30 Chapter 2. Linear Equations Dropping the first term on the left side of (1.27), then dividing both sides by T - t and sending t + T, we obtain (1.28) EI~T f(T)I 2 = O. By (1.19), and the relation Y(T) = GX(T) + Fg, we obtain (1.29) tiT[A1 GA1 + (B1 - GB1)G]X(T) + ~?T(B1 GB1)Fg = O, a.s. Thus, (1.21) follows. In the case (1.22) holds, for any x E ~'~ and g E ]R m (deterministic), by some choice of b, a, b and ~, (1.1) admits an adapted solution (X, Y, Z) _= (x, Gx + Fg, 0). Then, (1.21) implies (1.23). [] w Some Reductions In this section, we are going to make some reductions under condition (1.5). We note that (1.5) is very general. It is true if, for example, F = I E ]R m• which is the case in many applications. Now, we assume (1.5). By Theorem 1.2, if we want (1.1) to be solvable for all given data, we must have C1 -GC1 to be onto (and thus g > m). Thus, it is reasonable to make the following assumption: Assumption A. Let ~ = m and C1 - GC1 C ]R m• be invertible. Let us make some reductions under Assumption A. Set Y = Y - GX. Then Y(T) = Fg and (see (1.18)) dY = (.4X + BY + CZ + Db)dt + (AIX +/~IY + CIZ + DI~)dW - G(AX + BY + CZ + Db)dt - G(A1X + B1Y + C1Z + Dla)dW (2.1) = {[.4-GA+(B-GB)G]X+(B-GB)Y + (C - GC)Z + bb- GDb}dt + { [A1 -GA1 + (B1 -aB1)G]X -}- (gl -GB1)Y + (C1 - GCI)Z + b~3 - GD~a~dW. J Define (2.2) =[A1 - GA1 + (B1 - GB1)G]X + (BI - GB1)7 + (C1 - GC1)Z + DI~ - aDla. Since (C1 - GC1) is invertible, we have Z = (@1 - GC1)-~{-2- [A1 - GA1 + (B1 - aB )a]X (2.3) - (B1 - GB1)Y - (1)1~ - aDla)}. w Some reductions 31 Then, it follows that (2.4) where dx = (ix + B Y + c z + a)d, + + + + dr = (-XoX + BoY + CoZ + a)d, + ~dw, Z(O) = x, Y(T) = Fg, [ -A = A + BG - C(C1 - GC1)-~[-4~ - GA1 + (B~ - GB~)G], B = B - C(01 - GC~)-I (B~ - GB~), ] U C(01 - GC1) -1, -b = Db - C(dl - GC~)-1(/3~ - GD~a), ] A1 = A1 + B1G ) N~ = B~ - C~(01 - GC~)-l(/~ - GBI), (2.5) / U~ = C~(dl - GC1) -~, / v = Dla - C1(C1 - GCI )-1(/318 - GDla), I Ao = A - CA + (B - aB)c / - (~ GC)(C1 - GC1)-I[A1 - GA1 4- (B1 - GB~)G], I -Bo = B - GB - (C - GC)(C~ - GC~)-~ (Bx - GB~), / C0 ~ (0 - GC)(01 - GC1) -1, I,-~ = [g- GDb - (0 - CC)(Ol - GC~)-I(DI~ - GDla). The above tells us that under Assumption A, (1.1) and (2.4) are equivalent. Next, we want to make a further reduction. To this end, let us denote ) Bo ' (2.6) [~1__ (~01 -~01) ' ~1 __ (~/1) . Let q~(-) be the solution of the following: { d~(t) = ~a2(t)dt + 2~(t)dW(t), t >_ O, (2.7) ~(0) = I. 32 Chapter 2. Linear Equations Then (2.4) is equivalent to the following: For some y C ]R TM, x(t) (2.8) \ h(s) ,] t 9 [0, T], with the property that (2.9) b f ~(8) "~ "~1 ('~8) ) / Clearly, (2.9) is equivalent to the following: For some y 9 IR m and Z(-) 9 L~(0, T; Rm), it holds (;) (:) // = (0, I)~(T) + (0, I)V(T) ~(s)-l(g - Algl)Z(s)ds // + (0, I)~(T) ~(s)-lg-1 Z(s)dW(s). Thus, if we can solve the following: (2.11) { d(~) : ( ~(~) +~)dt+ (~1 (~) O, ]/'(T) : ~, with 7/being given by (2.10)~ then for such a pair y - Y(0) and 2(-) = Z(.), by setting (X,Y) as (2.8), we obtain an adapted solution (X,Y,Z) 9 A/f[0, T] of (2.4). The above procedure is reversible. Thus, by the equiv- alence between (2.4) and (1.1), we actually have the equivalence between the solvability of (1.1) and (2.11). Let us state this result as follows. w Solvability of linear FBSDEs 33 Theorem 2.1. Let F = I E ~m• and ~ = m. Then (1.1) is solvable for all b, a, b, ~, x and g satisfying (1.3) if and only if (2.11) is solvable for all c We note that by Theorem 1.2, F I and g = m imply Assumption A. Based on the above reduction, in what follows, we concentrate on the following FBSDE: (2.12) Ix(0) = 0, = (Ax + BY + cz)dt + (AIX + BaY + CIZ)dW(t), dY = (AX + BY + CZ)dt + ZdW(t), Y(T) = g. By denoting (2.13) A B .A1 ~ ( A1 B1) C1 ~-~ (C/1) 0 ~ we can write (2.12) as follows: (2.14) t E [0, T], { d (z) : {A(X) "~-Cz}dt-{- {'A1 (z) ~- 0, Y(T) : ?7. In what follows, we will not distinguish (2.12) and (2.14), and we will let { d~(t) = A~(t)dt + Al~(t)dW(t), t E [0, T], (2.15) ~(0) = I. If we call (X,Y) the state and Z the control, (2.12) is called a (lin- ear) stochastic control system. Then, the solvability of (2.12) becomes the following controllability problem: For give g E L~T (~; Rm), find a control Z E L~(0, T; Rm), such that some initial state (X(0), Y(0)) E {0} • IR m can be steered to the final state (X(T), Y(T)) E L~T (~t; R~) • {g} at the moment t = T, almost surely. This is referred to as the controllability of the system (2.12) from {0} • IR m to L~:T(~;~t n) • {g}. We note that g is an S-T-measurable square integrable random vector, and we need exactly control Y(T) to g. w Solvability of Linear FBSDEs In this section, we are going to present some solvability results for linear FBSDE (2.12). The basic idea is adopted from the study of controllability in control theory. For convenience, we denote hereafter in this chapter that 34 Chapter 2. Linear Equations H = L2r(~;~m) and 7/ L~(0, T;~r~) (which are Hilbert spaces to which the final datum g and the process Z(.) belong, respectively). w Necessary conditions First of all, we recall that if 9 is the solution of (2.15), then, 9 -1 exists and it satisfies the following linear SDE: ~ d0-1 = -o-l[r O-1~AqdW(t), t > O, (3.1) [ 0-1(0) = I. Moreover, (X, Y, Z) E AJ[0, T] is an adapted solution of (2.12) if and only if the following variation of constant formula holds: ( X(t)~ = O(t)(~)-[-O(t)f0 t 0(8)-1/C - .AlCl)Z(s)ds v(t) ] (3.2) + O(t) O(s)-lC1Z(s)dW(s), t e [0, T], for some y E ]R m with the property: AlCX,Z,s, s (3.3) + O(T) foTO(s)-lClZ(s)dW(s)}. Let us introduce an operator/(7 : 7/-+ H as follows: T K:Z = (0, I){O(T)fo O(s)-l(c- Axel)Z(s)ds (3.4) + O(T/~0 T o(s/-lClZ(s/~w(~/}. Then, for given g C H, finding an adapted solution to (2.12) is equivalent to the following: Find y C ]R "~ and Z C 7-/, such that (3.5) g = (O, I)O(T) ( ~ ) y + lCZ, and define (X, Y) by (3.2). Then (X, Y, Z) E AA[0, T] is an adapted solution of (2.12). Hence, the study of operators O(T) and /(7 is crucial to the solvability of linear FBSDE (2.12). We now make some investigations on 0(-) and K:. Let us first give the following lemma. L 1 rO T" IR n+m~ L~(O, T; ~n+,~), it Lemma 3.1. For any f E j=~ , , j and h E w Solvability of linear FBSDEs 35 holds { E~(t) = e ~t, (3.6) E{q~(t) ~ot~(s)-lf(s)ds} =- ~oteA(t-S)Ef(s)ds, t C [0, T]. E{ ~(t) ~otO(s)-l h(s)dW (s) } = O, Also, it holds that (3.7) E sup IO(t)[ 2k + E sup Io(t)-ll 2k < ~, Vk _> 1. O<_t<T O<_t<_T Proof. Let us first prove the second equality in (3.6). The other two in (3.6) can be proved similarly. Set (3.8) ~(t) = ~(t) ~(s)-lf(s)ds, t e [0, T]. Then ~(.) satisfies the following SDE: (3.9) ~ d~(t) = IMp(t) + f(t)]dt + Al~(t)dW(t), t C [0, T], [ ~(o) = o. Taking expectation in (3.9), we obtain (3.10) ~ d[E~(t)] = [AE~(t) + Ef(t)]dt, t E [0,T], [ E~(o) = o. Thus, (3.11) E~(t) = ~(~-s)Ey(s)ds, t 9 [0,T], proving our claim. Now, we prove (3.7). For any 40 9 IRn+m, process ~(t) ~ r satisfies the following SDE: (3.12) ~ d~(t) = A~(t)dt + Al~(t)dW(t), t 9 [0, T], [ ~(o) = ~o. Then, by It6's formula, Burkholder-Davis-Gundy's inequality and Gron- wall's inequality, we can show that (3.13) E sup I~(t)l 2k < Kl~012k, k > 1, O~t<_T for some constant K > 0. Thus, the first term on the left hand side of (3.7) is finite. Similarly, one can prove that the second term is finite as well. [] 36 Chapter 2. Linear Equations From (3.7), we see that/~ : 74 -+ H is a bounded linear operator. Now, applying (3.6) to (3.3), we obtain that (2.12) admits an adapted solution, then (3.14) Eg :(O,I){eATf~)yq-/oTeA(T-s)(c AIC1)EZ(s)ds}, for some y E IR m and EZ(.) E L2(0, T;IRm). This leads to the following necessary condition for the solvability of (2.12). Theorem 3.2. Suppose (2.12) is solvable for all g E H. Then rank {(O,I)(eAT (~) ,C- AICi,A(C- AiCI), (3.15) 9 ,r ~- m. Proof. Set C = C - AlCl and define f0 ~ &( s ~ e A(T-~) s)ds, Vu(.) 9 L2(O,T;~m). Then s : L2(0, T; ]R m) + IR n is a linear bounded operator. We claim that (3.16) 7"4(s = n(C) -]- n(r q- ''"-}- n(c4n+m lc). In fact, if x E 7-4(s • then, for any u(-) E L:(O,T;Rm), it holds 0 = xrs = xTeA(T-~)&(s)ds, which yields Consequently, xT eAsC : O, VS ~ [0, T]. dk T .As xT~4kC'=~[x e C~I~=o=O, k_O. This implies that A "1 • X E r" -'~,l,~(C) -}- ~T~(,,4C) q-''" q- ~'~(,Anq-m-lc)) , which results in n(~) + n(AC) + + n(w+m-l~) ; n(L). The above proof is reversible with the add of Calay-Hamilton's theorem. Thus, we obtain the other inclusion, proving (3.16). Then (3.15) follows easily. [] w Solvability of linear FBSDEs 37 We note that in the case g = AIC1, (3.15) becomes (3.17) det{(O,I)e~4T(Oi)}~O. This amounts to say that the FBSDEs (2.12) (with g = Algl) is solvable for all g E H implies that the corresponding two-point boundary value problem for ODEs: (3.18) \ Y(t) = A \ y(t)/' t e [0, T], Z(0) = 0, Y(T) = ~, admits a solution for all ~ E pro. Let us now present another necessary condition for the solvability of (2.12). Theorem 3.3. Let g = O. Suppose (2.12) is solvable for a11 g 9 H. Then, (3.19) det {(O,I)eAtgl} > O, Vt 9 [0,T]. Consequently, if (3.20) T = inf{T > 0 I det [(0, I)eaTC1] = 0} < co, then, for any T > T, there exists a g 9 H, such that (2.12) is not solvable. Remark 3.4. The above result reveals a significant difference between the solvability of FBSDEs and that of two-point boundary value problems for ODEs. We note that for (3.18) to be solvable for all ~ 9 ~m if and only if (3.16) holds. Sinee the function t ~-~ det { (O,I)eAt ( Oi ) } is analytic (and it is equal to 1 at t = 0), except at most a discrete set of T's, (3.16) holds. That implies that for any To 9 (0, c~), if it happens that (3.18) is not solvable for T = To with some ~ 9 ]R TM, then, at some later time T > To, (3.18) will be solvable again for all ~ 9 IR TM. But, in the above FBSDEs case, if T < c~, then for any T > T, we can always find a g 9 H, such that (2.12) (with g = 0) is not solvable. Thus, FBSDEs and the two-point boundary value problem for ODEs are significantly different as far as the solvable duration is concerned. Proo/ o/ Theorem 3.3. Suppose there exists an so 9 [0, T), such that (3.21) det } = O. Note that So < T has to be true. Then there exists an U 9 ]~m, [?~l : 1, such that (3.22) uT(0, I)eA(T-s~ = O. 38 Chapter 2. Linear Equations We are going to prove that for any r > 0 with So + ~ < T, there exists a g 9 L~%o+" (gt; IR m) c_ H, such that (2.12) has no adapted solutions. To this end, we let ~ : [0, T] + ~ be a Lebesgue measurable function such that (3.23) { /~(s)=+l, Vs9 So+r /~(s)=0, 8k 80 1{8 9 [8o,8~] I Z(s) = 1)1 = ~ , 1{8 9 [8o,8~] l Z(8) -1}1 = 8~ - so, 2 Vs 9 (so + e,T]; k_>l, for some sequence Sk $ so and Sk < T - ~. Next, we define ~0 t (3.24) ((t) = fl(s)dW(s), t 9 [0, T], and take g = ~(T)~? 9 L 2 (f~;lR "~) C H. Suppose (2.12) admits an .T'~0+E adapted solution (X, ]I, Z) 9 A4[0, T] for this g. Then, for some y 9 ~m, we have (remember C = 0) (3.25) -~/oTe'A(T-s)[.A1 (If:l) -[-ClZ(s)]dW(s)}. Applying ~T from left to (3.25) gives the following: (3.26) where (3.27) T ((T)=a+fo {7(s)+(J(s),Z(s))}dW(s), 7(.)=~T(o,I)eA(T_.)A 1 X(.) L~(U;C([O,T];P~)), y(.) 9 r [~T(o,I)eA(T ')CI] T is analytic, r 0. Let us denote ~0 t O(t) = a + [7@) + (r Z(s) )]dW(s), (3.28) Then, it follows that S d[0(t) - ((t)] = [7(0 + (r Z(t) ) -fl(t)]dW(t), (3.29) / [O(T) - ((T)] = O. t 9 [0, T]. t e [o, T], w Solvability of linear FBSDEs By ItS's formula, we have 0 =EIt?(t ) - ~(t)l 2 (3.30) ft T + E 1O'(s) + (r Z(s) ) -/3(s)12ds, Thus, (3.31) 13(s) - 7(s) = (r Z(s)), which yields (3.32) Elfl(s ) - "/(s)12ds = 0 o t e [o, T]. a.e. s E [0, T], a.s. E I(r Vk> 1. 39 w Criteria for solvability Let us now present some results on the operator /C (see (3.4) for defini- tion) which will lead to some sufficient conditions for solvability of linear FBSDEs. Now, we observe that (note 7 6 L2(f~; C([0, T]; ~{)) and (3.23)) fs 8~ _ O,(s)12ds E[~(s) 0 1/? /? (3.33) _> ~ Elfl(s) - 7(So)12ds - El~/(s) - 7(so)12ds o o > _ ~_Sk - SO E[ll _7(so)l 2+1 l+7(so)l 2]_o(sk_so), k _> l. On the other hand, since r is analytic with r = O, we must have (3.34) r = (s - So)~b(s), s e [0, T], for some r which is analytic and hence bounded on [0, T]. Consequently, /? /? (3.35) El (r Z(s) )12ds < K(Sk - s0) 2 ElZ(s)12ds. o o Hence, (3.32)-(3.33) and (3.35) imply sk - SOE[ll -~(s0)l ~ + I1 + ~(s0)lJ -o(sk - so) (3.36) <_ K(Sk - So) ~ ElZ(s)12ds, Vk > 1. J 80 This is impossible. Finally, noting the fact that det {(0, I)eAtC1 } I t=o = 1, we obtain (3.19). The final assertion is clear. [] It is not clear if the above result holds for the case C # 0 since the assumption C = 0 is crucial in the proof. [...]... (3. 17) and (3. 19) hold In this case, the adapted solution to (2.12) is unique (for any given g E H) Proof Theorems 3. 2 and 3. 3 tell us that (3. 17) and (3. 19) are necessary We now prove the sufficiency First of all, for any g E H, by (3. 17), we can find y E IRm, such that (3. 14) holds (note C = 0) Then we have (3. 67) g -(O,I)~.(T) ( ~ ) y E Af(E) Next, by (3. 46), there exists a Z C 7/, such that (3. 68)... over [0, T] and hence it follows from (3. 54) that 7/ = 0 This proves (3. 48) We now prove (3. 47) Suppose/CZ = 0 Again, we let (X(.), Y(-)) be defined by (3. 50) Then, for any ( E 7-/, by (3. 53) , we have 0 = E ( fo T r ~:Z ) (3. 63) : ~fo ~ i~(./, (0,1).'~-.~{.~ r~(./ 44 Chapter 2 Linear Equations This implies that (3. 64) (O,I)e~t(T-s){A1 ( X ( s ) ~ k Y(s) ] _~_ClZ(8)}=0, a.e.s E [0, T], a.s By (3. 19), we... lET = 0} ~ Af(E), (3. 47) N(tc) g { z e n I ~ z = 0} = {0) Proof First of all, by Lemma 3. 5, we see that 7~(~) is closed Also, by (3. 4) and Lemma 3. 1, 7~(/C) C_ A/(E) (since C = 4161) Thus, to show (3. 46), it suffices to show that H ( E ) ['-'l n(lc)" = {o} (3. 48) We now prove (3. 48) Take 7/E N'(E) Suppose O=E(~,~CZ) (3. 49) T = E (rl, (O,I)~(T) 42 Chapter 2 Linear Equations Denote (3. 50/ /x-(t) k Y(t)... we have (3. 51) {x(o) ~ \ Y(o) ) = v By It6's formula and Gronwall's inequality, we obtain (3. 52) E{IX(t)I 2 + IY(t)l 2} 0 Thus, (3. 40) implies T E{lZk(t)[2 + [Yk(t)]2 -b ft [Zk(s)12ds} < CE{[Xk(T)[ 2 + [Yk(T)[ 2 (3. 42) T + j~ (Ixk(s)L 2 + IYk(s)12)as}, t e [0,T] Using Gronwall's inequality, we obtain (3. 43) E{IXk(t)[2 + JYk(t)le + IZk(s)J2ds} < CE{[Xk(T)[ 2 + [Yk(T)t2}, t e [0, T] From the convergence (3. 38) and (3. 41), we see that Zk is bounded in 7/ Thus,Awe may assume that Zk + Z weakly in 7/ Then it is easy... (3. 38) in H, where (Xe, Yk) is the solution of the following: 9 (3. 39) { Xk (0) \ yk(0) ] = o Then, by It6's formula, we have \ Y~(s) j (3. 40) = E{IZk(T)[ 2 + IYk(T)[2 r ~l~l ~ f ~l~l ~} We note that (recall Cl = ( C 1 ) ) (3. 41) = ( ( I + c T c 1 ) z k , Z k ) + A1 ( X ~ ) 1 > ~lzk ]2 - C(IX~l 2 + Ivkl2), Xk 2 +2(cTAl (yk ) , Z k ) w Solvability of linear FBSDEs 41 for some constant C > 0 Thus, (3. 40) . all g C H if and only if (3. 17) and (3. 19) hold. In this case, the adapted solution to (2.12) is unique (for any given g E H). Proof. Theorems 3. 2 and 3. 3 tell us that (3. 17) and (3. 19) are necessary )12ds < K(Sk - s0) 2 ElZ(s)12ds. o o Hence, (3. 32)- (3. 33) and (3. 35) imply sk - SOE[ll -~(s0)l ~ + I1 + ~(s0)lJ -o(sk - so) (3. 36) <_ K(Sk - So) ~ ElZ(s)12ds, Vk > 1. J 80 This. formula, we have 0 =EIt?(t ) - ~(t)l 2 (3. 30) ft T + E 1O'(s) + (r Z(s) ) - /3( s)12ds, Thus, (3. 31) 13( s) - 7(s) = (r Z(s)), which yields (3. 32) Elfl(s ) - "/(s)12ds = 0 o t