High Dimensional Backward Stochastic Differential Equations, Forward-Backward Stochastic Differential Equations And Their Applications

The theory of Backward Stochastic Differential Equations (BSDEs) has been ex-tensively studied for the past two decades. This theory can be traced back to Bis-mut ([6]) who studied the linear case. In 1990, Pardoux-Peng ([37]) proved the well-posedness for nonlinear BSDEs. Since then, BSDEs have been extensively studied and used in many applied and theoretical areas, particularly in mathematical finance. The objective of this thesis is to improve and enrich the theory of BSDEs and FBSDEs.Although BSDEs have received intensive attention during the past decade, peo-ple have much less knowledge of high dimensional BSDEs compared to 1-dimensional BSDEs. The main difficulty for studying high dimensional BSDEs is that there has not a more general comparison theorem. The comparison theorem for real-valued BS-DEs turns out to be one of the classic results of this theory. It was originally stated by S. Peng ([41]) and then generalized by Pardoux-Peng ([38]) and El Karoui-Peng-Quenez ([16]). It allows to compare the solutions of two real-valued BSDEs whenever we can compare the terminal conditions and the generators. In 1994, under "Quasi-monotonicity" assumptions, Christel and Ralf ([12]) proved the comparison theorems for finite and infinite dimensional stochastic differential equations (SDEs in short). Using the similar idea, Zhou ([60]) obtained a comparison theorem for the multi-dimensional BSDEs in finite time intervals in 1999. In 2006, Hu-Peng ([21]) gived a necessary and sufficient condition under which the comparison theorem holds for multidimensional BSDEs and for matrix-valued BSDEs using the Backward Stochastic Viability Property(BSVP). Recently, in 2009, Wu-Xu ([52]) proved the comparison the-orem for high-dimensional FBSDEs, which is also under similar "Quasi-monotonicity" assumptions. In chapter 2, we will try to discuss the comparison theorem for high-dimensional BSDEs, without " Quasi-monotonicity" assumptions. We will also study the high-dimensional BSDEs with quadratic growth in variable z, and related PDEs.It has been noted that while in many situations the solvability of the original problems is essentially equivalent to the solvability of certain type of Forward Backward Stochastic Differential Equations (FBSDEs in short), these (mostly non-Markovian) FBSDEs are often beyond the scope of any existing frameworks. On the other hand, it is well known that the standard Lipschitz condition is not sufficient for the wellposedness of coupled FBSDEs. Therefore it is becoming increasingly clear that the theory now calls for new insights and ideas that can lead to a better understanding of the problem and hopefully to a unified solution scheme for the general FBSDEs.There have been three main methods to solve FBSDE (3.0.1):(i) Method of Contraction Mapping. This method, first used by Antonelli [2] and later detailed by Pardoux-Tang ([36]), works well when the duration T is relatively small; (ii) Four Step Scheme. This was the first method that removed restriction on the time duration for Markovian type FBSDEs, initiated by Ma-Protter-Yong ([30]). The trade-off is the requirement on the regularity of the coefficients so that a "decoupling" quasi-linear PDE has a classical solution; and (iii) Method of Continuation. This was a method that can treat non-Markovian FBSDEs with arbitrary duration, initiated by Hu-Peng [20] and Peng-Wu [45], and later developed by Yong [54]. The main assumption for this method is that the coefficients have to satisfy a set of so-called "monotonicity con-ditions", which is restrictive in a different way. We refer to the book (cf. [33]) for the detailed accounts for all three methods. We should remark that these three methods do not cover each other.In Chapter 3, we shall provide a systematic analysis for the general FBSDE (3.0.1), following the strategy in [15] and [59]. Our main device is a decoupling field u such that Yt = u(t, Xt) holds. We emphasize that the uniform Lipschitz continuity of u is crucial for our purpose. We shall provide a set of sufficient conditions for the existence of such decoupling field, which ultimately leads to the wellposedness of the original FBSDEs. We notice that all the existing frameworks in the literature could be analyzed by using our criteria, and in the linear case with constant coefficients, our conditions are also necessary.Optimal control theory has developed many applications in practical problems. For forward stochastic system, linear-quadratic control problem with Brownian motion as the noise source is the most popular optimal control problem and the classical theory has been well established. Recently the backward linear-quadratic stochastic control system has received more and more research attention (See e.g. Lim, Zhou ([1])).On the other hand, this kind of control problem naturally arises from financial investment problems. For instance, in financial market, when we consider the problem of hedging one contingent claim using injection and withdrawn of the fund case, the porfolio choice problem forms one kind of backward stochastic optimal control problem.This thesis is focused on high dimensional BSDEs, FBSDEs and their applications. We will give the comparison theorem of one kind of high dimensional BSDEs and obtain the existence and uniqueness of the solutions for high dimensional BSDEs whose coefficient is of quadratic growth in variable z and of linear growth in variable y. We also study the wellposedness of FBSDEs in a general non-Markovian framework. We show that all the existing frameworks could be analyzed using our new criteria.This thesis consists of five chapter. In the following, we list the main results of this thesis.Chapter 1:We introduce problems studied from Chapter 2 to Chapter 5.Chapter 2:We study the comparison theorem of one kind of high dimensional BSDEs and obtain the existence and uniqueness of high dimensional BSDEs whose coefficient is of quadratic growth in z and of linear growth in y. Theorem 2.3.4. (Comparison Theorem for high-dimensional BSDEs) As-sume f satisfies the Assumption 2.3. Moreover, for an (?)∈[O, T],ξ1ξ2∈L2(Ω,f,P), (Y1,Z1) and(Y2,Z2) in Lad2(Ω,C([O,(?)],(?)2))×Lad2(Ω,C((O,(?)),(?)2×d)) to the BSDE (2.2.3) with terminalξ1 andξ2 over time interval [O, (?)], then we can find a linear transformation Yt = AtYt such that:if ATξ1≥ATξ2, we haveTheorem 2.4.3 (Existence for high-dimensional BSDEs with quadratic growth) Assume Assumption 2.4 holds. Then BSDE (2.3.12) has a solution.Theorem 2.5.2 (Comparison Theorem for high-dimensional BSDEs with quadratic growth) Let (Y,Z) be a solution to (2.3.12) and (Y,Z) be a solution to the BSDE associated to the terminal conditionξand to the generator F belong toεx Ml for each l≥1. We assume that, for k = 1,2,…, n, P-a.s., If Fk verifies Assumption 2.4 and 2.5, then P-a.s., for each t∈[0, T], Ytk≤Ytk. If moreover, Y0k = Y0k, thenTheorem 2.5.3 (Uniqueness for high-dimensional BSDEs with quadratic growth) Let the Assumption 2.4 and 2.5 holds. Then BSDE (2.3.12) has a unique solution (Ytk, Ztk) such that Ytk belongs toεand Ztk belongs to Ml for each l≥1.Theorem 2.6.4 (Existence for related PDEs) (?)(t,x)∈[0,T]×Rm, u(t,x)= Ytt,x is a viscosity solution of PDE (2.5.40).Chapter 3:In this chapter we study the wellposedness of the FBSDE in a general non-Markovian framework. The main purpose is to build on all the existing methodology in the literature, and put them into a unified scheme. Our main device is a decoupling random field, and its uniform Lipschitz continuity in the spatial variable is crucial for the wellposedness of the original FBSDE. By analyzing a characteristic BSDE, which is a backward stochastic Reccati equation with quadratic growth in the Z component, we find various conditions under which such decoupling random field exists, which lead ultimately to the solvability of the original FBSDE. We show that all the existing frameworks could be analyzed using our new criteria. As a by product, we prove a comparison result for the decoupling field.Definition 3.2.2 Let Assumption 3.1 holds. We say u : [0,T]×R×Ω→R is a decoupling field of FBSDE if:(ⅰ) u(T,x)=g(x);(ⅱ) u is F-progressively measurable for each x∈R, and is uniformly Lipschitz continuous in x with a Lipschitz constant K> 0;(ⅲ) There exists a constantδ:=δ(K0,K)> 0, which depends only on the Lips-chitz constants K0 and K, such that for any 0=t1< t2≤T with t2-t1≤δ, and anyη∈L2(Ft1), the FBSDE(3.1.2) with initial valueηand terminal condition u(t2,·) has a unique solution, denoted as (Xt1,t2,η,u, Yt1,t2,η,u, Zt1,t2,η,u), which satisfies (1.2.3) for t∈[t1,t2]. Our first result is:Theorem 3.2.3 Let Assumption 3.1 hold. Assume FBSDE (3.0.1) has a decoupling field u, then it has a unique solution (X,Y,Z) and (1.2.3) holds on [0,T].Theorem 3.6.1 Let Assumption 3.1 and (3.3.14) hold. Assume one of (3.3,15), (3.3.17)-(3.3.20), (3.3.22)-(3.3.25) holds. Then FBSDE (3.0.1) has a decoupling filed u satisfying Consequently, FBSDE (3.0.1) admits a unique solution (X,Y,Z)€IL2 such thatCorollary 3.6.2 Let T be given. Assume Assumption 3.1 holds andσ=σ(t,x,y). If one of (3.3.7)-(3.3.9) holds, then FBSDE (3.0.1) has a decoupling filed u satisfying (3.5.1). Consequently, FBSDE (3.0.1) is wellposed.Corollary 3.6.3 Let Assumption 3.1 hold. If, for arbitrary coefficients defined in (3.1.15) and (3.1.14), or then, for any T, FBSDE (3.0.1) is wellposed.Remark 3.6.4(ⅰ)The work Antonelli [2] assumes that arbitraryσ3 and h defined in (3.1.15) and (3.1.14) satisfy condition (3.3.15) and T is small enough. This together with Proposition 3.2 implies condition (3.3.15). Of course, in this paper our arguments rely on this result.(ⅱ) The work Pardoux-Tang [36] essentially assumes, besidesσ3 and h satisfy condition (3.3.15), one of the following conditions:●weak coupling, that is, either 62, b3.σ2,σ3 are small or fl,h are small; ●strong monotonicity, that is, b1 is very negative or f2 is very negative. Recall (3.1.10). For fixed T, the first condition implies that the coefficients of y2 and y3 is small enough and thus the ODEs (3.1.16) has desired solutions on [0, T]. The second condition implies that the coefficient of y is very negative, which ensures that the solution to ODEs (3.1.16) will not blow up before T.(ⅲ) The works Hu-Peng [20], Peng-Wu [45], Yong [54] assume certain monotonic-ity condition, e.g. for some constantβ> 0. By some simple analysis, one sees immediately that Moreover, by setting Ax =0, we see that Then This implies (3.5.2) and thus the FBSDE is wellposed.(ⅳ) The work [59] is clearly a special case of Corollary 3.2.(ⅴ) We should note that our conditions (3.3.15), (3.3.17)-(3.3.20), (3.3.22)-(3.3.25) do not cover the results in [30] and [15]. However, in these cases by using the PDE arguments the deterministic decoupling function u is uniformly Lipschitz continuous.Theorem 3.7.2 (Stability) Assume both (b0,σ0, f0, g0) and (bn,σn,fn, gn), n≥0, satisfy the conditions in Theorem 3.12 uniformly. Let (X0,Y0,Z0), (Xn, Yn, Zn) be the corresponding solutions with initial values x, xn, and u. un be the corresponding random fields. Ifxn→x0 and (bn,σn, fn, gn)→(b0,σ0,f0, g0) in appropriate sense, thenTheorem 3.7.3 Lp-estimates Assume (b,σ,?,g) satisfy the conditions in Theorem 3.12, and (?)ε> 0, such that C1C3<εsmall enough, and Let(X,Y,Z) is the unique solution to the FBSDE (3.0.1) with initial ualue x, then (?)p≥2,we haveChapter 4:In this chapter,we study the linear-quadratic optimal control and a nonzero-sum differential game of backward stochastic differential equations. The optimal control and Nash equilibrium point are explicitly derived.Also the solvability of a kind Riccati equations is discussed.All these results develop that of Lim,Zhou ([1]) and Yu,Ji([58]).Theorem 4.3.1 The function u(t)= Rt﹣1BT(t)x(t)=Rt﹣1 BT(t)x(t),t∈[0.T],is￡the unique optimal Control for the BLQ problem (4.2.4-4.2.6),where (.r.t,pt,qt,kt) is the solution of the following FBSDEP:Theorem 4.5.1 The function (ut1,ut2)=((R1)﹣1(B1)Txt1,(R2)﹣1(B2)Txt2),t∈[O,T],is a Nash equilibrium point for the game problem (4.4.26-4.4.28),where(xt1,xt2,pt,qt,kt) is the solution of the following different dimensional FBSDEP:Chapter 5:In this paper,we present the model of corporate optimal investment with consideration of the difference between the day time and night time.In the model, the investor has three market activities of his choice: corporate investment,savings in a bank, and consumption. Let X(t) denote the total wealth at time t,π(t) represents the proportion of the wealth invested in the real project, then (1 -π(t))X(t) is the amount invested in non-risky bond, andIn the day time, the investor can choose investment with proportionπand con-sumption rate C1 to maximize his wealth, but at the night time, he cannot change his portfolio and only can choose a different consumption rate C2, the portfolio at the night time stays same as the optimal portfolio in the day time. So the investor wants to maximize the following utility of wealth by choosing his investment strategyπand consumption rate C1and C2.We consider one kind of special case called Hyperbolic Absolute Risk Aversion (HARA) case. For simplicity, we only consider the case where the entire time interval is T+N, T is the duration of the day time, N is the duration of the night time. For general case, the whole interval is n(T+N), we can get the explicit result by repeating the procedure with the same method.Let hereγand R are constants, whereγ> 0, R∈(0,1). We try to get explicit optimal decisionπ, consumption rates C1, C2 and the optimal value function in this case.Theorem 5.3.1 Under all the above assumptions, the optimal strategies to the optimal portfolio choice problem (2.21), (2.22), (3.1), (3.2) for the specific HARA case is given by where Pt satisfies ODEs (5.2.40) and (5.2.50).