Ω), limn→∞Eρn[φ]=Eρ[φ].
Theorem4.35Let {ρn}n=1∞be a sequence of convex and weakly compact sets of proba-bility measures satisfying supn Eρn[|X|1+α]<∞for some α>0and ρ be a convex and weakly compact set of probability measures. Then the following statements are equivalent.(ⅰ) ρn weakly converges to ρ.(ⅱ)W1(ρn,ρ)→0.
In Section4.5, we introduce the notion of duality, domination and optimal trans-portation from classical case to the sublinear expectation space. We first introduce the notion of Fenchel duality and related properties, which will be used in Chapter6when we study the dual game.
We also introduce the classical notion of Blackwell domination and we obtain a dominated theorem for sublinear expectations.
Theorem4.47If ρ1and ρ2are two convex and weakly compact sets of probability measures on (Ω,B(Ω)). Then the following statements are equivalent.(ⅰ) Eρ1[·] is dominated by Eρ2[·]. (ⅱ) ρ1(?)ρ2.
In the end of this section, we study the Kantorovich optimal transportation under sublinear expectation space. Let Ω1and Ω2be two complete separable metric spaces, Π(μ, v) be the set of probability measures defined on (Ω1(?)Ω2,B{Ω1(?)Ω2)) such that the marginal probability measures on Ω1and Ω2are μ and v respectively. Let ρ1and ρ2be two weakly compact and convex sets of probability measures defined on (Ω1, B(Ω1) and (Ω2,B(Ω2)) respectively. For a continuous function c on Ω1(?)Ω2,Φc denotes the set of all bounded continuous function pairs (φ,φ) satisfying φ(ω1)+φ(ω2)≥c(ω1,ω2),(?)ω1∈Ω1and (?)ω2∈Ω2.
Theorem4.49Let c: Ω1(?)Ω2→R be a continuous function, then we have
We also consider the following maximal covariance problem: Given a probability measure μ and ρ a set of probability measures. The maximal covariance function C(μ,ρ) is defined by
We give the main theorem in this section, which will be useful in Chapter5.
Theorem4.52Let {ρn} and ρ be the weakly compact and convex sets of probability measures, μ be a arbitrary probability measure. If W2(ρn,ρ)→0, then we have
(Ⅴ) In Chapter5, we introduce the notion of CMMV and the problem of maximal variation for martingales.
We first introduce the notion of CMMV which studied in De Meyer [26], and then generalize it to the framework of G-expectation. The main purpose of this chapter is to solve the following problem of maximal variation of martingales.
Let Mn{μ) be the set of pairs (F, X) where F:=(Fq)q=1,…,n is a filtration on a probability space (Ω, B(Ω), P), and X=(Xq)q=1,…,n is an F-martingale whose terminal value Xn is Blackwell dominated by μ. For a function M:△2(Rd)→R, we can define M-variation VnM(F,X) as Then the maximal M-variation VM(μ) is denned by
For one-dimensional case, we have the following theorem. Theorem5.11If M satisfies:(ⅰ) Positive homogeneity:(?)X∈L02(R),(?)α>0: M[αX]=αM[X].(ⅱ) Lipschitz continuity: There exists p∈[1,2) and K∈R such that for all X, Y∈
L02(R);|M[X]-M[Y]|≤K||X-Y||Lp.(ⅲ) Transportation invariance for constant: M [X+β]=M[X]+M[β],(?)β∈R. Then for all μ∈△2(R), we have
(3) If ρ>0and if, for all n,(Fn,Xn)∈Mn(μ) satisfies VnM(Fn,Xn)=VnM(μ), then the continuous time representation Πn of Xn defined as Πtn:=X[nt]n, converges in finite-dimensional distribution to the CMMV Πμ.
As for the multi-dimensional case, we first introduce a auxiliary function r defined by where cov(μ) denotes the covariance matrix of μ∈△02(Rd). Then we can define a set Γ as in Definition5.12.
We assume that function M:△2(Rd)→R satisfies the following hypotheses:
(H1) M≥0and has no degenerate directions:(?)x∈Rd, there exists μ∈△02(Rd) such that
(H2) M is K-Lipschitz for the Wasserstein distance of order p for some p∈[1,2).
(H3) M is positively homogenous:(?)X∈L2(Rd) and (?)λ>0, M[λX]=λM[X].
(H4) M is concave on△02(Rd).
(H5) r is quasiconvex, i.e.,(?)α∈R,{Y∈L2(Rd)|r(cov(Y))≤α} is convex in L2(Rd).
(H6) M[X+β}=M[X]+M[β],(?)β∈Rd.
Then we have the following theorem.
Theorem5.14Under hypotheses (H1)-(H6), we have
(Ⅵ) In Chapter6, we study a general class of repeated games with incomplete information on one side a la Aumann and Maschler [4].
This game was first introduced by De Meyer [25] as financial exchange game and then generalized by Gensbittel [45,47] in multi-dimensional context. The difference with Aumann-Maschler's model is that the state space and sets of actions of both players are allowed to be infinite. This chapter contains four sections.
In Section6.1, we study the linear game proposed in [45] and generalize the Cav(u) theorem in [45] from△(P) to△1(Rd), where P is a compact and convex subset of Rd. Let Vn(μ) and Vn(μ) denote the maximal and minimal payoffs for player1and player2respectively in the repeated game Γn(μ)(more details can be found in Section6.1). Let u(μ) and u(μ) denote the corresponding payoffs in the one-shot non-revealing game. Then we have the following Cav(u) theorem. Theorem6.5For all μ∈△1(Rd), we have
Furthermore, if we assume that (?)μ∈△2(Rd), the game Γ1(μ) always has the value V1(μ), i.e., V1(μ)=V1(μ)=V1(μ).Then we have the more precise Cav(u) theorem.
Theorem6.9If V1satisfies the following hypotheses:
(ⅰ) There exists μ0∈△2(Rd) such that V1(μ0)>0.
(ⅱ) V1([L+β)=V1([L])+V1([β]),(?)β∈Rd.
(ⅲ) For any α∈R,{X∈L2(Rd):supv∈△02(Rd):cov(v)=cov(X) V1(v)≤α} is convex in L2(Rd).
Then we have, for all μ∈△2(Rd), where ξ~N(0, Γ), Γ is given in Section5.3in Chapter5where V1is instead of M.
In Section6.2, we study a typical example of linear game called financial exchange game introduced by De Meyer [25]. We generalize the natural exchange mechanism in [25].
In the game Γn(μ)(see Section6.2), the hypotheses on the natural exchange mech-anism are the following:
(H1) Existence of the value:(?)μ∈△∞(R), the game Γ1(μ) has a value.
(H2) Bounded exchanges:(?)i,j:|Aij|≤K, where K is a constant.
(H3) Invariance with respect to the scale:(?)α>0,(?)L∈L2(R): V1([αL])=αV1([L]).
(H4) Transportation invariance with respect to the risk-less part of the risky asset:(?)β∈R:V1([L+β)=V1{[L])+V1{[β]).
(H5) Positive value of information:(?)L∈L2(R): V1([L])>0.
The following theorem gives the asymptotic characterizations of the value function and price process based on the results in Chapter5.
Theorem6.15If (H1)-(H5) is satisfied, then for all μ∈△2(R),(ⅰ)limn→∞1/nVn(μ)=V1([E(μ)]).
(ⅲ) If p>0and if, for all n,(Fn,Xn)∈Mn(μ) satisfies VnV1(Fn,Xn)=Vn(μ),then the continuous time representation Πn of Xn defined as Πtn:=X[nt]n converges in finite-dimensional distribution to the CMMV Πμ.
In Section6.3, we systematically study a game model with transaction costs, which does not satisfy the natural exchange mechanism in [25] but satisfies our general natural exchange mechanism in Section6.2. The explicit solution of Nash equilibrium of this game was obtained by the dual method, and we show that the price process posted by uninformed player converges to CMMV in finite distribution. This result confirms that CMMV is a very robust dynamic in stock market.
In Section6.4, we study a game model where both players are not risk-neutral, which generalize the results in De Meyer [26] while the uninformed player is risk-aversion but the informed player is risk-neutral. The results are very surprising:the Nash equilibrium in the game does not depend on the risk attitude of informed player, i.e., the equilibrium in the game where the informed player is risk-natural is also an equilibrium in the game when the informed player is risk-aversion or risk-seeking.
引文
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[24]De Meyer, B. (1999) From repeated games to Brownian games, Ann. Inst. Henri Poincare (B) Prob.,35(1),1-48.
[25]De Meyer, B. (2010) Price dynamics on a stock market with asymmetric informa-tion, Game Econ. Behav., (1),42-71.
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[28]De Meyer, B. and Rosenberg, D. (1999) "Cav u" and the dual game, Math. Oper. Res.,24(3),619-626.
[29]Dellacherie, C. and Meyer, P. (1978 and 1982) Probabilities and Potential A and B, North-Holland.
[30]Denis, L., Hu, M. and Peng, S. (2011) Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion paths, Potential Analy-sis,34(2),139-161.
[31]Denis, L. and Martini, C. (2006) A theoretical framework for the pricing of contin-gent claims in the presence of model uncertainty, Ann. Appl. Prob.,16(2),827-852.
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[41]Gao, F. and Jiang, H. (2010) Large deviations for stochastic differential equations driven by G-Brownian motion, Stoch. Proc. Appl.,120(11),2212-2240.
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[43]Gao, F. and Xu, M. (2012) The support of the solution for stochastic differential equations driven by G-Brownian motion, Acta Math. Sin., English Series,28(12), 2417-2430.
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[45]Gensbittel, F. (2010) Analyse asymptotique de jeux repetes a information incom-plete. Ph.D. thesis, manuscript, University Paris 1.
[46]Gensbittel, F. (2012) Covariance control problems over martingales with fixed ter-minal distribution arising from game theory, to appear in SIAM J. Contr. Opti-mizat.
[47]Gensbittel, F. (2012) Extensions of the Cav(u) theorem for repeated games with one-sided information, perprint.
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[50]Hu, F. (2011) Moment bounds for IID sequences under sublinear expectations, Sci. China Math.,54(10),2155-2160.
[51]Hu, F. and Zhang, D. (2010) Central limit theorem for capacities, C. R. Acad. Sci. Paris. Ser. I.348,1111-1114.
[52]Hu, M. (2009) Explicit solutions of G-heat equation with a class of initial conditions by G-Brownian motion, in arXiv:0907.2748vl.
[53]Hu, M. (2011) The independence under sublinear expectations, in arXiv:1107.0361.
[54]Hu, M., Ji, S., Peng, S. and Song, Y. (2012) Backward stochastic differential equations driven by G-Brownian motion, in arXiv:1206.5889.
[55]Hu, M., Ji, S., Peng, S. and Song, Y. (2012) Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion, in arXiv:1212.5403.
[56]Hu, M. and Peng, S. (2009) On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin., English Series 25(3),539-546.
[57]Hu, M. and Peng, S. (2009) G-Levy processes under sublinear expectations, in arXiv:0911.3533.
[58]Hu, Z. and Zhou, L. (2012) Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, in arXiv:1211.1090.
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[60]Huber, P. and Strassen, V. (1973) Minimax tests and the Neyman-Pearson Lemma for capacity, Ann. Stat.,1(2),252-263.
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