Theory Of Nonlinear Mathematical Expectations And Backward Stochastic Differential Equations

The basis of the classical probability theory is the assumption that prob-abilities or expectations are additive or linear. However, such additivity or linearity assumption is not reasonable in many areas of applications because many uncertain phenomena can not be well modeled using additive prob-abilities or linear expectations. For example, the famous Allais paradox shows that the Von Neumann&Morgenstein's axiomatic system of expected utilities which is the fundamental notion of the modern theory of economics, needs to be seriously modified. The paradox is essentially due to the linearity of the operator of mathematical expectations. More specifically, motivated by some problems in mathematical economics, statistics, quantum mechan-ics and finance, a number of papers have used non-additive probabilities or nonlinear expectations to describe and interpret the phenomena.Since the paper [1] on coherent risk measures, people are more and more interested in sublinaer expectation (or more generally, convex expectation). Roughly speaking, a sublinear expectation is a function defined on a linear space of random variables which satisfies monotonicity, constant preserving, sub-additivity and positive homogeneity. A sublinear expectation E can be represented as the upper expectation of a subset of linear expectations{Eθ: θ∈Θ}, that is E[X]=supθ∈ΘEθ[X](see [62]). In most cases, this subset is often treated as an uncertain model of probabilities{Pθ:θ∈Θ} and the notion of sublinear expectation provides a robust way to measure a risk loss X. Peng [59,60,61,62,63] initiated the notion of independent and identically distributed (ⅡD for short) random variables under sub-linear expectations. Peng also introduced the notion of G-normal distribution and G-Brownian motion as the counterpart of normal distribution and Brownian motion in linear case respectively. Under this framework, he proved a law of large numbers and a central limit theorem in [62]. G-expectation space is the most important sub-linear expectation space introduced by Peng [59], which take the role of Wiener space in classical probability. In G-expectation space, Peng introduced G-Ito integral, and get the formula of G-Ito integral (see [44,59]).Chapter1and Chapter2of this thesis will study the inequalities of a sequence of independent random variables and strong laws of large numbers in an upper expectation space respectively. Chapter3and Chapter4will study the invariance principles of G-Brownian motion for law of iterated logarithm and multiple G-Ito integral in G-expectation space respectively.Another important nonlinear expectation is g-expectation introduced by Peng [57] via a backward stochastic differential equation (for short BSDE) on finite time interval in which the generator is a given function g=g(ω,t,y,z): Ω×[0,T]×R×Rd→R. It was mainly during the last two decades that the theory of BSDEs took shape as a distinct mathematical discipline. The theory of BSDEs has important applications in stochastic optimal controls, stochastic differential game, financial mathematics, economics and the theory of partial differential equations. It has attracted a large number of mathe-maticians, economists and financial experts to research it.g-expectations and conditional g-expectations preserve most properties of the classical ex-pectations except the linearity. The nonlinearity of g-expectations can be characterized by their generator g. And g-expectation becomes a typical example of nonlinear expectations under which the time-consistency holds true, it is also a useful tool of studying nonlinear dynamic pricing as well as dynamic risk measures in finance(see [10,27,34]). Thus a theory of nonlin-ear martingales can be developed. Peng (1999) introduced the concepts of g-martingale, g-supermartingale/submartingale and proved the decomposi-tion theorem of Doob-Meyer's type (see[58]). Hu and Chen [33] generalized Peng's g-expectation and related properties. The last two chapters of this doctoral thesis study some fundamental problems about the theory of BSDEs and g-cxpectation space. This thesis consists six chapters. In the following, we list the main results of this thesis.(Ⅰ) In Chapter1, we study the inequalities for sum of random variables which are independent in an upper expectation space.Borel-Cantelli Lemma is one of the most fundamental theorems in clas-sical probability theory. Rcfs.[9,11,12] prove some Borcl-Cantclli Lemmas under capacity. In Chapter1, we first give a Borel-Cantelli Lemma in more weaker assumptions under capacity.Lemma1.2.5Borel-Cantelli Lemma. Let be a sequence of events in F and (V, v) be a pair of upper and lower probability generated by P.(2) For any n,m∈N*, then (3) Let v(·) be a lower continuous capacity, and for any n,m∈N*, then Suppose that is an independent random variables sequence in upper expectation space (Ω,F,P,E) with E[Xi]=ε[Xi]=0for i∈N*. The partial sum processes of {Xi}i-1∞is denoted by {Sn}n=1∞, that is Sn:=Σi=1nxi.Set Mn:=max(S1,…,Sn), mn:=min(S1,…,Sn), and the more general rank orders Then we establish the maximal inequalities, exponential inequalities, Marcin-kicwicz-Zygmund inequalities and their applications forTheorem1.3.1Maximal inequalities. Suppose that f is a nondecreasing function on R with f(0)=0; then for any n,j∈N*, in particular for any λ>0,Theorem1.4.1Exponential inequalities. Let be positive real numbers. If for any i∈N*,|Xi|≤Ci, then for any ε≥0,Theorem1.4.3Suppose|Xi|≤c<∞for i∈N*; then for r>1/2, n-rSn→0q.s..Theorem1.4.4Let be positive real numbers and|Xi|≤ci, i∈N*. Assume that Then for any r>0, we have n-rSn→0q.s.. Theorem1.5.2Marcinkiewicz-Zygmund inequalities. For p∈(1,2], we haveTheorem1.5.3For1<p≤2,‖Xj‖p<cj<∞, j∈N*. Then for any ε>0, we haveTheorem1.5.4For p>1,‖Xj‖p<cj<∞, j∈N*. Let us consider the following conditions for p∈(1,2]:(ⅰ)(?) and let r be a positive number such that pr>1;(ⅱ) where a is a positive number and let r be a positive number such that pr-1>α. If any one of the above conditions is true, then(Ⅱ) In Chapter2, we are interested in the strong law of large numbers for capacities, we develop new approaches to solve this problem. We obtain some new strong laws of large numbers for ca-pacities. They are natural extensions of the classical Kolmogorov's strong law of large numbers.Let (Ω,F,P,E) be an upper expectation space and X1,X2,…, Xn+1be real measurable random variables. Xn+1is said to be vertical independent of (X1,…,Xn) under E[·], if for each nonnegative measurable function φi(·) on R with i=1,…,n+1, we have {Xn} is said to be a sequence of vertical independent random variables, if Xn+1is independent of (X1,…, Xn) for each n∈N*.Then we give the following strong laws of large numbers for capacities.Theorem2.3.1Strong laws of large numbers. Let be a se-quence of vertical independent random variables in upper expectation space (Ω,F,P,E). Suppose supi≥1for some a>0, and (Ⅰ)andIf further assume v(·) is lower continuous, and for any subsequence of N*denoted by are mutually independent under v(·), and also there exist n0∈N*, c0>0such that Then we have (Ⅱ)It is of interest that using the above results, it is easy to obtain Strassen-type invariance principles (cf.[68]) of strong laws of large numbers for ca-pacities.Theorem2.3.4For any continuous function φ(·) on R, under the assump-tions of theorem2.3.1, we have In the last of Chapter2, we give an application of our strong laws of large numbers to Bernoulli-type experiments with ambiguity.(Ⅲ) In Chapter3, we investigate the invariance principle of G-Brownian motion for law of iterated logarithm adapting Peng's IID notion under G-expectations space. We also get some properties of random variables which are independent or identically distributed.Let (E,ε) denote the upper-lower expectations respective to G-expectation E[·].Proposition3.2.13Let X and Y be two random variables in G-expectation space (Ω), LG1(Ω),E), and both of them are continuous from Ω to R. If X=Y, then where φ(x)=IA(x), A is a finite union of closed intervals or a finite union of open intervals.Proposition3.2.14Let X and Y be two random variables in G-expectation space (Ω, Lg(Ω),E), and both of them are continuous from Ω to R. If X=Y, then where φ(x)=IA(x), A is a finite union of closed intervals or a finite union of open intervals.Proposition3.2.16In G-expectation space (Ω,LG1(Ω),E), let X,Y be two continuous random variables from Ω, to R, and Y is independent from X under E[·]. If A is a finite union of closed intervals and B is also a finite union of closed intervals, else if A is a finite union of open intervals and B is also a finite union of open intervals, then andLet {B(t)}t≥0denote the G-Brownian motion in G-expectation space, that is B(t)～N(0,[σ2, σ-2]). We only consider the case σ>0. Define Let C([0,1]) be the Banach space of continuous maps from [0,1] to R endowed with the supremum norm‖·‖, using the Euclidean norm in R. ζn is then a random variable with values in C([0,1]). For any/β≥0, define For any fixed ω, C(ζn(ω)) denotes the set of (norm-)limit points of {ζn(ω)}n=3∞Then C(ζn) is a random set. In this thesis, for shorten notations, we often omit the notation ω.Theorem3.3.1Let C(ζn) denote the cluster of sequence {ζn}n=3∞, thenMore precisely, the following invariance principle of G-Brownian motion for law of iterated logarithm hold.Theorem3.3.4If φ is a continuous map from C[0,1] to some Hausdorff space H, then we have(Ⅳ) In Chapter4, we introduce the multiple G-Ito integral of symmetric function in L2([0,T]n) and obtain the relationship be-tween Hermite polynomials and multiple G-Ito integrals as well. In order to introduce the definition of multiple G-Ito integral, we first introduce the following usual spaces of functions for n∈N*: whereFor any f on Sn:={(x1,…,xn)∈[0,T]n:0≤x1≤x2≤…≤xn≤T}(n∈N*), we define For we can define the (n-fold) iterated G-Ito integral by For convenience and completeness, we define J0T(c)=c,(?)c∈R.Noticing that for any g∈L2([0,T]m), we have Motivated by this, we give the following definition of multiple G-Ito integral.Definition4.3.1For each define And for each c∈R, define I0T(c):=0!J0T(c)=c.Let Hn(x) denote the nth Hermit3polynomial. We introduce some two-dimensional polynomial functions hn(x,y) as follows, for n∈N,(x,y)∈R2, Then we establish the relation between Hermite polynomials and multiple G-Ito Integrals. Theorem4.4.1Let g0=1. ForanyfeL2([0,T]),n∈N*,set gn(t1,t2,…,tn)=f{t1)f(h2)…f(tn). Then gn∈L2([0,T]n) and for n=0,1,2,…, where‖f‖T=[∫0Tf2(s)d(B)s]1/2is a nonnegative random variable and BT∫0T f(t)dBt. Immediately, we can give a general formula of∫0T∫0tn…∫0t2dBt1…dBtn-1dBtnCorollary4.3.3where [x] denotes the largest integer not greater than x.(Ⅴ)In Chapter5, we study multidimensional BSDEs. We introduce a new total order on Rn and obtain a necessary and sufficient condition for comparison theorem of multidimensional BSDEs under the new order.Definition5.3.1Let q∈Rn be any fixed nonvanishing vector. For any y1,y2∈Rn, we call y1bigger (or better) than y2under q, denote y(?)q y2, if (y1,q)≥(y2,q).Obviously, y1(?)q y2if and only if y1(?)q/|q|y2. So without loss of generality, we assume q to be a unit vector in the sequel. For any0≤u≤T, consider the following two BSDEs,Theorem5.3.3Suppose that g1and g2satisfy the following (A1)-(A3):(Al) For any (y,z)∈Rn×Rn×d,t→g(t,y,z) is continuous, P-a.s.,(A2) There exists a constant μ>0, such that for any t∈[0,T],(y,z),(y',z')∈Rn×Rn×d, we have (A3)g(t,0,0)∈ST2. Then, the following two statements (ⅰ) and (ⅱ) are equivalent:(i) For any such that then the unique solutions(Y1,Z1) and (Y2,Z2) in Su2×Hu2to BSDEs (0.2) over time interval [0, u] satisfy(ii) For any t∈[0, T],(y, z),(y', z')∈Rn×Rn×d, we havewhere C>0is a constant independent of (t, y, z).We also give some further results for special total order(?)9.(VI) In Chapter6, we consider the g-case. We study the prop-erties of g-martingale on infinite time interval and give a definition of g-potential with respect to classical potential. We establish a Riesz decomposition theorem of g-supermartingale. We also give some necessary and sufficient conditions under which the general-ized Peng's g-expectation is a sublinear expectation, without as-suming a priori that g is independent of y. Definition6.2.6A path right continuous nonnegative g-supermartingale {Xt}t≥0is called a g-potential, if {Xt}t≥0satisfyTheorem6.2.9The Riesz decomposition theorem of g-supermartingale. Let {Xt}t≥0be a right continuous g-supermartingale where g:Ω×R+×R×Rd→R satisfies the following conditions (Hl)-(H4):(H1) For (?)(y,z)∈R×Rd, g(·,·,y, z) is progressively measurable and (H2) g satisfies the Lipschitz condition: there exist non-random nonnegative functions v(t), u(t) such that dP x dt-a.s.,(?)(yi, zi)∈R×Rd, i=1,2,(H3)∫0∞v(s)ds<∞,∫0∞u2(s)ds<∞;(H4) dP×dt-a.s.,(?)y∈R, g(t,y,0)=0. Furthermore, g is super-additive, that is, dP×dt-a.s.,(?)(yi, zi)∈R×Rd,i=1,2we have If E[supt≥0Xt2]<∞, then Xt can be decomposed into the sum of a g-martingale and a g-potential, in other words, there exist g-martingale Yt, g-potential nt such that Xt=Yt+πt. Let where Θg={{θt}0≤t≤T|θ is Rd-valued, progressively measurable and dP×dt-a.s.,(?)(y,z]∈R×Rd,θt·z≤g(t,y,z)}. Then we get the necessary and sufficient conditions for the generalized Peng's g-expectation to be a sublinear expectation as the following theorem.Theorem6.3.8Let g satisfy (A1)-(A3):(Al) For all (y, z)∈R×Rd, g{·, y, z) is progressively measurable process;(A2) There exists a constant μ>0, such that P-a.s.,(?)t∈G[0,T],(?)(y,z),(y', z')∈R×Rd,|g(t,y,z)-g(t,y',z')|≤μ(|y-y|+|z-z'|).(A3)g(t,y,0)=0. Then the following conditions are equivalent:(ⅰ)ε[·]is a sublinear expectati on L(Ω,FT,P).(ⅱ)There exists a subset denoted by Q of (σ-additive) probability measures(ⅲ)g is independent of y and sublinear with respect to z.(ⅳ)εg[·] is sublinear on L(Ω,FT,P).