| The study for limit theory under nonlinear expectation space is inspired by the uncertainty and ambiguity issues in areas of Quantitative Economics, Financial Risk Measure and Quantum Mechanics, as well as the trends of the research about funda-mental theorems in classical Probability Theory and Mathematical Statistics.It was in 1970s the modern financial derivatives came into the world, and since then financial risk has never left financial market. For example, we could find that the risk, who is full of uncertainty, played very important roles both in the bankruptcy of Long Term Capital Management and the collapse of the century old Lehman Brothers. Actually, Knight (1921) had tried to make a classification about financial risk:some risks come from the uncertainty which can be quantified, i.e. market participants have a common sense about the behavior (probability distribution) of financial products, we call this uncertainty the Knightian Risk; the others come from the uncertainty which could not be quantified, i.e. risk managers don’t know the product’s probability distribution or market participants have different understandings about the same product, this part is the well-known Knightian Uncertainty or Ambiguity. The research about ambiguity is always a hot topic in Economics or Finance, such as the famous Ellsberg paradox from Ellsberg (1961), Nobel economics prize winners also use ambiguity to discuss some issues from Macroeconomics in their works Hansen, Sargent and Talarini (1999) and Hansen and Sargent (2001), and we can also find ambiguity in Epstein and Wang (1994) and Chen and Epstein (2002) which make some great contributions to Asset Pricing Theory. However, during the study, people sometimes found that the linear assumption about probability and expectation could not describe the natural sublinear behavior of risk any more.Peng (1997) introduced a new nonlinear expectation—g-expectation via Backward Stochastic Differential Equations (BSDE for short), which provides reasonable expla-nations for many uncertainty issues in finance. On the other hand, Artzner, Delbaen, Eber and Heath (1999) presented the concept of Coherent Risk Measure from the fi-nancial mathematics angle, i.e. a functional ρ(X) endowed on the contingent claims X, which is essentially a sublinear mathematical expectation. In fact, to study the sublin-ear behavior of risk is to study the sublinear mathematical expectation, and at the same time, sublinear expectation provides a robust approach to evaluate financial risk under uncertain environment.However, in previous study about financial issues using g-expectation or coherent risk measure, authors usually discuss finite financial products in a given time period, such as Barrieu and Karoui (2004), Chen and Kulperger (2006), Biagini, Fouque, Frittelli and Meyer-Brandis (2015). So how to describe contingent claims’ limit behaviors via these robust instruments under uncertain environment is becoming an interesting problem as the explosion of data and the urgency of prediction of future risk are increasing rapidly.On the other hand, nonlinear expectations and capacities also have great values in the theoretical study. Actually, Choquet (1953) firstly introduced the concept of Ca-pacity and Choquet integral many years ago, which are very important developments of classical linear probability theory. Inspired by random sets, Dempster (1967) de-fined upper and lower probabilities and corresponding expectations by different ways. Recently, Peng (2010) developed a further general framework of sublinear expectation— G-expectation, which is very different with g-expectation. Specially, as a very important development and continuation of classical probability theory, the limit theory under non-linear expectation space has attracted great interests and attentions from scholars and researchers, please see Walley and Fine (1982), Dow and Werlang (1994), Marinacci (1999), Epstein and Schneider (2003), Maccheroni and Marinacci (2005), Cooman and Miranda (2008), Chen and Wu (2011), Chen, Wu and Li (2013), Teran (2013), Zhang (2014), Chen and Chen (2014). Authors and scholars have presented many significant results under different conditions and spaces in these works, however, how to weak their assumptions and obtain the theorems for general sublinear expectation spaces is still an important and meaningful problem.In view of above problems, this doctoral dissertation conducts research on these issues and presents following results, we believe some of them are very interesting and innovative:1. Starting from a financial problem, we study limit behaviors of stock price under Knightian uncertainty environment via properties of g-expectation. Furthermore, we generalize this method to a general sublinear expectation space where the family of probabilities is no longer absolutely continuous.2. We conduct a deep research on limit, behaviors of Brownian motion under g-expectation space. And for the first time, we obtain a limit relation between the general sublinear expectation space and the p-expectation space, presenting a law of large num-bers connecting these two space, which is a very innovative result. We obtain two types of law of large numbers and their equivalent relation using this result. Moreover, we apply these results to some practical problems about financial risk measurement, and gain deeper understandings about limit behavior of stock price.3. With the help of limit theory under sublinear expectation space, we make a discussion about some famous hypotheses in Number Theory and obtain some new un-derstandings of them. Although the proofs under stochastic framework do not mean the settlement of these great hypotheses, we believe this discussion is very interesting as a return of probability theory to number theory, since some ideas of limit theorems in probability originated from number theory.4. Furthermore, we also obtain a law of large numbers under G-expectation space, and extend some convergence and stability results in classical stochastic analysis to G-expectation’s framework.Now we will introduce above results by chapters. These results come from seven academic papers I finished during my Ph.D. study, two of them have been officially pub-lished by SCI magazines.Chapter One Starting with a financial problem, we discuss the limit behavior of stock price in Knightian uncertainty environment under Chen and Epstein (2002)’s framework, and for the first time we obtain a law of large numbers for stock price in g-expectation space (Ω,F,εk,PK). Then we extend this method to a general sublinear expectation space (Ω,F, P, EP) where EP= supP∈P EP, obtaining a law of large number for a sequence of random variables satisfying exponential independence. More important, the probabilities in the family P need not to be absolutely continuous as the situation of g-expectation space.·1.1 Law of Large Numbers for Stock Price under Knightian UncertaintyEnvironmentIn this chapter, we use the following geometric brownian motion to characterize the price of a stock in financial market: where h,σ> 0 are constants,and So is a positive random variable.Definition 1. Consider the following family of probability measures from Chen and Epstein (2002) to describe the Knightian uncertainty environment in financial market, i.e. where k characterizes the degree of ambiguity in the market, which is usually called κ-Ignorance. Moreover, we set andIn fact, (Pκ,Pκ) is a pair of g-capacities induced by BSDE with generator g(t,y,z)= k|z|. Thus we keep the same symbol for different terminal time because of the consistence of g-expectations.Theorem 1. Let{Si}i=1∞ be the values of stock price process (0.0.9) at time t=1,2,. denote Sn=logSn-log So,κ:=h-1/2σ2+kσ and κ:=h-1/2σ2-kσ, then for any ε>0. andUsually, volatility is always positive in the market, however, we still give the math-ematical results for σ≤0.Corollary 1. If σ<0, denote{Si}i=1∞ is the values of solution of SDE (0.0.9) at time t=1,2,..., set Sn=log Sn-log S0, then for any ε>0 we have Corollary 2.If σ=0,set Sn=logSn-logS0,then for any￡>0,The following result provides a robust interval estimation for stock price in the future under Knightian uncertainty environment. Theorem 2.For any￡>0, 1.If σ≥0,we have 2.If σ<0,we have.1.2 Generalizations for Law of Larrge Numbers in some Sublinear Expec-tation SpacesWe will generalize the method in Theorem 1 to some sublinear expectation spaces, firstly we consider following equations in g-expectation space:Theorem 3.Consider above FBSDE system,assume generator g doesn’t contain y and has subadditivity and positive homogeneity with respect to z.Set εg and(Pg,Pg)are cor-responding g-expectation and g-capacity,Sn=Σi=1nXi=Σi=1n(φi(Xi)-φi-1(Xi-1)). If there exists measurable function φi such that for any i∈N we have εg[Xi]=εg[X1],-εg[-Xi]=-εg[-X1] and εg[Xi2]<∞,then for any ε>0, andUnder the framework of g-expectation, probabilities from Pκ are absolutely contin-uouw with each other.Now we consider a more general situation,for any nonempty set P of probability measures, we conduct the research under sublinear expectation space (Ω,F,P，EP),where EP[X]:=supP∈PEp[X].Definition 2.(Exponential Independence)Consider sublinear expectation space (Ω,F,P，EP),if for φ(x)=ex,we have then random variable Y is exponential independent with X under EP[·].Correspondingly, if for any i=1,2,…,Xi+1 is exponential independent,with ∑j=1iXj,{Xi)i=1∞ (?)H is called a sequence of exponential independent random variables.In fact,exponential independence comes from the definition of negative correlation under linear probability,please see Remark 1.10 in Chapter One.We have the following result under this independence condition；Theorem 4.Let {Xi)i=1∞ be a sequence of exponential independent random variables in sublinear expectation space(Ω,F,P，EP),if EP[Xi]=μ,-EP[-Xi]=μ and for any α> 0,we have supi∈N+EP[|Xi|1α]<∞.Note VP(A):=supP∈PP(A) and vP(A):=infP∈PP(A) are corresponding upper and lower capacities,denote Sn=∑i=1n Xi,then for any ε>0 andChapter Two This chapter is composed of two parts:Inspired by the results in chapter one,we conduct a deep study about the limit behavior of Brownian motion under g-expectation space,obtaining two types of law of large numbers,large deviation principle and central limit theorem about Brownian motion.On the other hand,different with previous works about limit theory which usually only focus on one probability space,we obtain the limit relation between a general sublinear expectation space(Ω,F,E)and g-expectation space (Ω,F,εg). We present a law of large number connect-ing these two expectation space, which means to study the limit behavior of Σi=1nXi/n in (Ω,F,E) could be turn to study the properties of Bn/n under (Ω,F,εg). In fact we obtain three types of law of large numbers for a sequence convolu-tionary independent random variables and their equivalent relation using this result. More important, the sublinear expectation E in this chapter need not to be in the form of max-expectation, and the assumption for independence is also more general.·2.1 The Limit Theory of Brownian Motion Under g-Expectation Space Definition 3. Consider following two BSDEs in (Ω,F,P), and where μ≤μ. Denote the corresponding g-expectations and g-capacities as (εg,Pg) and (εg, Pg), and we don’t make any distinctions on terminal time because of the consistence of g-expectations. Lemma 1. εg[ξ]=-εg[-ξ], for ξ∈L2(Ω,F,P).Denote (Ω,F,εg) is the g-expectation space induced by BSDE (0.0.10) and (0.0.11), and (Pg,Pg) are upper and lower g-capacities. Now we introduce the laws of large numbers, large deviation principle and central limit theorem for Brownian motion.Theorem 5. In g-expectation space (Ω,F,εg), we have following law of large numbers for Brownian motion, i.e. for any function φ∈Cb(R),Theorem 6. Note Pg is the g-capacity induced by BSDE (0.0.11), i.e. Pg(A)=εg[IA]=-εg[-IA], then for any ε> 0,Theorem 7. In g-expectation space (Ω,F,εg), the following two laws of large numbers are equivalent with each other:1. For any ε> 0,2. For any function φ∈Cb(R)Theorem 8. Note Pg and Pg are upper and lower g-capacities in (Ω,F,εg), i.e. Pg(A)= ε[Ia] and Pg(A)=εg[Ia]=-εg[-IA],then we have following large deviation principle for Brownian motion: 1. for any 2. for any 3. for any 4. for anyTheorem 9. In g-expectation space (Ω,F,εg), we have following central limit theorem for Brownian motion, i.e. for any x∈R: and where Φ(x) is the probability distribution function of normal distribution.·2.2 Limit Relation between General Sublinear Expectation Space and g-Expectation SpaceDefinition 4. Consider measurable space (Ω,F), denote H is a set of random variables on it. If the functional E:H'(-∞,+∞) defined on (Ω,F,H) satisfies following four properties, we call E a sublinear expectation,1. Monotonicity:E[X]≥E[Y] for X≥Y;2. Constant preserving:E[c]=c if c is a constant;3. Subadditivity:E[X+Y]≤E[X]+E[Y];4. Positive homogeneity:E[λX]=λE[X] for constant λ≥0.(Ω,F,E) is called a sublinear expectation space. For a given sublinear expectation E, we can define its dual expectation ε[X]=-E[-X]. Correspondingly, upper and lower capacities V(A)= E[IA] and v(A)=ε[IA], A∈F. This sublinear expectation E is more general than EP in chapter one, which is a max-expectation for a set P of probability measures.Definition 5. (Convolutionary Independence) In (Ω,F,E), if for any give function φ we have then random variable Y∈H is said to be convolutionary independent with X∈H with respect to function φ under E. Moreover, if for any i=1,2,...,Xi+1 is convolutionary independent with Σj=1iXj with respect to φ, then {Xi}i=1∞(?)H is called a convolutionary independent sequence with respect to φ.Convolutionary independence comes from convolutionary relation between random variables in classical probability theory, which is weaker than independence under linear expectation. This definition also relaxes restrictions of independent condition in Peng (2010), please see Remark 2.4,2.5 and 2.6 in Chapter Two.Theorem 10. (Limit Relation Between Two Space) Suppose that {Xi}i=1∞ is a convolutionary independent sequence with respect to any φ∈Cb2(R) under (Ω,F,E), assume E[Xi]=μ,ε[Xi]=μ, and supi∈N+E[|Xi|2]<∞. Denote Sn:=Σi=1nXi,and εg is the g-expectation induced by BSDE (0.0.10), then for any φ∈Cb(R) ,Theorem 10 means to study the limit behavior of Σi=1nXi/n in (Ω,F,E) could be turn to study the properties of Bn/n under (Ω,F,εg). Using the properties of g-expectation, we could get the following law of large numbers in general sublinear expectation space. Theorem 11. Suppose {Xi}i=1∞ satisfy the assumption of Theorem 10, set Sn:= Σi=1nXi, then for any φ∈Cb(R),Theorem 12. Suppose that {Xi}i=1∞ is a convolutionary independent sequence with respect to any nonnegative monotone function φ∈Cb2(R) under (Ω,F,E),assume E[Xi]=μ,ε[Xi]=μ, and supi∈N+E[|Xi|2]<∞. Denote v is corresponding capaci-ty in (Ω,F,E), i.e.v(A)= ε[IA], then for any ε> 0,Although Theorem 12 and 4 have similar results, they can’t be derived from each other, since they have different assumptions on expectations and independent conditions. In fact, they are proved by different methods. The following theorem provides a suffi-cient condition for the mutual equivalence of three laws of large numbers in sublinear expectation space.Theorem 13{Xi}i=1∞ (?)H is a sequence random variables in (Ω,F,E), assume it is a convolutionary independent sequence with respect to any φ∈Cb+(R).and E[Xi]=μ, ε[Xi]=μ, supi∈N+E[|Xi|2]<∞. Denote Sn:=Σi=1nXi, v(A)=ε[IA], A∈T, then following three laws of large numbers are equivalent with each other:Ⅰ. for any φ∈Cb(R)Ⅱ. for any ε>0,Ⅲ. for any φ∈Cb(R),We also introduce some applications of above results in coherent risk measures, and obtain more understandings about the limit behaviors of stock price, please see §2.2.5 and §2.5.Chapter Three This chapter is also composed of two parts:In the first part, we present a law of large numbers under a new nonlinear expectation induced by G-BSDE under G-expectation space. In the second part, with the help of limit theory under sublinear expectation space, we conduct a discussion about some famous hypotheses in Number Theory and obtain some new understandings of them.·3.1 Law of Large Numbers under G-Expectation SpaceThe study in this chapter is conducted under G-expectation’s framework introduced by Peng (2010). The following result is a lemma for Theorem 15, however, it can also be regarded as a p-moments law of large numbers in sublinear expectation space.Theorem 14. Let{Xi}i=1∞ be a sequence R-valued random variables in sublinear expectation space (Ω,H,E), and {Xi}i=1∞ LPG(Ω). p ∈ N. Assume for any i= 1,2,..., Xi+1d=Xi, Xi+1 is independent with X1,...,Xi and E[X1]=-E[-X1], denote Sn:= Σi=1nXi, then as n'∞,Definition 6. Consider the following backward stochastic differential equation driven by G-Brownian motion(G-BSDE for short), Define EtG[ζ]:=YtT,ζ and in particular when t=0, we have a nonlinear expectation EG[ζ［ induced by above G-BSDE.Now we introduce a law of large numbers under this nonlinear expectation.Theorem 15. Let {Xi}i=1∞ be a sequence R-valued random variables in sublinear expectation space (Ω,H,E), and {Xi}i=1∞ (?)LG2(ΩT).Assume for any i=1,2 Xi+1d=Xi, Xi+1 is independent with X1,...,Xi, and E[X1]= -E[-X1]. Denote Sn:=Σi=1nXi, for any φ∈Cb,Lip(R),· 3.2 Discussion of Number Theory Hypotheses via Limit Theory under Sublinear Expectation Space Definition 7. Define Mobius function on positive integers as follows,The Mertens function is defined as M(N)=Σ=1Nμ(n),which is very important in the study of Riemann Hypothesis. In fact, there is a famous proposition in number theory:Proposition 1. Let ζ(s) be Riemann-Zeta function, then the following two conjectures are equivalent:1. For any ε> 0, there exists a constant Cε such that|M(N)|< CεN1/2+ε.2. (Riemann Hypothesis) If ζ(s)=0 is true for some s with 0<Re(s)<1,then Re(s)=1/2.In number theory, Mobius function has a special randomness, please see Remark 3.7, Remark 3.8 and §3.2.3. So we regard μ(n), n=1,2,3... as an ⅡD sequence random variables in sublinear expectation space (Ω,F, P,E), and denote (V,v) are upper and lower capacities for P, then we have following results for Mertens function in stochastic sense,Theorem 16. Note M(N)=Σn=1Nμ(n), we haveTheorem 17. If{bN} is a increasing sequence of positive numbers such that n1/2/bN'0, then for any ε>0,Corollary 3. For any ε> 0, there exists a constant CE> 0 such thatRemark 1. Note the fact that for any ε> 0, there exists a constant Cε such that (?)< CεNε, thus above results imply that Riemann Hypothesis is true in s-tochastic sense under capacity v. More important, the framework of our discussion is more reasonable than assumptions of μ(n) in previous works of probabilistic number theory, please see Remark 3.7 and 3.8.Remark 2. Good and Churchhouse (1968) proposed the following conjecture about Mertens function, which hasn’t been proved or denied, i.e. By Theorem 16, we think this conjecture seems to be very doubtful.Remark 3. Odlyzko and te Riele (1985) proposed another conjecture about Mertens function, i.e. we give a proof for this in the sense of capacity, i.e. Theorem 18. Theorem 18. Denote M(N)=Σn=1Nμ(n); we haveChapter Four This chapter extends some classical results to the frame-work of G-expectation. We discuss almost periodic solutions and their asymptotic stability for stochastic differential equation driven by G-Brownian motion (G-SDE for short), the stability in capacity for G-SDE and stability of the solution of G-BSDE.·4.1 Almost Periodic Solutions and their Asymptotic Stability for G-SDE Theorem 19. Consider following G-SDE,dXt=AXtdt+F(t,Xt)dt+G(t,Xt)d(B)t+H(t,Xt)dBt, t∈R,(0.0.13) if the coefficients satisfy (H1) and (H2) in §4.1.2, and then G-SDE(0.0.13) has an unique square-mean almost period mild solution. Example 1. Consider following G-SDE satisfying Dirichlet condition, set where V(A)={x∈C1[0,1]| x’(r) is absolutely continuous on [0,1], x" ∈ L2[0.1], x(0)= x(1)=0}. Then it’s easy to check G-SDE(0.0.14) satisfies (H1) and (H2) in §4.1.2, and has an unique square-mean almost period mild solution.Theorem 20. Suppose G-SDE (0.0.13) satisfies the assumptions in Theorem 19, and then its unique square-mean almost period mild solution Xt* is asymptotically stable in square-mean sense.·4.2 Stability in Capacity for G-SDETheorem 21. Consider following G-SDE, if there exists a positive-definite function V(x, t)∈C2,1(Sh×[0,∞)) such that for any (x,t)∈Sh×[0,∞), then the trivial solution of G-SDE (0.0.15) is stochastically stable in v.Theorem 22. If there exists a positive-definite decrescent function V(x, t)∈C2,1(Sh× [0,∞)) such that is negative-definite, then the trivial solution of G-SDE (0.0.15) is stochastically asymp-totically stable in V.Theorem 23. If there exists a positive-definite decrescent radially unbounded function V(x,t)∈C2,1(Sh×[0,∞)) such that is negative-definite, then the trivial solution of G-SDE (0.0.15) is stochastically asymp-totically stable in the large in V.Remark 4. Consider following functions in above three theorems, we can get similar results for next G-SDE where h≠0,Example 2. Consider G-SDE (0.0.15), assume the coefficients satisfy following condi-tions uniformly for t in a neighbourhood of x=0, where f(t) and g(t) are bounded real-valued functions. Moreover, we assume there exist two positive constants H and K such that Set 0≤σ≤σ=1 in G-expectation space without loss of generality, then we have It’s easy to check V(x,t) satisfies the conditions in Theorem 22, so the trivial solution of G-SDE(0.0.15) is stochastically stable in v and stochastically asymptotically stable in V.●4.3 Stability for G-BSDE on CoefficientsIn this section, we consider following family of G-BSDEs dependent with δ(δ≥0),Theorem 24. Suppose the coefficients of G-BSDE (0.0.16) satisfy assumptions (A1), (A2) and (A3) in §4.2.3, then the solution Ytα is stabile in following sense, as δ'0,Theorem 25. Suppose the coefficients of G-BSDE (0.0.16) satisfy assumptions (A1), (A2) and (A4) in §4.2.3, then the solution (Ytδ, Ztδ, Ktδ) ∈(0, T) is stabile in following sense, as δ'0,... |
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