| Pardoux-Peng (1990) first studied the following backward stochastic differential equation (BSDE for short): and showed that the above BSDE has a unique solution. Based on further investigating the properties of BSDEs, Peng (1997) introduced the notions of g-expectations and conditional g-expectations via a class of BSDEs, more specific, suppose g satisfies (A1) the Lipschitz condition and (A2) g(t, y,0)≡0, define Sg[ξ]is called the g-expectation ofξandε9[ξ]Ft] is called the conditional g-expectation ofξwith respect to.Ft, where (yt)t∈[0,T] is the solution of the above BSDE corresponding to the generator g and terminal valueξ. It is important to note that g-expectations are the first dynamically consistent nonlinear expectations, but Choquet expectations, introduced by Choquet (1953) via capacities, still have not been proved to be dynam-ically consistent. A characteristic of the g-expectation is that it can be defined on a probability space, for overcoming this, Peng (2005) further proposed the general dy-namically consistent nonlinear expectations, especially the method of using nonlinear markov chains to construct dynamically consistent nonlinear expectations. Peng (2004) investigated a type of evaluation operators on stock market under volatility uncertainty, which were proved to be a dynamically consistent nonlinear expectations, but not g-expectations, this can be seen as the initial studies of G-expectations.Peng (2006) introduced the notions of G-normal distribution, G-expectation and G-Brownian motion, constructed the Ito's integral with respect to G-Brownian motion, obtained G-Ito's formula, the existence and uniqueness of G-SDE and G-BSDE. Un-der the G-expectation framework, Peng have proposed many interesting methods and obtained many interesting results, more important, it has many interesting problems. On the other hand, we emphasize that G-expectations are dynamically consistent and not g-expectations, but g-expectations can be seen as the special case of generalized G-expectations. Recently, Peng (2007,2008) studied the central limit theorem under sublinear expectations and obtained that the limit distribution exists and is just the G-normal distribution, this surprising result implies that the importance of G-normal distributions in sublinear expectations may be greater than normal distributions in linear expectations.In the following, we briefly recall part of the recent results of G-expectations. Denis-Hu-Peng (2008) and Hu-Peng (2009) studied the representation theorem of G-expectations and its application to G-Brownian motion paths; Bai-Buckdahn (2009) studied the application of G-expectations to risk measures; Xu-Zhang (2009) studied the Ito's integral with respect to G-martingales and the Levy characterization of G-Brownian motion; Gao (2009) studied the path properties of the solution to G-SDE; Hu-Peng (2009) studied G-Levy processes; Li-Peng (2009) studied stopping times and the extension of G-Ito's formula; Peng (2007), Soner-Touzi-Zhang (2010), Song (2010) and Hu Ying-Peng (2010) studied the G-martingale representation theorem; and so on.This doctoral thesis study some fundamental problems in dynamically consistent nonlinear expectations theory, especially in G-expectations theory.(I) In Chapter 1, we obtain necessary and sufficient conditions for g under which g-expectations equal to Choquet expectations on the whole space or a subset, under the assumption that g is deterministic. We also obtain a necessary and sufficient condition for g under which g-expectations can be dominated by Choquet expectations, under the assumptions that g is deterministic and convex.Chen-Chen-Davison (2005) first studied an interesting problem:if a g-expectation equals to a Choquet expectation, can we find the form of g? Under the assumption that the dimension of the Brownian motion is one, they showed that a g-expectation equals to a Choquet expectation if and only if the g-expectation is the classical linear expectation. A natural question is what is the condition on g for multi-dimensional Brownian motion case? More specific, whether the method used for 1-dimensional case can be used to deal with multi-dimensional case?The method in Chen-Chen-Davison (2005) depends on the form of g, for 1-dimensional case, the positive homogeneous function g is simple and just determined by two param-eters, but for multi-dimensional case, the positive homogeneous function is complex; on the other hand, for multi-dimensional case,ξ=y1+WT1 andη=y2+WT2 are not comonotonic, which is different from 1-dimensional case. All of this give a negative answer to the above question.In this Chapter, we give a new method to deal with this problem and obtain the following main result: Theorem 1.3.12. Suppose g is deterministic and satisfies (A1) and (A2). Then the g-expectation equals to the Choquet expectation if and only if g is independent of y and is linear in z, i.e., the g-expectation is the classical linear expectation.It is interesting that our new method also holds for 1-dimensional case and does not need the continuous assumption of g in t, which is important and can not be deleted in the method in Chen-Chen-Davison (2005). In addition, our method is more simple and direct, moreover, it can be used to deal with other problems, see our forthcoming papers.In Coquet-Hu-Memin-Peng (2002) and Delbaen-Peng-Rosazza Gianin (2009), the authors prove that the dynamically consistent nonlinear expectation must be the g-expectation under Brownian motion filtration. Hence, our result further implies that Choquet expectation can not be dynamically consistent in some sense. Thus, in the framework of dynamically consistent nonlinear expectations, we can not suppose comono-tonic additivity, and should study directly from expectations not capacities.The second main problem is to consider the equal relationship of g-expectations and Choquet expectations on some useful subsets in L2(FT). For this, we first recall some notations:Let H denote allξ∈L2(FT) such that there exist Lipschitz functions b(t, x) [0,T]×R→R andσ(t,x):[0,T]×R→Rd in x satisfyingξ=XT, where(Xt)t∈[0,T] is the solution of the following SDE:We also set: especially, H1 and H2 can be seen as the sets of European options. Chen-Sulem (2001) first discussed the equal relationship of g-expectations and Choquet expectations on H1, under the 1-dimensional Brownian motion case, they obtained a necessary and sufficient conditions for g, but for multi-dimensional Brownian motion case, Chen-Kulperger-Wei (2005) only gave a sufficient condition.The main reason is that in Chen-Kulperger-Wei (2005), the authors only consider the sign of z. In fact, z andσ(t,Xt) have a structure. We use this structure to obtain the second main result:Theorem 1.3.16. Suppose g is deterministic and satisfies (A1) and (A2). Then we have(i) the g-expectation equals to the Choquet expectation on H1 if and only if g is independent of y and is positively homogeneous in z, i.e., g(t,λz)=λg(t, z), (?)λ≥0;(ii) the g-expectation equals to the Choquet expectation on H2 if and only if g is independent of y and is homogeneous in z, i.e., g(t,λz)=λg(t, z), (?)λ∈R.It is interesting that our results give an explanation of positively homogeneous, homogeneous and linear functions.In this Chapter, we also consider the domination relation of g-expectations and Choquet expectations, and obtain the third main result: Theorem 1.4.3. Suppose g is independent of y, convex in z, deterministic and satisfies (A1) and (A2). Thenεg[ξ]≤Cg[ξ]for eachξ∈L2(FT) if and only if g is positively homogeneous and sub-additive in z.In this Chapter, we study the properties of g-expectations which are the first dynamically nonlinear expectations. We hope to know more properties of general dy-namically nonlinear expectations by g-expectations, in addition,g-expectations also provide a good example, this will be seen in Chapter 2 and 4.(Ⅱ) In Chapter 2, we obtain the explicit formula of odd power of G-normally distributed random variables. We also prove that the central limit theorem under convex expectations still holds.Peng (2006) introduced the notion of G-normal distribution, more specific, a ran-dom variable X under sublinear expectation E is called G-normally distributed, if u(t, x):= E[φ(x+t1/2+X)} satisfies the following G-heat equation: whereσ2=E[X2],σ2=-E[-X2], denoted by N(0, [σ2,σ2]). It is important that Peng (2006) gave the following formula for convex or concave functions: Ifφis convex, then we have E[[φ(X)]=EP[φ(σξ)], whereξis classically standard normal distribution under linear expectation Ep;Ifφis concave, then we have E[φ(X)]=EP[[φ(σξ)], whereξis classically standard normal distribution under linear expectation Ep.A natural question is how to calculate E[φ(X)] for neither convex nor concave function np, especially how to calculate E[X2n+1]?In this Chapter, we mainly consider the calculation of E[X2n+1]. We establish a relation between the solution of the G-heat equation and the solution of an ordinary differential equation under this case, and further obtain the formula of E[X2n+1]. For this, we denoteThe following is the first main result in this Chapter:Theorem 2.3.7. Suppose X is G-normally distributed with N(0, [σ2,1]),σ∈[0,1), gn and hn are defined above. Set kn=E[X2n+1], we have(1) Ifσ∈(0,1), then kn satisfies the following equation:(2) Ifσ=0, then kn satisfies the following equation: It is important to note that from the first formula in the above equation we can get cn, then by the second formula we can get kn. We hope our result can be used as an example for stochastic calculation.The second main problem is to consider the central limit theorem under convex expectations, this generalized the results of Peng (2007,2008) in sublinear expectations case. The following is our second main result:Theorem 2.4.13. Suppose (fΩ, H, E) is a sublinear expectation space, (Ω, H, E) is a convex expectation space, E is dominated by E. We also suppose that (Xi, Yi)i=1∞(?) H are independent and identically distributed under E and E, and satisfy: Then for eachφ∈Cb,Lip(R), we have where (ξ,η) is G-distributed under E, the corresponding G is defined as follows:It is surprising that the limit distribution under convex expectations is the same with the limit distribution under sublinear expectations. If the expectation in the above theorem is the g-expectation, we get the following third main result:Theorem 2.4.20. Suppose g(z) is a Lipschitz function with g(0)=0,μ=limδ↓0δ-1g(δ) andμ=-limδ↓0δ-1g(-δ) exist, we have(III) In Chapter 3, we study the measure theory based on a family probability measures, obtain the characterization of Banach spaces Ldp and Lcp, the generalized version of Kolmogorov's criterion for continuous modifi-cation of a stochastic process and the convergence theorem under sublinear expectations. We also obtain the representation theorem of the distribu-tion of a type of stochastic process under sublinear expectations, especially G-expectations, combining with the characterization of Lcp, we further estab-lish the paths description of such stochastic processes, especially G-Brownian motion. Peng (2006,2008) introduced the notion of G-Brownian motion and established the theory of related stochastic calculus, in this theory, many beautiful norms are introduced and the spaces are Banach spaces. More specific, we first recall some notations in Peng (2006,2008):LetΩ=C0d (R+) denote the space of all Rd-valued continuous functions on [0,∞) with initial value 0, (Bt)t≥0 be the canonical process. The space of bounded and Lips-chitz cylinder functions is defined as follows: Peng (2006,2008) constructed a dynamically consistent sublinear expectation on Lip(Ω), called G-expectation, the corresponding canonical process(Bt)t≥0 is called G-Brownian motion. For each given p≥1, we denote LG(Q) the completion of Lip(Ω) under the norm (E[|X|p)1/p.Since elements in LGp(Ω) are abstract, which gives some problems:whether all bounded continuous functions belong to LGp(Ω)? how to describe a sequence of random variables decreasing converging to 0 and the continuous process? A natural question is:can we give a characterization of LGp(Ω)?In this Chapter, we obtain the characterization of LGp(Ω). For dealing with this problem, we first give a simple and direct method to prove that G-expectations can be represented as the upper expectation of a weakly compact family of probability measures. More general, we obtain the following main result in this Chapter: Theorem 3.3.7. Let (Ω,Lip(Ω),E) be a sublinear expectation space, and E satisfy the following condition:there exist positive constants a,βandγsuch thatThen there exists a weakly compact family of probability measures P on (Ω,β(Ω)) such thatBased on P, we consider the corresponding measure theory. More specific, define the capacity (see Huber (1981)):Then we can define quasi-continuous functions and the version of quasi-continuous functions based on c. On the other hand, we define the following Banach spaces:for each p≥0, Lp:={X:X measurable and E[|X|P]=supP∈pEP[|X|p]<∞}.We prove that Lp is a complete metric space under natural distance. Furthermore, we denoteLdp the completion of all bounded measurable functions with respect to the distance in Lp.Lcp the completion of all bounded continuous functions with respect to the distance in Lp.In this Chapter, we obtain the following main results:Theorem 3.2.6. For each p> 0, we haveTheorem 3.2.28. For each p> 0, we have where q.c. denotes quasi-continuous.Theorem 3.2.22. (Kolmogorov Criterion) Let p> 0 and (Xt)t∈[0,1]d be a process satisfying for each t∈[0,1]d, Xt∈Lp. If there exist positive constants c andεsuch that then X has a continuous version X such that for each a∈[0,ε/p), Specially, paths of X are q.s. a-Holder continuous for each a<ε/p.Theorem 3.2.33. (Convergence Theorem) Let p be weakly compact, and{Xn}n=1∞(?) Lc1 satisfy Xn↓X, q.s.. Then we have E[Xn]↓E[X].Note that Lip(Ω) is a subspace of bounded continuous function space, then we have LGp(Ω) is a subspace of LCp. From this we can further prove all bounded continuous functions belong to LGp(Ω). The following is our main result:Theorem 3.3.10. (Paths Property) Let (Ω, Lip(Ω),E) be a sublinear expectation space satisfying the condition in Theorem 3.3.7.. Then for each p> 0, we have It is important that Theorems 3.2.6. and 3.2.22. do not need Q is a metric space. It is clear that G-expectations satisfy all the conditions in Theorem 3.3.7., thus Theorems 3.3.7.,3.3.10. and 3.2.33. hold for G-expectations case.It is interesting that Banach spaces Lp,Ldp and Lcp are same in the classical linear expectation case, but they are different spaces under sublinear expectation case. For example, we have conditional G-expectation in LGp(Ω)=Lcp, but whether we can define it in LP is still a interesting problem.Specially, we point out that [25] first clearly introduced the method of q(?)s. pathwise analysis, and our paper [26] first strictly prove this type of q.s. pathwise analysis.(IV) In Chapter 4, we first study G-Levy processes under sublinear ex-pectations, obtain the corresponding Levy-Khintchine formula, and further obtain that the distribution satisfy a nonlinear parabolic integro-partial dif-ferential equation, conversely, we can construct the G-Levy process from this integro-partial differential equation.Peng (2007) showed that every process (Bt)t>0 with independent and stationary increments satisfying E[|Bt|3]=o(t) is G-Brownian motion. On the other hand, we know there are many jump models in mathematical finance, then a natural question is, how to study processes with independent and stationary increments but not satisfy E[|Bt|3]=o(t), especially the pure jump processes?For dealing with this problem, we mainly consider the distribution of process (Xt)t>o with independent and stationary increments, more specific, to find the par-tial differential equation of u(t,x)::=E[φ(x+Xt)]. In this Chapter, we define G-Levy processes, this correspond to the sum of jump is finite, thus a G-Levy process can be decomposed as jump and continuous part. For the jump part, we can not use the method in Peng (2007), for dealing with this part, we mainly use Daniell-Stone theorem, obtain the Levy-Khintchine formula, and the corresponding integro-partial differential equation. From this integro-partial differential equation, we can also construct the corresponding G-Levy process. The following is our main results in this Chapter: Theorem 4.2.18. Let (Xt)t≥0 be a d-dimensional G-Levy process, define GX[f(·)] as follows:Then GX[f(·)] has the following Levy-Khintchine representation: Theorem 4.2.19. Let (Xt)t≥0 be a d-dimensional G-Levy process. For each hφE Cb.Lip(Rd), define u(t, x)=E[φ(x+Xt)]. Then u is the unique viscosity solution of the following integro-partial differential equation: where U represent GX.Theorem 4.2.23. For each given U satisfying Theorem 4.2.18., there exists a G-Levy process where U represent GX.In particular, if U only contains v, then we can define G-Poisson processes and G-Poisson distribution. Furthermore, our method still hold for independent processes.It is interesting that our method also hold for classical linear expectation case, and more simple and direct. In addition, the structure of integro-partial differential equation is new which is different from the one studied in Alvarez-Tourin (1996), Barles-Imbert (2008) and Jakobsen-Karlsen (2006), in our new integro-partial differential equation, the measures can be singular, this is more difficulty.
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