Representation For Operators And Jensen’s Inequality Under Backward Stochastic Differential Equations

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 whereNote 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 toInspired 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<x2, we haveMany 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<x2, and any φ∈ΠC which crosses the two points, we have φ(x)∈A for any x∈(x1,x2).Proposition3.6.9If f is a C-convex function, epi f={(x,y):f(x)<y} is a C-convex set. On the other hand, if epi f is a C-convex set, then f is a C-convex function.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+<C1,z> and for g=C|z|2+<C1,z>, 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 haveTheorem4.2.2(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 haveTheorem4.3.2(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 haveSuppose 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 notationsTheorem4.4.1Suppose 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-tiesLet 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)<c(B). A capacity μ is called2-alternating, if for all A,B∈B, μ(A∪B)+μ(AnB)≤μi(A)+μ(B). A capacity v is called2-monotone, if for all A,B∈B,μ(A∪B)+μ(A∩B)≥n(A)+μ(B). A capacity μ is called probability measure, if μ(A∪B)+μ(A∩B)=μ(A)+μ(B). We usually denote a probability measure by P.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.