Differential Equations and Their Applications Part 9 pps

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) = -Co < 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) + L2c(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)la(t,x ) - a(t,5)] 2
+ 2c(t) ( y - ~,
h(t, x, y, z) - h(t, 3, ~,-2) )
< -5{1~ -~1 ~ + ly-~l 2 + Iz-~12},
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)lx - 512 + d(t)ly - ~]2 + c(t)]z - -212
+ 2a(t)Lix - 512 + a(t)L2ix - 5} 2
(3.11) + 21c(t)lLly-~l{ix-~l+lY-Vl+lz 21}

< {&(t) +
2a(t)L + a(t)L 2 + Ic(t)]L}lx - 512
+ {d(t)+
3]c(t)]L + 2L2ic(t)i}iY- ~]2 + ~_lz _ .212.
Hence, to have (3.10), it suffices to have the following:
{
h(t) + (2L+
L2)a(t) + Lic(t)[ < -6,
(3.12) d(t) + (3L +
2L2)lc(t)[ ~ -5, Vt E
[0,T].
c(t) < -25,
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~
-Co(Co - 3L -
2L 2) ~ -5, Vt E [0,T],
and
(3.14)
c(t) = -Coe C~ <__ -Co <_ -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 =
Aoe A~ - CoL2e C~ > Ao - CoL2e C~ > 5,
(3.15)
and
a(t) + (2L +
L2)a(t) + LIc(t)l
(3.16) =
-Ao(Ao - 2L - L2)e A~ + LCoe c~
<_ -Ao(Ao - 2L - L 2) + LCoe C~ ~_ -~.
These are also possible by choosing A0 > 0 large enough. Hence, (3.9) and
(3.12) hold and 9 E/~(F; [0,T]). []
From the above, we obtain that any decoupled FBSDE is solvable. In
particular, any BSDE is solvable. Moreover, from Lemma 2.2, we see that
the adapted solutions to such equations have the continuous dependence
on the data.
The above proposition also tells us that decoupled FBSDEs are very
"close" to the trivial FBSDE since they can be linked by some direct strong
bridges of F0.
w FBSDEs with monotonicity conditions
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 that
(3.17)
(B(x-5),g(x)-g(5))
_>Plx-~l 2,

Vx,~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 - ~) < -~lx - ~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 -
~) _<
-~(lY - ~l 2 + Iz - ~12),
Vte [0, T], 9,9 E M, a.s.
Condition (3.17) means that the function
x F-+ BTg(x)
is uniformly
monotone on IR ~, and condition (3.18) implies that the function 9 ~-~
152 Chapter 6. Method of Continuation
-(BTh(t,O),Bb(t,O),Ba(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.
Proposition 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) = -25Coi < O,
A(T) = 5I > O,
r = (-Se~-~, 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) + Ka(t) + Kic(t)]}ixI 2 + {~(t)+ Kic(t)I + Ka(t)}ly[ 2
(3.23)
+ {c(_~_~ +Ka(t)}iz]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 + Ka(t) = -25C2e C~ + 2KCohe C~ + Khe T-t
< • - K) + Khe T < -6.
These two can be achieved by choosing Co > 0 large enough (independent
of 5 > 0). Next, we require
it(t) + Ka(t) + KIc(t)l = -he T-t A- Khe T-t + 25KCoe C~
(3.26)
< -6 + Khe T + 25KCoe 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,
c(t) c(t)I - -he~•

with 6, Ao > 0 being undetermined. Note that
c(0) = -6I < 0,
A(T) = AoI > 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) + Ka(t) + KIc(t)l = -hA2e A~ + 5KAoe A~ + Khe t
(3.31)
< -hAo(Ao - K) + 5Ke 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
- L2e T) >
(~.
We can choose Ao > 0 large enough (independent of 5 > 0) to achieve the
above two. Next, we require
e(t)

T + Ka(t) <_ Ka(t) <__ 5KAoe A~ ~_ 2~ - 6, (3.33)
and
(3.34)
d(t) + Klc(t)l + Ka(t) = -Se t + KSe t + KAoSe A~
<_ 5(Ke T + KAoe 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 that 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:
dX(t) = X(t)dt + dW(t),
(3.35) dY(t) = X(t)dt + Z(t)dW(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].
Remark 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 Properties of the Bridges
In order to find some more solvable FBSDEs with the aid of bridges, we
need to explore some useful properties that bridges enjoy.
Proposition 4.1. Let T > 0.
(i) For any F E H[0, T], the
set
BI(F; [0,
T]) is a convex cone whenever
it is nonempty. Moreover,

(4.1) BI(F; [0, T]) : BI(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 Bl(arl +•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])
is a convex cone. []
It is clear that 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 rl, 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 that 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)
bl 0
b2
~ 1
(hi)-~ (21 ~ ) (y) ( ) ( ) (Y)

1] ' h2 0 p '
(b3) ( 0 ~)(y) Crl =~ : ~ : 0,
h3 1
' gl = 92 = g3 = -x,
with A, v
E
~. Clearly, it holds
(4.5) F3 = F1 + F2.
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 that 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
means 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 [Ihl- h2llo(t) and Ilal -a2llo(t) similarly. For
gl,g2 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)
(4.8) + Ilhl - h21lo(t) + Ilgl - g211o.
Note that
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
(4.9) F2 = F1 + if,
for some "y 6 7-/[0, T].

Theorem 4.2.
Let
T > 0 and F e H[0,T].
Let 9 C Bs(F; [0, T]). Then,
there
exists
an e > 0, such that for any F' C H[0, T]
with
(4.10)
IIr- r'llo(t) <~, vte [0,T],
we have ~' C Bs(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
(4.12) IV - ~l = Ig'(x) - g'(e) -
g(x) +

g(e)l -< IIg' - gllol~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 +
~), ~' - ~)
_ _ + ' gllo}l~l 2-
>{8 21B(T)IIIg'
gllo-IC(T)lllY gllollg
~t
5 2
provided IIg' -g]lo is small enough. Similarly, we have the following:
A ^ -3)
(4.15)
+(r a'O I'( 0 ))
_<
-al~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.
Then, our assertion follows.
[]
The above result tells us that if the equation associated with F is solv-
able and F admits a strong bridge, then all the equations "nearby" are
solvable. This is a kind of stability result.
Remark 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'-~)} V0,0EM.
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.
Proposition 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 -flx, 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 general consideration
Let us start with the following linear FBSDE:
{ d {x(t)
={~4
X(t) bo(t) {ao(t) dW(t),
) (
)(
)}
(5.1)
\ Y(t) Y(t) + ho(t) dt +
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.
Lemma 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
\,(t)/ I) f x(t)
k Y(t)
/
Then we have the linear FBSDE for (~, 7) as follows:
d
r
+ ( 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 that of (5.1). By Theorem

3.7 of Chapter 2, we obtain that (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
conditions are the same as (5.2). This proves the result. []
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 that 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.
Proposition 5.2. Let To > 0 and F = (b, 0, h, g) with
(:)
(5.4) ~ h(t,O) = A , g(x) = Gx, V(t, 0) 6 [0, To] x M.
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 that
the resulting homogeneous equation only admits the zero solution. This is
equivalent to that (5.1) with the nonhomogeneous terms being zero only
160 Chapter 6. Method of Continuation
admits the zero solution. On the other hand, in this case, the solution of
(5.1) is given by
( )
with the condition
(5.6) 0 = (-a,5
(x(r)
We require that (5.6) leads to Y(0) = 0. Thus, it is necessary that the left
hand side of (5.2) is non-zero for t = T. Since T C (0, To] is arbitrary, we
must have (5.2). Then, by Lemma 5.1, we have r C $[0, T]. []
Let us now look at some class of nonlinear FBSDEs. Recall the semi-
norms [l" [10(t) defined by (4.8).

Theorem 5.3.
Let To > O, A c
]R (n+m)x(n+'~)
and F = (b,O,h,g) be
defined by (5.4). Suppose (5.2) holds for T = To and that
BS(F; [0, T]) ~ r
for all
T E (0, To].
Then for any T E (0, T], there exists an c > O, such that
for all/3 C ~ and
F - (b, ~, h, ~) E g[0, T]
with
(5.7)
Ilrllo(t)
< e, t E [0,T],
the
following FBSDE:
(X(t) (X(t) (~(t,O(t))~
}dt
d k y(t) ) = { (A + /3I) )+
k Y(t) \ h(t,
O(t)) ]
/'~(t,O(t))'~
dW(t), t E
[0, T],
(5.8) +\
z(t) ]
X(O) = x, Y(T) = GX(T) +
~(X(T)),
admits a unique adapted solution 0 -

(X, ]I, Z) E A4[0, T].
Proof.
We note that if
(5.9)
then,
(5.10)
~(t, 0) = eetg(t, e-z~0),
V(t, 0) E [0, T] x M,
Ilbllo(t) = Ilbllo(t), Vt c [0, T].
Similar conclusion holds for ~, h and ~ if we define ~, h and ~" similar to
(5.9). On the other hand, if O(t) _=
(X(t), Y(t), Z(t))
is an adapted solution
of (5.8) with /3 = 0, then ~)(t)z~
eZtO(t )
is an adapted solution of (5.8).
Thus, we need only consider the case/3 0 in (5.8). Then, by Theorems
2.1, 4.2 and Proposition 5.2, we obtain our conclusion immediately. []
We note that FBSDEs (5.8) is nonlinear and the Lipschitz constants
of the coefficients could be large. Also, (5.8) is not necessarily decoupled
nor with monotonicity conditions. Thus, Theorem 5.3 gives the unique
w Construction of bridges 161
solvability of a (new) class of nonlinear FBSDEs, which is not covered
by the classes discussed before. On the other hand, by Remark 4.3, we
see that condition (5.7) can be replaced by something like (4.16), or even
(4.17). This further enlarges the class of FBSDEs covered by (5.8).
We note that the key assumption of Theorem 5.3 is that F -= (b, 0, h, g)
given by (5.4) admits a strong bridge. Thus, the major problem left is
whether we can construct a (strong) bridge for F. In the rest of this section,
we will concentrate on this issue.

We now consider the construction of the strong bridges for F =
(b, 0, h, g) given by (5.4). From the definition of strong bridge, we can
check that r E BS(F; [0, T]) if it is the solution to the following differential
equation for some constants K, K, 5, s > 0,
(~(t) + ~ffr + r = -5I, t 9 [0, T],
(5.11) (i)(0) = ( K f~),
satisfying the following additional conditions:
(5.12) I
(I, GT)d2(T) (G) ~ sI.
On the other hand, we find that the solution to (5.11) is given by
(5.13) ~(t) =
e -Art ( K t
0 -:-
t 9 [0, T].
Thus, in principle, if we can find constants K,K, 5,~ > 0, such that (5.12)
holds with ~(t) given by (5.13), then we obtain a strong bridge 4(-) for F
and Theorem 5.3 applies.
w A one dimensional case
In this subsection, we are going to carry out a detailed construction of strong
bridges for a case of n m = d = 1 based on the general consideration of
the previous subsection. The corresponding class of solvable FBSDEs will
also be determined.
Let F = (b, 0, h, g) be given by
{
(;) (:)(:)
A _= ~ ,
(5.14)
g(x) = -gx,
for all (t, x, y, z) E [0, c~) • R 3, with A, #, g E R being constants satisfying
162

the following:
Chapter 6. Method of Continu'ation
1 3g#
_
92
(5.15) A,#,g > 0, ~ + ~ _> 0.
We point out that conditions (5.15) for the constants ,~, #, g are not neces-
sarily the best. We prefer not to get into the most generality to avoid some
complicated computation. Let us now carry out some calculations. First
of all
(5.16) eat=
(e; ~'t
3(11 e-)'t)) , Vt>O._
Thus, for all t _> O,
(5.17)
=( e ~' 0 3(1-e
~')
o ~ )
ICe
TM
~(~- e2~)
and
(5.18)
t t e2XS
3(e~S _ e2~S)
fo e-A%e-ASds = fo ( 3(e~S e
TM) 1+ ~(1-e~8) 2)ds
_ ~(e -1) . _xt 1~2
tt~ -
-2-~x (e -1) 2 2x3,-

We let K > 0 be undetermined and choose
3
(5.19) K= ~-~,
~=1.
Then, according to (5.13), we define
~(t) = (A(t)
B(t))
B(t) c(t)
w Construction of bridges
with
163
(5.20)
1 2~t -
= ~(e +2),
B(t)-K#,
~t_
2~,t, . 5#,
~t_l,
2
A [e e )-t-2)~2(e )
- #[K 5 ~ 2~t . #[T~ 5~e~ t , 5#
= -~-~e
. e
- z),
C(t) ~ - K#2 " :~t ,~2 5# 2 ~e~t 2~2
= -n-~ ~-~e -1) - ~ - )
5(i~2 +#2) . @2
A 2 \ 2A ] A 2 \ A ] ),2 \
_ ~(A2+~2).
-K- SZ ~

= ~(e 2;~t+2e xt+3)-K
A2+#2t.
A2
From (5.12), we need the following: (~ > 0 is undetermined)
(5.21)
A(t) >_0,
C(t) <-e, Vt9
A(T) - 2gB(T) + g2C(T) >_ e.
Let us now look at these requirements separately.
First of all, it is clear true that
A(t) > 0
for all t 9 [0, T].
C(t) <_ -e
for all t 9 [0, T], if and only if
N
ext,
A2 + #2 z~
(5.22) K > e +
(e
TM
+
2e at + 3) A2 t =
f(t),
t 9 [o, T].
Since
f"(t) >_ 0
for all t E [0, o0), the function
f(t)
is convex. Thus, (5.22)
holds if and only if

(5.23) K >/(o) v f(T).
164 Chapter 6. Method of Continuation
Finally, we need
< A(T) - 2gB(T) + g2C(T)
g# ( 2AT eAT
_
= ~(e 2~'~ + 2) + ~,~ + 2)
2 2
g # (e2~T 2e AT g2( 52 +
#2) T
(5.24) + -~- + + 3) - g2~ A 2
1 g# 392# 2
+ 25 52 + 45 7 - g2-~.
Thus, we need (note (5.23))
F(T)~=
1+~)
e 2~T
+ ~ ~(1 + 52
(5.25) 1 g# 3g2# 2
+ + ~
25 52 453
> g2~ > g2(f(0) V
](T)).
We now separate two cases (with
f(T)
and f(0), respectively). First of all,
for
f(T),
we want
0 <_ F(T) - g2f(T)

(5.26) ~ ( 2g#~ 2~T g# AT 1 g# C(1
+ g2) A if(T).
= 1 + -~ ] e + 2 ~ -e + 25 52 =
We see that T ~-~ F(T) is monotone increasing. Thus, to have the above,
it suffices to have
3 _ c(1 + g2).
(5.27) 0 < F(0) =
Hence, in what follows, we take
3
(5.28) ~ -
45(1 + g2)
Then, (5.26) holds. Next, we claim that under (5.15) and (5.28), the fol-
lowing holds.
(5.29)
F(T) - g2 f(O) >_ O.
In fact, by the choice of 6 and by (5.27),
(5.30)
F(O) - g2 f(O) = ff'(O) = O.
On the other hand,
1(1+~)2e2~T
g#(l+g#~e~T
g2(52+# 2)
(5.31)
F'(T) = -~ + ~-f ~] s 2
w Construction of bridges 165
Thus, by (5.15), it follows that
Then, by F"(T) > 0, together with (5.30) and (5.32), we must have (5.29).
Hence, we obtain (5.25). This shows that a strong bridge ~(t) has been
constructed with K, 5 and s being given by (5.19) and (5.28), respectively,
and we may take

(5.33) K = f(0) v
f(T).
It is interesting that the if(-) constructed in the above is not in
/3(Fo; [0, T]) for any T > 0 since )~(t) > 0. On the other hand, we note that
both
A(t)
and
B(t)
are independent of T. However, due to the fact that
K depending on T,
C(t)
depends on T. But, we claim that there exists a
constant co > 0, only depending on )~, #, g (independent of T), such that
-co-f(T)<_C(t)<
4)~(1+ 92), t e [0,T],
(5.34) 3
Co _<
C(T) <
4;~(1 + g2),
where
f(t)
is defined by (5.22). In fact, by (5.20), (5.22), (5.28) and (5.33),
we have
(5.35)
C(t) = f(t) - f(O) V f(T)
4),(1 + g2)"
Clearly,
C(t)
is convex. Thus,
3

(5.36)
C(t) <_ C(O) V C(T) -
4)~(1 + g2), kit e [0,T].
On the other hand, by the fact that
f(t)
is strictly convex and
limt ,oo
f(t)
= oo, we see that there exists a unique To > 0, only depending
on )~ and #, such that
3
(5.37)
C(t) > f(To) - f(O) V f(T)
4A(1 + g2)' t e [0, T].
This proves the first relation in (5.34). Next, we see easily that there exists
a unique T1 > To, such that f(T1) = f(0), and
f f(t) <
f(O), Vt E [0, 7"1],
(5.38)
f(t)
> f(0), Vt C (7"1, oo).
Hence, we obtain
(5.39)
C(T) > f(To) - f(O)
This proves the second relation in
(5.34).
4A(1 + g2)"
166 Chapter 6. Method of Continuation
Now, from Remark 4.3 and Theorem 5.3, we know that the following
FBSDEs is solvable on [0, T].

dX(t)
= {(/9 -
A)X(t) + #Y(t) +
b(t,
X(t),
Y(t),
Z(t)) }dt
+ ~(t, X(t), Y(t),
Z(t))dW(t),
(5.40)
dY(t)
= {/~Y(t)
+ -h(t, X(t), Y(t), Z(t))}dt + Z(t)dW(t),
X(O) : x, Y(T) = -gX(T) + y(X(T)),
where A, tt, g > 0 satisfying (5.15), ~ 6 lit, and F -= (b,~,h,y) 6 H[0,T]
satisfying
[ 2[B(T)III~II0 + IC(T)i[]YII02 < ~ A 1,
(5.41) sup 2(]A(t)l +
IB(t)i)li-bilo(t) +
2(IB(t)[ +
IC(t)I)II~ilo(t )
tE[0,T]
+ 2JB(t)lj]-~Jjo(t) + ]A(t)jljPJJo(t) 2 } < ~ A
1,
with A(.), B(.) and C(.) given by (5.20) and ~ > 0 given by (5.28). If we
use (4.17), then, (5.41) can be relaxed to the following:
{ 2B(T)~'~ +
C(T)(~ -
2g~)~ > -(~ A 1)[~[ 2, Vx, Z 6 IR,
(5.42) sup f2(A(t)~ +

B(t)T~)~ + 2 (B(t)'~ + C(t)~)~
tC[0,T] ~
+2B(t)~+A(t)~ 2}
<(~A1)[0"] 2, V0, SEM
If b, ~, h and ~ are differentiable, then, we see that (5.42) is equivalent to
the following:
2B(T)-ffx(x + C(T)(~(x) -
2g)yz(x) > -(s A 1), Vx 6 Ft,
]/ A(t) B(t) 0
|B(t)
C(t) 0 (V-6(t,O),V-h(t,O),V~(t,O))
\o
0 B(t) /
(5.43)
B(t) 0 )
+ A(t)V~(t, 8){V~(t,O)} T < s A1,
V(t,0) E[0, T]xM,
where Vb(t, 8) = (b~(t, 8),
by(t, 0),
b~(t, 8)) T, and so on. Some direct com-
putation shows that the first relation in (5.43) is equivalent to the following:
(5.44)
ix / c A 1 [ B(T)
-r(n =-V + c(n g)2 _ __
/ s A 1 (B(T)
< + c(n g)2.
B(T)
C(T) + g < -~ (z)
B(T)
C(T) + g' Vx E R.

w Construction of bridges 167
By (5.34), we know that C(T) is bounded uniformly in T, while, B(T)
-cr as T -~ cr (see (5.20)). Thus, by some calculation, we see that
/eAl
(5.45) -V~ > -r(T)$-oo, as T ~ cr
and y need only to satisfy the following:
(5.46) -r(T) < y~(x) < O, Vt 9 P~.
Clearly, the larger the T, the weaker the restriction of (5.46). The second
condition in (5.43) is also checkable (although it is a little more complicated
than the first on@). It is not hard to see that the choice of functions b and
are independent of T as ACt ) and B(t) do not depend on T. However, since
C(t) depends on T, by some direct calculation, we see that in order FBSDE
(5.40) is solvable for all T > 0, we have to restrict ourselves to the case that
h(t, 0) = h(t, y). Clearly, even with such a restriction, (5.40) is still a very
big class of FBSDEs, which are not necessarily decoupled, nor monotone.
Also, ~ is allowed to be degenerate. We omit the exact statement of the
explicit conditions on b, ~ and h under which (5.40) is solvable to avoid some
lengthy computation. Instead, to conclude our discussion, let us finally look
at the following FBSDE:
dX(t) = {(/3 - )QX(t) + #Y(t) + b(t, Z(t), Y(t), Z(t)) }dt
+ ~(t, X(t), Y(t), Z(t))dW(t),
(5.47) dY(t) = {flY(t)+ ho(t)}dt + Z(t)dW(t),
X(O) = x, Y(T) = -gX(T) + go,
with ),, #, g > 0 satisfying (5.15) and
{2(IA(t)[ + IB(t)[)I[bll0(t) + 21B(t)l II~[Io(t)
sup
(5.48) te[o,o~)
~t
+ IA(t)l N~ll0(t) 2 } < e
A

1.
This is a special case of (5.40) in which h =- ho and ~ = go. Then, by the
above analysis, we know that (5.47) is uniquely solvable over any finite time
duration [0, T]. Condition (5.48) can be carried out explicitly as follows:
{2(e
TM
+ 2) + 2it2 (e2Xt + e xt - 2)}llbIIo(t)
)~ ,
2
(5.49) + 2#~ (22~t + e At - 2)llffll0(t )
,
3
+
(e
TM
+ 2) l l0 (t) 2
< min{4)% 1-~g2 } , t 9 [0, oo).
It is clear that although (5.47) is a special case of (5.40), it is still very
general and in particular, it is not necessarily decoupled nor monotone.
Also, if we regard (5.47) as a nonlinear perturbation of (5.1) (with m =
168 Chapter 6. Method of Continuation
n = d = 1 and (5.14) holds), then the perturbation is not necessarily small
(for t not large).
Chapter 7
Forward-Backward SDEs with Reflections
In this chapter we study FBSDEs with boundary conditions. In the simplest
case when the FBSDE is decoupled, it is reduced to a combination of a well-
understood (forward) reflected diffusion and a newly developed reflected
backward SDE. However, the extension of such FBSDEs to the general
coupled case is quite delicate. In fact, none of the methods that we have

seen in the previous chapters seems to be applicable, due to the presence of
the reflecting process. Therefore, the route we take in this chapter to reach
the existence and uniqueness of the adapted solution is slightly different
from those we have seen before.
w Forward SDEs with Reflections
Let (9 be a closed convex domain in ~n. Define for any x E 0(9 the set of
inward normals to O at x by
(1.1)
.h/'x
= {~:
I '1
= 1, and (% x - y) < O, Vy e (9).
It is clear that if the boundary 0(.9 is smooth (say, C1), then for any x E 0(9,
the set Af~ contains only one vector, that is, the unit inner normal vector
at x. We denote BV([0, T]; lR n) to be the set of all llZn-valued functions of
bounded variation; and for ~ 9
BV([O,T]; ]Rn),
we denote 17/I(T) to be the
total variation of ~ on [0, T].
A general form of (forward) SDEs with reflection (FSDER, for short)
is the following:
/0 /0
(1.2)
X(t) = x + b(s, X(s))ds + a(s, X(s))dW(s) + ~(t).
Here the b and a are functions of (t,
x, w) 9
[0, T] • ~ x f~ (with w being
suppressed, as usual); and ~ 9
BVj:([O,T];]R'~),
the set of all {:Pt)t>0-

adapted processes ~? with paths in BV([0, T]; ~zm).
Definition
1.1. A pair of continuous, {~t}t>0-adapted processes (X, 7) 9
L~=([0, T]; ~n) • BV~:([0, T]; nZ ~) is called a solution to the FSDER (1.2) if
1) x(t) 9 (9, vt 9
[0,T], a.s.;
2)
rl(t) = f~ l{x(8)coo}7(s)dl~l(s),
where q'(s) 9
JY'X(s), 0 < s < t < T,
dlr/I-a.e.;
3) equation (1.2) is satisfied almost surely.
A widely used tool for solving an FSDER is the following (determinis-
tic) function-theoretic technique known as the
Skorohod Problem:
Let the
domain (9 be given,