Research About Non-additive Probabilities And Backward Stochastic Differential Equations

Non-additive set functions, as for example outer measures, appeared naturally earlier in the classical measure theory concerning countable additive set functions or more general finite additive set functions. The pioneer in the theory of non-additive set functions was G. Choquet [23] from 1953 with his theory of capacities. This theory had enormous influences on many parts of mathematics and different areas of sciences and technique. Non-additive set functions are used in mathematical economics, decision theory and artificial intelligence, called by various names, such as cooperative game, capacity or fuzzy measure. Recently, many authors have investigated different kinds of non-additive set functions, as upper probabilities, belief functions, 2-alternating capacities, null-additive set functions , and many other types of set functions (see [29], [31], [49], [84], [88] and so on). Among them is g-probability introduced via backward stochastic differential equations (BS-DEs for short). Pardoux and Peng [69] introduced a kind of equations called BSDEs and proved the existence and uniqueness of solutions. Since then, BSDE has not only developed rapidly in the fundamental theories (see [26], [50] and so on), but also become a powerful tool in applications of mathematical finance and mathematical economics (see [58], [59] and so on). Peng [71] introduced the notion of g-expectation via BSDEs. He showed that under suitable integrability assumptions on the generator g and the terminal valueξ, the g-expectation of a random variableξpreserves most basic properties of the classical mathematical expectation except linearity since g is a nonlinear functional. Via g-expectation, a non-linear probability: g-probability was introduced naturally. This dissertation focuses on the research about non-additive probabili-ties and BSDEs. Main results are as follows:1. Laws of large numbers of a sequence of pairwise negatively corre-lated random variables for upper probabilities and 2-alternating capacities are proved.2. We study the 2-alternating property of g-probability, prove Cheby-shev's inequality and laws of large numbers for g-probability, study the vari-ance, correlation and correlation coefficient of random variables under the framework of g-expectation.3. We establish an extended variational formula and prove some results by means of BSDEs.This doctoral thesis consists of three chapters, whose main contents are described as follows:In chapter one, we study laws of large numbers for capacities. We mainly consider a sequence of pairwise negatively correlated random variables. We prove weak and strong laws of large numbers for upper probabilities and 2-alternating capacities. A natural approach is to use the results in the classical probability theory to obtain our results for upper probabilities. As to 2-alternating capacities, because of its own sub-additivity of the induced Choquet integral, we can strengthen the results by establishing a non-additive version of Chebyshev's inequality and Boral-Cantelli lemma. We will see that the limit of averages is a number for 2-alternating capacities while it is in an interval for upper probabilities. Our results hold in a measurable space. Now we give our main theorems in this chapter.The next two results are strong and weak laws of large numbers of random variables for upper probabilities.Theorem 1.4.3 Let V be a continuous upper probability, C be the Choquet integral induced by V, {X_n}_n∈N be a sequence of random variables which are pairwise negatively correlated relative to V. Suppose there exists a constant M such that C[X_n²]≤M for all n. Denote S_n=(?).ThenTheorem 1.4.6 Under the same conditions in Theorem 1.4.3, for V (?) > 0, we can getThe next two results are weak and strong laws of large numbers of random variables for 2-alternating capacities.Theorem 1.5.3 Let V be a 2-alternating capacity, C be the Choquet integral induced by V, {X_n}_n∈N be a sequence of random variables which are pairwise negatively correlated relative to V. Suppose C[X_n] = 0 for all n, and there exists a constant M such that C[X_n²]≤M for all n. Denote (?)Then for (?) > 0,Theorem 1.5.9 Let V be a continuous 2-alternating capacity, C be the Choquet integral induced by V, {X_n}_n∈N be a sequence of random variables which are pairwise negatively correlated with respect to V. Suppose C[X_n] = 0 for all n, and there exists a constant M such that C[X_n²]≤M for all n. Denote (?). Then In chapter two, we investigate g-probability introduced via BSDEs. By Pardoux and Peng [69], we know there exists a unique adapted square inte-grable solution to the following BSDE:providing that the function g satisfies the square integrable assumption (H1) and the Lipschitz assumption (H2). If function g also satisfies assumption (H3): g(y,0,t) = 0 for any (y,t)∈R×[0,T], then y₀(ξ), denoted byε_g[ξ], is called g-expectation ofξ, and for an event A,ε_g[I_A], denoted by P_g(A), is called g-probability of A. It is well known that g-probability is a kind of capacities. This chapter surrounds the study of properties of g-probability and makes a research from four respects. We have BSDEs as the tool to study g-probability in all through. Thus it is more convenient to study than other capacities.Firstly, we show the relationship between g-probability and 2-alternating capacity. Most work on capacities has focused on the 2-alternating case. It is very difficult to establish an equivalence relation between 2-alternating capacity and g-probability. But when g is an odd function, We can show that a g-probability which is 2-alternating turns out to be additive.Theorem 2.2.6 Suppose function g satisfies assumptions (H2) and (H3), and g is an odd function. Then the following conditions are equivalent:(1) P_g is 2-alternating.(2) P_g is linear.Secondly, we consider Chebyshev's inequality which plays an important role in the classical probability theory. This draws out a natural question: under which conditions on g, does a g-probability satisfy Chebyshev inequality? In this chapter, we shall deal with this question. Under conditions (H2) and (H3) on g, we show that if g also satisfies assumption (H): for any (y,z,t), g(λy,λz,t)=λg(y,z,t) for allλ≥0, then Chebyshev's inequality holds for g-probability. Under the same conditions, Markov inequality and exponential inequality can be obtained in a similar way.Theorem 2.3.4 (Chebyshev's inequality) Supposeξ²âˆˆL²(Ω,F,P), function g satisfies assumptions (H2), (H3) and (H). Then Chebyshev's inequality for g-probability holds, that is for (?) > 0,Theorem 2.3.6 Let |ξ|^r∈L²(Ω,F,P), real number c> 0, and e^cη L²(Ω,F,P). Suppose that function g satisfies assumptions (H2), (H3) and (H). Then Markov inequality of g-probability holds,and exponential inequality of g-probability holds:Based on Chebyshev's inequality, if g also satisfies assumption (H4): g is sub-linear, we can prove one-sided chebyshev's inequality.Theorem 2.3.7 (one-sided Chebyshev's inequality) Suppose thatξ²âˆˆL²(Ω,F,P),ε_g[ξ]=0, function g satisfies assumptions (H2), (H3) and (H4). Then one-sided Chebyshev inequalities for g-probability holds:Thirdly, we study laws of large numbers of random variables in a broad sense for g-probabilities. In the BSDE theory, an important set of probability measures is defined by whereμis the Lipschitz constant in (H2). The key of our method lies in the given set P of probability measures. We study the relationship between laws of large numbers in a broad sense for g-probabilities and for probability measures in the given probability set P. We can draw the conclusion: we could study the corresponding laws of large numbers in a broad sense for probability measures in P in order to establish laws of large numbers for g-probabilities. We give our main theorems.Theorem 2.4.16 Let P_g be a g-probability, {X_n}_n∈N be a sequence of random variables in L²(Ω,F,P). Suppose that there exists a probability measure Q⁰ in V, such that {X_n}_n∈Nobeys weak (strong) law of large numbers in a broad sense under Q⁰, then {X_n}_n∈N obeys weak (strong) law of large numbers in a broad sense under P_g.Theorem 2.4.18 Let P_g be a g-probability, {X_n}_n∈N be a sequence of random variables in L²(Ω,F,P). Suppose that {X_n}_n∈N obeys weak (strong) law of large numbers in a broad sense under g-probability P_g, then for every probability measure Q in V, {X_n}_n∈N obeys weak (strong) law of large numbers in a broad sense under Q.In the classical probability theory, Chebyshev's inequality plays a fundamental role in proofs of various forms of laws of large numbers. Based on the Chebyshev's inequality for g-probability, we could establish another version of weak law of large numbers for g-probability.Theorem 2.4.23 Let {X_n}_n∈N be a sequence of random variables in L²(Ω,F,P). Suppose function g satisfies assumption (H2), (H3), and g is positively homogeneous. Denote (?) when n→∞,(?)ε_g[(S_n-ε_g[S_n])²]→0, then a weak law of large numbers of {X_n}_n∈N holds for g-probability, that is, for (?) > 0,At the end of this chapter, we introduce the definitions of variance, cor- relation and correlation coefficient of random variables under g-expectations and study their elementary properties. We investigate the similarities and differences of the properties between this non-additive case and the classical case.The definiton of variance of a random variable under g-expectation is as follows.Definition 2.5.13 Letξbe a random variable in L⁴(Ω,F,P). The variance ofξunder g-expectation is defined byThe next theorem suggests that when g satisfies assumptions (H2), (H3) and (H4), the g-expectation of the squared distance of its possible values of a random variable from other constants may be smaller than its variance. However, the variance or fluctuation around -ε_g[-ξ] is smaller than around any other constants which are outside interval [-ε_g[-ξ],ε_g[ξ]].Theorem 2.5.16 Letξbe a random variable in L⁴(Ω,F,P). Suppose function g satisfies (H2)-(H4). ThenWe introduce the defintion of correlation and give the correlation of random variables in a particular form.Definition 2.5.19 Letξ,ηbe random variables in L⁴(Ω,F,P). The covariance betweenξandηunder g-expectation is defined byDefinition 2.5.21 Letξ,ηbe random variables in L⁴(Ω,F,P). We callξandηare positive correlation under g-expectation if Cov_g(ξ,η)≥0, negative correlation under g-expectation if Cov_g(ξ,η)≤0, and no correlation under g-expectation if Cov_g(ξ,η)=0. Theorem 2.5.25 Suppose function g satisfies assumptions (H2)-(H4). LetΦ₁(X_T¹) andΦ₂(X_T²) be random variables defined by (2.25). SupposeΦ₁ andΦ₂ have the same monotonity andσ₁(t,X_t¹)≥0,σ₂(t,X_t²)≥0. ThenΦ₁(X_T¹)andΦ₂(X_T²) are positive correlation, that isDefinition 2.5.28 Letξ,ηbe random variables in L⁴(Ω,F,P). The correlation coefficient betweenξandηunder g-expectation is defined byWhen function g satisfies assumptions (H2)-(H4) and (H5): g(λz)≥λg(z) for allλ, we can prove that correlation coefficient between two random variables under g-expectation also lies between -1 and 1, and the linear relationship is equivalent to the value of correlation coefficient is 1.Theorem 2.5.31 Letξ,ηbe random variables in L⁴(Ω,F,P) such that D_g[ξ]≠0, D_g[η]≠0. Suppose g satisfies assumptions (H2)-(H5). Then(2)ρ_ξ,η=1(?)there exist constant a > 0, b, such that P(η=aξ+b)=1.In chapter three, we use a BSDE with quadratic growthwhereξand h(y, z, t) = (?)g(e^y,e^yz,t) satisfy suitable assumptions to prove an extended variational formulawhereε_g[·] is g-expectation introduced in Peng [71], n^Q(·) is a process satisfying a stochastic differential equation. The classical variational formula is a particular case here when g = 0. Theorem 3.3.2 Letξbe a bounded measurable random variable on (Ω,F), g be a function satisfying (H2) and (H3), n^Q(·) be a process satisfying SDE (3.4). Then the following results holds:(a) We have the extended variational formula(b) The infimum in the above equation is uniquely attained at Q^*, andWe prove the following variational representation on canonical space for certain functionals of Brownian motion showed in [9]where A is the set of all progressively measurable functions by means of BSDEs.Theorem 3.3.3 Let f be a bounded Borel measurable function mapping (C[0,1] : R) into R, (Y, Z) be the solution of BSDE (3.2) whenξ= -f(W). ThenTheorem 3.3.4 Let f be a bounded Borel measurable function mapping (C[0,1] : R) into R. Then we have the variational representation...