Forward-Backward Stochastic Differential Equations Driven By Markov Chain And Its Related Problems

Since the first introduction by Pardoux and Peng [63] in 1990, the theory of nonlin-ear backward stochastic differential Equations(BSDEs, for short) has been consumingly researched by many researchers and has been achieved abundant theoretical results. Now the theory of BSDEs has risen to be a powerful part of the stochastic analyst’s tool. It has found various achievements in applications, namely in finance, stochastic control, stochastic game and the partial differential Equation(PDEs, for short) theory.Based on the development of the theory of BSDE, theory of many different forms of BSDE also got rapid development. There are many theories of BSDE with jump processes recently. Furthermore there are results that the Brown motion in BSDE’s diffusion term was replaced by other process. In 2008, Cohen and Elliott [18] studied a BSDE driven by a continuous time, finite state Markov Chain. After then, many results such as comparison theorem about this kind of BSDE, nonlinear expected results and so on was achieved.Along with the rapid development of the BSDE theory, the theory of fully coupled forward backward stochastic differential equations(FBSDEs, for short) which is closely related to BSDE has been developing very rapidly. As we know now, to get results on the existence and the uniqueness of fully coupled FBSDE’s solutions on an arbitrary given time interval, there are mainly three approaches:method of contraction Mapping, four-step scheme, method of continuation.Using the method of continuation, this doctoral thesis mainly studies fully coupled FBSDE driven by a continuous time, finite state Markov Chain. We give results on the existence and the uniqueness of solutions of this kind of Equations at three cases:the stochastic differential Equation(SDE, in short) and the BSDE take same dimensions, the SDE and the BSDE take different dimensions and fully coupled FBSDE in stopping time duration. We also give comparison theorems and continuous results of the solution depending on parameters. The purpose of this paper is to improve the Semimartingale Theory and Stochastic Calculus, especially develop the theory of fully coupled FBSDE driven by jump processes.Next, let us introduce the main content and structure of this paper.In Chapter 1, we give background and preliminaries of the research, we also give a brief description of the main content of this article.In Chapter 2, at first, we study a continuous time Markov chain m={mt, t≥ 0} which has finite state and takes the unit column vectors in Rd. We mainly study the properties of the Markov chain m={mt, t≥0} in the homogeneous case. We consider three counting processes revelent to m={mt,t≥ 0), define the average transition rate function of the counting processes and give the simple calculus formula about the average transition rate function of the counting processes. Furthermore, we study the properties of the jump martingale generated by the Markov chain, get the conclusion that the Martingale is a local finite variation process. Next, we study the FBSDE driven by the martingale. The Equation is equivalent to a FBSDE driven by the Markov chain m={mt,t ∈ [0.T]}. Using the method of continuation, the Ito product rule of Semimartingales and iterative methods, with the help of results of BSDE on a finite state Markov chain, we prove the existence and the uniqueness of the solution of the FBSDE driven by the Martingale. We also give a comparison theorem and continuous results of the solutions depending on parameters.In Chapter 3, we study the fully Coupled FBSDE driven by a Martingale which is generated by a Markov chain:when SDE and BSDE have different dimensions meanwhile YT= ξ The Equation is equivalent to a FBSDE driven by the Markov chain m = {mt,t ∈ [0,T]}. We introduced a m x n full rank matrix G to overcome the problem caused by the different dimensions of SDE and BSDE. Using the method of continuation, the Ito product rule of Semimartingales and the fixed point principle, with the help of the theory of BSDE driven by a continuous time, finite state Markov Chain and the theory of Riccati Equations, we proved the existence and the uniqueness of this kind of Equations under different monotone assumptions. We also give continuous results of the solutions depending on parameters.In Chapter 4, we study the fully Coupled FBSDE driven by a Martingale which is generated by a Markov chain:when SDE and BSDE have different dimensions meanwhile YT=Φ(XT).The Equation is equivalent to a FBSDE driven by the Markov chain m={mt,t ∈ [0, T]}. Using the method of continuation and the Ito product rule of Semimartingales, with the help of results of BSDE on a finite state Markov chain and lemmas proved at Chapter 3, we prove the existence and the uniqueness of the solution of this kind of FBSDE. We also give continuous results of the solutions depending on parameters.In Chapter .5, we study fully Coupled FBSDE driven by the Martingale Mt which is generated hy a continuous time, finite state Markov chain m = {mr, t∈[0, T]} in stopping time duration. The Equation is equivalent to a FBSDE driven by the Markov chain m = {mt, t∈[0, T]}. Using the method of continuation, the Ito product rule of Semimartingales and the fixed point principle, furthermore stopping time technique, we prove the existence and the uniqueness of the solution of the FBSDE driven by the Martingale in stopping time duration. We also give a comparison theorem about initial value.Next, we give the main results of the dissertation.1. A finite state Markov Chain and Fully Coupled FBSDE driven by itIn Chapter 2, at first, we study a. continuous time Markov chain which has finite state and takes the unit column vectors in Rd. Furthermore, we study the properties of the jump martingale generated by the Markov chain. Next, we study the FBSDE driven by the martingale.We consider a continuous time, finite state Markov chain m = {mt, t≥0} on a probability space (Ω,F, P). Define the states of the Markov chain with the unit vectors ei in Rd, where d is the number of the states of the Markov chain. That is, for 0≤i≤d, we write ei = (0,…,1…,0)* for the ith unit column vector in Re, where "*" m means transpose. So its state space is the set S = {e1,…,ed}. By the Appendix B of Elliott, Aggoun and Moore [34], the Markov chain has the following representation: where Mt is a Martingale. We call it Martingale generated by the Markov Chain m= {mt,t≥0}.We mainly study the properties of the Markov chain m={mt, t≥0} in the homogeneous case. We consider three counting processes revelent to m = {mt, t≥0}: define the counting process Nm(t) for the total number of jumps of the chain m= {mt, t≥0} up to time t: define the counting process Nij(t) for the total number of jumps of the chain m = {mt,t≥0} from the state ei to the state ej up to time t; define the counting process Nj(t) for the total number of jumps of the chain m = {mt, t≥0} to the state ej up to time t.Suppose Pij(t)=P{mt=ej|m0=ei}, we have PtJ = ∑ei∈s P0iPij(t). Assume At=(aij(t)), t≥0 is the family of so-called Q-matrices of the chains. To above counting process, if E[N(t)] is differential, then we callR the average transition rate function of N(t). We give simple calculation formulas for the average transition rate functions of above three counting processes.Theorem 0.1. Rm(t) = -∑ei∈s Ptiaii.Theorem 0.2. Rij(t) = Ptiaij.Theorem 0.3. Rj(t) =∑ei∈s Ptiaij.There’s conclusion about m ={mt, t∈[0, T]}:Theorem 0.4. The Markov chain m = {mt, t≥0} has finite variation in any finite period of time.There are conclusions about Me:Theorem 0.5. The Martingale Mt has finite variation in any finite period of time.Theorem 0.6. The quadratic variation of the Martingale Mt can be said forTheorem 0.7. Suppose that h is a predictable process, thenWe consider the fully coupled FBSDE driven by M(t):where (X,Y,Z) ∈ (Rn,Rn,Rn×d). T > 0 is any fixed real number, we call it time duration and b, σ, f, Φ are functions of proper dimension. Equation(0.0.8) is equivalent to a FBSDE driven by the Markov chain m = {mt,t≥0} For u = (x, y, z) ∈ Rn×Rn×Rn×d,令 F(t,u) = (-∫(t,u), b(t,u), 0), G (t, u) = (0, 0, σ (t, u)).We write M2(0, T; Rn) for the adapted processes of satisfying the following condi-There are Assumptions:(A2.1)For each u = (x, y, z)∈ Rn ×Rn×Rn×d, F(·,u), G(·,u)∈ M2(0, T;Rn × Rn × Rn×d) and for each Rn×d) and for each x ∈ Rn,φ(x) ∈ L2(Ω,Fr;Rn); and there exists a constant c1 > 0 such that |F(t, u1)- F(t, u2)|≤c1|u1 - u2|, |G(t, u1) - G(t,u2)| ≤c1|u1-u2|, (?)u1 ∈ R2×R2× Rn×d, (?)u2 ∈ Rn × Rn × R×d; P - α.s.. α.e.t∈R+，and |φ(x1) -φ(x2)|≤c1|x1-x2|,P- α.s., (?)x1,x2∈Rn×Rn(A2.2)there exists a constant c2 > 0 such that (F(t, u1) - F(t, u2), u1 - u2] ≤ -c2|u1 - u2|2, P - a.s., a.e.t ∈R+, [G(t, u2) - G(t, u2), u1-u2]≤-c2|u1 - u2|2,P-α,s., α.e,t∈R+, (?)u1∈Rn ×Rn×Rn×d,(?)u2 ∈ Rn ×Rn × Rn×d; and (φ(x1) -φ(x2),x1-x2)≥c2|x1 - x2|2,Vx1∈Rn,(?)x2 ∈RnUsing the method of continuation, the Ito product rule of Semimartingales and iterative methods, with the help of results of BSDE on a finite state Markov chain, we can give conclusions about equation(0.0.8):Theorem 0.8. Suppose that Assumptions (A2.1) and (A2.2) hold, then there exist a unique adapted solution (X, Y. Z) for Equation (0.0.8).Furthermore, with the help of the results on BSDE, we can conclude the following conclusions:Theorem 0.9. Suppose that Assumptions (A2.1) and (A2.2) hold, then there exist a unique adapted solution (X, Y. Z) for Equation(0.0.8).Theorem 0.10. Suppose that Assumptions (A2.1) and (A2.3) hold, then there exists a unique adapted solution (X,Y,Z) for Equation(0.0.8).Theorem 0.11. Suppose that Assumptions (A2.1) and (A2.4) hold, then there exists a unique adapted solution (X,Y,Z) for Equation(0.0.8).Theorem 0.12. Suppose that Assumptions (A2.1) and (A2.3’) hold, then there exists a unique adapted solution (X,Y,Z) for Equation(0.0.8).Theorem 0.13. Suppose that Assumptions (A2.1) and (A2.4’) hold, then there exists a unique adapted solution (X,Y,Z) for Equation(0.0.8).We give a comparison theorem for the solutions of initial values below. We first give the following two fully coupled FBSDE driven by M(t): where i= 1.2.We have the following comparison theorem for the solutions of initial values:Theorem 0.14. Suppose that Equation(0.0.9) satisfy Assumptions (A2.1) and (A2.2), assume(X1,Y1,Z1) and (X2,Y2,Z2) are solutions of two Equations. If x1≥ x2, then Y01≥Y02Furthermore, we give result on continuity of parameter.Assume (fl,bl,σl,Φl),l ∈ R is a family of FBSDEs, which satisfy Assumptions (A2.1) and (A2.2), and solution is (Xl,Yl,Zl):There are Assumptions:(A2.5)● The family of (fl,bl,σl,Φl),l ∈ R are equi-Lipschitz with respect to (x,y,z) and x separately;● The function l →(fl,bl,σl,Φl) are continuous in their existing space norm sense respectively.Theorem 0.15. Assume (fl,bl,σl,Φl),l ∈ R is a family of FBSDE (2.22)satisfying Assumptions(A2.1), (A2.2) and (A2.5). Its solutions is denoted by (Xl,Yl,Zl). Then the function l→(Xl,Yl, Zl,XTl):R→M2(0,T;Rn × Rn × Rn×d) × L2(Ω,fT,P;Rn)is continuous. 2. Fully Coupled FBSDE driven by Markov chain:forward and backward Equations take different dimensions meanwhile YT= ξChapter 3 set on the same basic framework with Chapter 2, it is the generalization of Chapter 2. We study the fully Coupled FBSDE driven by a Martingale which is generated by a Markov chain:when forward and backward Equations take different dimensions meanwhile YT= ξ.Assume m={mt,t ∈ [0, T]} is a continuous time, finite state Markov chain and it takes values in unit vectors ei in Rd. where d is the number of states of the chain. Suppose {ft} is the completed natural filtration generated by the σ-fields ft=σ({ms,s≤ t},F ∈ fT: P(F)= 0), and f=fT. Consider stochastic processes defined on the filtered probability space (Ω,f,ft,P). Let At denotes the rate matrix for m at time t. By the Appendix B of Elliott, Aggoun and Moore [34], then we have this chain has below representation: where Mt is a martingale.We consider the fully coupled FBSDE driven by M(t): where(X,Y,Z) ∈ (Rn,Rm,Rm×d). T> 0 is any fixed real number and b,σ,f are func-tions with proper dimensions. Equation(0.0.11) is equivalent to a FBSDE driven by Markov chains.We give a m × n full rank matrix G, foru= (x, y, z) ∈ Rn ×Rm×Rm×d let where Gσ= (Gσl… Gσd). There are Assumptions:((A3.1) For each u = (x. y, z) E Rn × Rm × Rm×d,F(·, v), H(·, v) ∈ M2(0, T; Rn × Rm × Rm×d) and for each x ∈ Rn,ξ∈ L2(Ω,FT; Rn); and● F (t, v) is uniformly Lipschitz with respect to v;● H(t, v) is uniformly Lipschitz with respect to v.(A3.2)There exist constants c2, c’2. such that [F(t, v1) F(t, v2), v1 - v2] ≤-c2|G(x1-x2)|2- c2 (| G*(y1 - y2)|2 + |G*(z1-z2)|2), [H (t,v1) H(t,v2),vl-v2]≤-c2|G(x1-x2)|2 -c2 (|G*(y1-y2)|2+|G*(z1-z2)|2),P - a.s., a,.e.t ∈ R+, (?)v1 = (x1, y1, z1), v2 = (x2, y2,z2) ∈ Rn × Rm × Rm×d where c2 and c2 are given positive constants. (A3.3) There exist constants c2, c2, such that [F(t, v1) - F(t, v2), v1 - v2] ≥ c2| G(x1- x2)|2 +c2(| G* (y1 - y2)|2 + |G*(z1 - z2)|2), [H (t, v1) - H(t, v2), v1 - v2]≥|c2 |G(x1-x2)|2 + c2(| G* (y1-y2)|2 + |C*(z1- z2)|2), where c2 and c2 are given positive constants.Inspired by Peng and Wu [69], we also introduced a m. x n full rank matrix G to overcome the problem caused by the different dimensions of SDE and BSDE. Using the method of continuation, the 116 product rule of Semimartingales and the fixed point principle, with the help of the theory of BSDE driven by a continuous time, finite state Alarkov Chain and the theory of Riccati Equations, we proved the existence and the uniqueness of this kind of Equations under different monotone assumptions. Because of the property of the martingale generated by the finite state Markov Chain is different from the property of the Brownian motion, so the form of the monotone assumptions we got here is different from Peng and Wu [69].Theorem 0.16. Suppose that Assumptions (A3.1) and (A3.2) hold, then there exists a unique adapted solution (X, Y. Z) for Equation (0.0.11).Theorem 0.17. Suppose that Assumptions (A3.1) and (A3.3) hold, then there exists a unique adapted solution (X, Y; Z) for Equation (0. 0.4).We also study its revelent continuity result on parameter.3. Fully Coupled FBSDE driven by Markov chain: forward and backward Equations take different dimensions meanwhile YT = Φ(XT)Chapter 4 is the successor of Chapter 3.We study the fully Coupled FBSDE driven by Markov chain:as forward and backward Equations take different dimensions meanwhile YT=Φ(XT).We consider the fully coupled FBSDE driven by M(t): where(X,Y,Z)∈(Rn,Rm,Rm×d).T>0 is any fixed real number and b,σ,f,are func-tions with proper dimensions.There are Assumptions:(A4.1)For each v=(x,y,z)∈Rn×Rm×Rm×d,F(·,v),H(·,v)∈M2(0,T;Rn× Rm×Rm×d)and for each x∈Rn,Φ(x)∈L2(Ω,FT;Rn); and ●F(t,v)is uniformly Lipschitz with respect to v; ●H(t,v)is uniformly Lipschitz with respect to v; ●Φ(x)is uniformly Lipschitz with respect to x. (A4.2)There exist constants c2,c’2,c3 such that [F(t,v1)-F(t,v2),v1-v2]≤-c2|G(x1-x2)|2-c21(|G*(y2-y2)|2+|C*(z1-z2)|2), [H(t,v1)-H(t,v2),v1-v2]≤-c2|G(x1-x2)|2-c21(|G*(y1-y2)|2+|C*(z1-z2)|2), and (φ(x1) -φ(x2),G(x1- x2)) ≥ c3|G(x1- x2)|2, Vx1 ∈ Rn,Vx2 ∈ Nn. where c2,c’2 and c3 are given positive constants. (A4.3)There exist constants c2,c’2,c3 such that [F(t,u1)-F(t,u2),u1-u2|≥c2|G(x1-x2)|2+c21,(|G*(y1-y2)|2+[G*(z1-x2)|2), [H(t,u1) - H(t, u2),u1- u2]≥c2|C(x1- x2)|2+c21(|C*(y1- y2)|2 +[G*(z1- z2)|2), and where c2,c’2 and c3 are given positive constants.Using the method of continuation, the Ito product rule of Semimartingales, with the help of results of BSDE on a finite state Markov chain and lemmas proved at Chapter 3, we can give conclusions about equation(0.0.12):Theorem 0.18. Suppose that Assumptions (A4.1) and (A4.2) hold, then there exists a unique adapted solution (X, Y. Z) for Equation(0.0.12).Theorem 0.19. Suppose that Assumptions (A4.1) and (A4.3) hold, then there exists a unique adapted solution (X, Y. Z) for Equation(0.0.12).4. Fully Coupled FBSDE driven by Markov chain in Stopping time Du-rationIn Chapter 5, we study fully Coupled FBSDE driven by the Martingale Mt, which is generated by a continuous time, finite state Markov chain m = {mt, t ∈ [0; T]} in stopping time duration. The Equation is equivalent to a FBSDE driven by Markov chain m = {mt, t ∈ [0, T]}.Assume m = {mt, t ∈ [0, T]} is a continuous time, finite state Ma.rkov chain and it takes values in unit vectors e7 in Rd. where d is the number of states of the chain. Suppose {Ft} is the completed natural filtration generated by the σ-fields and F= FT. Consider stochastic processes defined on the filtered probability space (Ω,F.Ft. P). Suppose that Mt is a martingale generated by m = {mt, t ∈ [0, T]}. Assume τ= τ(ω) is a Ft-measurable stopping time and take values in [0, ∞]. We give the following notations: φ2 = {ut, 0 ≤ t≤τ, is α Ft -adapted process such t.ltat E[sup0<t<τ|ut|2] < ∞}, H2 = {ut,O < t < τ, is a Ft- adapted process such that L2 = {ξ,ξ is a FT- measurable random variable such that, E|ξ|2 <∞}.We consider the fully coupled FBSDE driven by M(t) where t > 0, (X, Y, Z) E (Rm, Rm, Rm×d). b,σ, f,φ are functions of proper dimensions.Set u = (x, y, z) ∈ Rm × Rm × Rm×d, let where σ=(σl…σd).There are Assumptions:(A5.1)For each v=(x,y,z)∈Rm×Rm×Rm×d,Φ(x)∈f2,b,σ are progressively measurable and (A5.2) .There exists a positive meanwhile deterministic bounded functionψ1(t),such that for v1=(x1,y1,z1)∈Rm×Rm×Rm×d,v2=(x2,y2,z2)∈Rm×Rm×Rm×d, |l(t,x1,y1,z1)-l(t,x2,y2,z2)|≤ψ1(t)[|x1-x2|+|y1-y2|+|z1-z2|],t≥0,l is equal to b,σ,f respectively and ∫0∞ψ1(t)dt<∫0∞ψ12(t)dt<∞..There exists a constant c>0,such that,|Φ(x1)-Φ(x2)|≤c|x1-x2|.(A5.3)There exist constants c2,c’2,c3,such that for each v1=(x1,y1,z1),v2= (x2,y2,z2),v=(x,t,z)=(x1-x2,y1-y2,z1-z2), [F(t,v1)-F(t,u2),v]≤-c2ψ1(t)|x|2-c2ψ1(t)(|y|2+|z|2), [H(t,v1)-H(t,v2),v]≤-c2ψ1(t)|x|2-c2ψ1(t)(|y|+|z|2),P-a.s.,a.e.t∈R+,(?)v1=(x1,y1,z1),u2=(x2,y2,z2)∈Rn×Rm×Rm×d, and where c2,c’2,c3 are given positive constants.By using the Ito product rule of the semimartingales and the fixed point theorem, furthermore stopping time technique,we prove the following conclusion: Theorem 0.20.Suppose that Assumptions(A5.1),(A5.2) and (A5.3)hold,then there exists a unique adapted solution(X,Y, Z)∈φ2×φ2×H2 for Equation(0.0.13). There is another Assumption:(A5.4)There exist constants c2,c’2,c3,and for each v1=(x1,y1,z1),v2=(x2,y2,z2), v=(x,y,z)=(1x-x2,y1-y2,z1-z2), [F(t,v1)-F(t,v2),v]≥c2ψ1(t)|x|2+c’2ψ1(t)(|y|2+|z|2), [H(t,v1)-H(t,v2),v]≥c2ψ1(t)|x|2+c’2ψ1(t)(|y|2+|z|2),P - a. s., a.e.t ∈ R+, (?)v1 = (x1, y1 z1), v2 = (x2, y2, z2) ∈ Rn × Rm×Rm×d, and where c2, c’2, c3 are given positive constants. We have the following conclusion too:Theorem 0.21. Suppose that Assumptions (A5.1), (A5.2) and (.45.4) hold, then there exists a unique adapted solution (X. Y. Z) ∈φ2×φ2×H2 for Equation (0.0.13).Next, we give a comparison theorem for the solutions of initial values.M,’e consider the fully coupled FBSDE driven by M(t): where i = 1, 20We have the following comparison theorem for the solutions of initial values:Theorem 0.22. Suppose that Equation(0.0.7) satisfying Assumptions (.45.1), (A5-) and (A5.0), suppose (X1, Y1, Z1) and (X2, Y2, Z2) are solutions of two Equations re-spectively. If .x1 ≥ x2, then Y01≥Y02 .