Differential Equations and Their Applications Part 6 ppt

[] Now, we come up with the following existence of nodal solutions to 3.1.. Then there exists at least one nodal solu- tion X, Y, Z of 3.1, with the representing function 8 being the sol

hold:

(3.3) Z(t) a(X(t),O(X(t)))XO~(X(t)),

then we call (X, Y, Z) a nodal solution of (3.1), with the representing func- tion O

Let us now make some assumptions

(H1) T h e functions a, b, h are C 1 with bounded partial derivatives and there exist constants A, # > 0, and some continuous increasing function

u : [0, oc) + [0, cx)), such that

(3.4) AI <_ a ( x , y ) a ( x , y ) T <_ #I, (x,y) 9 4 n • 4 ,

(3.5) Ib(x,y)[ ~ u(ly[), (x,y) 9 4 ~ x 4 ,

(3.6) inf h(x) _= 5 > 0, sup h(x) - 7 < ~z

T h e following result plays an important role below

L e m m a 3.2 Let (H1) hold Then the following equation admits a classical solution 0 9 C 2 + ~ ( 4 n ) :

(3.7) ~ t r ( O ~ a ( x , O ) a X ( x , O ) )

such that

+ ( b(x, 0), O~ ) -h(x)O + 1 = 0, x 9 4 ~

(3.8) - < O(x) < ~, x 9 4 ~

Sketch of the proof Let BR(O) be the ball of radius R > 0 centered

at the origin We consider the equation (3.7) in BR(0) with the homo- geneous Dirichlet boundary condition By [Gilbarg-Trudinger, Theorem 14.10], there exists a solution O R E C2+~(BR(0)) for some a > 0 By the

m a x i m u m principle, we have

(3.9) 0 <_ On(x) < ~, x C Bn(O)

Next, for any fixed x0 E 4 n, and R > [xo] + 2, by Gilbarg-Trudinger [1,

T h e o r e m 14.6], we have

(3.10) [Off(x)l _~ C, x 9 B l ( X 0 ) ,

where the constant C is independent of R > Ix0[ + 2 This, together with the boundedness of a and the first partial derivatives of a, b, h, implies

t h a t as a linear equation in 0 (regarding a(x, O(x)) and b(x, O(x)) as known

w Infinite horizon case 91 functions), the coefficients are bounded in C 1 Hence, by Schauder's interior estimates, we obtain t h a t

(3.11) 118Rllc2+~(Bl(x0)) < C, VR > Ix01 + 2

Then, we can let R + co along some sequence to get a limit function 8(x)

By the standard diagonalization argument, we may assume that 8 is defined

in the whole of ~'~ Clearly, 8'E C e + ~ ( ~ '~) and is a classical solution of (3.7) Finally, by the maximum principle again, we obtain (3.8) [] Now, we come up with the following existence of nodal solutions to (3.1) This result is essentially the infinite horizon version of the Four Step Scheme presented in the previous sections

T h e o r e m 3.3 Let (H1) hold Then there exists at least one nodal solu- tion (X, Y, Z) of (3.1), with the representing function 8 being the solution

of (3.7) Conversely, if (X, ]I, Z) is a nodal solution of (3.1) with the repre- senting function 8 Then 8 is a solution of (3.7)

Proof By Lemma 3.2, we can find a classical solution 8 E C 2 + ~ ( ~ n)

of (3.7) Now, we consider the following (forward) SDE:

(3.12) ~ dX(t) = b(X(t), 8(X(t)))dt + a(X(t), 8(X(t)))dW(t), t > O,

( x ( o ) = x

Since 8~ is bounded and b and a are uniformly Lipschitz, (3.12) admits

a unique strong solution X(t), t E [0, co) Next, we define Y(-) and Z(.)

by (3.3) Then, by Ith's formula, we see immediately t h a t (X, Y, Z) is an adapted solution of (3.1) By Definition 3.1, it is a nodal solution of (3.1) Conversely, let (X, ]I, Z) be a nodal solution of (3.1) with the represent- ing function 8 Since 8 is C 2, we can apply It6's formula to Y(t) = 8(X(t))

This leads to t h a t

dY(t) = [(b(X(t), 8(X(t))), 8,(X(t)) >

+ (8~(X(t)), a(X(t), 8(X(t)))dW(t) )

Comparing (3.13) with (3.1) and noting t h a t Y(t) = 8(X(t)), we obtain

t h a t

(3.14)

= h(X(t))8(X(t)) - 1, Vt > 0, P-a.s

Define a continuous function F : ~ n _+ ]R by

F(x) A= ( b(x, 8(x)), 8x (x) ) + l t r [Sxx (x)aa T (x, 8(x))]

(3.15)

- h(x)8(x) + 1

We shall prove t h a t F _= 0 In fact, process X actually satisfies the following FSDE

f dX(t) = b(X(t))dt + ~(X(t))dW(t),

(3.16)

X0~x,

t_>0;

where b(x) ~ b ( x , O ( x ) ) and ~(x) ~ a ( x , 0 ( x ) ) Therefore, X is a time- homogeneous Markov process with some transition probability density

p(t,x,y) Since both b and ~ are bounded and satisfy a Lipschitz con- dition; and since ~ A a a T is uniformly positive definite, it is well known (see, for example, Friedman [1,2]) that for each y 9 ~n, p(., ,y) is the fundamental solution of the following parabolic PDE:

and it is positive everywhere Now by (3.14), we have t h a t F(X(t)) = 0 for all t ~ 0, P-a.s., whence

(3.18) 0 = Eo,~ [F(X(t)) 2] =/Ft p(t,x,y)F(y)2dy, Vt > O

By the positivity of p(t, x, y), we have F(y) = 0 almost everywhere under the Lebesgue measure in IRn The result then follows from the continuity

T h e o r e m 3.3 tells us t h a t if (3.7) has multiple solutions, we have the non-uniqueness of the nodal solutions (and hence the non-uniqueness of the adapted solutions) to (3.1); and the number of the nodal solutions will be exactly the same as t h a t of the solutions to (3.7) However, if the solution

of (3.7) is unique, then the nodal solution of (3.1) will be unique as well Note t h a t we are not claiming the uniqueness of adapted solutions to (3.1)

w U n i q u e n e s s o f n o d a l s o l u t i o n s

In this subsection we study the uniqueness of the nodal solutions to (3.1)

We first consider the one dimensional case, that is, when X and Y are

b o t h one-dimensional processes However, the Brownian motion W(t) is still d-dimensional (d > 1) For simplicity, we denote

1

(3.19) a(x,y) = -~la(x,y)l 2, (x,y) 9 ~2

Let us make the some further assumptions:

(H2) Let m n = 1 and the functions a, b, h satisfy the following: (3.20) h(x) is strictly increasing in x E ~

w Infinite horizon case 93

(3.21) a ( x , y ) h ( x ) - (h(x)y - /0 1 1)./n 1 a y ( x , y + ~ ( ~ - y ) ) d ~ >_ ~1 > O,

[a(x,y)by(x,y + t3(~ - y))

- % ( x , y + 1 3 ( ~ - y ) ) b ( x , y ) ] d t 3 >_ O, y , ~ E [ 1 , 8 9 1 4 9 Condition (3.21) essentially says that the coefficients b, a and h should

be somewhat "compatible." Although a little complicated, (3.21) is still quite explicit and not hard to verify For example, a sufficient conditions for (3.21) is

a ( x , y ) h ( x ) - (h(x)y - 1 ) a v ( x , w ) > ~ > O,

(3.22) a(x, y)by(x, w) - ay(x, w)b(x, y) >_ 0,

It is readily seen t h a t the following will guarantee (3.22) (if (H1) is as- sumed):

(3.23) ay(x,y) = 0, by(x,y) >_ O, (x,y) e ]R x [1, 89

In particular, if both a and b are independent of y, then (3.21) holds auto- matically

Our main result of this subsection is the following uniqueness theorem

T h e o r e m 3.4 Let (H1)-(H2) hold Then (3.1) has a unique adapted solution Moreover, this solution is nodal

To prove the above result, we need several lemmas

L e m m a 3.5 Let h be strictly increasing and 0 solves

(3.24) a(x, 0 ) 0 ~ + b(x, 0)0~ - h(x)O + 1 = O, x e ~

Suppose XM is a local m a x i m u m of O and Xm is a locM minimum of O with O(Xm) ~ O(XM) Then Xm > XM

Proof Since h is strictly increasing, from (3.24) we see that 0 is not identically constant in any interval Therefore xm r XM Now, let us look

at XM It is clear t h a t O~(XM) = 0 and O~(XM) < O Thus, from (3.24) we obtain t h a t

1

(3.25) 9(XM) <_ h(xM -~"

Similarly, we have

1

h(xm)"

Since O(xm) <_ O(XM), we have

h(xm) h(XM) '

whence x M < X m because h ( x ) is strictly increasing [ ]

L e m m a 3.{} L e t ( H 1 ) - ( H 2 ) hold T h e n (3.24) a d m i t s a unique solution

Proof By L e m m a 3.2 we know t h a t (3.24) admits at lease one classical

solution 8 We first show t h a t 8 is m o n o t o n e decreasing Suppose not, assume t h a t it has a local m i n i m u m at xm Since 8 E C 1 and 8 is not

constant on any interval as we pointed out before, 8~ > 0 near Xm Using

L e m m a 3.5, one further concludes t h a t 8~ > 0 over (Xm, cx3) In other

words, 8 is m o n o t o n e increasing on (xm, co) T h e boundedness of 8 then leads to t h a t l i m ~ _ ~ 8(x) exists

Next we show t h a t

(3.28) lira 8~(x) = lira 8 z z ( x ) = O,

X '~ C:X:~ X - - + C X )

To see this, we first apply Taylor's formula and use the boundedness of 8x~

to conclude t h a t there exists M > 0 such t h a t for any x C (xm, o0) and

h > 0

8 ( x + h) - 8(x) - M h 2 < 8 x ( x ) h < 8(x + h) - 8(x) + M h 2

Since l i m ~ _ ~ 8(x + h) - 8(x) = 0, we have

- M h 2 < lim 8 ~ ( x ) h < ~ 8~(x)h < M h 2

Dividing h and letting h + 0 we derive lim~-~oo 8~(x) = 0 Further, note

t h a t

1

8 ~ - a ( x , 8 ~ [ - b ( x , 8 ) S x + h ( x ) 8 - 1]

T h e boundedness of 8, 8~, and 8~z and the assumption (H1) then show t h a t 8xz~ exists and is continuous and bounded as well Thus apply Taylor's expansion to the third order and repeat the discussion above one shows further t h a t limz-~oo 8~z(x) = 0 as well, proving (3.28) Consequently, by (3.24) we have

1

9 ~ h ( + c r "

On the other hand, by (3.20), we see t h a t

(3.30) lim 8(x) > 8(Xm) > - - > - -

9 -+~ - h ( x m ) h ( + c r

which contradicts (3.29) This means t h a t 8 has no local minimum Sim- ilarly one shows t h a t 8 can not have any local m a x i m u m either, hence it

m u s t be m o n o t o n e on ~ Finally, since

(3.31) ~ ( - c r h(-cx3 ~ > h(+cr - 8 ( + c r

it is necessary t h a t 8 is monotone d e c r e a s i n g

w Infinite horizon case 95 Next, let 0 and 0"be two solutions of (3.24) Then, w = 0"- 0 satisfies

(3.32) (h(x) - ~01 [ay(x,O -b t~W)Oxx -[-by(x,O § ~?.o)Ox]dl~)w

where

~o 1

c(x) = h(x) - [ay(x,O § ~ w ) h(x)9 - 1 - b(x,O)9~

a(x, e) + by(x, e + Zw)e~]dZ

a ( x , 9 ) h ( x ) - (h(x)9 - 1) f 3 a y ( x , 9 + j 3 ( O - 9))48

10~ I [a(x, O)by(x, 0 + ~ ( ~ - 0))

+

ay(x,130 + t3('0- O) )b(x,O)] dl3 >_ ~_

#

Here, we have used the fact t h a t O~(x) = -IO~(x)l (since 0 is decreasing

in x) and (3.21) as well as (3.7) From (H1), we also see t h a t a(x, "0) >_ 0 and b(x,'O) are bounded Thus, by the lemma that will be proved below,

L e m m a 3.7 Let w be a bounded classical solution of the following equa- tion:

(3.34) 5(x)wx~ + b(x)w~ - c ( x ) w = 0, x e R,

with c(x) > co > O, ~(x) >_ O, x C IR ~, and with 5 and b bounded T h e n

w(x) = o

Proof For any a > 0, let us consider ~ ( x ) w ( x ) - a[xl 2 Since w

is bounded, there exists some x~ at which ~ attains its global maximum Thus, ~ ( x ~ ) = 0 and ~"tx~ ~j~ _< 0, which means that

Now, by (3.34),

(3.36)

< 2 ( a ( x ) + k ~ ) ~ )

For any x C IR, by the definition of x~, we have (note the boundedness of

5 and b)

~(x) - ~lxl ~ < ~(x~) - ~1~1 ~

(3.37)

< ~- (23(~.) + 2~(~.)~ - I ~ l :) _< C~

co-

Sending a + 0, we obtain w(x) <_ O Similarly, we can show that w(x) > O

Proof of Theorem 3.4 Let (X, Y, Z) be any adapted solution of (3.1) Under (H1)-(H2), by Lemma 3.6, equation (3.24) admits a unique classical solution 0 with 0x _< 0 We set

Z(t) a(X(t),O(X(t)))VO~(X(t)),

By ItS's formula, we have (note (3.19))

= [0~ (X(t))b(X(t), Y(t)) + 0~ (X(t))a(X(t), Y(t))] dt + (a(X(t), Y(t))VOx(X(t)), dW(t) )

Hence, with (3.1), we obtain (note (3.24)) that for any 0 _< r < t < 0%

(3.40)

ElY(r) - Y(r)] 2 - ElY(t) - Y(t)] 2

= - E f f { 2 [ Y - Y][Ox(X)b(X,Y)+ Ozz(X)a(X,Y)

- h ( X ) Y + 1] + la(X,Y)Ox(X)- Z[2}ds

< -2E ft[~ _ Y] [0~(x)(b(X, Y) - b(X, f~))

+ Oxx (X) (a(X, Y) - a(X, Y)) - h(X)(Y - Y)] ds

-Ox~(X) fool ay(X,Y + ~ ( Y - Y))d~ + h(X)] }ds

= - 2 E frr t c(s)l~'(s ) - Y(s)lUds,

w Infinite horizon case 97 where (note the equation (3.24))

/o 1

c(s) = h(X) + IO.(X)I by(X, ~ + ~(Y - ~))d~

b(x,~)o~(x) - h ( x ) ~ + 1

+

a(x, #)

1

(3.41) - a(X, Y) {a(X, Y ) h ( X )

- [h(X)Y - 1] .~1 ay(X,F" + ~ ( Y Y))dfl

+ IOx (X)] [a(X, Y)by (X, Y + fl(Y - Y))

- b ( X , Y ) a y ( X , Y " +/3(Y - Y))] d/3 >_ _~

Denote ~o(t) = E l Y ( t ) - Y ( t ) ] 2 and a = ~ > 0 Then (3.40) can be written It

a 8

~o(r) _< ~o(t) - a ~(s)ds, 0 < r < t < oo

(3.42)

Thus,

,

> r t e [r, oo)

Integrating it over Jr, T], we obtain (note Y and Y" are bounded, and so is

~)

e-C~r _ e - a T f T

(3.44) a (fl(r) ~ e-aT I ~fl(s)ds < C T e -c'T, T > O

a 7,

Therefore, sending T -~ cx~, we see that 9~(r) = 0 This implies that (3.45) Y(r) = Y ( r ) =_ O(X(r)), r E [0, co), a.s co 9 f~

Consequently, from the second equality in (3.40), one has

(3.46) Z(s) = Z(s) = a ( X ( s ) , O ( X ( s ) ) ) T o , ( X ( s ) ) , Vs 9 [ 0 , ~ )

Hence, (X, Y, Z) is a nodal solution Finally, suppose (X, Y, Z) and

( X , Y , Z) are any adapted solutions of (3.1) Then, by the above proof,

we must have

(3.47) Y ( t ) = O(X(t)), ~'(t) = O(X(t)), t 9 [0, oo)

Thus, by (3.1), we see t h a t X(-) and )((.) satisfy the same forward SDE with the same initial condition (see (3.12)) By the uniqueness of the strong solution to such an SDE, X = )( Consequently, Y = Y and Z = Z This

Let us indicate an obvious extension of Theorem 3.4 to higher dimen- sions

T h e o r e m 3.8 Let (HI) hold and suppose there exists a solution 0 to (3.7) satisfying

(3.48)

L 1 a v (x, (1 - 13)O(x) + 130)0~.~i (x)

i , j = l

n

- E biu(x, (1 - ~ ) O ( x ) + ~O)O~,(x)]d~ > ~ > O,

i : 1

Then (3.1) has a unique adapted solution Moreover, this solution is nodal with 0 being the representing function

Sketch of the proof First of all, by an equality similar to (3.32), we can prove t h a t (3.7) has no other solution except O(x) Then, by a proof similar to t h a t of Theorem 3.4, we obtain the conclusion here [ ]

C o r o l l a r y 3.9 Let (H1) hold and both a and b be independent of y Then (3.1) has a unique adapted solution and it is nodal

Proof In the present case, condition (3.48) trivially holds Thus, The-

w T h e l i m i t o f f i n i t e d u r a t i o n p r o b l e m s

In this subsection, we will prove the following result, which gives a rela- tionship between the FBSDEs in finite and infinite time durations

T h e o r e m 3.10 Let (H1)-(H2) hold and let 0 be a solution of (3.7) with the property (3.48) Let (X, Y, Z) be the nodal solution of (3.1) with the representing function O, and (X K, y g , z K ) C J~[0, K] be the adapted solution of (3.1) with [0, ~ ) replaced by [0, K], and Y K ( K ) = g( X ( K ) ) for some bounded smooth function g Then

K ,.-+ o o

uniformly in t on any compact sets

To prove the above result, we need the following lemma

w Infinite horizon case 99

L e m m a 3.11 Suppose that

(3.50)

{ AI < (aiJ(t,x)) < ItI,

Ibi(t,x)l < C, l < i < n ,

c(t, x) >_ n > o,

I~o(~)1 _< M,

(t, z) 9 [0, o0) • R n

with some positive constants A, It, ~?, C and M Let w be the classical solu- tion of the following equation:

w t - ~ a i J ( t , x ) w ~ j - Ebi(t,x)w~, +c(t,x)w=O,

(t, x) 9 [o, ~ 1 • ~ ,

Then

(3.52) Iw(t,z)l <_ Me -nt, (t,x) 9 [0, c~) x An

Proof First, let R > 0 and consider the following initial-boundary value problem:

(3.53)

wtR- E a i J ( t ' x ) w ~ J - bi(t'x)wR~ +c(t'x)wn=O'

( t , x ) [ 0 , ~ ) • 9 BR,

wR[oBR = O,

WR[t=O = Wo(x)xR(x),

where BR is the ball of radius R > 0 centered at 0 and X R is some "cut- off" function T h e n we know t h a t (3.53) admits a unique classical solution

W R E C2+a'l+al2(BR • [0, OO)) for some a > 0, where C 2+a'1+~/2 is the space of all functions v(x, t) which are C 2 in x and C 1 in t with H51der continuous v , ~ s and vt of exponent a and a12, respectively Moreover, we have

(3.54) IwR(t,x)l < M, (t,x) e [ 0 , ~ ) • BR,

and for any Xo C ~ " and T > O, (0 < a ' < a)

(3.55) w R -~ w, in C 2A-~ ([0, T] x B 1 ( x o ) ) , as R ~ (20,

where w is the solution of (3.51) Now, we let r x) = Me -(n-~)t (~ > 0)