Linear, Degenerate BSPDEs In the case that 2.6 holds, we use the following estimate: 5.21 fit.. [] w Comparison Theorems In this section, we are going to present some comparison theore
Trang 1130 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
Trang 2w Comparison theorems 131 form (1.5) Let us denote (compare (3.1))
i s ltr[ADeu] - (a, D u ) - c u
A/[q A_ tr [BT Dq] - ( b, q ),
(6 1)
s ~ l t r [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
L e m m a 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 L e t : IR -+ [0, oc) be defined as follows:
(6.5) qo(r) = (6r 3 + 8 r 4 + 3r5) 2, 1 < r < 0,
Trang 3132
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) = - 2 r -
l i m ~ ( 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
T h e n by (1.3), we have
Chapter 5 Linear, Degenerate BSPDEs
a = b - V - B
uniformly,
E i R ~e(g(x))dx- E s ~e(u(t,x))dx
= E {V'e(u) [ - ~ V.(ADu) - V-(Bq) - (a, Du )
t
- e u - ( 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
_ < 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, D u ) = ~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)
Trang 4w Comparison theorems 133
We note that
(6.11)
and
fo ~ 99~ (r)rdr = ~ (u)u - (p~ (u),
(6.12) l i m [ ~ ( u ) u - ~ ( u ) ] = 2uI(._<o) + 2 u - = 0
e - + 0 Thus, let e + O in (6.10), we obtain
(6.13)
E /R I g ( x ) - i ~ d x - E /R lu(t,x)-12dxds
> E/Q~ { - I(u<<_o)lbl2n ' - V ( B b ) [ - 2 u - u - lu-I 2]
+ (v.a)lu-I 2 + 2 u - [ ~ + f]}dxds
where
A (6.14) I t = s u p [ - V ~ + V ( B b ) + Ib[' + 2 c + 1 ] < c o
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
Our main comparison result is the following
T h e o r e m 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
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,
v t e [0, T],
Trang 5134
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 - ] } d t
+ ( q - ~, d W ( 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
C o r o l l a r y 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 t a k e Z = s M = A4, f _ = 0 a n d y - 0 Then (~,~) = (0,0) is the unique adapted classical solution of (6.3) and (6.18) holds
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
L e m m a 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
Trang 6w Comparison theorems 135
Proof First, we assume t h a t 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) = ( p l ( t , x ) , " " ,pd(t,x)), V(t,x) 9 [O,T] x IR ~, a.s
p k ( t , x ) = ( D 2 ~ 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 t h a t and ~ are not necessarily smooth enough, we may make approximation
[]
P r o p o s i t i o n 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),
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
Thus,
(E - s x) = ~tr [AoD2~] + c0~ > 0
Next, we have the following
P r o p o s i t i o n 6.6 Let all the functions A, B, -5, b, -~, -] and -9 be determin- istic Let ~ be the solution of the following equation:
~t = - Z u - 7, (t, x) 9 [0, T] x IR n, (6.27)
(, ~l~=T = -9
Trang 7136 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)
T h e n similar to the proof of Proposition 6.5 and note ~ = 0, we can obtain
Note that in Proposition 6.6, B and b are arbitrary
Trang 8C h a p t e r 6
M e t h o d o f C o n t i n u a t i o n
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 t h a t 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 T h e B r i d g e
Recall t h a t 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 t h a t 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; W I , ~ ( ~ ; ~ ) )
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; W I ' ~ ( M ; I R n ) ) ,
a E L2_r(O,T;WI'~176215
h E L2(0, T; WL~176
g e n~%(~; Wl'~
Trang 9138 Chapter 6 Method of Continuation where the space L~(0, T; WI,~(M; ~'~)), etc are defined as in Chapter 1,
w Further, we let
(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 ~ n x d 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 + b o , ~ + a o , h + h o , g + g o ) 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 F B S D E 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
D e f i n i t i o n 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
Y(t) = g(X(T)) + go - I t {h(t, O(s)) + ho(s)}ds
T
.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,~,~
Trang 10w The bridge 139
D e f i n i t i o n 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
D e f i n i t i o n 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],
and either (1.8)-(1.9) or (1.8)'-(1.9)' hold:
(1.9)
X w
y ' h(t,O) h(t,8)] )
<_-5Ix-hi 2, VO, O e M , a.e t e [O,T], a.s
(1.8)' (O(T) g(x)-g(5) ' g(x)-g(5) ) > 0 , V x , h e l R n
y ' h(t,e) - h(t,~) )
+ ( ~(t) ( a(t'O) -a(t'z 2 -0) ) a(t,O)z-_a(t,-~) ) )
< - 5 { l y - ~ l 2 + I z - ~ 1 2 } , v e , ~ 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