Differential Equations and Their Applications Part 8 pps

130 Chapter 5. Linear, Degenerate BSPDEs In the case that (2.6) holds, we use the following estimate: (5.21) fit. ((O~-Z BT)D(OZu), BT D(O~u) ) dx for small enough e > 0 to get (5.22) /R ~ {((A - BBT)D(O~u), D(a~u) ) + 2 ~ C, z ((O~-ZBT)D(Oeu),BTD(O~u)) o<~<a > ~o/~ _ -~ . ((A - BBT)D(O~u), D(O~u) ) dx - Clul~. Then, we still have (5.14) and finally have (5.20) which is the same as (4.1). In the case (2.7) holds, we use the following estimate: (5.23) it. ( ( O~-Z BT)D(OZu), BT D(O~u) ) dx = fitJ(O"- Br)D(O u), c9~-a[BrD(Oau)] - (O~-5BT)D(O~u)) dx = - fit. { ((02(=-~)BT)D(O/3u)' BTD(O~u) ) + ((O(~-~)BT)D(O~u), BTD(OZu) ) + I(O~-aBT)D(OZu)I 2 } dx for c > 0 small enough to obtain (5.22) and finally to obtain (5.20). Note that in the case (2.6) or (2.7) holds, we have (2.10). Then, (4.2) follows from (4.1) easily. Finally, if in addition, (1.9) also holds, then, (4.3) follows from (4.2). This completes the proof of Lemma 4.1. [] w Comparison Theorems In this section, we are going to present some comparison theorems on the solutions of different BSPDEs. For convenience, we consider BSPDEs of w Comparison theorems 131 form (1.5). Let us denote (compare (3.1)) i s ltr[ADeu] - (a, Du)-cu A/[q A_ tr [BT Dq] - ( b, q ), (6. 1) s ~ ltr [AD2~] - ( ~, DE) -Uu, A ' T Mq=tr[B D~]-(b,~). We assume that (H),~ holds for {A, B, a, b, c] and {A, B, ~, b, U]. Consider the following BSPDEs: I du = -{ s + Mq + f }dt + < q'dW(t) )' (t,x) e [O,T] x ]1% n, (6.2) f d-~= -{-s + M~ +-]}dt + (~,dW(t) ), (t,x) E [0,T] x M% n, (6.3) [ult=T = #" Note that (6.2) and (3.2) are a little differenl since the operators s and J~4 are defined a little differently. However, by the discussion at the end of w we know that Theorems 2.1, 2.2 and 2.3 hold for (6.1). Throughout this section, we assume that the parabolicity condition (1.6), the symmetry condition (2.2) and (H),~ (for some m _> 1) hold for (6.2) and (6.3). Then by Theorem 2.3, for any pairs (f, g) and (f, ~) satisfying (2.14), there exist unique adapted weak solutions (u, q) and (~, ~) to (6.2) and (6.3), respec- tively. We hope to establish some comparisons between u and ~ in various cases. Our comparison results are all based on the following lemma. Lemma 6.1. Let (1.6), (2.2) and (H)m with m > 1 hold. Let (u,q) be the unique adapted weak solution of (6.2) corresponding to some (f, g) satisfying (2.14) for some A ~_ O. Then there exists a constant # 6 ~, such that E e lu(t, x)- 12, x (6.4) T Proof. We first assume that (f,g) satisfies (2.14) with A = 0. Let : IR -+ [0, oc) be defined as follows: r , r < -1, (6.5) qo(r) = (6r 3 + 8r 4 + 3r5) 2, 1 < r < 0, 0, r>_0. 132 We can directly check that ~ is C 2 and ~(0) =/(0) =/'(0) = O, (6.6) ~o(-1) = 1, ~o'(-1) = -Z, qo"(-1) = 2. Next, for any e > 0, we let ~o~(r) = e2~o(r). Then, it holds lim ~'e (r) = -2r- lim~(r) = Ir i2, ~-~o r (6.7) J'(r)l < C, Ve > o, r c IR; lim ~"(r) = ~ 2, r < O, s-+O [ O, r > O. Denote 1 (6.8) a = a - 7 V-A, Then by (1.3), we have Chapter 5. Linear, Degenerate BSPDEs a=b-V-B. uniformly, EiR ~e(g(x))dx- Es ~e(u(t,x))dx /o 1 = E {V'e(u) [ - ~ V.(ADu) - V-(Bq) - (a, Du ) t -eu-(g,q)-f] + ~o;'(u)lql=}dxds = E s { ( ADu, Du) +2 ( BrDu, q) +iq,2 ] - ~o'e (u)[ (a, Du ) +cu + ('b, q) +f] }dxds : E iQ, {~ ~~ ((A- BBT)Du'Du)+]BTDu +q-~ul2] 1 u + 7~(~)[-I~1~ ~ + 2 (B~D~,~) +2 (~,q)] - ( "d, Dg~e (u)) -~'~ (u)[cu + ( b, q } +f] } dxds >.s {_1,, So u _ < 7~o. (~)l'gl2u= + (Bb, D ~oy(r)rdr) + [~"(~)~ -/~(u)} (~, q) +(v. a)~(u) - ~o'E(u)[cu + f]}dxds. (6.10) (6.9) { ltr[AD2u]+(a, Du)= ~V.[ADu]+('d, Du), tr[BTDq] + (b,q) = V.(Bq) + (b,q}. Applying the It6's formula to ~o~ (u), we obtain (let Qt = [t, T] x IR n) w Comparison theorems 133 We note that (6.11) and fo ~ 99~ (r)rdr = ~ (u)u - (p~ (u), (6.12) lim[~(u)u - ~(u)] = 2uI(._<o) + 2u- = 0. e-+0 Thus, let e + O in (6.10), we obtain (6.13) E /R Ig(x)-i~dx- E /R lu(t,x)-12dxds >__ E/Q~ { - I(u<<_o)lbl2n ' - V.(Bb)[-2u-u- lu-I 2] + (v.a)lu-I 2 + 2u-[~ + f]}dxds _> - v + v.a- - where A (6.14) It=sup [-V.~+V.(Bb) + Ib[' +2c+ 1 ] <co. t~X,~O Then by Gronwall's inequality, we obtain (6.4) for the case )~ = 0. The general case can be proved by using transformation (2.11) and working on (v,p) for the transformed equations. [] Our main comparison result is the following. Theorem 6.2. Let (1.6), (2.2) and (H),~ hold for (6.2) and (6.3). Let (f,g) and (f,~) satisfy (2.14) with some ~ >_ O. Let (u,q) and (~,~) be Then for some adapted strong solutions of (6.2) and (6.3), respectively. #>0, (6.15) E/R ~ e -~ (~)[[u(t, x) - g(t, x)]-12dx < e"(~-*)E/~o e-~ ('>l[g(~) - Y(~)]-I :d~ + (M - M)~(s, x) + f(s, x) - ](s, x)]-[2dxds, vte [0, T], 134 In the case that Chapter 5. Linear, Degenerate BSPDEs g(x) - F(x) > 0, Vz ~ ~t", a.s. (6.16) (s - Z)~(t, x) + (M - M)~(t, x) + f(t, x) - f(t, x) >_ O, V(t, z) ~ [0, T] x IR n, a.s. it holds (6.17) u(t, x) >_ ~(t, x), V(t, x) E [0, T] x ~n, a.s. This is the case, in particular, if s = s M = ,44 and (6.18) ~ g(x) >_ -~(x),_ a.e. x E ]R n, a.s. L f(t, x) ___ f(t, x), a.e. (t, ~) ~ [o, T] • ~, a.s. Proof. It is clear that { d(u-~) -{Z:(u- ~) + M(q- ~) (6.19) + (s - -~)~ + (2vf - M)~ + f - ]}dt + ( q - ~, dW(t) ), Then, (6.15) follows from (6.4). In the case (6.16) holds, (6.15) becomes (6.20) EfR e-~(*>i[u(t,x) ~(t,z)]-12dz<_O, Vt E [0, T]. This yields (6.17). The last conclusion is clear. Corollary 6.3. Let the condition of Lemma 6.1 hold. Let I g(x) >_ 0, a.e. x E Rn, a.s. (6.21) f(t, x) > O, a.e. (t, x) E [0, T] x ~'~, a.s. and let (u, q) be an adapted strong solution of (6.2). Then (6.22) u(t, x) > 0, a.e. (t, x) E [0, T] x l& '~, a.s. [] Proof. We takeZ= s M = A4, f_= 0 andy- 0. Then (~,~) = (0,0) is the unique adapted classical solution of (6.3) and (6.18) holds. Consequently, (6.22) follows from (6.17). [] Let us make an observation on Theorem 6.2. Suppose (~,~) is an adapted strong solution of (6.3). Then (6.16) gives a condition on A, B, a, b, c, f and g, such that the solution (u, q) of the equation (6.2) satisfies (6.17). This has a very interesting interpretation (see Chapter 8). We now look at the cases that condition (6.16) holds. Lemma 6.4. Let A, B, -d, b and -~ be independent of x. Let -] and -~ be convex in x. Let (~,~) be a strong solution of (3.1). Then, ~ is convex in x almost surely. w Comparison theorems 135 Proof. First, we assume that f and g are smooth enough in x. Then, the corresponding solution (~,-9) is smooth enough in x. Now, for any 9 ~n, we define v(t,x) = ( D2~(t, xfihT1); p(t,x) = (pl(t,x),"" ,pd(t,x)), V(t,x) 9 [O,T] x IR ~, a.s. pk(t,x)=(D2~ k(t,xfi?,~), l <k<d, Then, it holds dv = [ s - -Mp - ((D2-f)rI,~)]dt + (p, dW(t) ), (6.23) v]t: T = ((D2~)~/' ~/). By Corollary 6.3 and the convexity of f and g (in x), we obtain ( D2~(t, x)r/, r/> = v(t, x) >_ O, (6.24) V(t, x) e [0, T] x Rn, y 9 ~tn, a.s. This implies the convexity of ~(t, x) in x almost surely. In the case that and ~ are not necessarily smooth enough, we may make approximation. [] Proposition 6.5. Let A, B, -5, b and -~ be independent of x. Let f and be convex in x and nonnegative. Let (~,~) be a strong solution of (6.3). Let A4 = A/[ and let A(t,x) = A(t) + Ao(t,x), (6.25) c(t,x) = -5(t) + co(t,x), (t,x) 9 [0, T] x IR ~, a.s. f(t,x) = f(t,x) + fo(t,x), g(x) = + go(x), with Ao(t,x)>_O, co(t,x)>_0, V(t,x) e[O,T]• n, a.s. (6.26) fo(t,x) > O, go(x) >_ O, Then (6.16) is satisfied and thus (6.17) holds. Proof. By Corollary 6.3 and Lemma 6.4, ~ is convex and nonnegative. Thus, (E - s x) = ~tr [AoD2~] + c0~ >__ 0. Then (6.16) follows. [] Next, we have the following. Proposition 6.6. Let all the functions A, B, -5, b, -~, -] and -9 be determin- istic. Let ~ be the solution of the following equation: ~t = -Zu - 7, (t, x) 9 [0, T] x IR n, (6.27) (, ~l~=T = -9 136 Chapter 5. Linear, Degenerate BSPDEs Further, we assume that g(t, x) is convex in x. Next, let (6.25) hold. Then (6.16) is satisfied and (6.17) holds. Proof. In the present case, (g, 0) is an adapted strong solution of (6.3). Then similar to the proof of Proposition 6.5 and note ~ = 0, we can obtain our assertion. [] Note that in Proposition 6.6, B and b are arbitrary. Chapter 6 Method of Continuation In this chapter, we consider the solvability of the following FBSDE which is the same as (3.16) of Chapter 1 (We rewrite here for convenience): dX(t) = b(t, X(t), Y(t), Z(t))dt + a(t, X(t), Y(t), Z(t))dW(t), (0.1) dY(t) = h(t, X(t), Y(t), Z(t))dt + Z(t)dW(t), x(o) = x, Y(T) = g(X(T)). Here, functions b, a, h and g are allowed to be random, i.e., they can depend on w E ~. For the notational simplicity, we have suppressed w and we will do so below. We have seen that for the case when all the coefficients are determin- istic, one can use the Four Step Scheme to approach the problem (see Chapter 4), which involving the study of parabolic systems; in the case of random coefficients, in applying the Four Step Scheme, we need to study the solvability of BSPDEs (see Chapter 5). In this chapter, we are going to introduce a completely different method to approach the solvability of (0.1). Such a method is called the method o/ continuation. w The Bridge Recall that S '~ is the set of all (n • n) symmetric matrices. In what follows, whenever A is a square matrix, (with A being a scalar), by A + A, we mean A+M. For any A E S n, by A >_ 5, we mean that A-5 is positive semidefinite. The meaning of A _< -5 is similar. For simplicity of notation, we will denote M = ~'~ x ][:~m X ~:~m• a generic point in M is denoted by 0 = (x,y,z) with x C ]R '~, y E IR m and z E ]R "~• The norm in M is defined by (1.1) 101 ~ {Ix12 + lYl 2 + Izl~} 1/=, VO = (x,y,z) 9 M, where Izl 2 ~ tr (zzT). Similarly, we will use 0 = (X, ]I, Z), and so on. Now, let T > 0 be fixed and let H[0, T] =L~:(0, T; WI'~ M ; ~ • ]R ~• • ~m) ) (1.2) • L~(a; WI,~(~;~)) Any generic element in H[0, T] is denoted by F =- (b, a, h, g). Thus, F - (b, a, h, g) 9 H[0, T] if and only if b E L~(O,T; WI'~(M;IRn)), a E L2_r(O,T;WI'~176215 h E L2(0, T; WL~176 g e n~%(~; Wl'~ 138 Chapter 6. Method of Continuation where the space L~(0, T; WI,~(M; ~'~)), etc. are defined as in Chapter 1, w Further, we let 2 nxd (1.3) ~/[0, T] = n~(0, T; ]R ~) x n3:(0 , T; R ) 2 . m x n~:(0, T; ~m) x n~- r (s ~ ). An element in ~[0, T] is denoted by 7 - (bo, ao, ho, go) with bo 9 L~(0, T; R~), ao 9 L~(0, T; ~nxd), ho 9 L~(0, T; ~m), go 9 L~ r (~t; ~m). We note that the range of the elements in H[0, T] and ~[0, T] are all in ~n x ~nxd x ~m x~m. Hence, for any F = (b,a,h,g) E H[0, T] and 7 = (bo, ao, ho,go) E ~/[0, T], we can naturally define (1.4) F+~/=(b+bo,~+ao,h+ho,g+go) E H[0, T]. Now, for any F - (b, a, h, g) C H[0, T], 7 - (b0, Cro, ho, go) E 7/[0, T] and x E ~n, we associate them with the following FBSDE on [0, T]: dX(t) = {b(t, O(t)) + bo(t)}dt + {a(t, O(t)) + ao(t)}dW(t), (1.5)r,~,~ dY(t) = {h(t, O(t)) + ho(t)}dt + Z(t)dW(t), x(0) = z, Y(T) = g(X(T)) + go, with O(t) -= (X(t), Y(t), Z(t)). In what follows, sometimes, we will simply identify the FBSDEs (1.5)r,~,~ with (F, 7, x) or even with F (since 7 and x are not essential in some sense). Let us recall the following definition. Definition 1.1. A process 0(-) - (X(.),Y(.),Z(.)) E M[0, T] is called an adapted solution of (1.5)r,~,~, if the following holds for any t C [0, T], almost surely. ~0 t X(t) = x + {b(t, O(s)) + bo(s)}ds + {a(t, O(s)) + ao(s)}dW(s), (1.6)r,~,~ rT Y(t) = g(X(T)) + go -It {h(t, O(s)) + ho(s)}ds T - f Z(s)dW(s). .It When (1.5)r,~,~ admits a unique adapted solution, we say that (1.5)r,~,~ is (uniquely) solvable. We see that (1.6)r,~,~ is the integral form of (1.5)r,~,~. In what follows, we will not distinguish (1.5)r,~,~ and (1.6)r,~,~. w The bridge 139 Definition 1.2. Let T > 0. A F E HI0, T] is said to be solvable if for any x C ~n and 7 E ~[0, T], equation (1.5)r,%~ admits a unique adapted solution O(.) E ~4[0,T]. The set of all F E H[0,T] that is solvable is denoted by S[0, T]. Any F E g[0, T] \ S[0, T] is said to be nonsolvable. Now, let us introduce the following notions, which will play the central role in this chapter. Definition 1.3. Let T > 0 and F - (b,a, h,g) e H[O,T]. A C 1 function (: = : [0, T] + S n+m, with A: [0, T] -~ S n, B: [0, T] -+ ~rn• c and C : [0, T] + S "~, is called a bridge extending from F, (defined on [0, T]), if there exist some constants K, 5 > 0, such that { c(T) _< A(t) __ o, vt Io, T], (1.7) ~(0) _~ K 00) , and either (1.8)-(1.9) or (1.8)'-(1.9)' hold: (1.8) ((I)(T) g(x) g(Z) ' g(x) g(~) )>51x-~?, vx,~e~". (1.9) X w y ' h(t,O) h(t,8)] ) <_-5Ix-hi 2, VO, OeM, a.e. te [O,T], a.s. (1.8)' (O(T) g(x)-g(5) ' g(x)-g(5) )>0, Vx,helR n. y ' h(t,e) - h(t,~) ) + ( ~(t) ( a(t'O) -a(t'z 2 -0) ) a(t,O)z-_a(t,-~) ) ) < -5{ly-~l 2 + Iz-~12}, ve,~e M, a.e.t e [0, T], a.s. If (1.7)-(1.9) (resp. (1.7) and (1.8)'-(1.9)') hold, we call 9 a type (I) (resp. type (II)) bridge emending from F (defined on [0, T]). The set of all type [...]... ho,go) E 7/[0, T] and x 9 IRn Thus, hereafter, we will refer to the FBSDE associated with F0 as the trivial FBSDE Now, let us present the following result .Pr~176176 _~ (3.2) B 3.1 Let T > 0 and Fo = {0,0,0,0} E H[0,T] Then, 9 BS(Fo; [O,T]) if and only if C(0) < 0, ~(t) < 0, A(T) > O, Yt e [0, T] Proof By Definition 1.3, we know that 9 9 BS(F0; [0,T]) if and only if (1.7)-(1.9) and (1 .8) '-(1.9)' hold... ~CE{l~l 2 + J~ol2 + ]/ {Ibo(t)J 2 + I~o(t)J 2 + I'ho(t)12}dt} Hence, (2.2) follows from (2.20) and (2.21) Case 2 Let ~ 6 Brz(Fi; [0, T]) (i = 1,2) now In this case, we still have (2.9), (2.11) and (2.12) Further, we have inequalities similar to (2.13) and (2.15) with I)~(T)I 2 and I.~(t)l 2 replaced by 0 and IY(t)l 2 + 12(t)] 2, respectively Thus, it follows that (2.22) Left side of (2.12) > -6EIJ((T)I... 2.1 Let T > 0 and F1,F2 E H[0, T] be linked by a bridge Then, Vl E $[0, T] if and only if F2 E S[0, T] The above theorem tells us that if the FBSDE associated with F1 is solvable, so is the one associated with F2, provided F1 and F2 are linked by a bridge In applications, if one wants to prove the solvability of the FBSDE associated with F2, he/she can start with a known solvable FBSDE F1, and try to... above system (2.1)~,x similar to Definition 1.1 It is clear t h a t (2.1)~ and (2.1)~,~ coincide with (1.5)r1,~,~ and (1.5)r2,~,~, respectively Let us assume that F1 C $[0,T], i.e., (2.1)~ is uniquely solvable for any 3` C //[0, T] and x E ]Rn We want to prove F2 E $[0,T], i.e., (2.1)~,~ is uniquely solvable for all 3` E 7-/[0, T] and x C ~ n The essence of the method of continuation is contained in the... be different in different places By taking the expectation and using Gronwall's inequality, we obtain (2 .8) fT < cE{r f +]o with some constant C : C(L,T) Next, applying Burkholder-DavisGundy's inequality to (2.7) (note (2 .8) ), one has that T (2.9) E sup r2(t)i 2 < C{J~J2 + Jo {IY(t)J2 + i2(t)12 f tE[O,T] + fo(t)l 2 + I~o(t)12}dt} On the other hand, by applying ItS's formula to IY(t)l 2, we have T I~(t)l... 2.1 We know t h a t it suffices to consider the case t h a t F1 and F2 are linked by a direct bridge Let us assume t h a t (1.5)rl,.r,z w Method of continuation 143 is uniquely solvable for any "y E ~/[0, T] and x E lRn This means that (2.1)~ is uniquely solvable By Lemma 2.3, we can then solve (2.1)~,~ uniquely for any a E [0, 1] In particular, (2.1)~,~, which is (1.5)r2,~,~, is 1 uniquely solvable... following implication C o r o l l a r y 2.4 Let F E HI0, T] with B(F; [0, T]) ~ r Then, for any 7 E 7-/[0, T] and x E ~'~, (1.5)r,.~,z admits at most one adapted solution Moreover, for any 7, ~ E 7-/[0, T] and x, ~ E lR '~, the stability estimate (2.2) holds for any adapted solutions 0(-) of (1.5)r,~,~ and 0(-) of (1.5)r,~,~ Proof We take F1 = F2 = F in Lemma 2.2 Then, (2.2) applies [] From Corollary 2.4... krr + ~-~, k _> 0 w A priori estimate In this subsection, we present a proof of the a priori estimate stated in Lemma 2.2 Proof Let O and O be two adapted solutions of (2.1)~,~ and (2.1)g,~, respectively Define ~'= ~ - ~ for ~ = X , Y , Z , O , bo,ao,ho,go, ~ = x - 5, and Ibi(t) = bi(t,O(t)) - bi(t,-O(t)), (2.6) ~i(t) = ai(t,O(t)) -ai(t,-O(t)), / h ~ ( t ) = hi(t,O(t)) - h~(t,-O(t)), i = 1,2, I,~i(T)... Method of Continuation (I) and type (II) bridges extending from F (defined on [0, T]) are denoted and BII(F; [0, T]), respectively Finally, we let by g~(r; [0, T]) B(F; [0, T]) BI(F; [0, T ] ) U / 3 I I ( F ; [0, T]), (1.10) Us(F; [0, T]) = BI(F; [0, T]) N BxI(F; [0, T]) Any element (~ E /3S(F; [0, T]) is called a strong bridge extending from F (defined on [0, T]) 1.4 Let T > 0 and F, F E HI0, T] We say... we obtain T (2.11) E sup IY(t)] 2 + E l J0 tE[0,T] < CE IZ(t)12dt IX(T)I + [~oI + 2 /o + We emphasize that the constants C appeared in (2.9) and (2.11) only depend on L and T Also, in deriving these two estimates, only the condition Fi 6 H[O,T] has been used (and we have not used the bridge yet) Now, we apply It6's formula to . (2.11) and working on (v,p) for the transformed equations. [] Our main comparison result is the following. Theorem 6.2. Let (1.6), (2.2) and (H),~ hold for (6.2) and (6.3). Let (f,g) and (f,~). ~(0) _~ K 00) , and either (1 .8) -(1.9) or (1 .8) '-(1.9)' hold: (1 .8) ((I)(T) g(x) g(Z) ' g(x) g(~) )>51x-~?, vx,~e~". (1.9) X w y ' h(t,O) h(t ,8) ] ) <_-5Ix-hi. the case that and ~ are not necessarily smooth enough, we may make approximation. [] Proposition 6.5. Let A, B, -5, b and -~ be independent of x. Let f and be convex in x and nonnegative.