| Since the fundamental work of Pardoux and Peng [47], nonlinear backward stochas-tic differential equations (BSDEs for short) have received quite a lot of attention due to their well applicability in for example, stochastic control, partial differential equations, and mathematical finance and so on. On the other hand, as one kind of nontrivial generalization of BSDEs, in this thesis we will concentrate on a new kind of equation, backward stochastic Volterra integral equations (BSVIEs for short) of To make it more precisely, let us explain the organization of this thesis.In Chapter1, we will give an introduction to the topics from Chapter2to Chapter6below, as well as some notations used there.Motivated by the non-uniqueness of adapted solutions for (23) in H2[0,T], in Chap-ter2we introduce the notion of symmetrical solution (S-solution for short) and study the corresponding wellposedness in [0, T]. Compared with existed results, we generalize and modify the results in [38], show the differences, relations between M-solutions in [69] and our notion here by several examples. At last we also derive one class of dynamic coherent risk measures with respect to process by means of S-solutions.Comparing with BSDEs, due to the complicated form of BSVIEs, together with the non-semigroup property, Yong [69] introduced four-step method to deal with the existence and uniqueness of M-solution on [0,T]. However, their arguments there are too sophisticated to apply in some other general case, such as infinite horizon case. Therefore, in Chapter3we develop a new and briefer method here to tackle such kind of problem. As to M-solution, we modify and generalize the results in [53] by giving one simple to show the gap there. For the adapted solutions in space H2Δ[0, T], the unique solvability of BSVIEs under more general stochastic non-Lipschitz conditions is shown, which improves and generalizes the results in [38],[66].Comparison theorem is one basic topic in stochastic equation, such as SDEs, BS-DEs. Therefore, in Chapter4we establish the comparison theorem systematically for backward stochastic Volterra integral equations (BSVIEs), by virtue of adapted solu-tions in H2Δ[0,T] and adapted M-solutions in H2[0,T]. Due to the general framework, some kind of monotonicity conditions play crucial roles to guarantee the comparison theorems for FSVIEs and BSVIEs to be true. For completeness, we also present com-parison theorems for SDEs, BSDEs and SVIEs, where duality principles work well in relevant proofs. Also, it is found that certain kind of Various counterexamples show that the assumed conditions are almost necessary in some sense.In Chapter5, we study the optimal control problem for forward-backward stochas-tic Volterra integral equations. Due to the dependence of g on Z(s, t) but not Z(t,s), some new features arise naturally, as well as some essential difficulties, see the details in Chapter1below. Therefore, the situation here is by no means a trivial generalization of FBSDEs case in [50]. As applications, we study the linear quadratic problem, together with two economic models which deepen the study in [7],[23] and [32]. In the last we give one special optimization problem for linear FBSVIEs, part of which improves the study in [45].Motivated by mean field BSDEs in [13] and [14], in Chapter6we introduce the notion of mean field BSVIEs and consider M-solutions in space HP[Q, T]. If p>2, then the generator and the nonlocal term are assumed to be sublinear growth with respect to Z(s, t) which is also shown to be necessary by an interesting counterexample. To our best, this is one new feature different from BSDEs case. We also establish two dual principles for MF-FSVIEs and MF-BSVIEs respectively, which lead to another new interesting result. By using one dual principle, the maximum principle for optimal control problem of MF-SVIEs is derived naturally.Now we give an overview over the main results of this dissertation.1Symmetrical solutions of backward stochastic Volterra integral equations and applicationsThis chapter is based on the following paper,TIANXIAO WANG AND YUFENG SHI, Symmetrical solutions of backward stochastic Volterra integral equations and applications, published in Dis. Cont. Dyn. Syst. Series-B,14,2010,251-274.As to BSVIE (23) Yong introduced M-solution in H2[0,T] as follows,Definition2.1.1Let S∈[0,T]. A pair of processes (Y(·),Z(·,·)∈H2[S,T] is the so-called M-solution of (23), if (23) holds in the Ito sense, for almost all the t∈[S, T], and, On the other hand, Example1.1.1in Chapter1suggests that the adapted solution in H2[0, T] for (23) is not unique. Therefore here we introduce the notion of symmetrical solutions (S-solutions for short) for BSVIEs as follows,Definition2.1.2Let S∈[0, T]. A pair of processes (Y(·), Z(·,·)∈*H2[S, T] is called the adapted S-solution of (23), if (23) holds in the Ito sense for almost t∈[S, T], and Z(t,s)=Z(s,t), t,s∈[S,T]. Notice that*H2[S,T] here contains H2[S,T] as subspace in Definition2.1.1, see for Example1.1.2below which shows the necessity of such modification in our setting.The main result in this chapter can be stated as,Theorem2.1.4Suppose g is Lipschitz in y, ζ and z, where L is Lipschitz function satisfying certain integrable condition, ψ(·)∈L2FT[O,T], then (23) admits a unique S-solution and for any S∈[0,T], Comparing with the work in Lin [38], we can use Theorem2.1.4to modify and generalize the arguments there. Moreover, examples in Remark2.2.4,2.2.5and2.2.8below also shows the difference between S-solutions and M-solutions. At last we will give some applications in the risk management. More precisely, we have the following result by defining p(t;ψ(·))=Y(t), with t∈[0,T], where (Y(·),Z(·,·)) is the unique S-solution of (24),Theorem2.3.7Suppose f(t,s,y)=η(s)y, where η(·) is a deterministic bounded function, then ρ(·) is a dynamic coherent risk measure with respect to process ψ. We refer to Definition2.3.1and2.3.2of Chapter2for the definition of coherent risk measures.2A new method to solve BSVIEs under weaker conditionsThis chapter is based on the paper ofYUFENG SHI AND TIANXIAO WANG, Solvability of general backward stochastic Volterra integral equations. Published in J. Korean. Math. Soc,49,2012,1901-1921.Our first aim in this chapter is to propose a briefer method to study adapted M-solutions. Inspired by the work in El Karoui and Huang [25], we introduce an equivalent norm in H2[0, T] as follows: where β is a positive constant and A*(t)=ft0α2p/2-p(s)ds, α(·)≥1is an adapted process. This leads to one important estimates for M-solutions as follows, with p∈[1,2). We refer to Lemma3.2.1for the detailed proof. After this, we can get the wellposedness of M-solution in [0, T] by contraction for one step.Theorem3.2.4Suppose that g satisfies|g(t,s,y,z,ζ)-g(t.s,y,z,ζ)|≤L(t,s)a(s)(|y-y|+|z-z|+|ζ-ζ|) where α is defined above, α(·) is deterministic, A*(·) is also bounded, then (23) admits a unique adapted M-solution in H2[0, T].If g is independent of Z(s, t), then by allowing α to be random, we have,Theorem3.2.7Suppose the conditions in Theorem3.2.4hold, g is independent of Z(s, t), α is adapted process, A*(·) is also bounded. Then BSVIE (23) admits a unique adapted solution in H2Δ[0,T].Recently Ren [53] considered the unique solvability of M-solutions under non-Lipschitz condition by adopting the method proposed by Anh and Yong in [6]. However, there is a gap in the splitting procedure in p.7of [53], see for example3.3.1below. In this chapter, we will bridge this gap by specifying some new assumptions which are weaker and more natural than the ones in [53]. By using the same norm as above, we can obtain the unique existence of M-solutions for BSVIE (23) under non-Lipschitz condition, which modifies and generalizes the results in [6],[53],[67],[68] and [69].Theorem3.3.3For (t, s)∈Δ, suppose that g satisfies where p is a increasing concave function from R+to itself such that∞. ψ(·)∈L2FTT(0, T;Rm), α(·) is deterministic, L(t, s) satisfies∞, then (23) admits a unique adapted M-solution in H2[0, T].3Comparison theorem of backward stochastic Volterra integral equationsThe chapter here is based on joint work with Prof. Jiongmin Yong,TIANXIAO WANG AND JIONGMIN YONG, Comparison theorem of backward stochas-tic Volterra integral equations. Submitted to Stochastic Process and their applicationIn this chapter, we will establish multi-dimensional comparison theorem for backward stochastic Volterra integral equations in a systematic way. For completeness, we will present the case of stochastic differential equations, see also [27],[51]. The following result is concerned about the nonlinear case of SDEs,for i=0,1Theorem4.2.2. Let bi,σ be Lipschitz in x,bi(s,0),σ(s,0)are bounded proceses. Suppose there is a b such that bx(t.x)exists and is uniformly bounded.If bx(t.x)∈Rn*+, σx(t,x)∈Rn×nd,(t.x)∈[0,T]×Rn,(26) and b0(t,x)≤b(t,x)≤b1(t,x).Then for any(s,xi)∈[0,T]×Rn with x0≤x1,the unique solutions Xi(·)三Xi(·；s,xi)of(25)satisfy X0(t)≤X1(t),with∈[s,T]. Conversely,if b0(t,x)=b(t,x)=b1(t.x),with(t,x)∈[0,T]×Rn,as,and(t,x)'(b(t,x),σ(t,x))is continuous.Then(26)is necessary for above comparison. Now we study the nonlinear case of BSDEs. Let us look at nonlinear n-dimensional BSDEs:For i=0,1,and F-stopping time τ-valued in[0,T],Theorem4.2.4.Let gi be Lipschitz in y and z,gi(s,0.0)is bounded.Suppose there is g such that gy(s,y,z)and gz(s,y,z)exist and are uniformly. Moreover,gy(s,y,z)∈Rn×n*+, gz(s,y,z)∈Rn×nd,(s,y,z)∈[0,T]×Rn×Rn,(28) and g0(s,y,z)≤g(s,y,z)≤g1(s,y,z).Then for any F-stopping time τ valued in(0,T], and any ζ0,ζ1∈L2Fτ(Ω；Rn)with ζ0≤ζ1,the solution(Yi(·):Zi(·))of(27)satisfy Y0(t)≤Y-1(t),with t∈[0,τ].Conversely,suppose g0(s,y,z)=g(s,y,z)=g1(s,y,z),(s,y,z)∈[0,T]×Rn×Rn,and(s,y,z)'g(s,y,z)is continuous.Then(28)js necessary for above comparison.The above result is a slight extension of a relevant one presented in[29],allowing g0(·)and g1(·)to the different for suffcient part.Finally,we point out that our proof is based on the duality and a corresponding result for linear FSDEs,which is different from that found in[29].Now let us turn to forward stochastic Volterra integral equation case ofProposition4.2.8.Suppose A0,A1are bounded,t'A0(t.s)is continuous on [S,T].(i)Suppose A0(t,s)∈Rn×n+,A1(s)=0with(t,s)∈△*.Then(29)admits a unique solution X(·) and it satisfies X(t)≥ψ(t)≥0with∈[0,T]. (ⅱ)Suppose A0(t,s)∈Rn×n*+,A1(s)∈Rn×nd,with(t,s)∈△*.Moreover,there exists a continuous nondecreasing function ρ:[0,T]'[0,∞)with ρ(0)=0such that|Ao(t,s)-A0(t’,s)|≤ρ(|t-t’|), t,t’∈[0,T],s∈[0,tΛt’], and A0(τ,s)-A0(t,s)∈Rn×n+,with0≤s≤t≤τ≤T.Then for any ψ(·)∈CF([0,T]；L2(Ω,Rn)),withψ(τ)≥ψ(t)≥0,0≤s≤t≤τ≤T,(29)admits a unique solution X(·)∈CF([0,T]；L2(Ω；Rn))and it satisfies:X(t)≥0,t∈[0,T].Now,returning to comparison theorems for BSVIEs, the theory of which is much more richer than that for BSDEs due to its cmnplicated structure.To this end,we first consider BSVIE(23)with the generator g(.)independent of Z(s,t).For i=0,1,Theorem4.3.2. Let(Yi,Zi)be the solution of (30). suppose g:△×R.Rn×Ω'Rn satisfies cortain measurability,(y,z)'g(t,s,y,z)is unigormly Lipschitz, y'g(t,s,y,z)is nondecreasing,such that g0(t,s,y,z)≤g(t,s,y,z)≤g1(t,s,y,z),(t,s,y,z)∈△×Rn×Rn, gz(t,s,y,z)exists and gz(t,s,y,z)∈Rn×nd.Then for any ψi(·)∈CFT([0,T]；L2([Ω;Rn)) satisfying ψ0(t)≤ψ1(t),the corresponding unique adapted solution(Yi(·),Zi(·,·))∈H2Δ[0,T]of BSVIE(30)satisfy Y0(t)≤Y-1(t).If we consider the following linear BSVIE: then Theorem4.3.2implies A(t,s)should be belong to Rn×n+.However,such require-ment can be dropped in the following.Theorem4.3.6. Let A and B be uniformly bounded,and for each s∈[0,T], t'H A(t,s)being continuous.Moreover,A(t,s)∈Rn×n*+,with(t,s)∈△,A(t,s) A(τ,s)∈Rn×n+,0≤t≤τ≤s≤T,and B(s)∈Rn×nd,with s∈[0,T].Then for any ψ(·)∈CFT([0,T]；L2(Ω；Rn))wit ψ(t)≥ψ(s)≥0,with0≤t≤s≤T,the adapted solution(Y(·),z(·,·))of linear BSVIE(31)satifies,Y(t)≥0,with t∈[0,T].The second case to be considered is that the generator g(·)depends on Z(s,t) and independent of Z(t,s). For such a.case,we are comparing adapted M-solution. Example4.3.8below suggests us that if linear BSVIEs are considered for comparison of adapted M-solutions,the following should be a proper form:Theorem4.3.9.Let A and C be uniformly bounded,and for each s∈[0,T],t'A(s,t)is continuous.Further,A(t,s)∈Rn×n*+,with(t,s)∈△,A(s,T)—4(s,t)∈Rn×n+, with s≤t≤τ≤T,s∈[0,T],and C(t)∈Rn×nd,with∈[0,T].Then the adapted M-solution(Y(·),Z(·,·))of linear BSVIE(32)with ψ(·)∈CFT(0,T；L2(Ω；Rn)),ψ(·)≥0satisfies EFt?Tt Y(s)ds≥0,t∈[0,T].This result corrects a relevant result in[67],[68].4Optimal control of forward backward stochastic Volterra integral equations and applicationsThis chapter is based onYUFENG SHI,TIANXIAO WANG AND JIONGMIN YONG,Optimal control problem of frward-backward stochastic Volterea integral equations and applications.preprintIn this chapter,we will study a stochastic maximum principle for FBSVIEs in a new framework.To make it more precise,we denote by ds.and Nv(t)=(?)Tt Zv(s,t)ds,with t∈[0,T].The state equation is described by and the cost fumctional byHere given s∈[0,T],λv is determined by v(s)=Ev(s)+(?)t0λ(s,r)dW(r).Our aim is to minimize the cost functional while the control domaiin is assumed to be convex. Theorem5.2.3. Let u(·) be an optimal control and (Xu(·), Yu(·), Zu(·,·)) be the corresponding M-solution of FBSVIE (33). Then we have,(?)v∈U,H(t, Xu(t), Yu(t)Zu(t,·), u(t), P(t), Q(t), R{·t))·(v-u(t))≥0, a.e., a.s. where and (P, Q, R) is the unique M-solution of FBSVIEDue to the absence of Ito formula, some new delicate and subtle skills are pro-posed here to get round this difficulty, and our approach here is by no means a trivial generalization of FBSDEs case in [50]. Compared with all the existed literature along this line, see for example [49] and [50], there are several new features appearing in our problem, see Chapter1next for details.After that, a linear quadratic (LQ in short) problem for FSBVIEs is investigated naturally, which can degenerate into the FBSDEs case in [50] in some sense. It is also remarkable that our linear quadratic problem for BSVIEs is also new in the literature.As an illustration of possible applications, we formulate two kinds of economic problems, stochastic input-output model and stochastic capital replacement model into a general FBSVIEs framework for the first time, which develops and deepen the research in[7],[23]and[32].As to the former one,the state equation is(33)with b=h1(t,s)v(s), σ=h2(t,s)u(s),ψ(t)=X(0),g=-h1(T,s)u(s)+f(t)Z(s,t)-f(t)[T-s]λ(s,T)[F(s)h2(T,s)+h1(T,s)]. and the cost functional is(34)with l=δ[y-[T-s][h2(T,s)f(s)+h1(T,s)]u],h(x)=-U(x).As to the appearance of λv(s,t)in the backward equation,we refer to Subsec-tion5.3.2below for details.Hence we have the following necessary condition for the existence of optimal solution,Theorem5.3.2. Suppose u(·)is an optimal solution,then,t∈[0,T], whereAs to stochastic capital replacement model,state equation is(33)with b=M(t-s)v(s),σ=N(t-s)v(s),ψ(t)=0,g=-h1(T,s)v(s)+f(t)Z(s,t)-f(t)[T-s]λ(s,T)[F(s)H2(t,s)+H1(t,s)]. aNd tHe cost functional is(34)with l=δ[y-[T-s][h2(T,s),(s)+h1(T,s)]v] α(t)[P’(s,x)-v],therefore,we have the following result,Theorem5.3.4.Let u(·)be an optimal solution,then for t∈[0,T], where L is adapted provess determined by this moddel.Particulary,by assuming p(t,x)=x—γ/2x2with γ>0,we can rewrite(35)asFinally,one special optimization problem is studied where the state equation be-comes(33)withψ=x0,b=(s)v(s),σ=β(s)v(s),g=-l1(s)x+l2(s)v+r(s)y+Κr(s)ζ,and cost functional is(34)with l=-l1(s)x-l2(s)v+2r(s)y,and h(x)=-h’(x), Theorem5.3.5. Let u is an optimal solution,h’(x)=x-γ/2x2,α,α-1,β,β-1are bounded process such that Ee-A(T)|Xu(T)|<∞, where A and W are defined by, then with u(·.)is given by,5Mean field backward stochastic Volterra integral equations and related topicsThis chapter is based on the foffowing joint work with Prof.Yufeng Shi and Prof. Jiongmin YongYUFENG SHI, TIANXIAO WANG AND JIONGMIN YONG,Mean field backward stochastic Volterra integral equations,.Accepted by Dis.Cont.Dyna.Syst Series B.In this chapter we will study the following form of backward equation, withTheorem6.3.2.Let θ and g satisfies Lipschitz condition and,|θ(t,s,y,z,z,y’,z’,z’,)|≤L(1+|y|_|z|+|z|2/q+|y’|+|z’|+|z’|2/q)|g(t,s,y,z,z,γ)|≤L(1+|y|+|z|+|z|2/q+|γ|), with2<q<∞. Then for any ψ(·)∈LqFT(0, T; Rn), MF-BSVIE (36) admits a unique adapted M-solution (Y(·), Z(·,·))∈Mq[0, T]. and the following estimate holds: When p∈(1,2], the adapted M-solutions for BSVIEs was discussed in [61]. It is possible to adopt the idea of [61] to treat MF-BSVIEs for p∈(1,2). On the other hand, if the map θ grow linearly with respect to z and z’, the adapted M-solution (Y(·), Z(·,·)) of (36) may not be in Mp[0, T] for p>2, even if ψ(·)∈LpFT(0,T;Rn). This can be seen from Example6.3.3.Next we will derive two dual principles for MF-FSVIEs and MF-BSVIEs respec-tively. Firstly, consider equation, we have the first dual principle for MF-SVIEs,Theorem6.4.1. Let ψ(·),ψ(·)∈L2F(0,T;Rn), X(·)∈L2F(0,T;Rn) is the solution to (37), and (Y(·),Z(·,·))∈M2[0, T] is the adapted M-solution of Next, different from the above, we want to start from the followng linear MF-BSVIE: Now, we are at the position to state and prove the following duality principle for MF-BSVIEs.Theorem6.4.2. Let ψ(·)∈L2FT(0,T;Rn),(Y(·), Z(·,·)) is the adapted M-solution of (38). Further, X(·)∈L2FT(0,T;Rn) satisfies ThenFrom the above two dual principles, we have the following interesting results, that is, twice adjoint of a linear MF-SVIE is itself, while twice adjoint of a linear MF-BSVIE is not necessarily itself. At last we study one optimal control problem for MF-SVIE of where The cost functional is defined where Dur aim is to minimize the cost functional. We haveTheorem6.5.1. Let b, σ, Tb, Tσ satisfy certain measurability, Lipschitz, linear growth condition, and (X(·), u(·)) be an optimal pair of this control problem. Then the adjoint equation (41) admits M-solution (Y(·), Z(·,·)) such that variational inequality (42) hold,... |
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