The Study Of Some Optimal Problems Under Sub-linear Operator Framework

IntroductionRecent years, the study of the nonlinear operator has been widely developed. For example, Artzner et al. has proposed the notion of the coherent risk measures; Peng has proposed the notion of g-expectation and further G-expectation(see [3],[42],[93],[97] and the references therein).Among all the theories of nonlinear expectation, the most widely studied is sub-linear expectation. Such as enumerated theories above, except the g-expectation, all others can be classified into the framework of sub-linear expectation. Although compared with the linear case, sub-linear expectation just mainly replace the linear-additive prop-erty into sub-additive property, the study becomes much more complicated. The major methods to study the sub-linear expectation comes from besides probability theory and stochastic analysis, and also functional analysis, convex analysis, partial differential e-quations and other disciplines. Different methods care different emphases, corresponding to the different research direction. This paper prefers to use the means from probability theory and functional analysis to solve optimization problems with constraints.The paper mainly studied three problems with constraints and the completeness of space L1(c).The first problem is,for a given unknown random variable ξ, when the expectation is sub-linear, if we have already known the partial information C, how to choose the best random variable such that it is C-measurable and reach the minimum of the error. Here, the constraint is to need to be C-measurable. In the linear case, the optimal random variable is just the conditional expectation of the unknown random variable ξ. When we consider the sub-linear case, there is not consensus of opinion about how to define conditional expectation. We find:though the conditional expectation defined by Artzner et al.[4] before is time consistent under some conditions, i.e. unbias, it is not the optimal estimate under the mean-square error sense. This tells us, maybe there will not exist an estimate both is unbias and attains the minimum of the mean square error. Based on this observation, we try to define a new conditional expectation from the point of minimizing the mean-square error and we also show a relationship between our new definition and the conditional expectation defined by Artzner et al..The second problem is a problem interested by financial mathematicians. In in-complete market, how to invest can decrease the risk of shortfall under the constraint portfolios. When both the risk measure and the wealth equation are linear, Karatzas and Shreve [66] have already solved by using the dual methods. We extend to the case, when the wealth equation is sub-linear, risk measure is convex. The main means we used are martingale method, analysis and terminal variation in g-expectation.The third problem is the hypothesis testing problem under sub-linear operators. The most important result is the Neyman-Pearson lemma. In linear case, the Neyman-Pearson lemma not only show the best test function exists but also give out its specific form such that people know how to find it. Under nonlinear case, many mathematicians have already done a lot of works. In1973. Huber and Strassen [58] studied hypothesis testing problem for capacities. Cvitanic and Karatzas [21] studied the min-max test by using convex duality method in2001. Ji and Zhou [61] studied hypothesis tests for g-probabilities in2008. Rudloff and Karatzas [114] studied composite hypothesis by using Fenchel duality in2010. Due to the results about finitely additive set function, we use the method we have created extending the Cvitanic and Karatzas’work to the case without reference probability measure. Furthermore, we studied the case that the best test function do not reach the significant level. The study of how to deal with the case without reference measure is hot but also difficult, need to point out. Though we abandon the strong condition of without reference measure, the cost is we loss the existence of the best test function. But generally speaking, our results are coincide well with the Cvitanic and Karatzas’work. For example, the Pa and Qa that we find to representing the best test function may also not be dominated by the corresponding sub-linear operators as theirs. Following, we have study the hypothesis testing problem when the space is restrict in continuous bounded functions. This problem comes from the thought about the G-expectation introduced by Professor Shige Peng [97]. Our framework is based on the view that we should study the sub-linear problem from the operator defined on functions proposed by Professor Peng. Need to point, part of our proof, in fact, is using the traditional method to reprove the Daniell-Stone theorem. It helps us to understand this theorem more deeply.Before the third question, we have studied the completeness of the L1(c) space. The importance of the completeness is self-evident. Based on there exists non-complete space when the operator is sub-linear, we want to know under which condition L1(c) will be complete. After completing our proof, we found Denis Hu and Peng [28] had already give a proof for our question in their unpublished part. Of course, our method is different from theirs. They mainly used the results from probability and we mainly used the results of functional analysis. From the different proof, we can understand the result more deeply.Let us introduce the main contents.In the first chapter, we studied the best estimate under mean-square error, the problem stated as following:问题0.3. Given a measurable space (Ω,F) and a sub-linear operator p on F. C is a sub σ-algebra of F and C is denoted as the set of all the bounded C-measurable functions, F is denoted as the set of all the bounded F-measurable functions. Then for any ξ∈F whether there exists a η∈C such that and if such a solution exits, whether it is unique or not.At first, by using the Komlos thcorem, we obtained the existence of our optimal solution.Theorem0.1.If the sub-linear operator ρ is continuous from above on F, then there exists η∈G solves our problem. G is all the C-measurable functions bounded by M.Then by using minimax theorem, we obtained the uniqueness of our optimal solu-tion.Theorem0.2.If the sub-linear operator ρ is continuous from above on F and proper, then the optimal solution exists and is unique in the P0-a.s. sense.We will call the unique optimal solution of our problem as the best estimate condi-tional expectationIn the third section,we show a result about’stable’coherent risk mea-sure.Theorem0.3. If ρ is a sub-linear operator continuous from above on F and the rep-resentation set P of ρ is ’stable’as definition in [4], then for a given ξ∈F, the ker(f) is just the set In the fourth section, we show a necessary and sufficient condition for the conditional expectation defined by Artzner et al. is also the best estimate under mean-square error sense.Theorem0.4. For a given ξ∈F, under the assumptions of proposition1.3, we have the ηess is the optimal solution of the problem (0.1) if and only ifwhere ηess denotes esssupP∈Ρ Ep[ξ|C]。In the fifth section, we studied the sufficient and necessary condition for the existence of the optimal solution when we only assume the operator is sub-linear.Theorem0.5. Given ξ∈F, if infη∈C ρ(ξ-η)2>0, then η is the optimal solution of the problem,(0.1) if and only if it is the bounded C-measurable solution of the following equationIn the last, we give a necessary condition for finding the optimal solution.Theorem0.6(Envelope theorem). Given ξ∈F, if there exists r￡R such that for any ξ∈B(ξ,r), we have where B(ξ,r) denotes as the ball centered in ξ and with radius r. Then the function V is Frechet-differentia.ble at ξ andIn the second chapter, we studied the optimal problem under incomplete market, our problem is:If the wealth equation isTo minimize the risk, i.e. After a sequence of discussions, we change our problem to where Xεχ:={ξ∈L2(0,T; R);ξ0(ξ)≤xand0≤ξ≤H}.then we prove the existence of the optimal solution:Theorem0.7. Under the Assumption (H1)-(H7) and the assumptions on φ, the optimal solution of (0.3) exists.In order to study the specific form that the optimal solution need to have, by using the Mazur-Orlicz theorem, we obtain:Then by using Ekeland’s variational principle, we haveTheorem0.8. We suppose (H5)-(H7) hold for the function g in Section2. Let ξ*be an optimal solution to (0.4). Then there exist h1∈R and h0∈R with h0≥0and|h0|2+|h1|2=1such that the following variational inequality holds where X0ξ-ξ*and yo are the corresponding solutions of (2.18) and (2.19) at time0with terminal value ξ-ξ*and-(ξ-ξ*).Then we haveTheorem0.9. We assume (H1)-(H8) hold.If ξ*is optimal to (0.4) with {X*(·), Z*(·))(resp.{y*(·), z*(·))) being the corresponding state of (2.18)(resp.(2.10)), then there exist h1∈R and h0∈R with h0≥0and|h0|+|h1|≠0such that where m(·)(resp.n(·)) is the solution of (2.22)(resp.(2.23)). In the third chapter, we studied the completeness of L1(c).Theorem0.10. If the sub-linear operator￡can be represented by a family of probability measures, then the L1(c) space is complete under the L1(c)-norm.In the fourth chapter, we studied the hypothesis testing problem under sub-linear operator, obtained the main result: Neyman-Pearson lemma.In the second section, we proposed our problem问题0.4. For a given significance level α∈(0,1), whether there exists a functional Xα such that where Xα is the set {X; Eμ[X)≤α, X∈[0,1], X∈χ}.In the third section, we give a necessary and sufficient condition for the existence of the optimal solutionTheorem0.11. The Xα is the optimal solution of the Problem,0.5if and only if Xα∈Xα and there exist nonnegative real numbers λ1, λ2and λ3, such that λ1g1(Xα)=0, λ2g2(Xα)=0, λ3g3(Xα)=0andIn the fourth section, we studied if the optimal solution exists, it should have which form.We give the necessary and sufficient condition for the first case happen.Theorem0.12. Denote B:={B∈F; Eμ[IB]>0, Ev(IB)=0} and if B is empty, we define (β:=0.then,(?)α∈(0,1), there exists Xα0such that Eμt[Xα0]<n and Ev[Xα0]=γα if and only if β>1-α.Then we divided into two cases to get our result.Theorem0.13. If there exists functional Xα0reaching γα such that Eμ[Xα0]<a. then Xα0must be able to be expressed as where HQα0and Gpα0are the Radon-Nikodym derivatives of Qα0∈Q and Pα0∈ρ with respect to K:=Pα0+Qα0/2. λ0=0, B is a random variable with values in the intcrval [0.1]. andTheorem0.14. If Xa is the second type optimal solution. then Xa must be able to be expressed aswhere HQα and Gpα are the Radon-Nikodym derivatives of Qα∈(?) and Pα∈ρ with respect to K:=Pα+Qα/2.B is a random variable with values in the interval [0,1].In the last section, we conclude this chapter and give some examples for the cases we discussed above.In the fifth chapter, we studied hypothesis testing problem in continuous bounded functions. From the point view of the operator defined on functions, we con-structed a new framework.In fact, it is coincide with from the view point of probability defined on sets. We can also get many important results. The benefit from our new view point is that we can get the Daniell-Stone theorem which can not be obtained from the old one.In the second section, we introduced our framework, and give the Radon-Nikodym theorem under our new framework.Theorem0.15(Radon-Nikodym theorem). If (Ω,H,L1) is a functional measurable space, L2is a σ-additive operator defined on (Ω,H),then there exist φ∈H and g∈H[0,1] such that L1(g)=0andIn the third section, we extended the σ-additive operator defined on a linear lattice to its responding linear a-lattice.Theorem0.16. If L is a σ-additive operator on linear lattice H, then there exists a unique extension L on σ(H) and L is also σ-additive.In the fourth section, we proved the Neyman-Pearson lemma in linear case. Theorem0.17(Neyman-Pearson lemma). Let L\and L2are two different σ-additive operators on a function measurable space (Ω,H). Take φ1:=dL1/dL and φ2:=dL2/dL, where L:=L1+L2/2, then for the hypothesis testing Problem5.1, we have(i) For any given significance level α∈(0,1), thcre exists a test function h(ω) and constant λ0≥0, such that L1(h)=α and(ii) The h(ω) determined by (i) is the most powerful test and all the most powerful tests must be the form of the (i).In the fifth section, when the space is restricted in bounded continu-ous functions or quasi-continuous functions, and operators are sub-linear, we studied if the optimal solution exists, the specific form it should have.Similar to the fourth chapter, we divided into two cases to get our results.Theorem0.18. If there exists functional hα0such that ρμ[hα0]<α reaching γα, then there exist two σ-additive operators Lα0m and Lα0n such that hα0can be expressed aswhere Hm and Gn are the Radon-Nikodym derivatives of Lα0m and Lα0n with respect to K:=Lα0m+Lα0n/2.λ0=0B is a random variable with values in the interval [0,1]. andTheorem0.19. If for any ρμ[f]<α, f∈Cb(Ω), we have Pv[f]<γα. If h is the optimal solution of the problem5.2, then h must be able to be expressed aswhere Hm and Gn are the Radon-Nikodym derivatives of Lαm∈M and Lαn∈N with respect to K:=Lαn+Lαm. B is a random variable with values in the interval [0.1].