倒向随机微分方程下的算子表示及Jensen不等式
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摘要
在对随机最优控制问题的研究过程中,Bismut于1973年首次提出了线性的倒向随机微分方程(简称BSDE)。然而直到1990年Pardoux-Peng[90]给出了一般形式的倒向随机微分方程,并证明了其解的存在唯一性,倒向随机微方程才在理论及应用方面取得了迅速发展。
BSDE的一般形式如下：其中(?)表示终端随机变量,T>0是固定的终端时刻,(Wt)t∈[0,T]表示d-维布朗运动,g是该BSDE的生成元。求BSDE的解是对固定的(?),T,g,求一对适应过程(Y.,Z.)使得上式成立。Pardoux-Peng[90]给出的假设是9关于y,z李普希兹连续,此后有很多研究致力于将该条件弱化,如改为局部李普希兹条件(Bahlali[1]等),连续性条件(Lepeltier-San Martin [73],Matoussi[80]等),一致连续性条件(Jia[53,56],Fan-Jiang[36],Hamadene[39]等),关于z平方增长(Kobylanski[68], Briand-Hu[9,10],Tevzadze[117], Briand-Elie[7], Dclbaen-Hu-Richou[23],Richou[104]等),关于z超平方增长(Delbaen-Hu-Bao[22], Richou[104]等),关于y多项式增长(Briand-Carmona[4]等),关于y超线性增长,关于z平方增长(Kobylanski-Lepeltier-Quenez-Torres[69]等),不连续(Jia[52],N'zi-Owo[83], Halidias-Kloeden[38]等)以及一些其他的条件(Royer[107]等)。另外,也出现了多种形式的BSDE,比如,带反射的、正倒向耦合的、驱动为相互独立的布朗运动与泊松跳过程,或者更一般的1evy过程、甚至一般的鞅等等,相关文献如,Tang-Li[116],Situ[112,113], Yin-Situ[121],Ouknine[85],Essaky-Ouknine-Harraj[35],Ren-Otmani[103],Morlais[81], Lejay[72],Bahlali-Essaky[2],Hamadene-Ouknine[40],Hassani-Ouknine[41],Essaky[33]等。另一个分支是对BSDE解的性质的研究,比如比较定理、逆比较定理、凸性、平移不变性等等,如Pcng[97,100]引入的g-期望的概念及其性质,参考文献如Briand-Coquet-Hu-Memin-Peng[6],Coquet-Hu-Memin-Peng[19],Jiang[64,65,66],Hu-Tang[48],Hu-Peng[47],Royer[108],Jia-Peng[58],Jia[55]等。BSDE在很多重要领域得到了应用,如经济、控制、金融等领域,具体说来,如随机最优控制、博弈理论、数理金融等等。相关文献如Peng[96,98],Wu-Yu[120],Chen-Li-Zhou[16],Liu-Pcng[76],Kohlmanm-Zhou[70],chen-Epstein[14]等。BSDE的另一方面重要应用是,提供了若干类二阶偏微分方程的概率解释,也即非线性Feynman-Kac公式,具体可参见Peng[94],Peng[98],Pardoux-Peng[91],E1Karoui[30],Barles-Buckdahn-Pardoux[3],Buckdahn-Hu[12],Briand-Hu[8],El Karoui-Kapoudjian-Pardoux-Peng-Qucnez[31],Pardoux[87],Padoux-Tang[93],Kobylanski[68], El Karoui-Peng-Quenez[32],N'zi-Ouknine-Sulem[82]等。
为叙述方便,我们将方程(1)记为(g,T,ξ),其解记为(Ytg,T,ξ,Ztg,τξ)t∈[0,T]另外将Ytg,T,ξ记为Et,τg[ξ]。假设对任意(y,z)∈R×Rd,g(·,y,z)是平方可积的。Tr[·]表示对称矩阵的迹。Gi表示矩阵G的第i行。
下面是本文的章节目录：
一、第一章引言；
二、第二章一致连续系数BSDE对二阶随机微分算子的不变表示及其在非线性半群上的应用；
三、第三章平方增长g-凸函数,C-凸函数及其相关关系；
四、第四章由带跳的BSDE的解来表示的随机积分-微分算子及f-凸函数的性质；
五、第五章关于容度的几个性质的新证明。
第二章证明了一类二阶随机微分算子可以表示为—(?)(?)FBSDE的解的极限,其中倒向方程的生成元只需一致连续；由非耦合的正倒向方程定义非线性半群,其无穷小生成元就是如上表示的二阶随机微分算子;证明了非线性半群的单调性与保序性；证明了耦合的FBSDE的一个比较性质。
(本章内容主要来自两篇文章Jia-Zhang[60],Zhang-Jia[123])
在本章中,我们总假设SDE是n维的,BSDE是1维的。
在g关于(y,z)一致连续且线性增长的条件下,Lepeltier-San Martin[73]证明了方程的解的存在性,然而解未必是唯一的,Jia-Peng[57]证明了此时解或者唯一或者有不可数多个,不唯一的例子,如g(y)=(?)|y|,T=1,ξ=0时,方程((?)|y|,1,0)的解就有无穷多个。本章第一个结论说明,不论解是否唯一,我们总是可以拿来与SDE的解一起,共同表示一类二阶随机微分算子。
首先引入下列关于SDE的假设。
假设b：Ω×[0,T]×Rn→Rn,σ：Ω×[0,T]×Rn→Rn×d是满足下列条件的函数：
(一致李普希兹条件)：|b(t,x1)-b(t,x2)|+|σ(t,x1)-σ(t,x2)|≤Kb|x1-x2|,(?)x1,x2∈Rn：
(线性增长条件)：|6(t,x)|+|σ(t,x)|≤Kb(1+|x|),(?)x∈Rn；其中Kb>0为一个常数。为简单起见,我们记此条件为正向的标准李普希兹条件,以区分9满足的标准李普希兹条件(即,g关于y,z一致李普希兹连续)。
设(Xst,x)s∈[0,T]为下列SDE的强解
其具体内容如下
定理2.2.1(表示定理Ⅰ)令g满足一致连续性条件,且关于t连续,6,σ满足以上假设且关于t连续。若φ∈C1,2([0,T]×Rn)且φ本身及其各阶导数关于x至多多项式增长,那么对每个x∈Rn,t∈[0,T),如下式子成立其中注意到此结论中解的选取是任意的。由此可以得到g在此条件下的一个逆比较定理以及一些BSDE的解的性质与生成元的性质之间的等价对应关系,比如,线性性、保常性。
接下来我们根据正倒向方程定义了两种非线性半群,这两种非线性半群是对由马尔科夫扩散过程生成的马氏半群的一种推广,马氏半群是给扩散过程取线性期望得到的,而此处的半群是对扩散过程取非线性期望(g-期望)得到的。上面的表示定理Ⅰ恰好给出了这两种半群的无穷小生成元。
在定义半群时,我们首先假定6,σ只依赖于x且关于x李普希兹连续,9只依赖于(y,z),故终端时刻T可以取值为任意正数。记H为Rn→R中至多多项式增长的连续函数的全体,Hb为Rn→R中有界一致连续函数的全体。根据g满足条件的不同,分为如下两种定义方式：
首先,当g关于(y,z)满足李普希兹条件时,通过εtf(x)：=y0g,t,f(Xt0,x)在H上定义族算子{εt；t≥0},并定义运算(εt(?)εs)f=εt(εsf)。则有εt(?)εs=εs(?)εt=εt+s。故{(εt)t≥0;(?)}构成一个交换半群。
其次,当g关于(y,z)一致连续,b,σ有界时,类似于Ma-Zhang2011年的文章[77],可以如下定义“节点集”0(t,x,f,s),
(?)(t,x,f.s)：={y：存在一个解y.g,s,f(Xst,x)满足ytg,s,f(Xst,y},其中f∈Hb。Ma-Zhang还证明了(?)(t,x,f,s)是一个有界区间,分别以u(t,x,f,s)和u(t,x,f,s)记其上界和下界,则有定义εt,sf(x)：=u(t,x,f,s)和εt,sf(x)：=u(t,x,f,s)。由于此时我们假设的g,b,σ都独立于时间t,故εt,sf(x)≡ε0,s-tf(x),可以简记εtf(x)=ε0,tf(x)。类似的,有εtf(x)=ε0,tf(x)=εr,t+τ f(x),对任意r≥0,t≥0。对固定t,εtf(x)和εtf(x)都是有界的一致连续函数,即属于Hb.同李普希兹情形,我们可以在两种算子上分别定义运算“。”,可以推出{(εt)t≥0；(?)}和{(εt)t≥0；(?)}都构成非线性半群。
由此可以看出,当9只是一致连续时,无穷小生成元与半群并不能完全一一对应,此时,可能有多个半群对应于同一个无穷小生成元。
有了以上的半群,我们下一问题着重讨论了当g满足李普希兹条件时半群的单调性质和比较性质。实际上,Herbst-Pitt[43]和Chen-Wang[17]证明了马氏半群单调的充要条件,以及两个马氏半群可以比较的充要条件。但是,他们的证明强烈依赖于马氏半群的线性性,所用的方法不能直接拿来用,所以为了得到这些结果,我们需要另辟蹊径。首先引入下面定义。
定义2.5.1令“≤”表示Rn中通常的半序。
(i).我们称一个定义在肿中取值于R的可测函数f是单调的,如果f(x)≤.f(x)对所有x≤x.我们以M来表示至多多项式增长的连续单调函数的集合。显然M(?)H。
(ii).对两个半群{εt}t≥0和{εt}t≥0,我们称εt≥εt,如果对所有f∈M,以及所有x≥x和t≥0,有εtf(x)≥εtf(x).另外,如果εt=εt,我们称εt单调。
我们证明了下面的两个定理成立。
定理2.5.1假定6,σ只依赖于x且关于x李普希兹连续,9只依赖于(y,z),且关于(y,z)李普希兹连续。则定义在H上的εt是单调的,当且仅当：
(2C1-i).对任意的i=1,...,n,若是x,x∈Rn,xi=xi,且对任意k≠i,xk≥xk,有bi(x)-bi(x)≥0成立；
(2C1-ii).对所有i=1,...,n,σ的第i行只依赖于xi。
由此定理,我们发现εt的单调性与g无关。
定理2.5.3假定n=d,假定6,σ,b,厅只依赖于x且关于x李普希兹连续,g,g只依赖于(y,z),且关于(y,z)李普希兹连续。σσ*(或者σσ*)是一致正定的,且b,b,σ,厅都有界。
如果εt或者εt是单调的,那么εt≥εt当且仅当
(2C3-i)σσ*≡σσ*并且σ和σ的第i行均只依赖于xi；
(2C3-ii)对所有x∈Rn,y∈R,K∈Rn且K≥0,
注意到第二个定理首次将SDE的漂移项系数与BSDE的生成元结合起来作为一个整体进行比较。由于半群对应唯一确定的PDE,如上的单调性定理以及保序性定理实际上可以转化成二阶拟线性抛物方程的单调性与保序性。在证明的过程中,我们还可以看到,在李普希兹条件下,半群对应唯一确定的PDE,但是未必对应唯一确定的正倒向方程。
特别的,取g,g：Rd→R,kj,pj,kj,pj∈R,(?)j=1,...,d。我们有以下结果。
定理2.5.4设b,σ,b,厅只依赖于x且关于x李普希兹连续,且条件(2C1-i)和(2C1-ii)对b,σ和b,厅成立,(σ)n×d的同一列中的元素符号相同,其符号可随x的不同而不同,σ的符号也具有这种性质。。那么εt≥εt当且仅当(2C3-i)和(2C3-ii)成立。此处(2C3-ii)意味着对任意i,只要x≥x,xi=xi,
这个结果不能被定理2.5.3所涵盖,因为此时我们并不假定n=d,以及σσ*一致正定。根据这个定理,我们可以将一类特殊的二阶拟线性抛物PDE部分的转化成二阶线性PDE。
注2.5.8考虑以下二阶拟线性PDE：其系数的条件同定理2.5.4,则我们有如下结论：如果f是至多多项式增长的非降连续函数,那么上述PDE等价于如果f是至多多项式增长的非增连续函数,以上PDE等价于
本章最后一个结果,受到半群保序性定理证明方法的启发,得到一类新的FBSDE的比较定理。该定理同样将SDE的漂移项系数和BSDE的生成元结合起来,据我们所知,在FBSDE的比较定理中,这是首例。首先我们所考虑FBSDE如下此处(Xt,Yt,Zt,Wt)∈Rn×R×Rd×Rd,b,σ,g和f都具有相应的维数。
以下是关于此定理的假设(主要来自[77])：系数(b,σ,g,f)可测且有界；σσ*一致正定；b,σ,g,f光滑且一二阶导数有界。
定理2.6.1假定bi,σi,gi,jfi(i=1,2)满足如上条件。设这两对FBSDE的初值分别为x1,x2,记其解为(Xi,yi,Zi)。如果
(i)σ1(σ1)‘三σ2(σ2)*,
(ii)对任意t∈[0,T],x∈Rn：y∈R,p∈Rn,
(iii)f1≥f2,
(iv)x1=x2,
那么Y01≥Y02。
第三章证明了平方增长的倒向随机微分方程的表示定理；得到了一个函数满足平方增长g-期望下的Jensen不等式的充要条件；基于常数C定义了一类C-仿射函数,并以此定义了C-凸和C-凹；证明了C-凸(凹)函数的类似于普通凸(凹)函数的性质；研究了这类C-凸(凹,仿射)函数与平方增长g-凸(凹,仿射)函数之间的相互关系。
(本章主要内容来自Jia-Zhang[63])
经典的数学期望下的Jensen不等式是现代概率论中的一个基本的不等式。在g满足标准李普希兹条件时,Jia-Peng [58]于2010年,首次将满足g-期望下的Jensen不等式的函数定义为“凸函数”,提出了g-凸的概念,即：一个函数h称为g-凸(凹)函数,如果对任意FT可测平方可积且使h(X)平方可积的随机变量X,以及任意t∈[0,T],Et,Tg[h(x)]≥(相应地,≤)h(Et,Tg[X]).如果h既是g-凸函数又是g-凹函数,则称为g-仿射函数。Jia-Peng还给出了一个光滑函数h是g-凸函数的一个简单的充要条件,即,Lgt,y,zh≥0,其中而一般的多项式增长的连续函数h是9-凸函数的充要条件是,h是下列方程的粘性下解：关于粘性解的概念,读者可参考[20]等。关于这种g-凸函数的性质,一个很重要的结果是在.g满足标准李普希兹条件时,g-凸函数一定是通常意义下的凸函数。
本章的主要内容是考虑平方增长条件下g-凸函数的性质。自Kobylanski在2000年的文章[68]中引入平方增长BSDE以来,平方增长的BSDE已经得到了比较好的研究。然而,就我们所知,除了Ma-Yao[78]给出了一个条件比较强的关于生成元的表示定理以及一个简单的Jensen不等式的结论外,再没有其它人涉猎平方增长条件下的表示定理以及Jensen不等式。本章将对这些问题进行深入探讨。
本章的核心结果是,证明9-凸函数应该满足的充要条件。在此证明中遇到了困难,主要原因是终端条件有界的要求与证明过程中引入的正向方程的解通常无界之间的矛盾。为解决此问题,我们引入停时以及可选停时定理等,停时的灵活运用成为这个证明过程的关键,也贯穿了整个证明过程的始终。最让我们感兴趣的是,此时的g-凸函数已经不再是通常意义下的凸函数了。为此,根据一个比较简单的依赖于常数C的ODE,我们将其解(通常不是直线)定义为新的“仿射函数”,称为C-仿射函数,并用类似于普通凸的方法定义了C-凸函数。这个函数体系有非常好的基本类似于通常凸的性质。比如C-凸函数具有连续性,拟凸性,以及可以表示为一族C-仿射函数的上包络等。值得一提的是,若是C=0,此时的0-凸框架恰好对应于通常的凸的框架。这类C-凸函数对于平方增长g-凸的意义,一如凸函数对于李普希兹条件下g-凸的意义。
本章中我们对9作如下假设：
存在两个常数Ky>0和Kz>0使得,A(t,y,z,y’,z,),
g(·,0,0)一致有界。为简单起见,称此假设为标准平方增长条件。
首先,如前,我们证明了g满足标准平方增长条件时,带有停时的表示定理成立。
定理3.3.1令9满足标准平方增长条件,b,σ满足正向的标准李普希兹条件。另外假设g,b,σ都关于t连续。设φ∈C1,2(R+×Rn),那么对每个(tnx)∈[0,T)×Rn,下面的极限成立其中Lg,b,σt,x的定义如前所述,ε要求充分小,Τε=ΤΛ(t+ε),Τ是一个使得X.t,x在[t,τ]上有界的停时。例如,(K0是一个正数且充分大)。
接下来关于g-凸的研究,我们总假设g(t,y,0)≡0,且g满足标准平方增长条件。
首先,有如下9-凸函数的定义。
定义3.4.1一个函数h：R→R被称为终端有界条件下的g-凸(相应地,g-凹)函数,如果对每个X∈L∞(FT),有,对任意t∈[0,T],
h(Etg,T[X])≤Et,Tg[h(X)].(相应地,h(Et,Tg[X])≥Et,Tg[h(X)])P-a.s.h被称为g-仿射如果它既是g-凸又是g-凹。
注意到终端的有界性,我们可以定义凸集上的g-凸函数,这对后面我们研究这一类g-凸函数提供了很大的方便。
定义3.4.3(凸集上的g-凸性)假定(?)是R上的一个凸集。对一个固定Eg[·],实值函数h被称为0上的g-凸(相应地,g-凹)函数,如果对每个X∈L∞(FT)使得Es,Tg[X](ω)∈(?),dPdt-a.s.我们有(相应地,h(Et,Tg[X])≥Et,Tg[h(X)])P-α.s,t∈[0,T].h被称为(?)上的g-仿射函数如果它既是(?)上的g-凸函数又是(?)上的g-凹函数。
下面两个定理是本章的主要结果。
定理3.4.1令h∈C2(R),那么以下两个叙述等价：
(ⅰ)h是终端有界条件下的9-凸函数(相应地,g-凹函数)；
(ⅱ)对每个y∈R,z∈Rd,Lgt,y,zh≥0(相应地,≤0)dPdt-a.s.
定理3.4.2假定g独立于ω且关于t连续。h是一个连续函数。以下论断等价：
(ⅰ)对每个(t,z)∈[0,T]×Rd,h是Lgt,y,zh=0的粘性下解；
(ⅱ)h是终端有界条件下的g-凸函数。
下面取一类g进行仔细研究：g=C|z|2+g1(t,y,z),其中如果C三0,任意9-凸函数是通常意义下的凸函数,g-仿射函数恰好是所有的线性函数。我们设C≠0,取φ∈C2(R)是一个g-仿射函数,则有解以上ODE,以Ⅱc来表示所有解的集合：,或φ(x)=x+b,或φ(x)=b,Aa,b∈R}.我们定义这类函数为C-仿射函数,通过这类函数,类似于普通凸函数的定义,得到了如下C-凸函数：
定义3.6.1(C-凸函数)定义在凸集D上的函数f称为C-凸函数,如果对任意φ∈Πc与f相交于两点x1,x2(不妨设x1 如上定义的C-凸函数有很多很好的性质,比如拟凸性,连续性,几乎处处可导,且在各个点的左右导数都存在,左导数不大于右导数等等。
此外,我们还定义了C-凸集。
定义3.6.7(R2中的C-凸集)一个集合A(?)R2被称为一个C-凸集,如果对任意两个点(x1,y1),(x2,y2)∈A,x1 关于f的上水平集epif={(x,y)：f(x) 命题3.6.9如果f是一个C-凸函数,epif是一个C-凸集。另一方面,如果epif是一个C-凸集,那么f是一个C-凸函数。
C-凸函数还有如下重要性质。
定理3.6.2以及定理3.6.4任意C-凸函数可以表示为一族C-仿射函数的上包络,反之,任意一族C-仿射函数的上包络都是C-凸函数。
接着,我们讨论了C-凸函数与g-凸函数之间的关系。首先,特定的一族函数g,C-凸函数与与g-凸函数一一对应,具体如下。
定理3.6.5任意C-凸函数是一个9-凸函数,其中g=C|z|2+G1z,(?)C1∈Rd。反之,设g=C|z|2+,(?)C1∈Rd,那么任意定义在D上的9-凸函数是定义在D一个C-凸函数。
另外,对更一般的g,如下结果成立。
定理3.6.6假定g=C|z|2+g1(t,y,z),其中C∈R,于是任意9-凸(相应地,凹,仿射)函数是C-凸(相应地,凹,仿射)函数。
最后的两个定理,是关于9-凸函数与9-凸函数或g-仿射函数之间的关系。
定理3.6.7假定Ⅰ是一个指标集,{fi：i∈Ⅰ}是g-凸函数的一个集合。那么f(x)=sup{fi(x)：i∈Ⅰ}也是一个g-凸函数。
定理3.6.8假定其中C∈R,则光滑的g-凸函数h可表示为一族g-仿射函数的上包络的必要条件是,对任意(t,y,z)成立。特别地,若C=0,且g独立于y,则上述条件是充要条件。
第四章由一列带跳的正倒向随机微分方程的解来表示二阶随机积分-微分算子；得到了带跳的BSDE的逆比较定理,以及解与生成元的一系列等价性质；证明了由带跳的BSDE构成的f-期望的Jensen不等式。
(本章主要内容来自Jia-Zhang[61,62])
本章中,我们将研究带跳的BSDE的表示定理和Jensen不等式。假定信息流(Ft)t∈[0,T]是由相互独立的d-维标准布朗运动(Wt)t∈[0,T]和定义在(R+×B)上的泊松随机测度μ。首先,本章内假设b,σ满足正向的标准李普希兹条件,f(ω,t,x,y,z,U)关于(y,z：U)致李普希兹连续,关于x至多多项式增长。对给定的(t,x)∈[0,T]×Rn,以X.t,x来表示以下SDE的解：并引入如下二阶随机积分-微分算子：关于积分的含义,将在第四章正文中阐述。
本章中我们将给出上述算子由带跳的正倒向方程的解的表示,以下是两个主要结果。
定理4.2.1(表示定理Ⅰ)假设f,b,σ关于t右连续。假设φ关于x有有界的三阶导数,记其公共界为Kφ。令1≤p≤2。则对每个(t,x,U(·))∈[0,T)×Rn×L2(B,B*,λ；Rn),以下式子成立另一个结论用到停时,也放宽了对φ的要求。
定理4.2.2(表示定理Ⅱ)假定生成元f,b,σ满足定理4.2.1中的条件。令φ∈C2。则对每个1≤p≤2,和我们有其中τε：=τ(?)(t+ε),τ是一个停时,使得Xt,x在[t,τ]上有界,比如τ：=inf{s>t|Xst,x-x|>N},ε充分小。
上面定理中的L2(B,B*,λ；Rn)和L∞2(B,B*,λ；Rn)的意义在第四章有详细介绍。
由表示定理,我们得到了关于带跳的BSDE的逆比较定理。
定理4.3.1(逆比较定理Ⅰ)令f1,f2独立于x,另外,设(?)(y,z,U),f1和f2都关于t∈[0,T)右连续,在Τ点连续,P-a.s.。对任意s∈[0,T],ξ∈L2(Fs),有则对任意(t,y,z,U(·))∈[0,T]×R×Rd×L2(B,B*,λ；R),有
定理4.3.2(逆比较定理Ⅱ)令.f1和f2除满足逆比较定理Ⅰ中所述条件外,还满足f(t,y,0,0)三0.若对任意ξ∈L2(FT),则有
假定.f独立于x,且对任意(t,y)∈[0,T]×R,f(t,y,0,0)≡0。类似于g-期望,Royer[108]由带跳的BSDE定义一类非线性期望f-期望。下面给出了f-凸函数的概念。
定义4.4.2对一个给定的f-期望Ef[·],一个函数h：R→R被称为f-凸(相应地,f-凹)函数,如果对每个X∈L2(FT),使得h(X)∈L2(FT),我们有(相应地,h被称为f-仿射函数如果它既是f-凸函数又是f-凹函数。下面记
定理4.4.1令f(t,y：z,U)关于(y,z,U)一致李普希兹连续且满足比较定理成立的条件,并关于t连续,对任意t,y,f(t,y,0,0)三0且h∈C2(R)。则以下两种叙述等价：
(i).h是f-凸的(相应地,f-凹)；
(ii).对每个t∈[0,T],y∈R,z∈Rd,U(·)∈L∞2(B,B*,λ；R),(相应地,≤0).
定理4.4.4令h∈C(R)至多多项式增长,.f(y,z,U)所需满足的条件比定理4.4.1稍强(具体见原定理),则以下两种叙述等价：
(i).h是f-凸的(相应地,f-凹)；
(ii).对每个z∈Rd,U(·)∈￡乙(B,B*,λ；R),h是Lft,y,z,Uφ=0的粘性下解。
第五章用一种新的方法证明了几个关于2-alternating容度的性质
(本章内容来自Jia-Zhang[59])
设Ω是一个基本集合,召是Ω上的σ-代数。我们称一个集函数c：B→[0,1]是一个容度如果它满足：
(C1).c(Ω)=1,c(Φ)=0；
(C2)(单调性).对任意A(?)B,A,B∈B,c(A)≤c(B)。
称容度μ为2-alternating,如果对任意A,B∈召,μ(A(?)B)+μ(A∩B)≤μ(A)+μ(B)。
称容度v为2-monotone,如果对任意4,B∈B,μ(A∪B)+μ(A∩B)≥μ(A)+μ(B)。称一个容度μ为概率测度,如果μ(A∪B)+μ(A∩B)=μ(A)+μ(B),我们以P来表示个概率测度。
根据Dcnnicbcrg [26]和Jia[54],简单推测可知如下结果成立。
定理5.2.1任意概率测度是2-alternating容度集合的极小元。反之,任意2-alternating容度集合的极小元都是概率测度。
定理5.2.2考虑空间(Ω,B)。B是Ω上的一个代数,且是一个有限集。c是定义在B上的一个2-alternating容度。取F1,...,Fn∈B使得F1(?)F2(?)...(?)Fn。则存在一个概率测度P,对任意i=1,...,n,P(Fi)=c(Fi)且P≤c。
定理5.2.3考虑空间(Ω,B)。召是Ω上的一个代数,且是一个有限集。μ是定义在B上的一个2-alternating容度,v是定义在B上的一个2-monotone容度。则μ≥v意味着存在一个概率测度P,使得μ≥P≥v。
Denneberg和Jia对上述相近问题从期望的角度上进行了证明,应用的分别是Choquet-期望和一般的次线性期望。本章中我们将完全从容度的角度上来证明上述结果。证明过程的关键是通过B上的集合A对一个2-alternating容度c进行如下变换文中证明了cA仍然是一个2-alternating容度,且cA≤c。这种变换还具有很多其他的很好的性质。特别的,对后两个定理的证明过程中,我们设计了一个循环程序,最终,通过构造的方法得出一个符合条件的概率测度。
During the research on stochastic control,in1973,Bismut first introduced the notion of backward stochastic differential equations(BSDEs,for short).However,until the generalized BSDEs were proposed by Pardoux-Peng[90]in1990,the BSDE theory just started its booming development.The BSDE is of the folloWing forms: where ξrepresents the terminal random variable,T>0is a fixed terminal time,(Wt)t∈[0,T] represents the d-dimensional Brownian motion and g is called the generator of the BSDE.The solution of BSDE is a pair of adapted processes(Y,Z.)that makes the above equality hold.Pardoux-Peng[90]assumed that g is Lipschitz continuous about y,z,and thereafter,a lot of papers were devoted to the relaxation of the conditions on g. Papers on local Lipschitz condition,for example,Bahlali[1] etc.;on continu-ous condition,Lepeltier-San Martin[73],Matoussi[80]etc.;on uniformly continuous condition,Jia[53,56],Fan-Jiang[36],Hamadene[39] etc.;on quadratic growth condi-tion in z,Kobylanski[68],Briand-Hu[9,10],Tevzadze[117],Briand-Elie[7],Delbaen-Hu-Richou[23],Richou[104]etc.;on super quadratic growth in z,Delbaen-Hu-Bao[22], Richou[104]etc.;on polynomial growth in y,Briand-Carmona[4]etc.;on discontinuous condition,Jia[52],N'zi-Owo[83],Halidias-Kloeden[38]etc.In addition.there are many other forms of BSDEs,for example,BSDE with reflection,coupled FBSDE,BSDE driven by Brownian motion and Poisson process,or even BSDE driven by a general martingale etc.,see Tang-Li[116],Situ[112,113],Yin-Situ[121],Ren-Otmani[103],Shen-Elliot [111],Ouknine[85],Hamadene-Ouknine[40],Essaky-Ouknine-Harraj[35],Morlais[81], Essaky[33],Bahlali-Essaky[2],Otmani[28],Otmani[29],Lejay[72]etc.There are also numerous papers studying the properties of solutions to BSDEs, for example the com-parison theorem, converse comparison theorem, convexity and translation invariance etc. Moreover, Peng [97,100] introduced g-expectation and deduced many properties about it. Papers on this part, for example, Briand-Coquet-Hu-Memin-Peng [6], Coquet-Hu-Memin-Peng [19], Jiang [64,65,66], Hu-Tang [48], Hu-Peng [47], Royer [108], Jia-Peng [58], Jia [55] etc. BSDE theory has wide range of application, for example, mathemat-ical finance, stochastic optimal control theory, game theory and nonlinear Feynman-Kac formula etc. Papers on finance and stochastic control, for example, Peng [96,98], Chen-Li-Zhou [16], Liu-Peng [76], Wu-Yu [120], Kohlmann-Zhou [70], Chen-Epstein [14], while papers on nonlinear Feynman-Kac formula, for example, Peng [94],Peng [98], Pardoux-Peng [91], El Karoui-Kapoudjian-Pardoux-Peng-Quenez [31], El Karoui [30], Barles-Buckdahn-Pardoux [3], Buckdahn-Hu [12], Briand-Hu [8], Pardoux [87], Padoux-Tang [93], El Karoui-Peng-Quenez [32], Kobylanski [68], N'zi-Ouknine-Sulem [82] etc.
As a matter of convenience, we denote equation (2) by (g,T, ξ), and its solution by (Ytg,T,ξ,Ztg,Tξ)t∈[0,T]. Moreover, we denote Ytg,T,ξ by Et,Tg[ξ]. Assume that for all (y,z)∈R÷Rd, g(·,y,z) is square integrable. Tr[·] denotes the trace of a symmetric matrix. Gi denotes the i-th row of matrix G.
We list the contents of this thesis as follows:
Chapter1Introduction;
Chapter2Invariant representation for second order stochastic differential operator by BSDEs with uniformly continuous coefficients and its applications in nonlinear semi-groups;
Chapter3Quadratic g-convex functions, C-convex functions and their relationships;
Chapter4Representation for second order integral-differential operator by BSDEs with jumps and properties of related f-convex functions;
Chapter5New proofs about several properties of capacities.
We now introduce the main results of each chapter.
(I) In Chapter2we prove that one kind of second order stochastic differential operator can be represented by a sequence of the solutions of FB-SDEs with uniformly continuous coefficients; and define nonlinear semigroups via decoupled FBSDEs; we prove the monotonicity and order-preservation of the nonlinear semigroups; and obtain a new comparison result for coupled FBSDE.
In this chapter, we suppose that dimensions of SDE and BSDE, are n and1, re-spectively. Let the coefficients bΩ x [0, T] x Rn→Rn and σ:Ω x [0, T] x Rn→Rnxd satisfy the following assumptions:
·(Uniformly Lipschitz Continuity):|b(t,x1)-b(t,x2)|+|σ(t,X1)-σ(t,X2)|≤Kb|x1-x2|,(?)x1,x2∈Rn;
·(lincar growth condition):|b(t,x)|+|σ(t,x)|≤Kb(1+|x|),(?)x∈Rn, where Kb>0is a fixed constant. For simplicity, we denote the condition by "forward standard Lipschitz condition". Let (Xst,x)s∈[0,T] be the strong solution of the following SDE:
Under the condition that g is uniformly continuous and of linear growth in (y,z), Lepeltier-San Martin [73] proved that there exits at least one solution. However, the solution may be not unique. For example, suppose g(y)=(?)|y|, T=1and ξ=0.In fact, there are uncountably many solutions of equation ((?)|y|,1,0). The first conclusion of this chapter shows that, no matter whether the solution is unique or not, we can always represent one kind of second order stochastic differential operator by the solutions of the SDEs and BSDEs. It can be specified as follows:
Theorem2.2.1[Representation Theorem I] Let g be uniformly continuous in (y, z) and continuous in t. Suppose b, σ satisfy the forward standard Lipschitz assump-tion. Assume that φ(t,x) E C1,2([0, T] x Rn). Moreover, we suppose that φ and its first and second order derivatives are all of at most polynomial growth. Then for each x∈Rn,t∈[0,T), we have where
Note that the solutions are arbitrarily chosen in this conclusion. With the help of this representation theorem, we can get a converse comparison theorem for g. Moreover, we can alos deduce some equivalent relationships between the properties of the solutions and those of the generator,for example,linearity, constant preservation,etc.
Then we define two kinds of nonlinear semgroups according to the decoupled FB-SDEs. The new semigroups generalize the Markovian semigroups generated by the Markovian diffusion processs. It is known that,Markovian semigroup is deduced by taking linear expectation over Markovian diffusion process,while the nonlinear semi-groups we defined are obtained through taking nonlinear expectation(g-expectation) over the diffusion processes.Representation Theorem I mentioned above just shows the infinitesimal generators of both the two nonlinear semigroups.
Suppose that b and σ depend on x only and are Lipschitz continuous in x.Further-more,we assume that g depends on(y,z)only.Therefore,the terminal time T can be any positive number.We denote by H the set of all real valued the continnuous functions defined on Rn,which are of at most polynomial growth and by Hb the set of real valued uniformly continuous functions defined on Rn.According to different conditions of g,we have two ways defining the semigroups.
First,when g is Lipschitz continuous in(y,z),for each f∈H,we define εtf(x):=y0g,t,f(xt0,x).Since εt∈H,we can define a binary operation over the operators by (εtoεs)f=εt(εsf).Then we have εtoεs=εsoεt=εt+s.Thus,{(εt)t≥0;o}forms a commutative semigroup.
Second,when g is uniformly continuous in(y,z)and both b and σ are bounded, according to Ma-Zhang[77],we define the so called "Nodal set" as follows:for any f∈Hb,
O(t,x,f,s):={y:there exists a solution y.g,s,f(Xst,x)such thatytg,s,f(Xst,x)=y} It is proved by Ma-Zhang that O(t,x,f,s)is a bounded interval.Thus we have O(t,x,f,s)=[u(t,x,f,s),u(t,x,f,s)]. Define εt,sf(x):=u(t,x,f,s)and εt,sf(x):=u(t,x,f,s).Since g,b,σ are all indepen-dent of t,εt,s f(x)≡ε0,s-tf(x).For simplicity,we denote εtf(x)=ε0,tf(x).Similarly εtf(x)=ε0,tf(x)=εr+trf(x),for all r≥0,t≥0.For fixed t,εtf and εtf are all bounded and uniformly continuous functions.Similar to the case of Lipschitz continuity, we can define binary operation "o" on Hb.It can be deduced that{(εt)t≥0lo} and {(εt)t≥;o}both constitute nonlinear semigroups.
We can see that,if g is uniformly continuous in y,z,the infinitesimal generator may not be one to one correspondent to semigroup. It is possible that several semigroups enjoy the same infinitesimal generator.
With the above semigroup, we are to discuss the monotonicity and order-preservation of the semigroup when g is Lipschitz continuous. In fact, Herbst-Pitt[43], Chen-Wang [17] deduced the necessary and sufficient conditions for a Markov semigroup to be mono-tone and for two semigroups to preserve order. However, their proof depends on the linearity of Markov semigroup heavily, and thus, can not be used here. To deduce these results, we have to find another way. We now introduce the following definition:
Definition2.5.1Suppose that "≤" denotes the usual semi-order in Rn.
(i). A measurable function f:Rn→R is called monotone, if for all x≤x, f(x)≤f(x). Denote by M the set of continuous monotone functions with at most polynomial growth. Then M∈H.
(ii). For two semigroups{εt}t≥0and {εt}t≥0, we write εt≥εt if for all f∈M, x≥x and t≥0, εt f(x)≥f(x). If in addition εt=εt, we callεt monotone.
The following two theorems hold.
Theorem2.5.1Suppose b, σ rely on x only and are Lipschitz continuous in x, g relies on (y, z) only and is Lipschitz continuous in (y, z). Then εt defined on H is monotone if and only if the following conditions hold:(2C1-i). bi(x)-bi(x)≥0for all x,x∈Rn with xi=Xi and xk≥xk for k≠i, i=1,...,n;(2C1-ii). σi depends on xi, only, for all i=1,..., n.
Note that, the monotonicity of Et has nothing to do with g.
Theorem2.5.3Assume n=d. Suppose b,σ,b,σ rely on x only and are Lipschitz continuous in x, g,g rely on (y,z) only and are Lipschitz continuous in (y.z). Suppose σσ*(or σσ*) is uniformly positive definite and b, b, σ, σ are all bounded. If one of εt and εt is monotone, then εt≥εt if and only if (2C3-i) σσ*=σσ*and both σi(x) and σi(x) depend only on xi;(2C3-ii) for all x∈Rn, y∈R, K∈Rn, K≥0, K*b(x)+g(y,σ*(x)K)≥K*b(x)+g(y,σ*(x)K).
By the one to one correspondence of the semigroup and the PDE, the above mono-tonicity and order-preservation theorem also hold for the PDEs. During the proof, we note that, a semigroup is linked to a unique PDE, but not necessarily to a unique pair of FBSDE.
In particular, we take g, g So we have the following result.
Theorem2.5.4Suppose b, σ and b, σ rely on x only and are Lipschitz continuous in x, and moreover that all the elements in the same column of (σ)n×d have the same sign, which can differ according to different x. Let (2C1-i) and (2C1-ii) hold for b, σ and b, σ. Then εt≥εt if and only if (2C3-i) and (2C3-ii) hold. Here (2C3-ii) means that for all i, whenever x≥x with xi=xi.
This result can not be covered by Theorem2.5.3, since we don't need the assump-tions of n=d and the positive definiteness of σσ*. According to this theorem, we can partly turn a special kind of second order quasilinear parabolic PDE into second order linear PDE.
Remark2.5.8Consider the following second order quasilinear PDE: with the coefficients satisfying the same conditions as those stated in Theorem2.5.4. We have the following results:if f is a nondecreasing function with at most polynomial growth, the above PDE is equivalent to while if f is a nonincreasing function with at most polynomial growth, the above PDE is equivalent to
Inspired by the proof for the order-preservation of semigroup, we deduce a new comparison theorem for FBSDEs, which connects the drift of SDE and the generator of BSDE for the first time.
We consider the following FBSDE where (Xt, Yt, Zt, Wt)∈Rn×R×Rd×Rd, with the dimensions of b, σ, g and f defined accordingly.
Here are the assumptions (mainly come from [77]): the coefficients (b,σ,g,f) are measurable and bounded; σσ*are uniformly positive defi-nite; b, σ, g, f are all smooth with bounded first and second order derivatives.
Theorem2.6.1Assume bi,σi,gi,fi (i=1,2) satisfy the above conditions. Suppose that the initial values are x1,x2respectively. Denote the solution by (Xi,Yi,Zi). If
(i)σ1(σ１)*≡σ2(σ2)',
(ii) p*b1(t, x, y,(σ1)*(t, x)p)+g1(t, x, y,(σ1)*(t, x)p)≥P*b2(t, x, y,(σ2)*(t,x)p)+g2(t,x,y,(σ2)*(t,x)p), for all t∈[0,T], x∈Rn, y∈R, p∈Rn,
(iii) f1≥f2,
(iv) x1=x2, then Y01≥Y02.
(II) In Chapter3we study the representation theorem for quadratic BSDEs; and obtain a necessary and sufficient condition for the Jensen's inequality to hold under quadratic g-expectation; we define a kind of C-affine functions for a fixed constant C, then define C-convex functions and C-concave functions; we prove the properties of C-convexity (resp. C-concavity); furthermore, we investigate the relationships between C-convex (resp. con-cave, affine) functions and quadratic g-convex (resp. concave, affine) func-tions.
Jensen's inequality plays an important role in classical probability theory. Jia-Pcng [58] firstly defined g-convex function as the function that satisfies the Jensen's inequality under g-expectation, i.e., h is called a g-convex (resp. concave) function, if for any FT-measurable, square integrable random variable X such that h(X) is also square integrable, and for any t∈[0,T], If h is both a g-convex function and a g-concave function, then h is a g-affine function. Jia-Peng also deduced a simple necessary and sufficient condition for a smooth function h to be g-convex, i.e., where Lgt,y,zφ:=1/2φ"(y)|z|2+g(t,φ(y),φ'(y)z)-φ'(y)g(t,y,z). For a general continuous function h of at most polynomial growth, it is g-convex if and only if h is a viscosity subsolution of the following PDE: For more details about viscosity solutions, the reader can refer to [20] etc. An interesting result is, when g is Lipschitz continuous in y, z,g-convex function is convex in the usual sense.
In this chapter, we assume that g is of quadratic growth in z and study the properties of g-convex function under this assumption (called quadratic g-convex function). The quadratic BSDE has been well studied since Kobylanski introduced the quadratic BSDE in [68]. However, as we known, the representation theorem and Jensen's inequality under quadratic growth condition have not been well studied. In this chapter, we will focus on these problems.
One of the major tasks in this chapter is to deduce the necessary and sufficient con-dition for a function to be quadratic g-convex. Because of the contradiction between the boundedness of the terminal condition and the unboundedness of the forward equation introduced in the proof, the proof is nontrivial. To solve this problem, we introduce stopping time and optional stopping time theorem which play an important role in the proof.
What's interesting about quadratic g-convex function is that it may not be convex. For a fixed constant C, we consider the following ODE Solve this ODE and define its solutions (which are not linear usually) as new "affinc functions", i.e., C-affine functions. Then we introduce the notion of C-convex func-tions. This kind of functions has many good properties similar to the classical convex functions, such as, continuity, quasi-convexity. Moreover, each C-convex function can be represented as the upper envelope of a family of C-affine functions. It is worthy to mention that, if C=0, the framework of C-convexity is just the classical one.
In this chapter, we make the following assumptions on g:
· there exit two constants Ky>0and Kz>0, such that (?)(t, y, z, y', z'),
· g(·,0,0) is uniformly bounded.
For simplicity, we call this assumption standard quadratic condition.
First, as before, we prove representation theorem with stopping time for g under the standard quadratic condition.
Theorem3.3.1Let g satisfy standard quadratic condition,b, σ satisfy the for-ward standard Lipschitz condition. Assume moreover that g, b, a are all continuous in t. Suppose that φ∈C1,2(R+x Rn). Then for each (t,x)∈[0,T) x R",nthe following holds: where Lg,b,σt,x is defined as above, τε:=τΛ (t+ε) for ε small enough and τ is a stopping time such that Xt,x is bounded on [t, τ]. For example,(K0is a positive number and is large enough).
From now on, in this chapter, we always assume that g(t, y.0)=0, and g satisfies standard quadratic condition.
First, we have the following definition. Definition3.4.1For a given Eg[·] a function h:R→R is said to be g-convex (resp. g-concave) under bounded terminal conditions, if for each X∈L∞(FT), one has h is called g-affine function if it is both g-convex and g-concave.
Definition3.4.3(g-convexity on a convex set) Suppose that O is a convex subset of R. For a given E9[·], real valued function h is said to be g-convex (rcsp. g-concave) on O, if for each X∈L∞(FT) such that E9s,T[X](ω)∈A, dP x dt-a.s., one has h is called g-affine if it is both g-convex and g-concave. We have the following two theorems.
Theorem3.4.1Suppose h∈C2(R), the following two statements are equivalent:
(i) h is g-convex (resp.g-concave) under bounded terminal conditions;
(ii) for each y∈R, z∈Rd,
Theorem3.4.2Suppose g is independent of ω and is continuous in t. h is a continuous function. Then the following statements are equivalent:
(i) for each (t,z)∈[0,T] x Rd, h is a viscosity subsolution of Lgt,y,zh=0;
(ii) h is g-convex under bounded terminal conditions.
We now choose a kind of g which are typical and study the properties of corre-sponding g-convex functions. Suppose g=C|z|2+gi(t,y,z), where lim|z|→∞|g1(t,y,z)|/|z|2=0and C∈R.
If C≡0, each g-convex function is convex and each g-affine function is linear. Suppose C≠0, thus any g-affine function φ∈C2(R) satisfies the following equality Solve the above ODE and denote by ΠC the set of solutions We call these functions C-affine. With the help of this kind of function, we can define the following C-convex function:
Definition3.6.1(C-convex function) A real valued function f defined on a convex set D R is called C-convex function, if for any φ∈ΠC such that there exist two different points of intersection x1 Many good properties hold for C-convex functions, for example, continuity, quasi-convexity, almost everywhere differentiability, the existence of the left and right deriva-tives, etc. Moreover, we can deduce that the left derivative is no lager than the right derivative at any point on D.
In addition, we can define C-convex set as follows.
Definition3.6.7(C-convexity on R2) A set A R2is called a C-convex set, if for any two points (x1,y1),(x2, y2)∈A, x1 Proposition3.6.9If f is a C-convex function, epi f={(x,y):f(x) Theorem3.6.2and Theorem3.6.4Any C-convex function can be represented as the upper envelope of a family of C-afine functions. On the other hand, the upper envelope of a family of C-affine functions is a C-convex function.
Now we consider the relationships between the C-convex functions and g-convex functions. We have the following results.
Theorem3.6.5Any C-convex function is a g-convex function with g=C|z|2+ and for g=C|z|2+, any g-convex function with domain D is a C-convex function on D.
Theorem3.6.6Suppose that g=C|z|2+g1(t, y, z), such that and C∈R. Thus all the g-convex (resp. concave, affine) functions are C-convex (resp. concave, affine) functions.
Theorem3.6.7Suppose I is an index set and{fi:i∈I} is a family of g-convex functions. Thus f(x)=sup{fi(x):i∈I} is also a g-convex function.
Theorem3.6.8Suppose thus the nec-essary condition for a smooth g-convex function h to be represented as the upper envelope of a family of g-affine functions is, for any (t,y,z), g1(t, h(y),h'(y)z)-h'(y)g1(t,y,z)=0. In particular, if C=0, g is independent of y, then the above condition is also a sufficient condition.
(Ⅲ) In Chapter4we represent the second order stochastic integral-differential operator by a sequence of FBSDEs with jumps; and obtain a converse comparison theorem and some equivalent properties of BSDE with jumps; we define the f-convex function under BSDE with jumps, and deduce the necessary and sufficient condition for a function to be f-convex.
In this chapter, we always assume that b, σ satisfy the forward standard Lipschitz condition, f(ω,t,x,y,z,U):Ω×[0,T]×Rn×R×Rd×L2(B,B*,λ;R)→R is uniformly Lipschitz continuous in (y, z, U) and with at most polynomial growth in x. For given [t, x)∈[0, T) x Rn, we denote by Xt,x the solution of the following SDE: and introduce the following stochastic integral-differential operator:
The meaning of the integral will be explained in the text of Chapter4.
In this chapter, the representation for the above operator by the solutions of FBSDE with jumps will be given. Here are the two main results.
Theorem4.2.1(Representation Theorem Ⅰ) Suppose that f,b,σ are right con-tinuous in t. Suppose that φ has bounded three order derivatives. For any1≤p≤2and we have
Theorem4.2.2(Representation Theorem Ⅱ) Suppose that f, b, σ satisfy the same conditions as those in Theorem4.2.1. Assume that φ∈C2. Thus for each1≤p≤2, and we have where τε:=σ Λ (ι+ε), τ is a stopping time, such that Xt,x is bounded on [t,τ). For example,we can take τ:=inf{s> t:|Xst,x-x|>N}.
The meaning of the integral in Lf,b,σ,Ut,L2(B,B*,λ;Rn) and L∞2(B, B*, λ;Rn) will be explained in Chapter4.
With the help of the representation theorems, we can deduce the following converse comparison theorem.
Theorem4.3.1(Converse Comparison Theorem Ⅰ) Suppose that f1,f2are all independent of x. Moreover,(?)(y, z, U), both f1and f2are all right continuous in t∈[0,T) and right continuous in T, P-a.s.. For any s∈[0,T], ξ∈L2(FS), we have Thus for any (t, y, z, U(·))∈[0,T]×R×Rd×L2(B,B,λ;R), we have
Theorem4.3.2(Converse Comparison Theorem Ⅱ) Suppose that f1and f2satisfy the same conditions in Converse Comparison Theorem I. Moreover, f(t, y,0,0)≡0. If for any ξ∈L2(FT), we have
Suppose that f is independent of x and f(t,y,0,0)≡0for all (t,y)∈[0,T] x R. Similarly to g-expectation, Royer [108] introduced f-expectation by solutions of BSDE with jumps. Here is the definition of f-convex function. Definition4.4.2For a given f-expectation Ef[·], a function h:R→R is called f-convex (resp.f-concave) function, if for each X∈L2(FT), such that h(X)∈L2(FT), we have h is called f-affine function if it is both f-convex and f-concave.
We now introduce the following notations
Theorem4.4.1Suppose that f(t,y,z,U) satisfy uniform Lipschitz condition, and f(t,y,0,0)≡0, h∈C2(R), thus the following two statements are equivalent:
(i). h is a f-convex function (resp.,f-concave function);
(ii). for each t∈[0,T], y∈R, z∈Rd, U(·)∈L∞2,(B,B*, λ;R),
Theorem4.4.4Suppose h∈C(R) is of at most polynomial growth, f(y, z, U) satisfies some more strengthen condition, thus the following two statements are equivalent:
(i). h is f-convex function (resp.f-concave);
(ii). for each z∈Rd, U(·)∈L∞2, B*, λ; R)1, h is a viscosity subsolution of PDE Lf t,y,z,Uφ=0.
(Ⅳ) In Chapter5we prove several properties of2-alternating capaci-ties
Let Ω be a basic set, B is a σ-algebra on Ω. A set function c:B→[0,1] is called a capacity, if it satisfies:
(C1). c(Ω)=1,c((?))=0;
(C2)(monotonicity). for all AC B,A,B∈B, c(A) According to Denneberg [26] and Jia [54], it can be deduced easily that the following three results hold:
Theorem5.2.1Any probability measure is a minimal member of the set of2-alternating capacities. On the other hand, any minimal member of the2-alternating capacities is probability measure.
Theorem5.2.2Suppose B defined on Ω is finite and that c is a2-alternating capacity. Take F1,..., Fn∈B such that F1∈F2∈...∈Fn. Then there exists a probability measure P such that P(Fi)=c(Fi) for i=1,..., n and P≤c.
Theorem5.2.3Suppose B defined on Ω is finite. Let μ be a2-monotone capacity and v be a2-alternating capacity. The μ≥v implies that there exists a probability measure P such that μ≥P≥v.
Denneberg and Jia prove similar results by ways of expectations. They mainly use Choquet expectation and general sublinear expectation respectively. In this chapter, we'll prove the above results by capacity only. The key of the proof is to transform a2-alternating capacity c to a new capacity μA, by the subset A∈B. The transformation is as follows:
It is proved that cA is still a2-alternating capacity and cA≤c. Moreover, there are many other interesting properties. In particular, we can create a probability measure by a2-alternating capacity.
BSDE的一般形式如下：其中(?)表示终端随机变量,T>0是固定的终端时刻,(Wt)t∈[0,T]表示d-维布朗运动,g是该BSDE的生成元。求BSDE的解是对固定的(?),T,g,求一对适应过程(Y.,Z.)使得上式成立。Pardoux-Peng[90]给出的假设是9关于y,z李普希兹连续,此后有很多研究致力于将该条件弱化,如改为局部李普希兹条件(Bahlali[1]等),连续性条件(Lepeltier-San Martin [73],Matoussi[80]等),一致连续性条件(Jia[53,56],Fan-Jiang[36],Hamadene[39]等),关于z平方增长(Kobylanski[68], Briand-Hu[9,10],Tevzadze[117], Briand-Elie[7], Dclbaen-Hu-Richou[23],Richou[104]等),关于z超平方增长(Delbaen-Hu-Bao[22], Richou[104]等),关于y多项式增长(Briand-Carmona[4]等),关于y超线性增长,关于z平方增长(Kobylanski-Lepeltier-Quenez-Torres[69]等),不连续(Jia[52],N'zi-Owo[83], Halidias-Kloeden[38]等)以及一些其他的条件(Royer[107]等)。另外,也出现了多种形式的BSDE,比如,带反射的、正倒向耦合的、驱动为相互独立的布朗运动与泊松跳过程,或者更一般的1evy过程、甚至一般的鞅等等,相关文献如,Tang-Li[116],Situ[112,113], Yin-Situ[121],Ouknine[85],Essaky-Ouknine-Harraj[35],Ren-Otmani[103],Morlais[81], Lejay[72],Bahlali-Essaky[2],Hamadene-Ouknine[40],Hassani-Ouknine[41],Essaky[33]等。另一个分支是对BSDE解的性质的研究,比如比较定理、逆比较定理、凸性、平移不变性等等,如Pcng[97,100]引入的g-期望的概念及其性质,参考文献如Briand-Coquet-Hu-Memin-Peng[6],Coquet-Hu-Memin-Peng[19],Jiang[64,65,66],Hu-Tang[48],Hu-Peng[47],Royer[108],Jia-Peng[58],Jia[55]等。BSDE在很多重要领域得到了应用,如经济、控制、金融等领域,具体说来,如随机最优控制、博弈理论、数理金融等等。相关文献如Peng[96,98],Wu-Yu[120],Chen-Li-Zhou[16],Liu-Pcng[76],Kohlmanm-Zhou[70],chen-Epstein[14]等。BSDE的另一方面重要应用是,提供了若干类二阶偏微分方程的概率解释,也即非线性Feynman-Kac公式,具体可参见Peng[94],Peng[98],Pardoux-Peng[91],E1Karoui[30],Barles-Buckdahn-Pardoux[3],Buckdahn-Hu[12],Briand-Hu[8],El Karoui-Kapoudjian-Pardoux-Peng-Qucnez[31],Pardoux[87],Padoux-Tang[93],Kobylanski[68], El Karoui-Peng-Quenez[32],N'zi-Ouknine-Sulem[82]等。
为叙述方便,我们将方程(1)记为(g,T,ξ),其解记为(Ytg,T,ξ,Ztg,τξ)t∈[0,T]另外将Ytg,T,ξ记为Et,τg[ξ]。假设对任意(y,z)∈R×Rd,g(·,y,z)是平方可积的。Tr[·]表示对称矩阵的迹。Gi表示矩阵G的第i行。
下面是本文的章节目录：
一、第一章引言；
二、第二章一致连续系数BSDE对二阶随机微分算子的不变表示及其在非线性半群上的应用；
三、第三章平方增长g-凸函数,C-凸函数及其相关关系；
四、第四章由带跳的BSDE的解来表示的随机积分-微分算子及f-凸函数的性质；
五、第五章关于容度的几个性质的新证明。
第二章证明了一类二阶随机微分算子可以表示为—(?)(?)FBSDE的解的极限,其中倒向方程的生成元只需一致连续；由非耦合的正倒向方程定义非线性半群,其无穷小生成元就是如上表示的二阶随机微分算子;证明了非线性半群的单调性与保序性；证明了耦合的FBSDE的一个比较性质。
(本章内容主要来自两篇文章Jia-Zhang[60],Zhang-Jia[123])
在本章中,我们总假设SDE是n维的,BSDE是1维的。
在g关于(y,z)一致连续且线性增长的条件下,Lepeltier-San Martin[73]证明了方程的解的存在性,然而解未必是唯一的,Jia-Peng[57]证明了此时解或者唯一或者有不可数多个,不唯一的例子,如g(y)=(?)|y|,T=1,ξ=0时,方程((?)|y|,1,0)的解就有无穷多个。本章第一个结论说明,不论解是否唯一,我们总是可以拿来与SDE的解一起,共同表示一类二阶随机微分算子。
首先引入下列关于SDE的假设。
假设b：Ω×[0,T]×Rn→Rn,σ：Ω×[0,T]×Rn→Rn×d是满足下列条件的函数：
(一致李普希兹条件)：|b(t,x1)-b(t,x2)|+|σ(t,x1)-σ(t,x2)|≤Kb|x1-x2|,(?)x1,x2∈Rn：
(线性增长条件)：|6(t,x)|+|σ(t,x)|≤Kb(1+|x|),(?)x∈Rn；其中Kb>0为一个常数。为简单起见,我们记此条件为正向的标准李普希兹条件,以区分9满足的标准李普希兹条件(即,g关于y,z一致李普希兹连续)。
设(Xst,x)s∈[0,T]为下列SDE的强解
其具体内容如下
定理2.2.1(表示定理Ⅰ)令g满足一致连续性条件,且关于t连续,6,σ满足以上假设且关于t连续。若φ∈C1,2([0,T]×Rn)且φ本身及其各阶导数关于x至多多项式增长,那么对每个x∈Rn,t∈[0,T),如下式子成立其中注意到此结论中解的选取是任意的。由此可以得到g在此条件下的一个逆比较定理以及一些BSDE的解的性质与生成元的性质之间的等价对应关系,比如,线性性、保常性。
接下来我们根据正倒向方程定义了两种非线性半群,这两种非线性半群是对由马尔科夫扩散过程生成的马氏半群的一种推广,马氏半群是给扩散过程取线性期望得到的,而此处的半群是对扩散过程取非线性期望(g-期望)得到的。上面的表示定理Ⅰ恰好给出了这两种半群的无穷小生成元。
在定义半群时,我们首先假定6,σ只依赖于x且关于x李普希兹连续,9只依赖于(y,z),故终端时刻T可以取值为任意正数。记H为Rn→R中至多多项式增长的连续函数的全体,Hb为Rn→R中有界一致连续函数的全体。根据g满足条件的不同,分为如下两种定义方式：
首先,当g关于(y,z)满足李普希兹条件时,通过εtf(x)：=y0g,t,f(Xt0,x)在H上定义族算子{εt；t≥0},并定义运算(εt(?)εs)f=εt(εsf)。则有εt(?)εs=εs(?)εt=εt+s。故{(εt)t≥0;(?)}构成一个交换半群。
其次,当g关于(y,z)一致连续,b,σ有界时,类似于Ma-Zhang2011年的文章[77],可以如下定义“节点集”0(t,x,f,s),
(?)(t,x,f.s)：={y：存在一个解y.g,s,f(Xst,x)满足ytg,s,f(Xst,y},其中f∈Hb。Ma-Zhang还证明了(?)(t,x,f,s)是一个有界区间,分别以u(t,x,f,s)和u(t,x,f,s)记其上界和下界,则有定义εt,sf(x)：=u(t,x,f,s)和εt,sf(x)：=u(t,x,f,s)。由于此时我们假设的g,b,σ都独立于时间t,故εt,sf(x)≡ε0,s-tf(x),可以简记εtf(x)=ε0,tf(x)。类似的,有εtf(x)=ε0,tf(x)=εr,t+τ f(x),对任意r≥0,t≥0。对固定t,εtf(x)和εtf(x)都是有界的一致连续函数,即属于Hb.同李普希兹情形,我们可以在两种算子上分别定义运算“。”,可以推出{(εt)t≥0；(?)}和{(εt)t≥0；(?)}都构成非线性半群。
由此可以看出,当9只是一致连续时,无穷小生成元与半群并不能完全一一对应,此时,可能有多个半群对应于同一个无穷小生成元。
有了以上的半群,我们下一问题着重讨论了当g满足李普希兹条件时半群的单调性质和比较性质。实际上,Herbst-Pitt[43]和Chen-Wang[17]证明了马氏半群单调的充要条件,以及两个马氏半群可以比较的充要条件。但是,他们的证明强烈依赖于马氏半群的线性性,所用的方法不能直接拿来用,所以为了得到这些结果,我们需要另辟蹊径。首先引入下面定义。
定义2.5.1令“≤”表示Rn中通常的半序。
(i).我们称一个定义在肿中取值于R的可测函数f是单调的,如果f(x)≤.f(x)对所有x≤x.我们以M来表示至多多项式增长的连续单调函数的集合。显然M(?)H。
(ii).对两个半群{εt}t≥0和{εt}t≥0,我们称εt≥εt,如果对所有f∈M,以及所有x≥x和t≥0,有εtf(x)≥εtf(x).另外,如果εt=εt,我们称εt单调。
我们证明了下面的两个定理成立。
定理2.5.1假定6,σ只依赖于x且关于x李普希兹连续,9只依赖于(y,z),且关于(y,z)李普希兹连续。则定义在H上的εt是单调的,当且仅当：
(2C1-i).对任意的i=1,...,n,若是x,x∈Rn,xi=xi,且对任意k≠i,xk≥xk,有bi(x)-bi(x)≥0成立；
(2C1-ii).对所有i=1,...,n,σ的第i行只依赖于xi。
由此定理,我们发现εt的单调性与g无关。
定理2.5.3假定n=d,假定6,σ,b,厅只依赖于x且关于x李普希兹连续,g,g只依赖于(y,z),且关于(y,z)李普希兹连续。σσ*(或者σσ*)是一致正定的,且b,b,σ,厅都有界。
如果εt或者εt是单调的,那么εt≥εt当且仅当
(2C3-i)σσ*≡σσ*并且σ和σ的第i行均只依赖于xi；
(2C3-ii)对所有x∈Rn,y∈R,K∈Rn且K≥0,
注意到第二个定理首次将SDE的漂移项系数与BSDE的生成元结合起来作为一个整体进行比较。由于半群对应唯一确定的PDE,如上的单调性定理以及保序性定理实际上可以转化成二阶拟线性抛物方程的单调性与保序性。在证明的过程中,我们还可以看到,在李普希兹条件下,半群对应唯一确定的PDE,但是未必对应唯一确定的正倒向方程。
特别的,取g,g：Rd→R,kj,pj,kj,pj∈R,(?)j=1,...,d。我们有以下结果。
定理2.5.4设b,σ,b,厅只依赖于x且关于x李普希兹连续,且条件(2C1-i)和(2C1-ii)对b,σ和b,厅成立,(σ)n×d的同一列中的元素符号相同,其符号可随x的不同而不同,σ的符号也具有这种性质。。那么εt≥εt当且仅当(2C3-i)和(2C3-ii)成立。此处(2C3-ii)意味着对任意i,只要x≥x,xi=xi,
这个结果不能被定理2.5.3所涵盖,因为此时我们并不假定n=d,以及σσ*一致正定。根据这个定理,我们可以将一类特殊的二阶拟线性抛物PDE部分的转化成二阶线性PDE。
注2.5.8考虑以下二阶拟线性PDE：其系数的条件同定理2.5.4,则我们有如下结论：如果f是至多多项式增长的非降连续函数,那么上述PDE等价于如果f是至多多项式增长的非增连续函数,以上PDE等价于
本章最后一个结果,受到半群保序性定理证明方法的启发,得到一类新的FBSDE的比较定理。该定理同样将SDE的漂移项系数和BSDE的生成元结合起来,据我们所知,在FBSDE的比较定理中,这是首例。首先我们所考虑FBSDE如下此处(Xt,Yt,Zt,Wt)∈Rn×R×Rd×Rd,b,σ,g和f都具有相应的维数。
以下是关于此定理的假设(主要来自[77])：系数(b,σ,g,f)可测且有界；σσ*一致正定；b,σ,g,f光滑且一二阶导数有界。
定理2.6.1假定bi,σi,gi,jfi(i=1,2)满足如上条件。设这两对FBSDE的初值分别为x1,x2,记其解为(Xi,yi,Zi)。如果
(i)σ1(σ1)‘三σ2(σ2)*,
(ii)对任意t∈[0,T],x∈Rn：y∈R,p∈Rn,
(iii)f1≥f2,
(iv)x1=x2,
那么Y01≥Y02。
第三章证明了平方增长的倒向随机微分方程的表示定理；得到了一个函数满足平方增长g-期望下的Jensen不等式的充要条件；基于常数C定义了一类C-仿射函数,并以此定义了C-凸和C-凹；证明了C-凸(凹)函数的类似于普通凸(凹)函数的性质；研究了这类C-凸(凹,仿射)函数与平方增长g-凸(凹,仿射)函数之间的相互关系。
(本章主要内容来自Jia-Zhang[63])
经典的数学期望下的Jensen不等式是现代概率论中的一个基本的不等式。在g满足标准李普希兹条件时,Jia-Peng [58]于2010年,首次将满足g-期望下的Jensen不等式的函数定义为“凸函数”,提出了g-凸的概念,即：一个函数h称为g-凸(凹)函数,如果对任意FT可测平方可积且使h(X)平方可积的随机变量X,以及任意t∈[0,T],Et,Tg[h(x)]≥(相应地,≤)h(Et,Tg[X]).如果h既是g-凸函数又是g-凹函数,则称为g-仿射函数。Jia-Peng还给出了一个光滑函数h是g-凸函数的一个简单的充要条件,即,Lgt,y,zh≥0,其中而一般的多项式增长的连续函数h是9-凸函数的充要条件是,h是下列方程的粘性下解：关于粘性解的概念,读者可参考[20]等。关于这种g-凸函数的性质,一个很重要的结果是在.g满足标准李普希兹条件时,g-凸函数一定是通常意义下的凸函数。
本章的主要内容是考虑平方增长条件下g-凸函数的性质。自Kobylanski在2000年的文章[68]中引入平方增长BSDE以来,平方增长的BSDE已经得到了比较好的研究。然而,就我们所知,除了Ma-Yao[78]给出了一个条件比较强的关于生成元的表示定理以及一个简单的Jensen不等式的结论外,再没有其它人涉猎平方增长条件下的表示定理以及Jensen不等式。本章将对这些问题进行深入探讨。
本章的核心结果是,证明9-凸函数应该满足的充要条件。在此证明中遇到了困难,主要原因是终端条件有界的要求与证明过程中引入的正向方程的解通常无界之间的矛盾。为解决此问题,我们引入停时以及可选停时定理等,停时的灵活运用成为这个证明过程的关键,也贯穿了整个证明过程的始终。最让我们感兴趣的是,此时的g-凸函数已经不再是通常意义下的凸函数了。为此,根据一个比较简单的依赖于常数C的ODE,我们将其解(通常不是直线)定义为新的“仿射函数”,称为C-仿射函数,并用类似于普通凸的方法定义了C-凸函数。这个函数体系有非常好的基本类似于通常凸的性质。比如C-凸函数具有连续性,拟凸性,以及可以表示为一族C-仿射函数的上包络等。值得一提的是,若是C=0,此时的0-凸框架恰好对应于通常的凸的框架。这类C-凸函数对于平方增长g-凸的意义,一如凸函数对于李普希兹条件下g-凸的意义。
本章中我们对9作如下假设：
存在两个常数Ky>0和Kz>0使得,A(t,y,z,y’,z,),
g(·,0,0)一致有界。为简单起见,称此假设为标准平方增长条件。
首先,如前,我们证明了g满足标准平方增长条件时,带有停时的表示定理成立。
定理3.3.1令9满足标准平方增长条件,b,σ满足正向的标准李普希兹条件。另外假设g,b,σ都关于t连续。设φ∈C1,2(R+×Rn),那么对每个(tnx)∈[0,T)×Rn,下面的极限成立其中Lg,b,σt,x的定义如前所述,ε要求充分小,Τε=ΤΛ(t+ε),Τ是一个使得X.t,x在[t,τ]上有界的停时。例如,(K0是一个正数且充分大)。
接下来关于g-凸的研究,我们总假设g(t,y,0)≡0,且g满足标准平方增长条件。
首先,有如下9-凸函数的定义。
定义3.4.1一个函数h：R→R被称为终端有界条件下的g-凸(相应地,g-凹)函数,如果对每个X∈L∞(FT),有,对任意t∈[0,T],
h(Etg,T[X])≤Et,Tg[h(X)].(相应地,h(Et,Tg[X])≥Et,Tg[h(X)])P-a.s.h被称为g-仿射如果它既是g-凸又是g-凹。
注意到终端的有界性,我们可以定义凸集上的g-凸函数,这对后面我们研究这一类g-凸函数提供了很大的方便。
定义3.4.3(凸集上的g-凸性)假定(?)是R上的一个凸集。对一个固定Eg[·],实值函数h被称为0上的g-凸(相应地,g-凹)函数,如果对每个X∈L∞(FT)使得Es,Tg[X](ω)∈(?),dPdt-a.s.我们有(相应地,h(Et,Tg[X])≥Et,Tg[h(X)])P-α.s,t∈[0,T].h被称为(?)上的g-仿射函数如果它既是(?)上的g-凸函数又是(?)上的g-凹函数。
下面两个定理是本章的主要结果。
定理3.4.1令h∈C2(R),那么以下两个叙述等价：
(ⅰ)h是终端有界条件下的9-凸函数(相应地,g-凹函数)；
(ⅱ)对每个y∈R,z∈Rd,Lgt,y,zh≥0(相应地,≤0)dPdt-a.s.
定理3.4.2假定g独立于ω且关于t连续。h是一个连续函数。以下论断等价：
(ⅰ)对每个(t,z)∈[0,T]×Rd,h是Lgt,y,zh=0的粘性下解；
(ⅱ)h是终端有界条件下的g-凸函数。
下面取一类g进行仔细研究：g=C|z|2+g1(t,y,z),其中如果C三0,任意9-凸函数是通常意义下的凸函数,g-仿射函数恰好是所有的线性函数。我们设C≠0,取φ∈C2(R)是一个g-仿射函数,则有解以上ODE,以Ⅱc来表示所有解的集合：,或φ(x)=x+b,或φ(x)=b,Aa,b∈R}.我们定义这类函数为C-仿射函数,通过这类函数,类似于普通凸函数的定义,得到了如下C-凸函数：
定义3.6.1(C-凸函数)定义在凸集D上的函数f称为C-凸函数,如果对任意φ∈Πc与f相交于两点x1,x2(不妨设x1
此外,我们还定义了C-凸集。
定义3.6.7(R2中的C-凸集)一个集合A(?)R2被称为一个C-凸集,如果对任意两个点(x1,y1),(x2,y2)∈A,x1
C-凸函数还有如下重要性质。
定理3.6.2以及定理3.6.4任意C-凸函数可以表示为一族C-仿射函数的上包络,反之,任意一族C-仿射函数的上包络都是C-凸函数。
接着,我们讨论了C-凸函数与g-凸函数之间的关系。首先,特定的一族函数g,C-凸函数与与g-凸函数一一对应,具体如下。
定理3.6.5任意C-凸函数是一个9-凸函数,其中g=C|z|2+G1z,(?)C1∈Rd。反之,设g=C|z|2+
另外,对更一般的g,如下结果成立。
定理3.6.6假定g=C|z|2+g1(t,y,z),其中C∈R,于是任意9-凸(相应地,凹,仿射)函数是C-凸(相应地,凹,仿射)函数。
最后的两个定理,是关于9-凸函数与9-凸函数或g-仿射函数之间的关系。
定理3.6.7假定Ⅰ是一个指标集,{fi：i∈Ⅰ}是g-凸函数的一个集合。那么f(x)=sup{fi(x)：i∈Ⅰ}也是一个g-凸函数。
定理3.6.8假定其中C∈R,则光滑的g-凸函数h可表示为一族g-仿射函数的上包络的必要条件是,对任意(t,y,z)成立。特别地,若C=0,且g独立于y,则上述条件是充要条件。
第四章由一列带跳的正倒向随机微分方程的解来表示二阶随机积分-微分算子；得到了带跳的BSDE的逆比较定理,以及解与生成元的一系列等价性质；证明了由带跳的BSDE构成的f-期望的Jensen不等式。
(本章主要内容来自Jia-Zhang[61,62])
本章中,我们将研究带跳的BSDE的表示定理和Jensen不等式。假定信息流(Ft)t∈[0,T]是由相互独立的d-维标准布朗运动(Wt)t∈[0,T]和定义在(R+×B)上的泊松随机测度μ。首先,本章内假设b,σ满足正向的标准李普希兹条件,f(ω,t,x,y,z,U)关于(y,z：U)致李普希兹连续,关于x至多多项式增长。对给定的(t,x)∈[0,T]×Rn,以X.t,x来表示以下SDE的解：并引入如下二阶随机积分-微分算子：关于积分的含义,将在第四章正文中阐述。
本章中我们将给出上述算子由带跳的正倒向方程的解的表示,以下是两个主要结果。
定理4.2.1(表示定理Ⅰ)假设f,b,σ关于t右连续。假设φ关于x有有界的三阶导数,记其公共界为Kφ。令1≤p≤2。则对每个(t,x,U(·))∈[0,T)×Rn×L2(B,B*,λ；Rn),以下式子成立另一个结论用到停时,也放宽了对φ的要求。
定理4.2.2(表示定理Ⅱ)假定生成元f,b,σ满足定理4.2.1中的条件。令φ∈C2。则对每个1≤p≤2,和我们有其中τε：=τ(?)(t+ε),τ是一个停时,使得Xt,x在[t,τ]上有界,比如τ：=inf{s>t|Xst,x-x|>N},ε充分小。
上面定理中的L2(B,B*,λ；Rn)和L∞2(B,B*,λ；Rn)的意义在第四章有详细介绍。
由表示定理,我们得到了关于带跳的BSDE的逆比较定理。
定理4.3.1(逆比较定理Ⅰ)令f1,f2独立于x,另外,设(?)(y,z,U),f1和f2都关于t∈[0,T)右连续,在Τ点连续,P-a.s.。对任意s∈[0,T],ξ∈L2(Fs),有则对任意(t,y,z,U(·))∈[0,T]×R×Rd×L2(B,B*,λ；R),有
定理4.3.2(逆比较定理Ⅱ)令.f1和f2除满足逆比较定理Ⅰ中所述条件外,还满足f(t,y,0,0)三0.若对任意ξ∈L2(FT),则有
假定.f独立于x,且对任意(t,y)∈[0,T]×R,f(t,y,0,0)≡0。类似于g-期望,Royer[108]由带跳的BSDE定义一类非线性期望f-期望。下面给出了f-凸函数的概念。
定义4.4.2对一个给定的f-期望Ef[·],一个函数h：R→R被称为f-凸(相应地,f-凹)函数,如果对每个X∈L2(FT),使得h(X)∈L2(FT),我们有(相应地,h被称为f-仿射函数如果它既是f-凸函数又是f-凹函数。下面记
定理4.4.1令f(t,y：z,U)关于(y,z,U)一致李普希兹连续且满足比较定理成立的条件,并关于t连续,对任意t,y,f(t,y,0,0)三0且h∈C2(R)。则以下两种叙述等价：
(i).h是f-凸的(相应地,f-凹)；
(ii).对每个t∈[0,T],y∈R,z∈Rd,U(·)∈L∞2(B,B*,λ；R),(相应地,≤0).
定理4.4.4令h∈C(R)至多多项式增长,.f(y,z,U)所需满足的条件比定理4.4.1稍强(具体见原定理),则以下两种叙述等价：
(i).h是f-凸的(相应地,f-凹)；
(ii).对每个z∈Rd,U(·)∈￡乙(B,B*,λ；R),h是Lft,y,z,Uφ=0的粘性下解。
第五章用一种新的方法证明了几个关于2-alternating容度的性质
(本章内容来自Jia-Zhang[59])
设Ω是一个基本集合,召是Ω上的σ-代数。我们称一个集函数c：B→[0,1]是一个容度如果它满足：
(C1).c(Ω)=1,c(Φ)=0；
(C2)(单调性).对任意A(?)B,A,B∈B,c(A)≤c(B)。
称容度μ为2-alternating,如果对任意A,B∈召,μ(A(?)B)+μ(A∩B)≤μ(A)+μ(B)。
称容度v为2-monotone,如果对任意4,B∈B,μ(A∪B)+μ(A∩B)≥μ(A)+μ(B)。称一个容度μ为概率测度,如果μ(A∪B)+μ(A∩B)=μ(A)+μ(B),我们以P来表示个概率测度。
根据Dcnnicbcrg [26]和Jia[54],简单推测可知如下结果成立。
定理5.2.1任意概率测度是2-alternating容度集合的极小元。反之,任意2-alternating容度集合的极小元都是概率测度。
定理5.2.2考虑空间(Ω,B)。B是Ω上的一个代数,且是一个有限集。c是定义在B上的一个2-alternating容度。取F1,...,Fn∈B使得F1(?)F2(?)...(?)Fn。则存在一个概率测度P,对任意i=1,...,n,P(Fi)=c(Fi)且P≤c。
定理5.2.3考虑空间(Ω,B)。召是Ω上的一个代数,且是一个有限集。μ是定义在B上的一个2-alternating容度,v是定义在B上的一个2-monotone容度。则μ≥v意味着存在一个概率测度P,使得μ≥P≥v。
Denneberg和Jia对上述相近问题从期望的角度上进行了证明,应用的分别是Choquet-期望和一般的次线性期望。本章中我们将完全从容度的角度上来证明上述结果。证明过程的关键是通过B上的集合A对一个2-alternating容度c进行如下变换文中证明了cA仍然是一个2-alternating容度,且cA≤c。这种变换还具有很多其他的很好的性质。特别的,对后两个定理的证明过程中,我们设计了一个循环程序,最终,通过构造的方法得出一个符合条件的概率测度。
During the research on stochastic control,in1973,Bismut first introduced the notion of backward stochastic differential equations(BSDEs,for short).However,until the generalized BSDEs were proposed by Pardoux-Peng[90]in1990,the BSDE theory just started its booming development.The BSDE is of the folloWing forms: where ξrepresents the terminal random variable,T>0is a fixed terminal time,(Wt)t∈[0,T] represents the d-dimensional Brownian motion and g is called the generator of the BSDE.The solution of BSDE is a pair of adapted processes(Y,Z.)that makes the above equality hold.Pardoux-Peng[90]assumed that g is Lipschitz continuous about y,z,and thereafter,a lot of papers were devoted to the relaxation of the conditions on g. Papers on local Lipschitz condition,for example,Bahlali[1] etc.;on continu-ous condition,Lepeltier-San Martin[73],Matoussi[80]etc.;on uniformly continuous condition,Jia[53,56],Fan-Jiang[36],Hamadene[39] etc.;on quadratic growth condi-tion in z,Kobylanski[68],Briand-Hu[9,10],Tevzadze[117],Briand-Elie[7],Delbaen-Hu-Richou[23],Richou[104]etc.;on super quadratic growth in z,Delbaen-Hu-Bao[22], Richou[104]etc.;on polynomial growth in y,Briand-Carmona[4]etc.;on discontinuous condition,Jia[52],N'zi-Owo[83],Halidias-Kloeden[38]etc.In addition.there are many other forms of BSDEs,for example,BSDE with reflection,coupled FBSDE,BSDE driven by Brownian motion and Poisson process,or even BSDE driven by a general martingale etc.,see Tang-Li[116],Situ[112,113],Yin-Situ[121],Ren-Otmani[103],Shen-Elliot [111],Ouknine[85],Hamadene-Ouknine[40],Essaky-Ouknine-Harraj[35],Morlais[81], Essaky[33],Bahlali-Essaky[2],Otmani[28],Otmani[29],Lejay[72]etc.There are also numerous papers studying the properties of solutions to BSDEs, for example the com-parison theorem, converse comparison theorem, convexity and translation invariance etc. Moreover, Peng [97,100] introduced g-expectation and deduced many properties about it. Papers on this part, for example, Briand-Coquet-Hu-Memin-Peng [6], Coquet-Hu-Memin-Peng [19], Jiang [64,65,66], Hu-Tang [48], Hu-Peng [47], Royer [108], Jia-Peng [58], Jia [55] etc. BSDE theory has wide range of application, for example, mathemat-ical finance, stochastic optimal control theory, game theory and nonlinear Feynman-Kac formula etc. Papers on finance and stochastic control, for example, Peng [96,98], Chen-Li-Zhou [16], Liu-Peng [76], Wu-Yu [120], Kohlmann-Zhou [70], Chen-Epstein [14], while papers on nonlinear Feynman-Kac formula, for example, Peng [94],Peng [98], Pardoux-Peng [91], El Karoui-Kapoudjian-Pardoux-Peng-Quenez [31], El Karoui [30], Barles-Buckdahn-Pardoux [3], Buckdahn-Hu [12], Briand-Hu [8], Pardoux [87], Padoux-Tang [93], El Karoui-Peng-Quenez [32], Kobylanski [68], N'zi-Ouknine-Sulem [82] etc.
As a matter of convenience, we denote equation (2) by (g,T, ξ), and its solution by (Ytg,T,ξ,Ztg,Tξ)t∈[0,T]. Moreover, we denote Ytg,T,ξ by Et,Tg[ξ]. Assume that for all (y,z)∈R÷Rd, g(·,y,z) is square integrable. Tr[·] denotes the trace of a symmetric matrix. Gi denotes the i-th row of matrix G.
We list the contents of this thesis as follows:
Chapter1Introduction;
Chapter2Invariant representation for second order stochastic differential operator by BSDEs with uniformly continuous coefficients and its applications in nonlinear semi-groups;
Chapter3Quadratic g-convex functions, C-convex functions and their relationships;
Chapter4Representation for second order integral-differential operator by BSDEs with jumps and properties of related f-convex functions;
Chapter5New proofs about several properties of capacities.
We now introduce the main results of each chapter.
(I) In Chapter2we prove that one kind of second order stochastic differential operator can be represented by a sequence of the solutions of FB-SDEs with uniformly continuous coefficients; and define nonlinear semigroups via decoupled FBSDEs; we prove the monotonicity and order-preservation of the nonlinear semigroups; and obtain a new comparison result for coupled FBSDE.
In this chapter, we suppose that dimensions of SDE and BSDE, are n and1, re-spectively. Let the coefficients bΩ x [0, T] x Rn→Rn and σ:Ω x [0, T] x Rn→Rnxd satisfy the following assumptions:
·(Uniformly Lipschitz Continuity):|b(t,x1)-b(t,x2)|+|σ(t,X1)-σ(t,X2)|≤Kb|x1-x2|,(?)x1,x2∈Rn;
·(lincar growth condition):|b(t,x)|+|σ(t,x)|≤Kb(1+|x|),(?)x∈Rn, where Kb>0is a fixed constant. For simplicity, we denote the condition by "forward standard Lipschitz condition". Let (Xst,x)s∈[0,T] be the strong solution of the following SDE:
Under the condition that g is uniformly continuous and of linear growth in (y,z), Lepeltier-San Martin [73] proved that there exits at least one solution. However, the solution may be not unique. For example, suppose g(y)=(?)|y|, T=1and ξ=0.In fact, there are uncountably many solutions of equation ((?)|y|,1,0). The first conclusion of this chapter shows that, no matter whether the solution is unique or not, we can always represent one kind of second order stochastic differential operator by the solutions of the SDEs and BSDEs. It can be specified as follows:
Theorem2.2.1[Representation Theorem I] Let g be uniformly continuous in (y, z) and continuous in t. Suppose b, σ satisfy the forward standard Lipschitz assump-tion. Assume that φ(t,x) E C1,2([0, T] x Rn). Moreover, we suppose that φ and its first and second order derivatives are all of at most polynomial growth. Then for each x∈Rn,t∈[0,T), we have where
Note that the solutions are arbitrarily chosen in this conclusion. With the help of this representation theorem, we can get a converse comparison theorem for g. Moreover, we can alos deduce some equivalent relationships between the properties of the solutions and those of the generator,for example,linearity, constant preservation,etc.
Then we define two kinds of nonlinear semgroups according to the decoupled FB-SDEs. The new semigroups generalize the Markovian semigroups generated by the Markovian diffusion processs. It is known that,Markovian semigroup is deduced by taking linear expectation over Markovian diffusion process,while the nonlinear semi-groups we defined are obtained through taking nonlinear expectation(g-expectation) over the diffusion processes.Representation Theorem I mentioned above just shows the infinitesimal generators of both the two nonlinear semigroups.
Suppose that b and σ depend on x only and are Lipschitz continuous in x.Further-more,we assume that g depends on(y,z)only.Therefore,the terminal time T can be any positive number.We denote by H the set of all real valued the continnuous functions defined on Rn,which are of at most polynomial growth and by Hb the set of real valued uniformly continuous functions defined on Rn.According to different conditions of g,we have two ways defining the semigroups.
First,when g is Lipschitz continuous in(y,z),for each f∈H,we define εtf(x):=y0g,t,f(xt0,x).Since εt∈H,we can define a binary operation over the operators by (εtoεs)f=εt(εsf).Then we have εtoεs=εsoεt=εt+s.Thus,{(εt)t≥0;o}forms a commutative semigroup.
Second,when g is uniformly continuous in(y,z)and both b and σ are bounded, according to Ma-Zhang[77],we define the so called "Nodal set" as follows:for any f∈Hb,
O(t,x,f,s):={y:there exists a solution y.g,s,f(Xst,x)such thatytg,s,f(Xst,x)=y} It is proved by Ma-Zhang that O(t,x,f,s)is a bounded interval.Thus we have O(t,x,f,s)=[u(t,x,f,s),u(t,x,f,s)]. Define εt,sf(x):=u(t,x,f,s)and εt,sf(x):=u(t,x,f,s).Since g,b,σ are all indepen-dent of t,εt,s f(x)≡ε0,s-tf(x).For simplicity,we denote εtf(x)=ε0,tf(x).Similarly εtf(x)=ε0,tf(x)=εr+trf(x),for all r≥0,t≥0.For fixed t,εtf and εtf are all bounded and uniformly continuous functions.Similar to the case of Lipschitz continuity, we can define binary operation "o" on Hb.It can be deduced that{(εt)t≥0lo} and {(εt)t≥;o}both constitute nonlinear semigroups.
We can see that,if g is uniformly continuous in y,z,the infinitesimal generator may not be one to one correspondent to semigroup. It is possible that several semigroups enjoy the same infinitesimal generator.
With the above semigroup, we are to discuss the monotonicity and order-preservation of the semigroup when g is Lipschitz continuous. In fact, Herbst-Pitt[43], Chen-Wang [17] deduced the necessary and sufficient conditions for a Markov semigroup to be mono-tone and for two semigroups to preserve order. However, their proof depends on the linearity of Markov semigroup heavily, and thus, can not be used here. To deduce these results, we have to find another way. We now introduce the following definition:
Definition2.5.1Suppose that "≤" denotes the usual semi-order in Rn.
(i). A measurable function f:Rn→R is called monotone, if for all x≤x, f(x)≤f(x). Denote by M the set of continuous monotone functions with at most polynomial growth. Then M∈H.
(ii). For two semigroups{εt}t≥0and {εt}t≥0, we write εt≥εt if for all f∈M, x≥x and t≥0, εt f(x)≥f(x). If in addition εt=εt, we callεt monotone.
The following two theorems hold.
Theorem2.5.1Suppose b, σ rely on x only and are Lipschitz continuous in x, g relies on (y, z) only and is Lipschitz continuous in (y, z). Then εt defined on H is monotone if and only if the following conditions hold:(2C1-i). bi(x)-bi(x)≥0for all x,x∈Rn with xi=Xi and xk≥xk for k≠i, i=1,...,n;(2C1-ii). σi depends on xi, only, for all i=1,..., n.
Note that, the monotonicity of Et has nothing to do with g.
Theorem2.5.3Assume n=d. Suppose b,σ,b,σ rely on x only and are Lipschitz continuous in x, g,g rely on (y,z) only and are Lipschitz continuous in (y.z). Suppose σσ*(or σσ*) is uniformly positive definite and b, b, σ, σ are all bounded. If one of εt and εt is monotone, then εt≥εt if and only if (2C3-i) σσ*=σσ*and both σi(x) and σi(x) depend only on xi;(2C3-ii) for all x∈Rn, y∈R, K∈Rn, K≥0, K*b(x)+g(y,σ*(x)K)≥K*b(x)+g(y,σ*(x)K).
By the one to one correspondence of the semigroup and the PDE, the above mono-tonicity and order-preservation theorem also hold for the PDEs. During the proof, we note that, a semigroup is linked to a unique PDE, but not necessarily to a unique pair of FBSDE.
In particular, we take g, g So we have the following result.
Theorem2.5.4Suppose b, σ and b, σ rely on x only and are Lipschitz continuous in x, and moreover that all the elements in the same column of (σ)n×d have the same sign, which can differ according to different x. Let (2C1-i) and (2C1-ii) hold for b, σ and b, σ. Then εt≥εt if and only if (2C3-i) and (2C3-ii) hold. Here (2C3-ii) means that for all i, whenever x≥x with xi=xi.
This result can not be covered by Theorem2.5.3, since we don't need the assump-tions of n=d and the positive definiteness of σσ*. According to this theorem, we can partly turn a special kind of second order quasilinear parabolic PDE into second order linear PDE.
Remark2.5.8Consider the following second order quasilinear PDE: with the coefficients satisfying the same conditions as those stated in Theorem2.5.4. We have the following results:if f is a nondecreasing function with at most polynomial growth, the above PDE is equivalent to while if f is a nonincreasing function with at most polynomial growth, the above PDE is equivalent to
Inspired by the proof for the order-preservation of semigroup, we deduce a new comparison theorem for FBSDEs, which connects the drift of SDE and the generator of BSDE for the first time.
We consider the following FBSDE where (Xt, Yt, Zt, Wt)∈Rn×R×Rd×Rd, with the dimensions of b, σ, g and f defined accordingly.
Here are the assumptions (mainly come from [77]): the coefficients (b,σ,g,f) are measurable and bounded; σσ*are uniformly positive defi-nite; b, σ, g, f are all smooth with bounded first and second order derivatives.
Theorem2.6.1Assume bi,σi,gi,fi (i=1,2) satisfy the above conditions. Suppose that the initial values are x1,x2respectively. Denote the solution by (Xi,Yi,Zi). If
(i)σ1(σ１)*≡σ2(σ2)',
(ii) p*b1(t, x, y,(σ1)*(t, x)p)+g1(t, x, y,(σ1)*(t, x)p)≥P*b2(t, x, y,(σ2)*(t,x)p)+g2(t,x,y,(σ2)*(t,x)p), for all t∈[0,T], x∈Rn, y∈R, p∈Rn,
(iii) f1≥f2,
(iv) x1=x2, then Y01≥Y02.
(II) In Chapter3we study the representation theorem for quadratic BSDEs; and obtain a necessary and sufficient condition for the Jensen's inequality to hold under quadratic g-expectation; we define a kind of C-affine functions for a fixed constant C, then define C-convex functions and C-concave functions; we prove the properties of C-convexity (resp. C-concavity); furthermore, we investigate the relationships between C-convex (resp. con-cave, affine) functions and quadratic g-convex (resp. concave, affine) func-tions.
Jensen's inequality plays an important role in classical probability theory. Jia-Pcng [58] firstly defined g-convex function as the function that satisfies the Jensen's inequality under g-expectation, i.e., h is called a g-convex (resp. concave) function, if for any FT-measurable, square integrable random variable X such that h(X) is also square integrable, and for any t∈[0,T], If h is both a g-convex function and a g-concave function, then h is a g-affine function. Jia-Peng also deduced a simple necessary and sufficient condition for a smooth function h to be g-convex, i.e., where Lgt,y,zφ:=1/2φ"(y)|z|2+g(t,φ(y),φ'(y)z)-φ'(y)g(t,y,z). For a general continuous function h of at most polynomial growth, it is g-convex if and only if h is a viscosity subsolution of the following PDE: For more details about viscosity solutions, the reader can refer to [20] etc. An interesting result is, when g is Lipschitz continuous in y, z,g-convex function is convex in the usual sense.
In this chapter, we assume that g is of quadratic growth in z and study the properties of g-convex function under this assumption (called quadratic g-convex function). The quadratic BSDE has been well studied since Kobylanski introduced the quadratic BSDE in [68]. However, as we known, the representation theorem and Jensen's inequality under quadratic growth condition have not been well studied. In this chapter, we will focus on these problems.
One of the major tasks in this chapter is to deduce the necessary and sufficient con-dition for a function to be quadratic g-convex. Because of the contradiction between the boundedness of the terminal condition and the unboundedness of the forward equation introduced in the proof, the proof is nontrivial. To solve this problem, we introduce stopping time and optional stopping time theorem which play an important role in the proof.
What's interesting about quadratic g-convex function is that it may not be convex. For a fixed constant C, we consider the following ODE Solve this ODE and define its solutions (which are not linear usually) as new "affinc functions", i.e., C-affine functions. Then we introduce the notion of C-convex func-tions. This kind of functions has many good properties similar to the classical convex functions, such as, continuity, quasi-convexity. Moreover, each C-convex function can be represented as the upper envelope of a family of C-affine functions. It is worthy to mention that, if C=0, the framework of C-convexity is just the classical one.
In this chapter, we make the following assumptions on g:
· there exit two constants Ky>0and Kz>0, such that (?)(t, y, z, y', z'),
· g(·,0,0) is uniformly bounded.
For simplicity, we call this assumption standard quadratic condition.
First, as before, we prove representation theorem with stopping time for g under the standard quadratic condition.
Theorem3.3.1Let g satisfy standard quadratic condition,b, σ satisfy the for-ward standard Lipschitz condition. Assume moreover that g, b, a are all continuous in t. Suppose that φ∈C1,2(R+x Rn). Then for each (t,x)∈[0,T) x R",nthe following holds: where Lg,b,σt,x is defined as above, τε:=τΛ (t+ε) for ε small enough and τ is a stopping time such that Xt,x is bounded on [t, τ]. For example,(K0is a positive number and is large enough).
From now on, in this chapter, we always assume that g(t, y.0)=0, and g satisfies standard quadratic condition.
First, we have the following definition. Definition3.4.1For a given Eg[·] a function h:R→R is said to be g-convex (resp. g-concave) under bounded terminal conditions, if for each X∈L∞(FT), one has h is called g-affine function if it is both g-convex and g-concave.
Definition3.4.3(g-convexity on a convex set) Suppose that O is a convex subset of R. For a given E9[·], real valued function h is said to be g-convex (rcsp. g-concave) on O, if for each X∈L∞(FT) such that E9s,T[X](ω)∈A, dP x dt-a.s., one has h is called g-affine if it is both g-convex and g-concave. We have the following two theorems.
Theorem3.4.1Suppose h∈C2(R), the following two statements are equivalent:
(i) h is g-convex (resp.g-concave) under bounded terminal conditions;
(ii) for each y∈R, z∈Rd,
Theorem3.4.2Suppose g is independent of ω and is continuous in t. h is a continuous function. Then the following statements are equivalent:
(i) for each (t,z)∈[0,T] x Rd, h is a viscosity subsolution of Lgt,y,zh=0;
(ii) h is g-convex under bounded terminal conditions.
We now choose a kind of g which are typical and study the properties of corre-sponding g-convex functions. Suppose g=C|z|2+gi(t,y,z), where lim|z|→∞|g1(t,y,z)|/|z|2=0and C∈R.
If C≡0, each g-convex function is convex and each g-affine function is linear. Suppose C≠0, thus any g-affine function φ∈C2(R) satisfies the following equality Solve the above ODE and denote by ΠC the set of solutions We call these functions C-affine. With the help of this kind of function, we can define the following C-convex function:
Definition3.6.1(C-convex function) A real valued function f defined on a convex set D R is called C-convex function, if for any φ∈ΠC such that there exist two different points of intersection x1
In addition, we can define C-convex set as follows.
Definition3.6.7(C-convexity on R2) A set A R2is called a C-convex set, if for any two points (x1,y1),(x2, y2)∈A, x1
Now we consider the relationships between the C-convex functions and g-convex functions. We have the following results.
Theorem3.6.5Any C-convex function is a g-convex function with g=C|z|2+
Theorem3.6.6Suppose that g=C|z|2+g1(t, y, z), such that and C∈R. Thus all the g-convex (resp. concave, affine) functions are C-convex (resp. concave, affine) functions.
Theorem3.6.7Suppose I is an index set and{fi:i∈I} is a family of g-convex functions. Thus f(x)=sup{fi(x):i∈I} is also a g-convex function.
Theorem3.6.8Suppose thus the nec-essary condition for a smooth g-convex function h to be represented as the upper envelope of a family of g-affine functions is, for any (t,y,z), g1(t, h(y),h'(y)z)-h'(y)g1(t,y,z)=0. In particular, if C=0, g is independent of y, then the above condition is also a sufficient condition.
(Ⅲ) In Chapter4we represent the second order stochastic integral-differential operator by a sequence of FBSDEs with jumps; and obtain a converse comparison theorem and some equivalent properties of BSDE with jumps; we define the f-convex function under BSDE with jumps, and deduce the necessary and sufficient condition for a function to be f-convex.
In this chapter, we always assume that b, σ satisfy the forward standard Lipschitz condition, f(ω,t,x,y,z,U):Ω×[0,T]×Rn×R×Rd×L2(B,B*,λ;R)→R is uniformly Lipschitz continuous in (y, z, U) and with at most polynomial growth in x. For given [t, x)∈[0, T) x Rn, we denote by Xt,x the solution of the following SDE: and introduce the following stochastic integral-differential operator:
The meaning of the integral will be explained in the text of Chapter4.
In this chapter, the representation for the above operator by the solutions of FBSDE with jumps will be given. Here are the two main results.
Theorem4.2.1(Representation Theorem Ⅰ) Suppose that f,b,σ are right con-tinuous in t. Suppose that φ has bounded three order derivatives. For any1≤p≤2and we have
Theorem4.2.2(Representation Theorem Ⅱ) Suppose that f, b, σ satisfy the same conditions as those in Theorem4.2.1. Assume that φ∈C2. Thus for each1≤p≤2, and we have where τε:=σ Λ (ι+ε), τ is a stopping time, such that Xt,x is bounded on [t,τ). For example,we can take τ:=inf{s> t:|Xst,x-x|>N}.
The meaning of the integral in Lf,b,σ,Ut,L2(B,B*,λ;Rn) and L∞2(B, B*, λ;Rn) will be explained in Chapter4.
With the help of the representation theorems, we can deduce the following converse comparison theorem.
Theorem4.3.1(Converse Comparison Theorem Ⅰ) Suppose that f1,f2are all independent of x. Moreover,(?)(y, z, U), both f1and f2are all right continuous in t∈[0,T) and right continuous in T, P-a.s.. For any s∈[0,T], ξ∈L2(FS), we have Thus for any (t, y, z, U(·))∈[0,T]×R×Rd×L2(B,B,λ;R), we have
Theorem4.3.2(Converse Comparison Theorem Ⅱ) Suppose that f1and f2satisfy the same conditions in Converse Comparison Theorem I. Moreover, f(t, y,0,0)≡0. If for any ξ∈L2(FT), we have
Suppose that f is independent of x and f(t,y,0,0)≡0for all (t,y)∈[0,T] x R. Similarly to g-expectation, Royer [108] introduced f-expectation by solutions of BSDE with jumps. Here is the definition of f-convex function. Definition4.4.2For a given f-expectation Ef[·], a function h:R→R is called f-convex (resp.f-concave) function, if for each X∈L2(FT), such that h(X)∈L2(FT), we have h is called f-affine function if it is both f-convex and f-concave.
We now introduce the following notations
Theorem4.4.1Suppose that f(t,y,z,U) satisfy uniform Lipschitz condition, and f(t,y,0,0)≡0, h∈C2(R), thus the following two statements are equivalent:
(i). h is a f-convex function (resp.,f-concave function);
(ii). for each t∈[0,T], y∈R, z∈Rd, U(·)∈L∞2,(B,B*, λ;R),
Theorem4.4.4Suppose h∈C(R) is of at most polynomial growth, f(y, z, U) satisfies some more strengthen condition, thus the following two statements are equivalent:
(i). h is f-convex function (resp.f-concave);
(ii). for each z∈Rd, U(·)∈L∞2, B*, λ; R)1, h is a viscosity subsolution of PDE Lf t,y,z,Uφ=0.
(Ⅳ) In Chapter5we prove several properties of2-alternating capaci-ties
Let Ω be a basic set, B is a σ-algebra on Ω. A set function c:B→[0,1] is called a capacity, if it satisfies:
(C1). c(Ω)=1,c((?))=0;
(C2)(monotonicity). for all AC B,A,B∈B, c(A)
Theorem5.2.1Any probability measure is a minimal member of the set of2-alternating capacities. On the other hand, any minimal member of the2-alternating capacities is probability measure.
Theorem5.2.2Suppose B defined on Ω is finite and that c is a2-alternating capacity. Take F1,..., Fn∈B such that F1∈F2∈...∈Fn. Then there exists a probability measure P such that P(Fi)=c(Fi) for i=1,..., n and P≤c.
Theorem5.2.3Suppose B defined on Ω is finite. Let μ be a2-monotone capacity and v be a2-alternating capacity. The μ≥v implies that there exists a probability measure P such that μ≥P≥v.
Denneberg and Jia prove similar results by ways of expectations. They mainly use Choquet expectation and general sublinear expectation respectively. In this chapter, we'll prove the above results by capacity only. The key of the proof is to transform a2-alternating capacity c to a new capacity μA, by the subset A∈B. The transformation is as follows:
It is proved that cA is still a2-alternating capacity and cA≤c. Moreover, there are many other interesting properties. In particular, we can create a probability measure by a2-alternating capacity.
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