| In this paper, we want to discuss some properties of BSDE with Constraints and give some applications about it. BSDE is a powerful theory in stochastic analysis and has many connections with other mathematical fields such as optimal control and math-ematical finance.Given a probability space (Ω,F, P) and Rd-valued Brownian motion W(t), we con-sider a sequence{(Ft);t∈[0,T]} of filtrations generated by Brownian motion W(t) and augmented by P-null sets. P is the cr-field of predictable sets of Ω×[0,T]. We use LT2(Rd) to denote the space of all FT-measurable random variables ξ:Ω'Rd for which and use HT2(Rd) to denote the space of predictable process φ:Ω×[0, T]'Rd for whichSnk denotes the space of (Ft)-progressively measurable processes φ:[0, T]×Ω'Rn with E(supo≤t≤T||φ||k)<∞, k∈N.Sci2denotes the space of RCLL, increasing,(Ft)-adapted processes A:[0, T]×Ω'[0,∞) with A(0)=0, E(A2(T))<∞.A BSDE on finite time interval [0, T] is defined as follows where ξ∈LT2(Ω) and*is the transpose of a vector in Rd. To make BSDE be well defined, the generator function g {ω, t,y,z):Ω×[0, T]×R×Rd'R is always assumed satisfy uniform Lipchize property:(A1)|g(ω,t,y2,z2)-g(ω,t,y1,z1)|≤M(||z2-z1||+|y2-y1|),(?)(ω,t)∈Ω×[0,T] and (A2) g(·,0,0)∈HT2(R). Sometimes, we will further assume (A3) g(·,·,0)=0.In classic BSDE analysis, solutions can go anywhere in the value spaces and the existence and uniqueness of adapted solutions have been proved in E.Pardoux, S.G.Peng [9] under assumptions (Al) and (A2). In our paper, we will put constraints on (y(t), z(t)) to confine trajectories in prescribed tubes and analyze its properties and give some applications of it.Our constraints are formulated as same as those in S.G.Peng [54] by a function φ(t, y, z):[0, T]×R×Rd'R+satisfying exactly the same assumptions with g.The idea for us to consider BSDE with constraints conies from hedging and super-pricing contingent claims in incomplete markets. In general case, when there are con-straints and ambiguity in the market, wealth processes obtained by solving BSDEs are g-super-martingales, thus in the Constrained case, we only consider g-super-solutions of BSDE. We say a triple (y(t), z(t). K(t)) a solution of CBSDE with terminal value ξ if (C) φ(t,y(t),z(t))=0, holds. For any given ξ∈LT2(Ω), we denote Hφ(ξ) as the set of triples (y(t), z(t). K(t)) satisfying Constrained condition (C) and (0.0.11). Note that for a given ξ∈LT2(Ω), Hφ(ξ) maybe empty or contain more than one elements. If it is not empty, for the purpose of analysis, we are mainly interested in the smallest one in the sense of definition (1.1) in chapter2, which is denoted as εtg,φ(ξ) and called as gr-solution sometimes. As noted before, the prototype of the smallest solution of CBSDE comes from super-pricing contingent claims in incomplete market. In H.M.Soner, N.Touzi [23], inspired by the same idea, the author considered some kind of stochastic target problem instead. Roughly speaking, given a stochastic variable ξ as a target, the author want to find a stochastic path to arrive it at terminal time, as long as some constraints be considered. In this framework, a new kind of dynamic programming principle was obtained.The study of BSDE with constraints is a meaningful topic, it has closed connections with many other interesting topics such as Viability theory, Singular control problems, Nonlinear optimization etc..Viability theory can be described simply by asking solutions of some differential equation or inclusion stay in some prescribed set K, namely keeping y(t)∈K for all t∈[0, T] if y(t) are solutions of differential equations or inclusions. Some important concepts and tools were introduced in Viability theory which will be used to analysis BSDE with Constraints later. Almost all existing literatures about Stochastic Viabil-ity theory by now are studied only in the framework of Forward case, see example for J.P.Aubin [36]. Backward stochastic viability was first considered in R.Buckdahn, M.Quincampoix, A.Rascanu [51] for the y(t.) part of solutions of BSDEs, that is the author defined a Backward Stochastic Viability Property (BSVP) via asking y(t)∈Γ, where (y(t), z(t)) is a solution of (1.1.1) and Γ is a closed convex set. In the respect of the z(t) part, corresponding Viability theory was considered in Z.WU, Z.Y.YU [71]. As noted in R.Buckdahn, M.Quincampoix, A.Rascanu [51], if BSVP is held for some set F, then the set must be convex. Constrained BSDE can be thought as an extension of BSVP since we ask y(t)∈Γt where Γt:={(y, z)|φ(t,y,z)=0} is dependent on time and are more flexible for solutions to survive.In our analysis of Constrained BSDE, it is always convenient to take terminal value ξ as a variable in a Banach space and consider εtg,φ(ξ) as a function defined on a subset of it. In this point of view, many methods and tools in non-smooth analysis can be adopted to study optimal problem involving constraints via BSDE approach. For optimization on Banach space with non-smooth function and constraints, there are fruitful results existed in the deterministic case. In stochastic case and in the framework of BSDE, our paper wants to do some work about it.Continuous property is important in many optimization analysis and then we will study continuous properties of gΓ-solutions in chapter2. These results are very helpful for us to continue our study of CBSDEs. We first prove that εtg,φ(ξ) is lower continuous on the domain of definition in LT2(Ω). This result will help us to analyze the existence of an optimal solution and investigate necessary and sufficient conditions. Secondly, we prove that εtg,φ(ξ) is continuous from below, that is, if {ξn}1∞is an increasing sequence of variables converging to some ξ with Hφ(ξ)≠(?), then εtg,φ(ξn) converges to εtg,φ(ξ) increas- ingly.This result is meaningful since if we define a risk measure by p(.):=εtg,φ(一),then it satisfies the important Fatou property.see E.R.Gianin[8]and F.Delbaen,S.G.Peng, E.R.Gianin[13]for more reference.For many proof in our paper,our analysis are heavily based on a penalization method to construct9Γ-solutions.We let gn=g+nφ for any n=1,2,…and consider BSDEs driven by gn, then by comparison theorem of BSDE,{yn,n=1,2,...)is an increasing sequence and converges to the smallest g-super-solution according to S.G.Peng[54]if Hφ(ξ)is not empty.Since only increasing process with bounded variance is added into BSDEs,the continuous properties hold only in Semi sense(continnuous from below and lower semi-continuous).Formally, we claim them below,Theorem0.1.(continuous from below)Suppose g(t,y,z):[0,T]×R×Rd'R and φ(t,y,z,):[0,T]×R×Rd'R+satisfying assumptions(A1) and (A2),{ξn∈LT2(P),n1,2,…)is n sequence converging to ξ∈LT2(p) increasingly a.s.,thenViewing ε0g,φ(·)as a function defined on LT2(Ω),we define its effective domain as D:={ξ∈LT2(R)|-∞<ε0g,φ(ξ)<∞]-and assume that D is norn closed in LT2(R). For any k∈R,at the same time,we define the k-level set of ε0g,φ(∈)as then we haveTheorem0.2.(lower semi-continuous property) Suppose φ(t,y,z):[0,T]×R×Rd'R+and g(t,y,z):[0,T]×R×Rd'R assumptious (A1),(A2),then for any k∈R,Ak is closed in LT2(R)-norm and hence ε0g,φ(ξ) is lower semi-continuous.Except for those results stated avove,we find out that9Γ-solution is continuous in the interior of D in convex case.This is based on a well-known result in convex analysis that convex functions are continuous in the interior of its domain of definition if and only if they are lower semi-continuous.We have the following result.Theorem0.3.Suppose g(t,y,z) and φ(t,y,z)are convex functions satisfying(Ai),i=1,2,then ε0g,φ(ξ)is continuous in D under norm. By penalization method, the convexity of gr-solution can be verified easily when the coefficients of CBSDE are convex. But in non-convex case, continuity will fail as illustrated by counter-example in chapter2.At last, we turn to investigate continuous dependence properties of εtg,φ(ξ) when t≠0. This time, we take a sequence {ξn,n=1,2...} from D and have Theorem0.4. Suppose φ(ω,t,y,z):Ω×[0,T]×R×Rd'R+and g(ω,t,y,z) Ω×[0, T]×R×Rd'R both are convex functions satisfying assumptions (Ai), i=1.2,3, then when E|ξn-ξ|2'0as n'∞for ξn,ξ∈D.We also have to remark that we can even drop the assumption of the increasing property of{ξn∈LT3(P), n=1,2,...} in Theorem2.1in fact, that is we only need to assume ξ≤ξ and ξn'ξa.s.. Corollary0.1. Suppose g(t,y, z) and φ(t, y. z) satisfying assumptions {Ai),i=1.2,3and (AA) holds. If {ξ∈LT2(R), n=1,2,...} is a sequence converging to ξ∈LT2(R) with norm and ξ≤ξa.s.for any n,then we haveIt is difficult to determine the effective domain of definition D as defined before for a CBSDE generally. To avoid senseless, we only consider problems in the domain of definition and suppose it is norm-closed in FT2(R) throughout this paper. Although it is very hard and complex to analyze the definition domain of a CBSDE, we can still assure that it is big enough to contain L∞(F), the space of (P)-essentially bounded variables on some probability space (Ω,F, P) under some assumptions, see Chapter2in our paper or S.G.Peng, M.Y.Xu [57] for more details.The role of K(t) in CBSDE is to put y(t) part upward due to a comparison theorem of BSDE. An interesting constraint for BSDE is to ask y(t) be above a given barrier L(t). In this simple case, as illustrated in S.G.Peng, M.Y.Xu [58], the smallest solution is the same as the solution of BSDE Reflected by L(t) just because the Skorohod condition in RBSDE guarantee that the increasing process acts in a minimal way. RBSDE was first studied in N.E1Karoui etc.[43] and is a powerful tool to solve optimal stoping problems. An optimal stopping time is obtained as the first time the increasing process added in RBSDE begins to take effect. Interestingly, many papers also invested connections between optimal stopping and singular control problems. But an careful reader may find that singular controls are LCRL by definition and is different from those increasing processes added in CBSDEs which are RCLL. Although there is difference, they both act as a similar role to solve optimal stopping times.It will be interesting to solve an optimal stopping as well as constraints are con-sidered in the system. We want to study this kind of problem in the framework of BSDE with constraints. To make the problem be well defined, we ask the reward pro-cess L(t)∈L∞(Ft) for any t∈[0, T] and thus ε0g,φ(L(τ)) makes sense for every stopping time τ. This kind of study was inspired by F.Riedel, X.Cheng [17] where the author stud-ied optimal stopping problem under g-expectation and S.G.Peng, M.Y.Xu [57] where the author studied Reflected BSDE with constraints.With a view to ambiguity or model uncertainty in the financial market, optimal stopping problems under nonlinear expectations have been studied more and more by reserchers in recent years. In discrete time case, F.Riedel [16] studied the optimal stop-ping problem with time consistent multiple priors. V.Kratschmer, J.Schoenmakers [64] considered the optimal stopping for more general dynamic utility functionals satisfying nice properties such as time consistency and recursiveness but without strict comparison. In continuous time case, an optimal stopping problem was considered under ambiguity in F.Riedel, X.Cheng [17]. In that paper, the authors developed a theory of Snell envelope under g-expectation which is used to characterize ambiguity in the market. E.Bayraktar, S.Yao [6] investigated optimal stopping problem under more general framework. Given a stable family of F-expectations (whose definition can be found in F.Coquet, Y.Hu, J.Memin [18]){εi}i∈I defined well on a common domain, the authors considered the optimal problems and where S0,T denotes all the stopping times valued in [0, T] and{(Yt+Hti),i∈I} are model-dependent reward processes.When both ambiguity and constraints were considered, we will study a stopping time under gΓ-expectation which is defined via gr-solution. Compare with solving optimal stopping under g-expectation, the main difficulties in Constrained case come from two directions. One is the lack of perfect continuity of CBSDE solutions as showed above and the other is the lack of strict comparison theorem as explained by a counter-example later. But with the help of continuous properties obtained in Chapter2, we can also prove that the Snell-envelope of the reward process L(t) is still a gΓ-super-martingale for gΓ-expectation (g-expectation with Γ-constraints) under mild assumptions. Along with the classical approach to solve optimal stopping problems, a next step for us is prove a RCLL modification of V(t). Recall that the strict comparison proposition plays a crucial role in the proof under linear or g-expectation but it does not hold any more with constraints. Thus we have to find a new way to get over this difficulty. Fortunately, we observe that, by the penalization method, V(t) is a limit of an increasing sequence of RCLL gn-super-martingale and by a limit theorem in S.G.Peng [54], the existence of RCLL modification of V(t) is obtained. To overcome difficulty of continuity, we assume that L(t) is nondecreasing for t or work in convex case. Under these assumptions, we get the following result,Theorem0.5. Suppose g,φ:[0, T]×R×Rd'R satisfy assumptions A(ⅰ),ⅰ=1,2,3and L(t) is an adapted nonnegative process bounded in L∞{FT) and nondecreasing in time or when εtg,φ(·) is convex, then τ*is an optimal stopping for problem (0.0.13) and τ*:=inf{t≥0:X(t)=V(t)}.BSDEs Reflected both from below and above has been studied largely by authors, see S.G.Peng. M.Y.Xu [58], S.Hamadene, J.P.Lepeltier [60] for examples. Inspired by J.Cvitanic, I.Karatzas [33], where the author investigate connections between two-sided Reflected BSDE and stoping games (Dynkin’s game), we consider Dynkin’s game under g-expectation with constraints in chapter4. We define lower and upper value process for g-expectation as and respectively, where R(τ,σ):=L(τ)1(τ≤σ)+U{σ)1(σ<τ) and Tt are stopping times taking values between t, and T, the finite termination of problem, and εtg(·) is the g-expectation induced by BSDE.Dynkin’s game can be interpreted as two players struggle to find best stopping for their own benefit. More explicitly, player A has the right to chose stopping time τ and player B has the right to chose σ. When τ and σ have been chosen, player B should pay player A the amount of money R(τ, σ). The aim of player A is to maximize it and player B is to minimize it. A solution of such game is a pair of stoping times [τ*,σ*) where equilibrium holds in the sense that no further advantage can be obtained when one change his strategy. In mathematical formulation, it means for any τ,σ∈Tt. Generally,(τ*,σ*) is called a saddle point.Although the formulation of our problem is different from those considered under linear expectation as in S.Hamadene, J.P.Lepeltier [60] and J.Cvitanic, I.Karatzas [33], the saddle point and value function are same with those in J.Cvitanic, I.Karatzas [33]. This is not meaningless because there is an integral part added in the reward process in linear case, for more details see arguments in chapter4.We will also consider stopping game under g-expectation in Constrained case, this time we define lower and upper value process as, and respectively, where R{τ,σ):=L(τ)1(τ≤σ)+U(σ)1(σ<τ) and%are stopping times taking values between t and T and εtg,φ(·) is the gT-expectation induced by CBSDE.Just as optimal stoping, imperfectness of continuity in CBSDE is a major difficulty for our analysis and we make up this by assuming the time monotonicity of the lower barrier L(t).In Constrained case, we got a similar result as follows, Theorem0.6. Let g and φ satisfy assumptions (A1), A(2) and (A3), L(t) and U(t) are nonnegative continuous processes satisfying L(t)≤B,U(t)≤B for some B>0for any t∈[0, T], then there is a pair of stopping times (τ*,σ*) which is a saddle point for the Dynkin’s game defined by (0.0.16) and (0.0.17) if L(t) is increasing in time.In chapter5, we study a constrained optimization problem by a terminal perturba-tion method. Our arguments are heavily based on well-known results about non-smooth analysis. We take a BSDE approach to model the wealth process. Our formulation of BSDE for an investment help us to solve optimal portfolio problem via a terminal perturbation method. This kind of method was used in T.R.Bielecki, H.Jin, S.R.Pliska, X.Y.Zhou [63] and developed in S.L.Ji, S.G.Peng [62] to handel optimal investment prob-lem with initial value constraint, the authors find a necessary condition, i.e. a maximum principle for an optimal strategy by the use of Elkland variational method. In our pa-per, under more general constraint assumptions on coefficients, we use general results in nonsmooth analysis to obtain similar conditions for optimality.Suppose the investor has initial value x. he invests in a market according to with φ(t, yt, zt)=0.His strategy (zt, Ct) is determined by the terminal value of wealth among all possible values. The admissible set is defined by then our problem is to minimize where ρ(·) is a function defined on LT2(R), usually it is a risk measure, but it can be more general Lischitze function in our paper.Based on elementary facts in nonsooth analysis, we find a necessary condition for (0.0.18)Theorem0.7. When both ρ(·) and ε0g,Tφ(·) are Lipschitz on their domain of definition. If ξ*is a solution of (0.0.18) and0(?)°ε0g.φ(ξ*),then for some nonnegative real number λ≥0, there are some ζ∈(?)°ρ(ξ*),η∈(?)°ε0g.φ(ξ*),such that ζ+λη=0.When there is no constraints, i.e.φ=0, then εtg,Rφ(ξ) is a solution of BSDE. It is well known that, by the priori estimate of BSDE,εtg,Rφ(ξ) is Lipschitz. Furthermore, by the strict comparison property of BSDE,0(?)εtg,Rφ(ξ) for any ξ∈LT2(R). Thus in the non-constrained (except for constraint on initial value) just like in S.L.Ji, S.G.Peng [62], all the assumptions satisfied in our theorem and thus our result can be thought as a generalization of the result in S.L.Ji, S.G.Peng [62]. |
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