Differential Games,Numerical Approximations And Applications In Optimal Premium Policy Of Forward-Backward Stochastic Systems

In this paper, we will discuss stochastic differential games, numerical approximations and applications in optimal premium policy of forward-backward stochastic differential equations (FBSDEs). This paper consists of four parts. In the first part, we study the forward-backward mon-zero sum stochastie differential games with impulse controls, and derive its maximum prin-ciple and verification theorem. In the second part, we consider the linear quadratic non-zero sum differential game problem with asymmetric information, and obtain the explicit form of its Nash equilibrium points under four classes of specific asymmetric information respectively. In the third part, we present a new numerical scheme for a class of coupled forward-backward stochastic differential equations, and prove the convergence of the scheme and its rate of con-vergence. In the fourth part, we address a new class of dynamic optimal premium problems of an insurance firm to the cash-balance management driven under a FBSDE formulation, we explicitly derive the optimal premium policy with the associated optimal objective functional and also give numerical simulations to illustrate the theoretical results.Next, we introduce the content and structure of this paper.In the first chapter, we make a brief review about historical literatures related to researches in this paper.In the second chapter, we study a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the game systems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls. Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibrium point of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investment portfolio and optimal impulse consumption strategy are obtained explicitly. The content of this chapter is included in the following paper:D. Chang and Z. Wu, Stochastic maximum "principle for non-zero sum, differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, Vol.11 (2015),27-40.In the third chapter, we study linear quadratic non-zero sum differential game problem with asymmetric information. Compared with the existing literature, a distinct feature is that the information available to players is asymmetric. Nash equilibrium points are obtained for several classes of asymmetric information by stochastic maximum principle and technique of completion square. The systems of some Riccati equations and forward-backward stochastic filtering equations are introduced and the existence and uniqueness of the solutions are proved. Finally, the unique Nash equilibrium point for each class of asymmetric information is repre-sented in a feedback form of the optimal filtering of the state, by virtue of the solutions of the Riccati equations. The results of this chapter are from the following paper:D. Chang and H. Xiao, Linear quadratic nonzero sum differential games with asymmetric information, Mathematical Problems in Engineering, vol.2014, Article ID 262314,11 pages, 2014.In the fourth chapter, a new numerical scheme for a class of coupled forward-backward stochastic differential equations (FBSDEs) is proposed by branching particle systems in a random environment. First, by the Four Step Scheme, we introduce a partial differential equation (PDE) used to represent the solution of the FBSDE system. Then, infinite and finite particle systems are respectively constructed to obtain the approximate solution of the PDE. The location and weight of each particle are governed by stochastic differential equations derived from the FBSDE system. Finally, a branching particle system is established to define the approximate solution of the FBSDE system. The branching mechanism of each particle depends on the path of this particle itself during its short lifetime e= n-2α, where n is the number of initial particles and α<1/2 is a fixed parameter. The convergence of the scheme and its rate of convergence are obtained. The results of this chapter are included in the following paper:D. Chang, H. Liu and J. Xiong, A branching particle system approximation for a class of FBSDEs, Journal of Mathematical Analysis and Applications, submitted.In the fifth chapter, we consider an optimization problem faced by an insurance firm who can control its cash-balance dynamics by adjusting the underlying premium rate. The firm’s objective is to minimize the total deviation of its cash-balance process to some pre-set target levels, the running cost of the premium policy and the generalized recursive utility by selecting an appropriate premium policy. Our problem has three distinguishable features:(1) both full and partial information cases are investigated; (2) the state is subjected to terminal constraint; (3) the optimization problem is under a forward-backward stochastic differential equation formulation by introducing a generalized stochastic recursive utility. The optimal premium policy with the associated optimal objective functional are completely and explicitly derived. Finally, we also give numerical simulations to illustrate the theoretical results. The content of this chapter is included in the following paper:D. Chang and Z. Wu, Optimal premium policy driven by FBSDEs under full and partial information, working paper.In the following, we show the main results of this paper. 1. Stochastic Maximum Principle for Non-Zero Sum Differential Games of FBSDEs with Impulse ControlsConsider the following FBSDE dynamic of the non-zero sum differential game involving impulse controls: where b:[0, T] × Rn × Rk1 × Rk2→ Rn, σ:[0, T] × Rn × Rk1 × Rk2→ Rn×d, f:[0, T] × Rn × Rm × Rm×d × Rk1 × Rk2→Rm, g: Rn→ Rm are measurable mappings, C1:[0,T]→ Rn×d1, C2:[0,T]→Rm×d1, D1:[0, T]→ Rn×d2, D2:[0,T]→ Rm×d2 are continuous functions. v1(·) and v2(·) are the continuous control processes of Player 1 and Player 2.η1 (·) and η2(·) are the impulse control processes of Player 1 and Player 2.Then we introduce a cost functional: Ji(v1(·),v2(·),η1(·),η2(·)) where φi:Rn→ R,ri:Rm→ R (i= 1,2) and hi:[0,T] × Rn × Rm × Rm×d × Rk1 × Rk2→ R (i= 1,2) are given measurable mappings.Suppose each player hopes to maximize her/his cost functional Ji(v1(·),V2(·),η1(·),η2(·)) by selecting an appropriate admissible control (vi(·),ηi(·)) (i = 1,2), then the problem is to find a pair of admissible controls (v1(·),ζ1(·), v2(·),ζ2(·) ∈ A1× A2 such thatWe call the problem a forward-backward non-zero sum stochastic differential game involv-ing the impulse controls.The main result of this part are the following theorems. Theorem 2.1. Let (H2. 1) and (H2.3) hold. Let (v1(·),ζ1(·), v2(·)ζ2(·)) ∈ A1 × A2 is a Nash equilibrium point of the aforementioned game and (x(·), y(·), z(·)) is the corresponding state trajectory, and (pi(·), qi(·), ki(·)) (i = 1. 2) is the solution of the adjoint equation (2.2.6). Then. (?)vl ∈ U1, v2 ∈ U2, η1 (·) ∈ K1 and η2(·) ∈ K2, we haveH1v1, (t, x(t), y(t), Z(t), v1(t), v2(t), p1(t), q1 (t), k1(t))(vl - v1 (t)) ≤0 a. e., a. s., (2.2.10) E{∑j≥1[(l1ζ1(Tj,ζ1j) +q1*(Tj)C1(Tj) -p1*(Tj)C2(Tj))(η1j -ζ1j)]} ≤0, (2.2.11) H2v2(t,x(t),y(t),Z(t),v1(t),v2(t),p2(t),q2(t),k2(t))(v2-v2(t)) ≤0 a. e., a. s., (2.2.12) E{∑j≥1[(l2ζ2(Tj, ζ2j) + q2*(Tj)D1(Tj) p2*(Tj)D2(Tj))(η2j -ζ2j)]}≤0. (2.2.13)Theorem 2.2. Let (H2.1)-(H2.3) hold. Assume that the functions φi, γi, ηi → li (t, ηi) and (x, y, z,v1, v2) → Hi (t,x,y,z,v1,v2, pi,qi,ki) (i= 1, 2 are concave. Moreover, for K ∈Rm×n and ξ ∈ L2(Ω,_FT,P;Rm) yv1,v2,η1,η2(T) = Kxvl,v2,η1,η2(T) +ξ, (vl(·),η1(·),v2(·),η2(·)) ∈ A1 × A2. Let (pi, qi, ki) (i = 1, 2) be the solutions of the adjoint equation associated with (v1, ζ1,v2,ζ2) E A1× A2. Then (v1,ζ1, v2, ζ2) is a Nash equilibrium point of the forward-backward non-zero .sum stochastic differential game involving the impulse controls, if it satisfies (2.2.10), (2.2.11), (2.2.12) and (2.2.13).2. Linear Quadratic Non-Zero Sum Differential Games with Asymmetric Informa-tionIn this part, we study the linear quadratic (LQ, for short) non-zero sum differential game with asymmetric information. For simplicity; we only study the case of two players. Consider the following SDE and cost functionals of the form Here a, b1, b2, c, e, g1, g2 and 93 are bounded and deterministic functions in t, l1 and l2 are bounded, nonnegative and deterministic functions in t, m1 and m2 are bounded, positive and deterministic functions in t, and r1 and r2 are two nonnegative constants. Hereinafter, we omit all dependence on time variable t of all processes or deterministic functions if there is no risk of ambiguity from the context for the notational simplicity; u1(·) and u2(·) are the control processes of Player 1 and Player 2, respectively. We always use the subscript 1 (resp. the subscript 2) to characterize the control variable corresponding to Player 1 (resp. Player 2) and use the notation xu1,u2 to denote the dependence of the state on the control variableLet Ft denote the full information up to time t and gti(?)Ft be a given sub-filtration which represents the information available to Player i (i= 1,2) at time t ∈ [0, T]. If gti(?) Ft and gti≠ Ft, we call the available information partial or incomplete for Player i. If gt1≠gt2, we call the available information asymmetric for Player 1 and Player 2. In the sequel, we shall study the following four classes of asymmetric information:(ⅰ) gt1= Ft1,2 and gt2= Ft2,3 i.e., the two players possess the common partial information Ft2;(ⅱ) gt1= Ft1,2 and gt2= Ft2, i.e., Player 1 possesses more information than Player 2;(ⅲ) gt1= Ft and gt2= Ft2, i.e., Player 1 possesses the full information and Player 2 possesses the partial informaion;(ⅳ) gt1= Ft1,2 and gt2= Ft3, i.e., the two players possess the mutually independent information.Suppose each player hopes to minimize her/his cost functional Ji(u1(·),u2(·)) by selecting a suitable admissible control ui(·) (i= 1,2). In this study, the problem is, under the setting of asymmetric information, to look for (u1(·), u2(·)) ∈u1×u2 which is called the Nash equilibrium point of the game, such that We call the problem above an LQ non-zero sum differential game with asymmetric information. For simplicity, we denote it by Problem (LQNZSDG).The results of this part are given by the following theorems.Theorem 3.1. (u1,u2) is a Nash equilibrium point for Problem (LQNZSDG) if and only if(u1,u2) has the form denoted by (3.3.8) and (x,(y1, z11,z12,z13), (y2,z21,z22,z23)) satisfies FBSDE (3.3.9).We consider four classes of asymmetric information, respectively. Case 1:gt1=ft1,2 and gt2=Ft2,3.Theorem 3.1 can be rewritten as follows:Theorem 3.2. (u1,u2) is a Nash equilibrium, point for Problem (LQNZSDG) if and only if (u1,u2) has the following form:where (x, (y1, z11, z12,z13); (y2,z21,z22,z23)) is a solution of the following FBSDEwe derive the Nash equilibrium point which is represented in the feedback of the optimal filters x,x and x of the state x.Theorem 3.4. Under the assumption (H3.3), Problem (LQNZSDG) has a unique Nash equi-librium point denoted bywhere x, x and x are as shown in (3.3.32), (3.3.40) and (3.3.45) respectively, and γi and Ti (i= 1,2,3) are uniquely determined by the systems of (3.3.37) and (3.3.43), respectively. Case 2:gtl= Ft1,2 and gt1= Ft2.We can obtain the following theorems. Theorem 3.5.(v1,v2) is a Nash equilibrium point for Problem (LQNZSDG) if and only ifwhere (x, (y1, z11, z12, z13), (y2,z21,z22,z23)) is a solution of the following FBSDETheorem 3.6.If(H3.3) holds, then Problem (LQNZSDG) has a unique Nash equilibrium point denoted by where x and x are shown in (3.3.32) and (3.3.40), respectively. Case 3:gtl= Ft and gf2= Ft2. We have the following theorem.Theorem 3.7. [v1,v2) is a Nash equilibrium point for Problem (LQNZSDG) if and only ifwhere (x, (y1, z11,z12, z13), (y2, z21,z22,z23)) is a solution of the following FBSDETheorem 3.8. Under the assumptions (H3.3), Problem (LQNZSDG) has a unique Nash equilibrium point denoted bywhere x and x are shown as (3.3.32) and (3.3.52), respectively.Case 4:gtl= Fg1,2 and gt2= Ft3. We present the following theorem.Theorem 3.9. (v1, v2) is a Nash equilibrium point for Problem (LQNZSDG) if and only ifwhere (x, (y1, z11, z12,z13), (y2, z21, z22,z23)) is a solution of the following FBSDETheorem 3.11. Under the assumptions (H3.3) and (H3.4), Problem (LQNZSDG) has a unique Nash equilibrium point denoted by here Ex, x and x are shown in (3.3.59), (3.3.61) and (3.3.63) respectively, and γi and n (i = 1, 2, 3) are uniquely determined by the systems of (3.3.37) and (3.3.43) with e(·) replaced by O, respectively.3. A branching particle system approximation for a class of FBSDEsWe consider the following forward-backward stochastic differential equations (FBSDEs) in the fixed duration [0, T]: where b : Rd × Rk → Rd, σ : Rd → Δd×1,g:Rd×Rk×k×l→Rk and f:Rd→RK.In what follows, we make the following hypothesis: Hypothesis (H4.1): g has the following form: for z = (z1,…,z1), b(x, y), σ(x), g(x, g, z), f(x), C(x, y) and D(x, y) are all bounded and Lipschitz continuous maps with bounded partial derivatives up to order 2.Rely on the idea of the Four Step Scheme, we know the solutions to the above forward-backward SDE have the relation Y(t) = u(t, X(t)), Z(t) = (?)xu (t, X (t)) σ (X(t)), where u(t, x) is a solution to the PDE with aij = (σσ*)ij, σ = (σ1,…σl) and bi being the i-th coordinate of b.For 0≤t≤T, assume v (t, x) = u (T - t, x). Note thatThe nonlinear parabolic partial differential equation (4.1.2) can be written as:We construct an infinite particle system {X,(t) : i ∈ N} with locations in Rd and time varying weights {A,(t) : i ∈ N} governed by the following equations: for 0 < t ≤ T, i = 1,2,.with i.i.d initial values {Xi(0), Ai(0), i∈N}, where {Bi(t), i∈N} are independent standard Brownian motions andWe can obtain the following theorem.Theorem 4.2. The solution to particle system (3.1.3) is unique and its density function is the solution to partial differential equation (3.1.3).Next we introduce a finite partMe system to get the approximation solution: for fixed 5 > 0, t (0, T],where i = 1, 2, …… n. The initial values are given as Xin,δ(0) = x,Ain,δ(0)=1.Theorem 4.3. The convergence of vn,δ(t) to v(t) is bounded byNote that KT of Theorem 4.3 above is of exponential growth as T increases, and hence, the error of the approximation grows exponentially fast. To avoid this drawback of the numerical scheme, we introduce a branching particle system to modify the weights of the particles at the time-discretization steps.For fixed δ>0, ε=n-2α, 0 < α < 1. there are n particles initially, each with weight 1 at locations Xin,δ,ε(0),i= 1, 2,……, n which are i.i.d random variables in Rd. Assume the time interval is[0,T] and N*=[T/∈]which is the largest integer not greater than T/∈.Define ∈(t)=j∈ for i∈≤t<(j+1)∈.In the time interval [j∈,(j+1)∈),j≤N*,there are mjn particles alive and their locations and weights are determined as following:for i=1,2,……,mjn, where the initial values are defined as:Xin,δ,∈,(0)=x,Ain,δ,∈(0,0)=1,m0n=n.We define the unnormailized approximate filter as following:We can get the following theorem. Theorem 4.5. For any f∈[0,T],δ>0,∈=n-2α and 0<α<1/2,there exists a constant Kδ,such that Eρ12(Vn,δ,∈(t),Vδ(t))≤ KTn-(1-2α)+Kδ,T,n-2α.We define i.e.the smooth density of Vn,δ,∈(t), as the numerical approximation of u(t,x) and un,δ,∈(t,x)=vn,δ,∈(T-t,x)as the numerical approximation of u(t,x).Then,we have the following corollary: Corollary 4.1. For any t∈[0,T]0<α<1/2,there exists a constant Kδ,such thatWe apply the Euler Scheme to approximate X(t) of the FBSDE(4.1.1). Define the numerical solution Xn,δ,∈(t)satisfying:Theorem 4.6.The convergence of Xn,δ,∈(t) to X(t) is bounded by Kδ,T(n-(1-2α)V n-2α)+ KTδ.Then,by the result of the Four Step Scheme,we define Yn,δ,∈(t)=un,δ,∈(t,Xn,δ,ε)as the the numerical solution of Y(t)in FBSDE(4.1.1).We have the following theorem:Theorem 4.7. The convergence of Yn,δ,∈(t)to Y(t)is bounded by4.Optimal premium policy driven by FBSDEs under full and partial information4.1.Optimal premium policy under full informationConsider now an insurance firm whose cash-balance process Xtv is governed bywhere the control v is admissible in the senseDefinition 5.1.An R-ualued premium policy v={vt}0≤t≤T is called admissible,if vt is FvW -adapted and E∫0Tvt4dt<+∞ for each 0≤t≤T.we consider an additional constraint on the terminal state variable XTv by EXTv=w0 (5.1.2)We consider recursive utility or risk measure Ytv satisfying:We assume the cost functional is of the form where β is discounting factor,Ut is dynamic preset target,R is preset target for recursive utility,Lt,Nt,M and Q are weighting factors.We state the optimal premium problem with full information(OPFI,for short)as follows Problem(OPFI). Seek a v∈UF such that J[v]=infv∈vF J[v]subject to(5.1.1),(5.1.2) and(5.1.3).The main result of this problem is the following theorem.Theorem 5.1.Let Hypothesis(H5.1)holds.If vt=-Nt-1eβt(pt-Btqt) is the optimal premium policy of Problem (OPFI),then it can be represented as where(X,Y,Z,p,q,k),λt1,λt2,ψt,αt1,αt2, and αt3 are the solutions of (5.1.10),(5.1.15), (5.1.16),(5.1.17), (5.1.21),(5.1.22) andc (5.1.23),respectively.Theorem 5.2.Let(H5.1)holds.Then the optimal premium policy of Problem(OPFI)is where Xt satisfies (5.1.24), λt1,λt2,ψt,αt1,αt2 and αt3 are the solutions of (5.1.15), (5.1.16), (5.1.17), (5.1.21), (5.1.22) and (5.1.23), respectively. Moreover, under (H5.2), the optimal cost functional is given by (5.1.35).4.2 Optimal premium policy under partial informationWe suppose that the policymaker can only get information from the stock, and consider the system and where the cash-balance process Xtv is the underlying factor which is partially observed through the signal (observation) St with instantaneous volatility ht. We give a definition of admissible control. SetDefinition 5.2. A control v is called admissible, if v ∈Vad0 is gt-adapted. The set of all admissible controls is denoted by Vad.The optimal premium problem with partial information (OPPI, for short) can be stated as follows:Problem (OPPI). Seek a v ∈ Vad such that J[v] = infv∈Vad J[v] subject to (5.2.41), (5.2.42) and (5.1.2).The main result of this problem is the following theorem.Theorem 5.3. Let, Hypothesis (H5.1) holds. Suppose that v is an optimal premium policy of Problem (OPPI) and (X, Y, Z) is the corresponding optimal state. Then the FBSDE admits a unique solution (p, q, k)∈LFW2 (0, T; R) such that e-βt Ntvt +E[pt|gt] - BtE[qt|gt] = 0, (5.2.44) with gt=△σ{Ss:0≤s≤t}.Theorem 5.4.Suppose that Hypothesis(H5.1)holds.Let v∈Uad satisfy e-βt Ntvt+E[pt|gt]一BtE[qt|gt]=0, where(X,Y,Z,p,q,k)is a solution to (5.2.48).Then v is an optimal premium policy of Problem (OPPI),Theorem 5.5.Let Hypothesis(H5.1)holds.If vt=eβtNt-1(Btqt-pt) is the optimal control of Problem(OPPI),then it can be represented as where(X,Y,Z),(p,q,k),αt1,αt2,αt3,λt1,λt2 and ψt are the solutions of(5.2.56) with v=v, (5.2.61),(5.1.15),(5.1.16),(5.1.17),(5.1.21),(5.1.22) and (5.1.23),respectively.Theorem 5.6.Let (H5.1) and (H5.3) hold.Then the optimal premium policy of Problem (OPPI)is where Xt satisfies(5.2.56).λt1,λt2,ψt,αt1,αt2 and αt3 are the solutions of (5.1.15),(5.1.16), (5.1.17),(5.1.21),(5.1.22) and (5.1.23),respectively.Moreover,under(H5.2),the optimal cost functional is given by (5.2.66).