STOCHASTIC ANALYSIS AND APPLICATIONS
Vol. 22, No. 5, pp. 1295–1314, 2004
Shooting Methods for Numerical Solution of
Stochastic Boundary-Value Problems
Armando Arciniega and Edward Allen*
Department of Mathematics and Statistics, Texas Tech University,
Lubbock, Texas, USA
ABSTRACT
In the present investigation, numerical methods are developed for
approximate solution of stochastic boundary-value problems. In
particular, shooting methods are examined for numerically solving
systems of Stratonovich boundary-value problems. It is proved
that these methods accurately approximate the solutions of stochastic boundary-value problems. An error analysis of these methods is
performed. Computational simulations are given.
Key Words: Shooting methods; Numerical solutions; Stochastic
boundary-value problems; Ito and Stratonovich stochastic differential equations.
*Correspondence: Edward Allen, Department of Mathematics and Statistics,
Texas Tech University, Lubbock, Texas 79409-1042, USA; Fax: (806) 742-1112;
E-mail: eallen@math.ttu.edu.
1295
DOI: 10.1081/SAP-200026465
Copyright # 2004 by Marcel Dekker, Inc.
0736-2994 (Print); 1532-9356 (Online)
www.dekker.com
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Arciniega and Allen
Mathematics Subject Classification:
65C30; 60H10.
1. INTRODUCTION
Methods for numerically solving stochastic initial-value problems
have been under much study (see, for example, Refs.[3–5,8,9] and the references therein). However, the theory and numerical solution of stochastic
boundary-value problems have received less attention. In the present
investigation, shooting methods are applied to numerically solve systems
of Stratonovich boundary-value problems. The following linear stochastic system with boundary conditions is the object of interest in the present
investigation:
8
k
X
>
< d~
uðtÞ ¼ ðA~
uðtÞ þ ~
aðtÞÞdt þ
ðBi~
uðtÞ þ ~
bi ðtÞÞ dWi ðtÞ; 0 t 1
>
:
uð0Þ þ F1~
uð1Þ ¼ ~
0;
F0~
i¼1
ð1:1Þ
where ~
u, ~
a, ~
bi 2 Rn and A, F0 , F1 , Bi are n n matrices, and where the
stochastic integrals for this problem are understood in the sense of
Stratonovich integrals. In addition, it is assumed that
b0
F
F0 ¼
0
and
F1 ¼
0
b1
F
b1 is a ðn lÞ n matrix of rank
b0 is a l n matrix of rank l and F
where F
n l. Also, 0 < l < n. Ocone and Pardoux[6] and Zeitouni and Dembo[10]
have established existence and uniqueness of solutions to Eq. (1.1). As
u 2 Rn is a solution to Eq. (1.1) if ~
u is Stratonovich
defined in Ref.[10], ~
integrable and satisfies for all t 2 ½0; 1:
8
Z t
Z tX
k
>
<~
uðtÞ ~
uð0Þ ¼
ðA~
uðsÞ þ ~
aðsÞÞds þ
ðBi~
uðsÞ þ ~
bi ðsÞÞ dWi ðsÞ;
0
0 i¼1
>
:
F0~
uð0Þ þ F1~
uð1Þ ¼ ~
0:
ð1:2Þ
System (1.1) is an anticipative problem as the solution at any position
is dependent on the Brownian motion beyond that position. However,
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1297
Ocone and Pardoux[6] and Zeitouni and Dembo[10] show that solutions to
Eq. (1.1) can be defined as standard Stratonovich-type stochastic
integrals. In the present investigation, shooting methods are applied to
numerically solve Eq. (1.1) and error analyses are performed. A much
simpler two-dimensional form of Eq. (1.1) was investigated by Allen and
Nunn[1] who studied shooting and finite-difference numerical schemes for
approximating the solution.
A shooting method for numerically approximating the solution of
Eq. (1.1) is now described. To describe this shooting method, consider
the stochastic initial-value system:
d~
um ðtÞ ¼ ðA~
um ðtÞ þ ~
aðtÞÞdt þ
k
X
ðBi~
bi ðtÞÞ dWi ðtÞ;
um ðtÞ þ ~
ð1:3Þ
i¼1
0, and ~
um ð0Þ for m ¼
for m ¼ 1; 2; . . . ; n l þ 1, where ~
u1 ð0Þ ¼ ~
2; 3; . . . ; n l þ 1 are chosen to be n l linearly independent vectors in
the null space of F0 . The stochastic system (1.3) is a system of initial-value
problems rather than a boundary-value problem. In effect, (1.1) is
replaced by system (1.3). The ~
um ðtÞ obtained using (1.3) can be combined
to form a solution to Eq. (1.1). To see this, let
~
uðtÞ ¼
nlþ1
X
ð1:4Þ
lm~
um ðtÞ;
m¼1
where the lm , 1 m n l þ 1, satisfy
nlþ1
X
ð1:5Þ
lm ¼ 1
m¼1
and the linear system of rank ðn lÞ:
nlþ1
X
m¼1
lm ðF0~
um ð0Þ þ F1~
um ð1ÞÞ ¼
nlþ1
X
lm F1~
0:
um ð1Þ ¼ ~
ð1:6Þ
m¼1
In the next section, it is shown that the solution ~
uðtÞ, of (1.1) can be written
as given by (1.4) where the ~
um ðtÞ, for m ¼ 1; 2; . . . ; n l þ 1, satisfy (1.3).
As system (1.3) is a stochastic initial-value system, standard numerical methods such as Euler’s or Milstein’s method (see for example
Refs.[3,4]) can be applied to approximate the solution of (1.3) at discrete
times. Using (1.5) and (1.6), the values of lm can be calculated uniquely
and the solution ~
uðtÞ can be approximated by combining the approximate
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Arciniega and Allen
solutions using (1.4). This approach is described in the third section
along with an error analysis. Finally, computational results are given to
illustrate the procedure.
2. SOLVABILITY USING THE SHOOTING METHOD
In this section, it is verified that the shooting method procedure
yields the solution to the original stochastic boundary-value problem
(1.1). Clearly, if ~
uðtÞ is given by (1.4), then ~
uðtÞ satisfies (1.1) as
d~
uðtÞ ¼
nlþ1
X
lm d~
um ðtÞ
m¼1
¼
nlþ1
X
lm ðA~
um ðtÞ þ ~
aðtÞÞdt þ
m¼1
k
X
!
ðBi~
bi ðtÞÞ dWi ðtÞ
um ðtÞ þ ~
i¼1
¼ ðA~
uðtÞ þ ~
aðtÞÞdt þ
k
X
ðBi~
uðtÞ þ ~
bi ðtÞÞ dWi ðtÞ;
i¼1
where (1.5) has been applied, that is,
nlþ1
X
lm ¼ 1:
m¼1
Also, the boundary conditions are satisfied as:
F0~
uð0Þ þ F1~
uð1Þ ¼
nlþ1
X
lm ðF0~
um ð0Þ þ F1~
um ð1ÞÞ
m¼1
¼
nlþ1
X
lm F1~
0;
um ð1Þ ¼ ~
m¼1
as ~
u1 ð0Þ ¼ ~
0 and ~
um ð0Þ are linearly independent
null space
Pnlþ1 vectors in the
~
~
l
F
ð1Þ
¼
0
.
The next
of F0 for m ¼ 2; 3; . . . ; n l þ 1, and
u
m
1
m
m¼1
theorem shows that the lm , 1 m n l þ 1, satisfying Eqs. (1.5) and
(1.6) can be uniquely determined. Thus, the solution of the original
stochastic boundary-value problem (1.1) is obtained using (1.4) by
combining the solutions ~
um ðtÞ of the initial-value problems (1.3).
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1299
Theorem 2.1. The lm , 1 m n l þ 1 uniquely exist and satisfy
Eqs. (1.5) and (1.6).
Proof. For the original stochastic boundary-value problem:
8
d~
uðtÞ ¼ ðA~
uðtÞ þ ~
aðtÞÞdt
>
>
>
k
<
X
þ
ðBi~
uðtÞ þ ~
bi ðtÞÞ dWi ðtÞ; 0 t 1
>
>
i¼1
>
:
F0~
uð0Þ þ F1~
uð1Þ ¼ ~
0;
ð2:1Þ
the solution ~
uðtÞ uniquely exists, which is proved in Refs.[6,10]. Consider
the corresponding stochastic initial-value problem:
8
d~
u1 ðtÞ ¼ ðA~
u1 ðtÞ þ ~
aðtÞÞdt
>
>
>
k
<
X
þ
ðBi~
bi ðtÞÞ dWi ðtÞ; 0 t 1
u1 ðtÞ þ ~
ð2:2Þ
>
>
i¼1
>
:
~
0;
u1 ð0Þ ¼ ~
It is well-known that ~
u1 ðtÞ is uniquely determined for this problem
(see, for example, Refs.[2–4]). Subtracting Eqs. (2.1) and (2.2) yields:
8
k
X
>
<
ðBi ~
d~
wðtÞ ¼ A~
wðtÞdt þ
wðtÞÞ dWi ðtÞ; 0 t 1
ð2:3Þ
i¼1
>
:
F0 ~
wð0Þ þ F1 ~
wð1Þ ¼ F1~
u1 ð1Þ;
where ~
wðtÞ ¼ ~
uðtÞ ~
u1 ðtÞ. As ~
uðtÞ and ~
u1 ðtÞ uniquely exist, then ~
wðtÞ also
uniquely exists. In particular, ~
wð0Þ ¼ ~
uð0Þ exists. Consider, next
"
#
b0 ~
b0
F
wð0Þ
F
~
F0 ~
wð0Þ ¼
wð0Þ ¼
~
0
0
and
wð1Þ ¼
F1 ~
0
b
F1
~
wð1Þ ¼
Then,
wð0Þ þ F1 ~
wð1Þ ¼
F0 ~
"
~
0
:
b1 ~
F
wð1Þ
b0 ~
wð0Þ
F
b
wð1Þ
F1 ~
#
~
0
:
¼
b1~
u1 ð1Þ
F
wð0Þ is in the null space
Thus F0 ~
wð0Þ ¼ ~
0 and F1~
wð1Þ ¼ F1~
u1 ð1Þ. Hence ~
of F0 . However, the null space of F0 and the range of F1 are identical,
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Arciniega and Allen
i.e., NullðF0 Þ ¼ RangeðF1 Þ. Furthermore, dimðRangeðF1 ÞÞ ¼ n l ¼
dimðNullðF0 ÞÞ. Therefore, one can let
~
wð0Þ ¼
nlþ1
X
lm~
vm ð0Þ;
m¼2
where ~
vm ð0Þ ¼ ~
um ð0Þ are linearly independent vectors in the range of F1
and flm gnlþ1
are to be determined. However, as ~
wð0Þ 2 NullðF0 Þ,
m¼2
flm gnlþ1
uniquely
exist.
Consider
now
the
stochastic
initial-value
problem:
m¼2
8
k
X
>
>
>
wðtÞ ¼ A~
wðtÞdt þ
ðBi~
wðtÞÞ dWi ðtÞ; 0 t 1
> d~
<
i¼1
ð2:4Þ
nlþ1
X
>
>
>
>
lm~
wð0Þ ¼
vm ð0Þ:
:~
m¼2
Therefore, (2.4) has a unique solution. As (2.3) and (2.4) have unique
solutions, the solutions agree by the above argument. Now consider
~
um ðtÞ ~
u1 ðtÞ that solves the stochastic initial-value problem:
vm ðtÞ ¼ ~
8
k
X
>
< d~
vm ðtÞ ¼ A~
vm ðtÞdt þ
ðBi~
vm ðtÞÞ dWi ðtÞ; 0 t 1
ð2:5Þ
i¼1
>
:
F0~
0;
vm ð0Þ ¼ ~
for m ¼ 2; 3; . . . ; n l þ 1. Then, it is clear that
~
wðtÞ ¼
nlþ1
X
lm~
vm ðtÞ:
m¼2
Thus,
wð1Þ ¼ F1~
u1 ð1Þ, then l2 ; l3 ; . . . ; lnlþ1
Pnlþ1 as F1~
~
~
l
ð1Þ
¼
F
ð1Þ.
Finally, notice that
v
u
m m
1 1
m¼2
satisfy
~
uðtÞ ¼ ~
wðtÞ þ ~
u1 ðtÞ:
Therefore,
~
uðtÞ ¼ ~
u1 ðtÞ þ
nlþ1
X
lm~
u1 ðtÞ þ
vm ðtÞ ¼ ~
m¼2
Thus,
~
uðtÞ ¼
nlþ1
X
m¼1
lm~
um ðtÞ;
nlþ1
X
m¼2
lm ð~
um ðtÞ ~
u1 ðtÞÞ:
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Shooting Methods for Numerical Solution
1301
where
F1
nlþ1
X
lm ð~
um ð1Þ ~
u1 ð1ÞÞ ¼ F1~
u1 ð1Þ;
m¼2
nlþ1
X
lm ¼ 1;
m¼1
0 for each m ¼ 1; 2; . . . ; n 1 þ l: Thus, lm , m ¼
and F0~
vm ð0Þ ¼ ~
1; 2; . . . ; n l þ 1, uniquely exist and satisfy (1.5) and (1.6).
&
3. NUMERICAL SOLUTION AND ERROR ANALYSIS
In this section, error analyses for numerical solution of system (1.3)
are performed. Then, it is verified that Eq. (1.4) yields correspondingly
accurate approximations to ~
uðtÞ in the original stochastic boundary-value
problem (1.1). Two numerical methods are considered to numerically
solve system (1.3), namely, Euler’s method and Milstein’s method. The
approximate solutions obtained by numerically solving (1.3) are then
combined to approximate the solution of (1.1) using (1.4).
To perform an error analysis, it is useful to convert the Stratonovich
system (1.3) to its corresponding Ito system. The Ito form of system (1.3)
is given by (see Ref.[4]):
ðA~
um ðtÞ þ ~
aðtÞÞ þ
d~
um ðtÞ ¼
þ
k
X
k
1X
bj ðtÞÞ dt
ðB2j~
um ðtÞ þ Bj~
2 j¼1
ðBi~
bi ðtÞÞdWi ðtÞ:
um ðtÞ þ ~
ð3:1Þ
i¼1
System (3.1) is solved numerically rather than (1.3). By solving (3.1) one
solves (1.3) as the two systems are equivalent. To solve (3.1) numerically,
select a positive integer N 2 and partition the interval ½0; 1 into
0 ¼ t0 < t1 < < tN ¼ 1;
where tp ¼ ph for each p ¼ 0; 1; . . . ; N . It is assumed that the step size h is
fixed, so that the common distance between the discrete times is h ¼ N1 .
For example, the Euler approximations to system (3.1) are stochastic
ORDER
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Arciniega and Allen
processes satisfying the iterative scheme (see Refs.[3,4]):
~
um;pþ1
k
1X
bj ðtp ÞÞ
¼~
um;p þ h ðA~
um;p þ ~
aðtp ÞÞ þ
ðB2~
um;p þ Bj~
2 j¼1 j
þ
k
X
pffiffiffi
ðBi~
bi ðtp ÞÞ hZi ;
um;p þ ~
ð3:2Þ
i¼1
for each m ¼ 1; 2; . . . ; n l þ 1. In the above scheme, ~
um;p denotes the
approximation to the exact solution at the pth time step. That is
~
um ðtp Þ, for each p ¼ 0; 1; . . . ; N 1, m ¼ 1; 2; . . . ; n l þ 1. Also,
um;p ~
the random increments Zi are independent normal random variables with
mean zero and variance unity, i.e., Zi 2 N ð0; 1Þ (see Ref.[4]). After (3.2) is
solved numerically for each m ¼ 1; 2; . . . ; n l þ 1, the approximate ^lm ,
for m ¼ 1; 2; . . . ; n l þ 1, are calculated using
nlþ1
X
^
lm ðF0~
um;N þ F1~
um;N Þ ¼
m¼1
nlþ1
X
^lm F1~
0
um;N ¼ ~
m¼1
with
nlþ1
X
^
lm ¼ 1
m¼1
corresponding to (1.5) and (1.6). As a result,
~
uðtp Þ nlþ1
X
^
lm~
um;p
m¼1
corresponding to (1.4).
The theorem below is a well-known result concerning the strong
convergence of Euler’s method for stochastic differential equations (see
Refs.[3,4,8]). To be consistent with existing literature, the following
notation is used in the present investigation. In particular, denote
k
1X
~
bj ðtÞÞ
f ðt;~
um ðtÞÞ ¼ ðA~
um ðtÞ þ ~
aðtÞÞ þ
ðB2~
um ðtÞ þ Bj~
2 j¼1 j
and
Gðt;~
um ðtÞÞ ¼
k
X
i¼1
ðBi~
bi ðtÞÞ:
um ðtÞ þ ~
ORDER
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Then, the system
~ ðtÞ
d~
um ðtÞ ¼ ~
f ðt;~
um ðtÞÞdt þ Gðt;~
um ðtÞÞdW
is equivalent to (3.1). In the above system,
~
f ¼ ffi g is an n-vector-valued function,
G ¼ fgi;j g is an n k-matrix-valued function,
~ ¼ fWi g is a k-dimensional Wiener process,
W
and the solution ~
um is an n-dimensional process. The above system can be
expressed as
d~
um ðtÞ ¼ ~
f ðt;~
um ðtÞÞ dt þ
k
X
~
um ðtÞÞ dWi ðtÞ;
gi ðt;~
i¼1
where the ~
gi are the columns of the matrix G and the Wi are the indepen~.
dent scalar Wiener processes forming the components of W
Theorem 3.1. Consider the system of Ito stochastic differential
equations,
~ ðtÞ
f ðt;~
um ðtÞÞdt þ Gðt;~
um ðtÞÞdW
d~
um ðtÞ ¼ ~
~ 2 Rk .
for m ¼ 1; 2; . . . ; n l þ 1, where ~
f 2 Rn , G 2 Rnk , and W
~
Suppose f and G satisfy uniform growth and Lipschitz conditions in
the second variable, and are Hölder continuous of order 12 in the first
variable. Specifically, there exists a constant Km > 0 for each m ¼
vm 2 Rn ,
1; . . . ; n l þ 1 such that for all s; t 2 ½0; 1, ~
um ;~
jj~
f ðt;~
um ðtÞÞ ~
f ðt;~
vm ðtÞÞjj þ jjGðt;~
um ðtÞÞ Gðt;~
vm ðtÞÞjj
vm jj
Kjj~
um ~
ð3:3Þ
2
jj~
f ðt;~
um ðtÞÞjj þ jjGðt;~
um ðtÞÞjj2 K2 ð1 þ jj~
um jj2 Þ
ð3:4Þ
1
jj~
f ðs;~
um ðtÞÞ ~
f ðt;~
um ðtÞÞjj þ jjGðs;~
um ðtÞÞ Gðt;~
um ðtÞÞjj Kjjs tjj2 :
ð3:5Þ
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Arciniega and Allen
Then, there exists a positive constant Cm such that
Ejj~
um ðtp Þ ~
um;p jj2 Cm h
for each m ¼ 1; 2; . . . ; n l þ 1, where jjjj is the Euclidean norm.
In Theorem 3.1, the Eqs. (3.3) and (3.4) guarantee existence and
uniqueness of solutions of the Ito stochastic differential equations and
equation (3.5) guarantees the convergence of the Euler method (see
Refs.[2–4]).
A second method to numerically approximate the solution of system
(3.1) is the Milstein method (see Ref.[4]):
~
um;pþ1
k
1X
bj ðtp ÞÞ
¼~
um;p þ h ðA~
um;p þ ~
aðtp ÞÞ þ
ðB2~
um;p þ Bj~
2 j¼1 j
þ
k
k
X
X
pffiffiffi
bj 1 ;
ðBi~
bi ðtp ÞÞ hZi þ
Iðj1 ;j2 Þ Bj2 Bj1~
um;p þ ~
um;p þ Bj2~
j1 ;j2
i¼1
ð3:6Þ
where
Z
8
>
>
I
¼
ðj
;j
Þ
< 1 2
>
>
:I
ðj1 ;j1 Þ
tnþ1
tn
Z
s1
tn
dWsj21 dWsj12 ;
j1 6¼ j2
ð3:7Þ
1
ðDWj1 Þ2 h :
¼
2
The last term in Eq. (3.6) differentiates Milstein’s method from Euler’s
method. Notice that Milstein’s method is complicated for general problems, due to the evaluation of Iðj1 ;j2 Þ . If ~
bj ðtÞ ¼ ~
0 for j ¼ 1; 2; . . . ; k and
Bj2 Bj1 ¼ Bj1 Bj2 for j1 ; j2 ¼ 1; 2; . . . ; k, then Milstein’s method becomes:
k
pffiffiffi
X
~
um;p þ h ðA~
um;p þ ~
aðtp ÞÞ þ
Bi~
um;p hZi
um;pþ1 ¼ ~
i¼1
þ
k X
k
1X
2
j1
ðDWj1 ÞðDWj2 ÞBj2 Bj1~
um;p ;
ð3:8Þ
j2
and the evaluation of Iðj1 ;j2 Þ is not required. The following theorem states
that Milstein’s method has second-order strong convergence in the mean
ORDER
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Shooting Methods for Numerical Solution
1305
square error as compared with Euler’s method, which has first-order
strong convergence (see Refs.[3,4]).
Theorem 3.2. Under the hypotheses of Theorem 3.1 for the Milstein
approximations (3.6), the following estimate holds:
Ejj~
um ðtp Þ ~
um;p jj2 Cm h2
for each m ¼ 1; 2; . . . ; n l þ 1.
In the next two theorems, it is shown that for the shooting method
the errors between the exact and approximate solutions are small.
Theorem 3.3. The error in estimating ~
l is of the same order as the error
in estimating the solutions of the initial-value problems.
Proof. Consider the equations
nlþ1
X
lm F1~
0;
um ð1Þ ¼ ~
ð3:9Þ
m¼1
and,
l1 ¼ 1 nlþ1
X
lm :
ð3:10Þ
m¼2
Substituting Eq. (3.10) into Eq. (3.9) and rearranging terms yields
nlþ1
X
lm F1 ð~
um ð1Þ ~
u1 ð1ÞÞ ¼ F1~
u1 ð1Þ:
ð3:11Þ
m¼2
But
F1 ¼
0
b1
F
b1 is a ðn lÞ n matrix of rank n l. Thus, the equation in (3.11)
where F
is a linear system of the form
A~
l ¼~
b;
ð3:12Þ
where A is an ðn lÞ ðn lÞ matrix, ~
l and ~
b are vectors of length
~
b
a1 ; . . . ;~
anl , where
u1 ð1Þ and A ¼ ½~
n l. In particular, b ¼ F1~
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Arciniega and Allen
b1 ½~
~
ai ¼ F
uiþ1 ð1Þ ~
u1 ð1Þ. Then,
~
l ¼ A1~
b:
ð3:13Þ
In numerical solution of the initial-value problem (3.1) the ~
um ð1Þ, for
m ¼ 2; . . . ; n l þ 1 are being approximated by ~
u^m ð1Þ. Therefore, the linear
system to be solved is an approximation of (3.12). Call this linear system
^
^
b~
b;
l¼ ~
A
ð3:14Þ
^
b ¼½~
b1~
b1 ½~
a^1 ;...;~
where A
a^nl , ~
a^i ¼ F
b ¼F
u^1 ð1Þ. Howu^iþ1 ~
u^1 ð1Þ, and ~
^
b and ~
b are perturbations
ever, notice by Theorems 3.1 and 3.2, that A
to A and ~
b for small time step h. It is well-known (see, e.g., Ref.[7]) that
#
"
#"
^
b jj jj~
bjj
jjAjjjjA1 jj
jjA A
b ~
^
~
~
~
þ
:
jjl ljj jjljj
b jj
jjAjj
jj~
bjj
1 jjA1 jjjjA A
Hence,
0 "
^
ljj @E
Ejj~
l ~
jj~
ljjjjA1 jj
#2 11=2
1=2
A
b jj2
EjjA A
b jj
1 jjA1 jjjjA A
0 "
#2 11=2
1=2
1
~
jj
l
jjjjAjjjjA
jj
^
A
bjj2
þ @E
Ejj~
b ~
:
b jjÞ
jj~
bjjð1 jjA1 jjjjA A
^
ljj is proportional to the error obtained in estiThus, the error Ejj~
l ~
mating ~
um ð1Þ using, for example, Euler’s method or Milstein’s method.
This completes the proof of the theorem.
&
Theorem 3.4. Let ~
uðtÞ be the exact solution of the boundary-value
problem (1.1) and ~
u^ðtÞ the approximate solution obtained using the
initial-value system (3.1). Then,
Ejj~
uðtp Þ ~
u^ðtp Þjj1=2
!1=2
nlþ1
X
Ejlm j
m¼1
^ 1=2
ljj
þ Ejj~
l ~
nlþ1
X
!1=2
Ejj~
um ðtp Þ ~
um;p jj
m¼1
nlþ1
X
m¼1
!1=2
Ejj~
um;p jj
:
ORDER
REPRINTS
Shooting Methods for Numerical Solution
1307
Proof. Consider
~
u^ðtp Þ ¼
nlþ1
X
^
lm~
uðtp Þ:
um;p ~
m¼1
Then,
~
uðtp Þ ~
u^ðtp Þ ¼
nlþ1
X lm~
um ðtp Þ ^lm~
um;p
m¼1
¼
nlþ1
X
lm ð~
um ðtp Þ ~
um;p Þ þ
nlþ1
X
m¼1
ðlm ^lm Þ~
um;p : ð3:15Þ
m¼1
As a result,
Ejj~
uðtp Þ ~
u^ðtp Þjj1=2
nlþ1
X
E
2
E4
!1=2
jlm jjj~
um ðtp Þ ~
um;p jj
m¼1
!1=4
jlm j2
nlþ1
X
m¼1
þ E4
!1=4 3
5
jj~
um ðtp Þ ~
um;p jj2
m¼1
nlþ1
X
!1=4
jlm ^lm j2
m¼1
00
@@E
þE
!1=2
jlm ^lm jjj~
um;p jj
m¼1
nlþ1
X
2
nlþ1
X
nlþ1
X
nlþ1
X
!1=4 3
5
jj~
um;p jj2
m¼1
!1=2
jlm j2
111=2 0 0
AA @E@
m¼1
nlþ1
X
!1=2 111=2
AA
jj~
um ðtp Þ ~
um;p jj2
m¼1
0 0
nlþ1
X
^ 1=2
ljj @E@
þ Ejj~
l ~
jj~
um;p jj2
!1=2 111=2
AA
m¼1
nlþ1
X
!1=2
Ejlm j
m¼1
^ 1=2
ljj
þ Ejj~
l ~
nlþ1
X
!1=2
Ejj~
um ðtp Þ ~
um;p jj
m¼1
nlþ1
X
m¼1
!1=2
Ejj~
um;p jj
;
ð3:16Þ
ORDER
REPRINTS
1308
Arciniega and Allen
using the Cauchy-Schwarz inequality and the inequality:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u N
N
X
uX
t
jai j2 jai j:
i¼1
i¼1
&
This completes the proof.
Note by Theorems 3.1–3.4, the numerical method is convergent with
accuracy determined by the order of accuracy of the methods used to
approximately solve the initial-value problems.
4. COMPUTATIONAL RESULTS
In this section, computational results are given to test the numerical
method developed in the present investigation. A description of an
interesting first problem is presented here (see Ref.[1].) Consider the
second-order two-point stochastic boundary-value problem:
8 00
j ðxÞ ¼ ð1 þ jðxÞÞdx þ jðxÞ dWðxÞ
>
>
<
jð0Þ ¼ 0
>
>
:
jð1Þ ¼ 0:
ð4:1Þ
Letting y1 ðxÞ ¼ jðxÞ and y2 ðxÞ ¼ j0 ðxÞ, Eq. (4.1) becomes:
8
dy1 ðxÞ ¼ y2 ðxÞdx
>
>
>
>
>
< dy2 ðxÞ ¼ ð1 þ y1 ðxÞÞdx þ y1 ðxÞ dWðxÞ
ð4:2Þ
>
y1 ð0Þ ¼ 0
>
>
>
>
:
y1 ð1Þ ¼ 0:
Now, letting u1 ðtÞ ¼ y1 ðxÞ, u2 ðtÞ ¼ y2 ðxÞ, with t ¼ x, Eq. (4.2) becomes:
8
d~
uðtÞ ¼ ðA~
uðtÞ þ ~
aðtÞÞdt
>
>
>
>
2
<
X
ðBi~
uðtÞ þ ~
bi ðtÞÞ dWi ðtÞ;
þ
>
>
i¼1
>
>
:
uð0Þ þ F1~
uð1Þ ¼ ~
0;
F0~
0t1
ð4:3Þ
ORDER
REPRINTS
Shooting Methods for Numerical Solution
1309
where
0
;
1
0 0
F1 ¼
;
1 0
~
aðtÞ ¼
A¼
0
1
~
b1 ¼ ~
0;
1
;
0
B1 ¼
~
b2 ¼ ~
0;
0 0
;
1 0
and
F0 ¼
1 0
;
0 0
B2 ¼ 0:
Thus, problem (4.3) has the form
#
#!
#
" # "
"
#"
#"
8 "
u1 ðtÞ
0
0 0 u1 ðtÞ
0 1 u1 ðtÞ
>
>
>
d
¼
dt þ
dW1 ðtÞ
þ
>
>
< u2 ðtÞ
1
1 0 u2 ðtÞ
1 0 u2 ðtÞ
# "
# " #
"
#"
#"
>
>
1 0 u1 ð0Þ
0 0 u1 ð1Þ
0
>
>
>
þ
¼
:
0 0 u2 ð0Þ
1 0 u2 ð1Þ
0
ð4:4Þ
To solve this problem, consider two different solutions to a corresponding stochastic initial-value problem. Consider ~
um ðtÞ, m ¼ 1; 2 where
0
0
~
u1 ð0Þ ¼
and ~
u2 ð0Þ ¼
:
0
1
The ~
um ðtÞ for m ¼ 1; 2 solve the stochastic initial-value problem:
8
0
0 0
0 1
>
< d~
~
~
um ðtÞ ¼
þ
um ðtÞ dWðtÞ
um ðtÞ dt þ
1
1 0
1 0
>
:
for m ¼ 1; 2 where ~
um ð0Þ is given above :
ð4:5Þ
Notice that Eq. (4.5) is solved for both ~
u1 ðtÞ and ~
u2 ðtÞ using different initial
conditions but the same Wiener process. Euler and Milstein methods are
used for comparison in numerically solving the corresponding stochastic
u2 ðtÞ are numerically solved to time
initial-value problems. After ~
u1 ðtÞ and ~
t ¼ 1, they are combined to approximate the solution of the stochastic
boundary-value problem (4.4). The numerical results are shown below.
For this problem, the Euler and Milstein methods are identical. Also, the
Ito form and the Stratonovich form of this problem are the same. Table
4.1 presents approximations of Eðu1 ð1=2ÞÞ and Eðu21 ð1=2ÞÞ using both, the
Euler and Milstein methods. The approximate values are based on 100,000
independent trials. Figure 4.1 illustrates the average of the approximate
solution with 100,000 independent trials, using both, the Euler and Milstein
ORDER
REPRINTS
1310
Arciniega and Allen
Table 4.1. Approximate values of Eðu1 ð1=2ÞÞ and Eðu21 ð1=2ÞÞ.
Euler shooting
Number of intervals in t (h)
2
4
8
16
Milstein shooting
Eu
Eu2
Eu
Eu2
0.1251
0.1169
0.1155
0.1153
0.0156
0.0138
0.0135
0.0135
0.1251
0.1169
0.1155
0.1153
0.0156
0.0138
0.0135
0.0135
Methods. Two particular trajectories of the solutions are also shown.
Absolute errors of the numerical solution at time t ¼ 0:5 are shown in
Figure 4.2 for Euler and Milstein methods.
As a second example, consider the following two-point stochastic
boundary-value problem:
8
2
X
>
<
ðBi~
d~
uðtÞ ¼ ðA~
uðtÞ þ ~
aðtÞÞdt þ
uðtÞ þ ~
bi ðtÞÞ dWi ðtÞ; 0 t 1
>
:
F0~
uð0Þ þ F1~
uð1Þ ¼ ~
0
i¼1
ð4:6Þ
Figure 4.1. Illustration of the average and two trajectories of the solution.
ORDER
REPRINTS
Shooting Methods for Numerical Solution
1311
Figure 4.2. Illustration of the absolute errors for the first example.
where
2
1
3
2
2
1
0
3
7
7
6
6
~
aðtÞ ¼ 4 0 5; A ¼ 4 1 3 1 5;
0 1 4
1
2
3
2
2 1 0
1 0
1 6
7
6
B2 ¼ 4 1 2 1 5; F0 ¼ 4 0 0
10
0 1 2
0 0
~
0;
b1 ¼ ~
2
B1 ¼
0
3
7
0 5;
0
1
0 0
3
7
1 0 5;
0 1
2
3
0 0 0
6
7
F1 ¼ 4 0 1 1 5;
0 1 1
1 6
40
10
0
~
b2 ¼ ~
0:
As in the first example, the corresponding initial-value problem is
solved numerically using three different initial conditions:
2 3
2 3
2 3
0
0
0
~
u1 ð0Þ ¼ 4 0 5; ~
u2 ð0Þ ¼ 4 1 5; and ~
u3 ð0Þ ¼ 4 0 5:
0
0
1
The three numerical solutions are then combined to approximate the
solution to the original boundary-value problem (4.6). Table 4.2 presents
ORDER
REPRINTS
1312
Arciniega and Allen
Table 4.2. Approximate values of Eðu1 ð1=2ÞÞ and Eðu21 ð1=2ÞÞ.
Euler shooting
Number of intervals in t (h)
2
4
8
16
32
64
Milstein shooting
Eu
Eu2
Eu
Eu2
0.4493
0.5256
0.5729
0.5988
0.6118
0.6183
0.2019
0.2771
0.3305
0.3620
0.3786
0.3872
0.4491
0.5262
0.5741
0.6002
0.6134
0.6200
0.2017
0.2777
0.3317
0.3637
0.3806
0.3893
approximations of Eðu1 ð1=2ÞÞ and Eðu21 ð1=2ÞÞ using both the Euler and
Milstein methods. The approximate values are based on 100,000 independent trials. Notice that only the approximations of the first component of the vector are given. Although the exact values are unknown,
the best estimates for Eðu1 ð1=2ÞÞ and Eðu21 ð1=2ÞÞ are, respectively,
0:6248 and 0:3960. Notice that Milstein’s method appears to converge
Figure 4.3. Illustration of the average and two trajectories of the solution.
ORDER
REPRINTS
Shooting Methods for Numerical Solution
1313
Figure 4.4. Illustration of the average and two trajectories of the solution.
slightly faster than Euler’s method for this example. Figure 4.3 illustrates
the average of the approximate solution with 100,000 independent trials,
using Euler’s Method. Two particular trajectories of the solutions are
also shown.
Figure 4.4 illustrates the average of the approximate solution with
100,000 independent trials, using Milstein’s Method. Two particular
trajectories of the solutions are also shown.
5. CONCLUSION
Numerical methods were used to numerically solve a stochastic
boundary-value system. In particular, shooting methods were examined for numerically solving systems of Stratonovich boundary-value
problems. It was proved that these methods accurately approximate
the solutions of stochastic boundary-value problems. Error analyses
of these methods were performed. Computational simulations were
given.
ORDER
REPRINTS
1314
Arciniega and Allen
ACKNOWLEDGMENTS
The research was supported by the Texas Advanced Research
Program Grant ARP 0212-44-1582 and the National Science Foundation
Grant DMS-0201105.
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and Breach Publishers: Amsterdam, 1995.
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6. Ocone, D.; Pardoux, E. Linear stochastic differential equations with
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Inc.: London, 1972.
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Vol. 451, 63–106.
9. Talay, D.; Turbano, L. Expansion of the global error for numerical
schemes solving stochastic differential equations. Stoch. Anal. Appl.
1990, 8 (4), 483–509.
10. Zeitouni, O.; Dembo, A. A Change of variables formula for
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stochastic boundary value problems. Probab. Theory Rel. Fields
1990, 84, 411–425.