Differential Equations and Their Applications Part 9 pps

To this end, we let L be the common Lipschitz constant for b, a, h and g... w Some solvable FBSDEs 151 These two are possible if Co > 0 is large enough.. Again, we let L be the common Li

150 Chapter 6 Method of Continuation where Ao, Co > 0 are undetermined constants We first check that this

E B~(Fo; [0,T]) In fact,

(3.8)

c(0) = - C o < 0, a ( T ) = Ao > O, i~(t) = - A ~ e A~ < O, t E [0, T],

e(t) = - C ~ e C~ < 0, t E [0, T]

Thus, by Proposition 3.1, we see that 9 E B~(Fo; [0, T]) Next, we show that 9 E B~(F; [0, T]) for suitable choice of Ao and Co To this end, we let

L be the common Lipschitz constant for b, a, h and g We note that (3.8) implies (1.7) Thus, it is enough to further have

(3.9) a ( T ) + L 2 c ( T ) >_ 5,

and

(3.10)

a(t)lx - ~12 + e(t)ly - ~12 + c(t)lz - 212

+ 2a(t) ( x - ~ , b ( t , x ) - b(t,5) ) + a ( t ) l a ( t , x ) - a(t,5)] 2

+ 2c(t) ( y - ~, h(t, x, y, z) - h(t, 3, ~,-2) )

< -5{1~ -~1 ~ + l y - ~ l 2 + I z - ~ 1 2 } ,

VtE [0,T], x , ~ E ~ n, y , ~ E IR '~, z,.2 E ~ re• a.s

Let us first look at (3.10) We note that

Left side of (3.10) _< a ( t ) l x - 512 + d(t)ly - ~]2 + c(t)]z - -212

+ 2 a ( t ) L i x - 512 + a ( t ) L 2 i x - 5} 2

(3.11) + 21c(t)lLly-~l{ix-~l+lY-Vl+lz-.21}

< {&(t) + 2 a ( t ) L + a ( t ) L 2 + Ic(t)]L}lx - 512

+ { d ( t ) + 3]c(t)]L + 2 L 2 i c ( t ) i } i Y - ~]2 + ~ _ l z _ 212

Hence, to have (3.10), it suffices to have the following:

{ h(t) + ( 2 L + L2)a(t) + Lic(t)[ < - 6 ,

(3.12) d(t) + (3L + 2L2)lc(t)[ ~ - 5 , Vt E [0,T]

c(t) < - 2 5 ,

Now, we take a(t) and c(t) as in (3.7) and we require

(3.13) d(t)+(3L + 2L2)lc(t)l = -Co(Co - 3L - 2L2)e C~

- C o ( C o - 3L - 2L 2) ~ - 5 , Vt E [0,T], and

(3.14) c(t) = - C o e C~ < - C o <_ -25~ Vt E [0, T]

w Some solvable FBSDEs 151 These two are possible if Co > 0 is large enough Next, for this fixed Co > 0,

we choose Ao > 0 as follows We want

a ( T ) + c ( T ) L 2 = A o e A ~ - CoL2e C~ > Ao - C o L 2 e C~ > 5,

(3.15)

and

a(t) + (2L + L 2 ) a ( t ) + LIc(t)l

(3.16) = - A o ( A o - 2 L - L2)e A ~ + L C o e c~

<_ - A o ( A o - 2 L - L 2) + L C o e C~ ~_ - ~

These are also possible by choosing A0 > 0 large enough Hence, (3.9) and

From the above, we obtain that any decoupled F B S D E is solvable In particular, any BSDE is solvable Moreover, from Lemma 2.2, we see t h a t the adapted solutions to such equations have the continuous dependence

on the data

T h e above proposition also tells us that decoupled FBSDEs are very

"close" to the trivial F B S D E since they can be linked by some direct strong bridges of F0

w F B S D E s w i t h m o n o t o n i c i t y c o n d i t i o n s

In this subsection, we are going to consider coupled FBSDEs which satisfy certain kind of monotonicity conditions Let F = (b, a, h, g) E HI0, T] We introduce the following conditions:

(M) Let m _> n There exists a matrix B E IR mx'~ such that for some /~ > 0, it holds t h a t

(3.17) ( B ( x - 5 ) , g ( x ) - g ( 5 ) ) _>Plx-~l 2, V x , ~ E ~ ~, a.s

(3.18)

( B T [h(t, 0) - h(t, 9 ) ] , x - 5 ) + ( B [b(t, 9) - b(t, 9)], y - ~)

+ ( B [ ~ ( t , 0) - o ( t , ~ ) ] , z - ~ ) < - ~ l x - ~12,

Vt E [0, T], 9,9 E M, a.s

(M)' Let m < n There exists a matrix B E ~,mxn such that for some /~ > 0, it holds that

(3.17)' ( B ( x - x ) , g ( x ) - g ( x ) ) ~ O, Vx,xE ]R n, a s

(3.18)'

( B T [h(t, 9) - h(t, 9 ) ] , x - -2) + ( B [b(t, 0) - b(t, 9)], y - ~)

+ ( B [~(t, 0) - ~ ( t , ~ ) ] , z - ~) _< - ~ ( l Y - ~l 2 + Iz - ~12),

V t e [0, T], 9,9 E M, a.s

Condition (3.17) means t h a t the function x F-+ B T g ( x ) is uniformly

monotone on IR ~, and condition (3.18) implies that the function 9 ~-~

152 Chapter 6 Method of Continuation

- ( B T h ( t , O ) , B b ( t , O ) , B a ( t , O ) ) is monotone on the space M The mean-

ing of (3.17)' and (3.18)' are similar Here, we should point out that (3.17) implies m _> n and (3.17)' implies m ~ n Hence, (M) and (M)' overlaps only for the case m = n

We now prove the following

P r o p o s i t i o n 3.4 Let T > 0 and F - (b, or, h, g) r H[0, T] satisfy (M) (resp (M)') Then, (3.6) holds Consequently, F E S[0, T]

Proof First, we assume (M) holds Take

{ ~ ( t ) = ( A ( t ) B(t) T'~

\ B(t) C(t) J

(3.19) A(t) = a ( t ) I - 5eT-tI, t E [0, T],

B ( t ) B, C(t) = c(t)I =_ -25CoeC~ I,

with 5, Co > 0 being undetermined Since

(3.20)

C(0) = - 2 5 C o i < O,

A ( T ) = 5I > O,

0 _2~C~eCo t < O,

h(t)]xI 2 + c(t)iy] 2 + c(t)]z] 2 + 2La(t)lxl(]xI + lY] + Izl)

(3.22) + 2LIc(t)I ]yi(]xI + ly] + IzI) + L2a(t)(ixI + lYl + Izl) 2

(2t3 - 5)Ix] 2 - 5(lyl 2 + Izl2), V(t,O) C [0, T] • M

It is not hard to see that under (3.17)-(3.18), (3.21) implies (1.8) and (3.22) implies (1.7) and (1.9)' (Note (1.8) implies (1.8)') We see that the left hand side of (3.22) can be controlled by the following:

{/L(t) + K a ( t ) + Kic(t)]}ixI 2 + { ~ ( t ) + Kic(t)I + Ka(t)}ly[ 2

(3.23)

+ {c(_~_~ + K a ( t ) } i z ] 2 ,

for some constant K > 0 Then, for this fixed K > 0, we now choose 5 and

Co First of all, we require

(3.24) c(t) + Ka(t) = -SCoe C~ + KSe T-t ~_ -5Co + K(~e T ~ -5,

(3.21)

and

a(T) + 23 + c ( T ) L 2 > 5,

by Proposition 3.1, we see that 9 E BS(F0;[0, T]) Next, we prove 9 C Bs(F; [0, T]) for suitable choice of 5 and Co Again, we let L be the common

Lipschitz constant for b, a, h and g We will choose 5 and Co so that

w Some solvable FBSDEs 153 and

(3.25) d(t) + KIc(t)l + K a ( t ) = - 2 5 C 2 e C~ + 2 K C o h e C~ + K h e T - t

< • - K ) + K h e T < - 6

These two can be achieved by choosing Co > 0 large enough (independent

of 5 > 0) Next, we require

it(t) + K a ( t ) + KIc(t)l = - h e T - t A- K h e T - t + 2 5 K C o e C~

(3.26)

< - 6 + K h e T + 2 5 K C o e c ~ <_ 2~ - 6,

and

(3.27) a ( T ) + 2~ + c ( T ) L 2 = 5 + 2~ - 25CoeC~ L 2 > 6

Since/~ > 0, (3.26) and (3.27) can be achieved by letting 5 > 0 be small enough (note again that the choice of Co is independent of 5 > 0) Hence,

we have (3.21) and (3.22), which proves 9 e BS(F; [0, T])

Now, we assume (M)' holds Take (compare (3.19))

A ( t ) B ( t ) T ' ~

( B(t) c(t) ) '

O(t)

(3.28) A ( t ) a ( t ) I = 5AoeA~ Vt e [0, T],

B ( t ) B ,

with 6, Ao > 0 being undetermined Note that

c ( 0 ) = - 6 I < 0,

A ( T ) = A o I > O,

Thus, by Proposition 3.1, we have 9 C /~S(Fo; [0, T]) We now choose the constants 5 and Ao In the present case, we will still require (3.21) and the following instead of (3.22):

it(t)lxl 2 + d(t)lY] 2 + c(t)lzl 2 + 2La(t)lxl(Ixl + lYl + N )

(3.30) + 2LIc(t)l[Y](Ixl + lyl + Izl) + L2a(t)(]xl + lyl + IzlY

<_ -51xl 2 + (2/~ - 5){lyl 2 + Iz12}, v ( t , o ) e [0,T] • M

These two will imply the conclusion 9 E BS(F; [0, T]) Again the left hand side of (3.30) can be controlled by (3.23) for some constant K > 0 Now,

we require

it(t) + K a ( t ) + KIc(t)l = - h A 2 e A~ + 5 K A o e A~ + K h e t

(3.31)

< - h A o ( A o - K ) + 5 K e T < - 6 ,

154 Chapter 6 Method of Contim~ation and

a(T) + c ( T ) L 2 = 5Aoe A~ - 5L2 e t

(3.32)

> 5(Ao - L 2 e T) > (~

We can choose Ao > 0 large enough (independent of 5 > 0) to achieve the above two Next, we require

e(t)

T + K a ( t ) <_ K a ( t ) < 5 K A o e A~ ~_ 2~ - 6,

(3.33)

and

(3.34) d(t) + Klc(t)l + K a ( t ) = - S e t + KSe t + K A o S e A~

<_ 5 ( K e T + K A o e A~ <_ 2~ - 5

These two can be achieved by choosing 5 > 0 small enough Hence, we obtain (3.21) and (3.30), which gives 9 9 gS(F; [0,T])

It should be pointed out t h a t the above FBSDEs with monotonicity conditions do not cover the decoupled case Here is a simple example Let n = m = 1 Consider the following decoupled FBSDE:

d X ( t ) = X ( t ) d t + d W ( t ) ,

(3.35) d Y ( t ) = X ( t ) d t + Z ( t ) d W ( t ) ,

X(O) = x, Y ( T ) = X ( T )

We can easily check that neither (M) nor (M)' holds But, (3.35) is uniquely solvable over any finite time duration [0, T]

R e m a r k 3.5 From the above, we see that decoupled FBSDEs and the FBSDEs with monotonicity conditions are two different classes of solvable FBSDEs None of them includes the other On the other hand, however, these two classes are proved to be linked by direct bridges to the trivial FBSDE (the one associated with Fo = (0,0, 0, 0)) Thus, in some sense, these classes of FBSDEs are very "closer" to the trivial FBSDE

w P r o p e r t i e s o f t h e B r i d g e s

In order to find some more solvable FBSDEs with the aid of bridges, we need to explore some useful properties t h a t bridges enjoy

P r o p o s i t i o n 4.1 Let T > 0

(i) For any F E H[0, T], the s e t BI(F; [0, T]) is a convex cone whenever

it is nonempty Moreover,

(4.1) BI(F; [0, T]) : B I ( F + 3'; [0, T]), V3' 9 7-/[0, T]

(ii) For any F1, F2 9 H[0, T], it holds

(4.2) t31(F1;[O,T])NBx(F2;[O,T]) C_ N B l ( a r l +•F2;[0,T])

c~,f~>0

w Properties of the bridges 155

Proof (i) The convexity of BI(F; [0, T]) is clear since (1.7)-(1.9) are lin- ear inequalities in q~ Conclusion (4.1) also follows easily from the definition

of the bridge

(ii) The proof follows from (2.13), (2.15) and the fact that Be(F; [0, T])

It is clear t h a t the same conclusions as Proposition 4.1 hold for

BII(F; [0, T]) and BS(P; [0, T]).'

As a consequence of (3.2), we see that if r l , F2 E HI0, T], then

(4.3) Br(aF1 +/3F2; [0, T]) = r for some ct,/3 > 0,

Be(F1; [0, T]) f')Bx(P2; [0, T]) = r This means t h a t for such a case, F1 and F2 are not linked by a direct bridge (of type (I)) Let us look at a concrete example Let Fi = (bi, ai, hi, gi) E

H[0, T], i = 1,2,3, with

(4.4)

( h i ) - ~ ( 2 1 ~ 1] ) ( y ) ' ( h2 ) - - ( 0 p ) ( Y ) ' ( b 3 ) ( 0 ~ ) ( y ) Crl =~ : ~ : 0,

with A, v E ~ Clearly, it holds

By the remark right after Corollary 2.4, we know that B(F3; [0, T]) = r Thus, it follows from (3.5) and (4.3) that F1 and F2 are not linked by

a direct bridge However, we see that the FBSDE associated with F~ is decoupled and thus it is uniquely solvable (see Chapter 1) In w we will show t h a t for suitable choice of s and v, F2 E S[0, T] Hence, we find two elements in S[0, T] that are not linked by a direct bridge This m e a n s F 1 and F2 are not very "close"

Next, for any hi, b2 E L~(O, T; WI,~176 IRn)), we define

(4.6)

IIb~ - b211o(t)

esssup sup

wCf~ 0,OEM

Ibl(t,O;w) - hi(t,0; w) - b2(t,O;w) + b2(t,O;w)l

l0 - 0l

We define [ I h l - h2llo(t) and Ilal - a 2 l l o ( t ) similarly For g l , g 2 e

L2r (f~; Wl'~(~n; IRm)), we define

(4.7)

Ilal - g2110

= esssup sup

wE~ x,~ER ~

I g I(x; a)) 91 (X; CO) g2(X; W) -~ g2(X; W) I

i x - ~ l

156 Chapter 6 Method of Continuation

Then, for any Fi = (hi, ai, hi, gi) 6 g [ o , T] (i = 1, 2), set

Ilrl - r211o(t) = lib1 - b211o(t) + I1Ol - a211o(t)

Note t h a t I1 IIo(t) is just a family of semi-norms (parameterized by t E [0, T]) As a matter of fact, lit1 - r211o(t) = o for all t E [0, T] if and only if

for some "y 6 7-/[0, T]

T h e o r e m 4.2 L e t T > 0 and F e H[0,T] L e t 9 C Bs(F; [0, T]) T h e n , there exists an e > 0, such that for any F' C H[0, T] w i t h

(4.10) I I r - r'llo(t) <~, v t e [0,T],

we h a v e ~' C B s ( r ' ; [0,T])

Proof Let F = ( b , a , h , g ) and F' = ( b ' , a ' , h ' , g ' ) Suppose 9 6 B~(F; [0, T]) Then, for some K, 5 > 0, (1.7)-(1.9) and (1.8)'-(1.9)' hold Now, we denote (for any 0, 0 E M)

[ ~ = x - ~ , 0"= 0 - 0,

l ~ = b(t, O) - b(t,-0), 3 = a ( t , O) - a(t,-0),

(4.11) I ~ h = h(t,O) - h(t,O), "~ = g(x) - g(g), ,

IN,= b'(t,O) - b'(t,O), "d' = ~ (t,O) - a (t,-O),

I

I h' = h'(t,O) - h'(t,0), ~' = g'(x) - g'(g)

Then one has

( 4 1 2 ) IV - ~l = I g ' ( x ) - g ' ( e ) - g(x) + g ( e ) l -< IIg' - g l l o l ~ l

Similarly, we have

(4.13) { I t' -'bl -< lib'/bllo(t)10"l,

Ih' - hi -< Ilh - hllo(t)101

w Properties of the bridges

Hence, it follows that

157

(4.14)

> ~1~1 = + 2 < B ( T ) ~ , ~' - ~> + < C(T)('f + ~), ~' - ~)

_ _ + ' 2 g l l o } l ~ l

->{8 21B(T)IIIg' gllo-IC(T)lllY gllollg ~ t

5 2

provided IIg' - g ] l o is small enough Similarly, we have the following:

A ^ - 3 )

_< - a l ~ l 2 + 2 ( A ( t ) ~ + B ( t ) r ~ , ~ ' - ~)

+ 2 ( B ( t ) ~ + C(t)~,h' -h>

+ 2 (B(t)T~,~ ' ~) + (A(t)(~' + "~),~' - ~)

< { - 5 + 2(IA(t)l + ]B(t)])llb' - bllo(t)

+ 2(IB(t)l + Ic(t)l)llh' - hllo(t )

+ 21B(t)lll# - ~llo(t) + IA(t)lIla' + allo(t)lJ# - ~llo(t)}lol 2

T h e a b o v e result tells us that if the equation associated with F is solv- able a n d F admits a strong bridge, then all the equations "nearby" are solvable This is a kind of stability result

R e m a r k 4.3 We see from (4.14) and (4.15) that the condition (4.10) can

158 Chapter 6 Method of Continuation

be replaced by

2(IB(T)I + IC(T)IIIg' + gli0)iig'- glIo < 6,

sup {2(IA(t)l + IB(t)l)tl b ' - blio(t)

(4.16) tE[O,T]

+ 2(IB(t)] + ]C(t)I)lih'-hilo(t)

+ [2[B(t)l + [A(t)I[IW + al]o(t)] I]W- allo(t)} < 5, where 6 > 0 is the one appeared in the definition of the bridge (see Defi- nition 1.3) Actually, (4.16) can further be replaced by the following even weaker conditions:

2 ( B ( T ) s - ~) + ( C ( T ) ( ~ + ~),~' - ~> > 2,

Vx,5 E ~'~, sup {2 ( A ( t ) ~ + B(t)T~,b '

(4.17) tc[O,T]

I

+ 2 ( B(t)~ + C(t)~,h' - h) +2 (B(t)TF, 8' - 8 >

+ ( A ( t ) ( 8 ' + 8 ) , 8 ' - ~ ) } V 0 , 0 E M The above means that if the perturbation is made not necessarily small but

in the right direction, the solvability will be kept This observation will be useful later

To conclude this section, we present the following simple proposition

P r o p o s i t i o n 4.4 Let T > O, F - (b,a,h,g) 6 H[0, T] and 9 6

Bi(r; [0,T]) Let f E R and

f ~(t) = e2Zto(t), t E [0, T], (4.18)

= (b - f l x , a,h - fly, g) 6 H[0,T]

Then, ~2 9 BI(F; [0, T])

The proof is immediate Clearly, the similar conclusion holds if we replace BI(F; [0, T]) by Bn(F; [0, T]), B(F; [0, T]) or BS(F; [0, T])

w Construction of Bridges

In this section, we are going to present some more results on the solvability

of FBSDEs by constructing certain bridges

w A g e n e r a l consideration

Let us start with the following linear FBSDE:

{ d { x ( t ) ) ={~4 ( X(t) )( bo(t) )} {ao(t) dW(t),

t 9 [0, T],

X(O) = x, Y(T) = GX(T) + go,

w Construction of bridges 159 where ,A E IR (n+m)• G E IR mxn, ")' - (bo,~o, ho,go) C 7/[0, T] (see (1.3)) and x E IRn We have the following result

L e m m a 5.1 Let T > O, Then, the two-point boundary value problem (5.1) is uniquely solvable for all V E 7/[0, T] if and only if

Proof Let

Then we have the linear FBSDE for (~, 7) as follows:

+ ( bo(t)

ho(t)Gbo(t) ) }dt

(5.3)

+ z(t) -a~o(t)

[(0) = x, ~(T) = go,

Clearly, the solvability of (5.3) is equivalent to t h a t of (5.1) By Theorem 3.7 of Chapter 2, we obtain t h a t (5.3) is solvable for all V E 7/[0, T] if and only if (3.16) and (3.19) of Chapter 2 hold In the present case, these two

Now, let us relate the above result to the notion of bridge From Theorem 2.1, we know that if F1 and F2 are linked by a bridge, then F1 and F2 have the same solvability On the other hand, for any given F, Corollary 2.4 tells us t h a t if F admits a bridge, then, the FBSDE associated with F admits at most one adapted solution The existence, however, is not claimed The following result tells us something concerning the existence This result will be useful below

P r o p o s i t i o n 5.2 Let To > 0 and F = (b, 0, h, g) with

(:)

Then F E S[0,T] for all T E (0, To] if/3(F; [0, T]) r r for all T E (0, T0]

Proof Since B(F; [0, T]) ~ r by Corollary 2.4, (5.1) admits at most one solution By taking V = (bo,ao,ho,go) = 0 and x = 0, we see t h a t the resulting homogeneous equation only admits the zero solution This is equivalent to t h a t (5.1) with the nonhomogeneous terms being zero only