rounding error in numerical solution of stochastic...
TRANSCRIPT
Rounding Error in Numerical Solution of StochasticDifferential Equations
Armando Arciniega* and Edward Allen
Department of Mathematics and Statistics, Texas Tech University,
Lubbock, Texas, USA
ABSTRACT
The present investigation is concerned with estimating the rounding error
in numerical solution of stochastic differential equations. A statistical
rounding error analysis of Euler’s method for stochastic differential
equations is performed. In particular, numerical evaluation of the
quantities EjXðtnÞ2 Ynj2
and E½FðYnÞ2 FðXðtnÞÞ� is investigated, where
X(tn) is the exact solution at the nth time step and Yn is the approximate
solution that includes computer rounding error. It is shown that rounding
error is inversely proportional to the square root of the step size. An
extrapolation technique provides second-order accuracy, and is one way
to increase accuracy while avoiding rounding error. Several compu-
tational results are given.
281
DOI: 10.1081/SAP-120019286 0736-2994 (Print); 1532-9356 (Online)
Copyright q 2003 by Marcel Dekker, Inc. www.dekker.com
*Correspondence: Armando Arciniega, Department of Mathematics and Statistics,
Texas Tech University, Lubbock, Texas 79409-1042, USA; E-mail: aarcinie@
math.ttu.edu.
STOCHASTIC ANALYSIS AND APPLICATIONS
Vol. 21, No. 2, pp. 281–300, 2003
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1. INTRODUCTION
The study of stochastic differential equations plays a prominent role in a
range of application areas. When a differential equation model for some
physical phenomenon is formulated, preferably the exact solution can be
obtained. However, even for ordinary differential equations, this is generally
not possible and numerical methods must be used. Numerical solution of
stochastic differential equations has been studied by many researchers (see,
for example, Refs.[3,5,8] and the references therein). In the present
investigation, rounding error in Euler’s method for stochastic differential
equations is analyzed and computationally tested.
Rounding error is present in any numerical scheme, and can lead to
unsatisfactory results. The following deterministic example illustrates that
rounding error can be of significant importance. Consider the initial value
problem
dxdt¼ xðtÞ2 1:07
tþ0:07
� �2
2 2ð1:07Þ2
ðtþ0:07Þ3; 0 # t # 1
xð0Þ ¼ e21 þ 1:070:07
� �2:
8><>: ð1:1Þ
The exact solution to this problem is
xðtÞ ¼ e t21 þ1:07
t þ 0:07
2
:
Absolute errors of the numerical solution at time t ¼ 1 are shown in Fig. 1
using the first-order Euler’s method. In addition, the absolute errors of the
second-order method obtained by extrapolating (through Richardson
extrapolation[6]) the approximate values are also shown. (This will be
referred to as the extrapolated Euler method.) Notice that when the step size
gets sufficiently small, the errors exhibit a random behavior due to
accumulation of rounding errors and the error does not decrease at the
theoretical rate. However, the errors in the extrapolated Euler method are
much smaller than the errors in Euler’s method for larger step size. This
indicates that Richardson extrapolation may be used to obtain accurate results
before rounding error becomes significant. (Of course, in addition to using
higher-order numerical methods, increasing the number of digits in the
calculations can also reduce the rounding errors.)
In the next section, statistical analyses of rounding error for numerical
solution of stochastic differential equations are given for mean square error
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and for expectation of functions of the solution. It is also shown how
Richardson extrapolation can alleviate the rounding error with regard to
approximation of the expectation of functions of the solution.
2. ANALYSES OF ROUNDING ERROR
2.1. Introduction
Consider an Ito process X ¼ {XðtÞ : 0 # t # T} satisfying the stochastic
differential equation
dXðtÞ ¼ f ðt;XðtÞÞdt þ gðt;XðtÞÞdWðtÞ; 0 # t # T
Xð0Þ ¼ X0;
(ð2:1Þ
Figure 1. Illustration of the error reduction possible by extrapolating Euler’s method.
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where X(t) satisfies the equivalent Ito stochastic integral equation
XðtÞ ¼ X0 þ
Z t
0
f ðs;XðsÞÞds þ
Z t
0
gðs;XðsÞÞdWðsÞ; 0 # t # T : ð2:2Þ
Select a positive integer N $ 2 and partition the interval [0,T ] into
0 ¼ t0 , t1 , · · · , tN ¼ T;
where tn ¼ nh for each n 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 TN:
An Euler approximation to (2.1) is a stochastic process satisfying the
iterative scheme
Y0 ¼ X0;
Yn ¼ Yn21 þ f ðtn21;Yn21Þðtn 2 tn21Þþ gðtn21;Yn21ÞðWðtnÞ2Wðtn21ÞÞ
(
ð2:3Þ
for each n ¼ 1; . . .;N; where Yn denotes the approximation to the exact solution
at the nth time step. That is Yn < XðtnÞ: Denote the random increments of the
Wiener process W ¼ {WðtÞ : t $ 0} by DWn ¼ WðtnÞ2Wðtn21Þ: It is well
known that these increments are independent normal random variables with
mean zero and variance tn 2 tn21; for each n ¼ 1; . . .;N (see, for example,
Ref.[2]). If h ¼ tn 2 tn21; equation (2.3) takes the form
Y0 ¼ X0;
Yn ¼ Yn21 þ hf ðtn21;Yn21Þþ gðtn21;Yn21ÞðDWnÞ
(ð2:4Þ
for each n ¼ 1; . . .;N: The following theorem is a well known result concerning
the strong convergence of Euler’s method for stochastic differential equations
(see Ref.[3] or Ref.[5]).
Theorem 2.1. Suppose the functions f and g satisfy uniform growth and
Lipschitz conditions in the second variable, and are Holder continuous of
order 12
in the first variable. Specifically, there exists a constant K . 0 such
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that for all s; t [ ½0; T�; x; y [ R;
j f ðt; xÞ2 f ðt; yÞj þ jgðt; xÞ2 gðt; yÞj # Kjx 2 yj ð2:5Þ
j f ðt; xÞj2þ jgðt; xÞj
2# K 2ð1 þ jxj
2Þ ð2:6Þ
j f ðs; xÞ2 f ðt; xÞj þ jgðs; xÞ2 gðt; xÞj # Kjs 2 tj12: ð2:7Þ
Then, there exists a positive constant C1 ¼ C1ðTÞ such that
EjXðtnÞ2 Ynj2# C1h:
The error formula given in Theorem 2.1 depends linearly on the step size h.
Consequently, reducing the step size should give correspondingly greater
accuracy to the numerical values. However, neglected in the result of Theorem
2.1 is the effect that rounding error plays in the choice of the step size. As h
becomes smaller, more calculations are necessary and more rounding error is
expected. In practice then, the difference-equation
Y0 ¼ X0;
Yn ¼ Yn21 þ hf ðtn21; Yn21Þ þ gðtn21; Yn21ÞðDWnÞ
(ð2:8Þ
for each n ¼ 1; . . .;N; is not used to calculate the approximation to the
solution X(tn) at the point tn. Instead, the following equation is used
Y0 ¼ Y0 þ ~e0;
Yn ¼ Yn21 þ hf ðtn21; Yn21Þ þ gðtn21; Yn21ÞðDWnÞ þ ~en
8<: ð2:9Þ
for each n ¼ 1; . . .;N; where en denotes the rounding error in performing
function evaluations, multiplications, and summations in the nth step.
Statistical rounding error analyses as described in Refs.[4,7] are performed
in the present investigation. At each iteration, one assumes that the rounding
errors en are normally distributed with mean zero and that en is independent of
em for n – m: In particular, it is assumed that Eð ~enÞ ¼ 0 and Eð ~e2nÞ #
~Cd2 for
each n, where d is proportional to the unit roundoff error. That is, d/ b2t;where b is the base of the computer system and t is the number of digits in the
floating-point representation. Now notice that the numerical scheme for Yn in
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(2.9) can be written in the form
Y0 ¼ Y0 þ e0;
Yn ¼ Yn þ en
(ð2:10Þ
where en is the accumulated error due to rounding. To see this, first notice that
Y0 ¼ Y0 þ e0 ¼ Y0 þ ~e0: Therefore,
Y1 ¼ Y0 þ hf ðt0; Y0Þ þ gðt0; Y0ÞðDW0Þ þ ~e1
¼ Y0 þ ~e0 þ hf ðt0; Y0 þ ~e0Þ þ gðt0; Y0 þ ~e0ÞðDW0Þ þ ~e1
¼ Y0 þ hf ðt0; Y0Þ þ gðt0; Y0ÞðDW0Þ þ e1 þ ~e1
¼ Y1 þ e1 þ ~e1 ¼ Y1 þ e1;
where
Y1 ¼ Y0 þ hf ðt0; Y0Þ þ gðt0; Y0ÞðDW0Þ;
e1 ¼ e1 þ ~e1; and
e1 ¼ ~e0 þ h½ f ðt0; Y0 þ ~e0Þ2 f ðt0; Y0Þ�
þ ðDW0Þ½gðt0; Y0 þ ~e0Þ2 gðt0; Y0Þ�:
Considering e1, it follows that
e 21 ¼ ~e 2
0 þ h2½ f ðt0; Y0 þ ~e0Þ2 f ðt0; Y0Þ�2
þ ðDW0Þ2½gðt0; Y0 þ ~e0Þ2 gðt0; Y0Þ�
2
þ 2 ~e0h½ f ðt0; Y0 þ ~e0Þ2 f ðt0; Y0Þ�
þ 2 ~e0ðDW0Þ½gðt0; Y0 þ ~e0Þ2 gðt0; Y0Þ�
þ 2hðDW0Þ½ f ðt0; Y0 þ ~e0Þ
2 f ðt0;Y0Þ�½gðt0; Y0 þ ~e0Þ2 gðt0; Y0Þ�:
However, the random increments DWn are normally distributed with
mean zero and variance h, i.e., DWn [ Nð0; hÞ for each n. Therefore,
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from the preceding equation,
Eðe21Þ ¼Eð ~e2
0Þ þ h2E{½ f ðt0; Y0 þ ~e0Þ2 f ðt0; Y0Þ�2} þ hE{½gðt0; Y0 þ ~e0Þ
2 gðt0; Y0Þ�2} þ E{2 ~e0h½ f ðt0;Y0 þ ~e0Þ2 f ðt0; Y0Þ�}:
Applying the Lipschitz condition (2.5) on f and g, and using
the Cauchy–Schwarz inequality on the third term of the last equation
leads to
Eðe21Þ # Eð ~e2
0Þ þ K 2½h2Eð ~e20Þ þ hEð ~e2
0Þ� þ 2hKEð ~e20Þ
# ~Cd2½1 þ 2hK þ hðh þ 1ÞK 2�:
For notational convenience, denote r1h by
r1h ¼ ½2K þ ðT þ 1ÞK 2�h:
Then,
Eðe21Þ #
~Cd2ð1 þ r1hÞ:
Since Eð ~e0Þ ¼ 0 and Eð ~e21Þ #
~Cd2; then
Eðe21Þ ¼ E½ð ~e1 þ e1Þ
2� ¼ Eð ~e21Þ þ Eðe2
1Þ #~Cd2 þ ~Cd2ð1 þ r1hÞ
¼ ~Cd2½1 þ ð1 þ r1hÞ�:
Similarly, the second step leads to
Y2 ¼ Y2 þ e2
and
Eðe22Þ # ð1 þ r1hÞEðe2
1Þ # ð1 þ r1hÞ½Eð ~e21Þ þ Eðe2
1Þ�
# ð1 þ r1hÞ½ ~Cd2 þ ~Cd2ð1 þ r1hÞ�
¼ ~Cd2½ð1 þ r1hÞ þ ð1 þ r1hÞ2�:
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Hence,
Eðe22Þ ¼ E½ð ~e2 þ e2Þ
2� ¼ Eð ~e22Þ þ Eðe2
2Þ
# ~Cd2 þ ~Cd2½ð1 þ r1hÞ þ ð1 þ r1hÞ2�
¼ ~Cd2½1 þ ð1 þ r1hÞ þ ð1 þ r1hÞ2�:
Continuing in this manner, the nth step leads to
Yn ¼ Yn þ en
and
Eðe2nÞ # ð1 þ r1hÞEðe2
n21Þ
# ð1 þ r1hÞ½Eð ~e2n21Þ þ Eðe2
n21Þ�
# ð1 þ r1hÞ½ ~Cd2 þ Eðe2n21Þ�
# ~Cd2½ð1 þ r1hÞ þ ð1 þ r1hÞ2 þ · · · þ ð1 þ r1hÞn�:
Thus,
Eðe2nÞ ¼ E½ð ~en þ enÞ
2�
¼ Eð ~e2nÞ þ Eðe2
nÞ
# ~Cd2½1 þ ð1 þ r1hÞ þ ð1 þ r1hÞ2 þ · · · þ ð1 þ r1hÞn�
# ~Cd2 ð1 þ r1hÞn
r1h#
~Cd2
r1her1T ¼ C
d2
h;
where C ¼~Ce r1T
r1and r1 ¼ ½2K þ ðT þ 1ÞK 2�: This proves the following
result:
Theorem 2.2. Let Y0; . . .; YN be the approximations obtained using (2.9) or
(2.10). If en are independently distributed random variables with Eð ~enÞ ¼ 0 and
Eð ~e2nÞ #
~Cd2; then the accumulated error en for any n satisfies
Eðe2nÞ # C
d2
h
for some positive constant C, where d is proportional to the unit roundoff error.
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Theorem 2.2 will be useful in the analysis of rounding error for functional
expectations. First, however, a mean square convergence result for Euler’s
method with rounding error is formulated. The proof is similar in structure to the
proof of Theorem 7.2 in Ref.[3]
2.2. Rounding Error for Mean Square Convergence
(Strong Convergence)
Theorem 2.3. Let Y0; . . .; YN be the approximations obtained using (2.10).
Let Eð ~enÞ ¼ 0;Eð ~e2nÞ ,
~Cd2; and suppose the hypotheses of Theorem 2.1 are
satisfied. Then, there exist positive constants C1, C2, and C3 such that
EjXðtnÞ2 Ynj2# C1h þ C2 þ
C3
h
d2:
Proof. First, notice that conditions (2.5)–(2.7) guarantee a unique solution
X(t) to (2.1). Denote gn by
gn ¼ EjXðtnÞ2 Ynj2:
Then,
g0 ¼ EjXðt0Þ2 Y0j2¼ EjX0 þ ~e0 2 X0j
2¼ Ej ~e0j
2# ~Cd2:
Next, define Y(t) by
YðtÞ ¼ Yn21 þ ~en þ
Z t
tn21
f ðtn21; Yn21Þds
þ
Z t
tn21
gðtn21; Yn21ÞdWðsÞ: ð2:11Þ
Notice that
Yðtn21Þ ¼ Yn21 þ ~en:
Now,
XðtÞ2 YðtÞ ¼ Xðtn21Þ2 Yn21 2 ~en
þ
Z t
tn21
½ f ðs;XðsÞÞ2 f ðtn21; Yn21Þ�ds
þ
Z t
tn21
½gðs;XðsÞÞ2 gðtn21; Yn21Þ�dWðsÞ:
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and
dðXðtÞ2 YðtÞÞ ¼ ½ f ðt;XðtÞÞ2 f ðtn21; Yn21Þ�dt þ ½gðt;XðtÞÞ
2 gðtn21; Yn21Þ�dWðtÞ:
Applying Ito’s formula to the preceding equation, one obtains
dððXðtÞ2 YðtÞÞ2Þ
¼ {2ðXðtÞ2 YðtÞÞðf ðt;XðtÞÞ2 f ðtn21; Yn21ÞÞ
þ ðgðt;XðtÞÞ2 gðtn21; Yn21ÞÞ2}dt
þ 2ðXðtÞ2 YðtÞÞðgðt;XðtÞÞ2 gðtn21; Yn21ÞÞdWðtÞ:
Integrating the preceding equation from tn21 to tn and taking expectations
where, as before,
gn ¼ EjXðtnÞ2 Ynj2;
and noticing that
EjXðtn21Þ2 Yðtn21Þj2¼ gn21 þ Eð ~e2
nÞ
leads to
gn ¼ gn21 þ Eð ~e2nÞ þ
Z tn
tn21
E{2ðXðsÞ2 YðsÞÞðf ðs;XðsÞÞ2 f ðtn21; Yn21ÞÞ
þ ðgðs;XðsÞÞ2 gðtn21; Yn21ÞÞ2}ds # gn21 þ ~Cd2
þ
Z tn
tn21
{EjXðsÞ2 YðsÞj2þ Ej f ðs;XðsÞÞ2 f ðtn21; Yn21Þj
2
þ Ejgðs;XðsÞÞ2 gðtn21; Yn21Þj2}ds ð2:12Þ
as Eð ~e2nÞ #
~Cd2 for each n. Invoking the Lipschitz (2.5) and Holder (2.7)
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conditions, one has
j f ðs;XðsÞÞ2 f ðtn21; Yn21Þj2#3{j f ðs;XðsÞÞ2 f ðs;Xðtn21ÞÞj
2
þ j f ðs;Xðtn21ÞÞ2 f ðtn21;Xðtn21ÞÞj2
þ j f ðtn21;Xðtn21ÞÞ2 f ðtn21; Yn21Þj2}
þ ðs 2 tn21Þ þ jXðtn21Þ2 Yn21j2}
# 3K 2 jKðsÞ Xðtn1Þj2þ ðs tn1Þ
n
þjXðtn1Þ Ynj2o
ð2:13Þ
Similarly,
jgðs;XðsÞÞ2 gðtn21; Yn21Þj2# 3K 2{jXðsÞ2 Xðtn21Þj
2
þ ðs 2 tn21Þ þ jXðtn21Þ2 Yn21j2}: ð2:14Þ
Now, consider
EjXðsÞ2 Xðtn21Þj2
¼ E
Z s
tn21
f ðt;XðtÞÞdt þ
Z s
tn21
gðt;XðtÞÞdWðtÞ
��������2
# 2 E
Z s
tn21
f ðt;XðtÞÞdt
��������2
þE
Z s
tn21
gðt;XðtÞÞdWðtÞ
��������2
" #
# 2{ðs 2 tn21Þ
Z s
tn21
Ej f ðt;XðtÞÞj2dt þ
Z s
tn21
Ejgðt;XðtÞÞj2dt}
# 2{½ðs 2 tn21Þ þ 1�
Z s
tn21
K 2Eð1 þ jXðtÞj2Þdt}:
However, from Theorem 3.8 of Ref.,[3]
EjXðtÞj2# ð1 þ EjX0j
2ÞeLt
for some constant L. Therefore,
EjXðsÞ2 Xðtn21Þj2# K1ðs 2 tn21Þ; ð2:15Þ
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where K1 depends only on K, T, and X0. Substituting (2.13), (2.14), and (2.15)
into (2.12) leads to
gn # gn21 þ ~Cd2 þ
Z tn
tn21
{EjXðsÞ2 YðsÞj2
þ 6K 2½ðK1 þ 1Þðs 2 tn21Þ þ gn21�}ds
# gn21ð1 þ 6K 2hÞ þ 3K 2ðK1 þ 1Þh2
þ
Z tn
tn21
EjXðsÞ2 YðsÞj2ds þ ~Cd2: ð2:16Þ
Applying the Bellman–Gronwall inequality: if a(t) and b(t) are measurable
bounded functions such that for some ~L . 0;
aðtÞ # bðtÞ þ ~L
Z t
0
aðsÞds;
then
aðtÞ # bðtÞ þ ~L
Z t
0
e~Lðt2sÞbðsÞds
to (2.16) with
aðtÞ ¼ EjXðtÞ2 YðtÞj2
and
bðtÞ ¼ ðEjXðtn21Þ2 Yn21j2Þð1 þ 6K 2hÞ þ 3K 2ðK1 þ 1Þh2 þ ~Cd2
on ½tn21; tn� leads to
gn # gn21ð1 þ 6K 2hÞ
þ
Z tn
tn21
e tn2s½gn21ð1 þ 6K 2hÞ þ 3K 2ðK1 þ 1Þh2 þ ~Cd2�ds
þ 3K 2ðK1 þ 1Þh2 þ ~Cd2
# ½gn21ð1 þ 6K 2hÞ þ 3K 2ðK1 þ 1Þh2 þ ~Cd2�eh: ð2:17Þ
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Iterating (2.17) with g0 ¼ Ej ~e0j2
leads to
gn # rn2Ej ~e0j
2þ ð3K 2ðK1 þ 1Þh2 þ ~Cd2Þeh 1 2 rn
2
1 2 r2
; ð2:18Þ
where r2 ¼ ð1 þ 6K 2hÞeh: Now, Ej ~e0j2# ~Cd2 implies that rn
2Ej ~e0j2# C2d
2;where C2 ¼ ~CeTð1þ6K 2Þ: Therefore,
gn # C2d2 þ h½3K 2ðK1 þ 1Þ�
heh
r2 2 1rn
2 þ~Cd2 eh
r2 2 1rn
2: ð2:19Þ
However,
rn2 # r
Th
2 ¼ ½ð1 þ 6K 2hÞeh�Th # eTð1þ6K 2Þ;
and
heh
r2 2 1¼
h
1 þ 6K 2h 2 e2h#
1
6K 2:
Also,
~Cd2 eh
r2 2 1rn
2 ¼~Cd2
h
heh
r2 2 1rn
2:
Substituting all this into (2.19) leads to
gn # C1h þ C2 þC3
h
d2; ð2:20Þ
where C1 ¼ ðK1þ1Þ2
eTð1þ6K 2Þ and C3 ¼~C
6K 2 eTð1þ6K 2Þ: This completes the proof
of the theorem. A
2.3. Rounding Error for Functional Expectation
(Weak Convergence)
Let Yn and Yn be the approximations of (2.1) using Euler’s method with
and without rounding error, respectively. If F is a smooth function, it is
possible to obtain an expansion of the form
E½FðYnÞ2 FðXðtnÞÞ� ¼ c1h þ Oðh2Þ;
where X(tn) denotes the exact solution at the nth step and c1 is a constant
independent of h (see Ref.[9]). This expansion justifies the Richardson
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extrapolation technique described in the next section, and it indicates how
Euler’s method can be modified to a weak second-order scheme.
Now, consider an equation of the form
E½FðYnÞ2 FðXðtnÞÞ� ¼ E½FðYnÞ2 FðYnÞ þ FðYnÞ2 FðXðtnÞÞ�
¼ E½F0ðYnÞðYn 2 YnÞ� þ c1h þ Oðh2Þ
# ðEðF0ðYnÞÞÞ1=2ðEðYn 2 YnÞ
2Þ1=2 þ c1h þ Oðh2Þ:
ð2:21Þ
Since
Yn ¼ Yn þ en;
it follows from Theorem 2.2 that
Eðe2nÞ
� �1=2¼ ðEðYn 2 YnÞ
2Þ1=2 #Cd2
h
1=2
:
Substituting this into (2.21) leads to
E½FðYnÞ2 FðXðtnÞÞ� # MCd2
h
1=2
þc1h þ Oðh2Þ; ð2:22Þ
where
M ¼Y[RmaxðF0ðYÞÞ1=2:
This proves the following result:
Theorem 2.4. Let Yn and Yn be the approximations of (2.1) using Euler’s
method with and without rounding error, respectively. Let F be a smooth
function satisfying the following expansion
E½FðYnÞ2 FðXðtnÞÞ� ¼ c1h þ Oðh2Þ;
where X(tn) denotes the exact solution at the nth step and c1 is a constant
independent of h. Then,
E½FðYnÞ2 FðXðtnÞÞ� # MCd2
h
1=2
þc1h þ Oðh2Þ
for some positive constants C and M.
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3. ALLEVIATION OF ROUNDING ERROR THROUGH
HIGHER-ORDER METHODS (EXTRAPOLATION)
A consequence of Theorem 2.4 is the justification of Richardson
extrapolation between values corresponding to two different step sizes. The
extrapolation technique is described in Refs.[5,9] or Ref.[6] That is, an
approximation is first calculated with step size h, and then a second
approximation is calculated with step size h/2. For example, if Euler’s method
is used with n equal time steps h, then
E½FðYnÞ2 FðXðtnÞÞ� ¼ MCd2
h
1=2
þc1h þ Oðh2Þ:
Next, Euler’s method is used with 2n time steps of equal length h/2 so that
E½FðYh=2
2n Þ2 FðXðtnÞÞ� ¼ M2Cd2
h
1=2
þc1
h
2þ Oðh2Þ:
A combination of the two preceding equations yields
E½2FðYh=2
2n Þ2 FðYnÞ2 FðXðtnÞÞ� ¼ ~MCd2
h
1=2
þOðh2Þ;
where ~M ¼ Mð2ffiffiffi2
p2 1Þ: This result implies that through Richardson
extrapolation, the method error may be made sufficiently small before the
rounding error dominates as the step size h is decreased. The computational
results described in the next section support this supposition.
4. COMPUTATIONAL RESULTS
In this section, computational results are given that support the theoretical
results in the present investigation. The first example is the stochastic version
of the deterministic example given in the introduction. Consider the stochastic
initial value problem
dXðtÞ ¼ XðtÞ2 1:07tþ0:07
� �2
2 2ð1:07Þ2
ðtþ0:07Þ3
dtþ
ffiffi2
p
10dWðtÞ; 0 # t # 1
Xð0Þ ¼ e21 þ 1:070:07
� �2:
8>><>>: ð4:1Þ
It is desired to estimate EFðXð1ÞÞ ¼ EXð1Þ: The expectation of the solution to
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this problem is
EXðtÞ ¼ e t21 þ1:07
tþ 0:07
2
and is obtained by application of Ito’s formula. Therefore,
EXð1Þ ¼ 2:
Table 1 presents the absolute errors in Euler’s method and those in the
extrapolated Euler method. The numerical values are based on 100,000
independent trials. As in the deterministic case, when the step size gets
sufficiently small, the errors exhibit a random behavior due to accumulation of
rounding errors. However, the errors in the extrapolated Euler method are
much smaller than the errors in Euler’s method for larger step size h. This
indicates that Richardson extrapolation may be used to obtain accurate results
before the rounding error dominates as the step size h is decreased. Figure 2
illustrates graphically the numerical values given in Table 1. Notice that Fig. 2
is similar to Fig. 1.
The next example presents the error reduction possible by extrapolating
Euler’s method for a system. To illustrate, consider the following stochastic
system
dX1ðtÞ ¼ 2dW1ðtÞ þ X22ðtÞdW2ðtÞ; 0 # t # 1
dX2ðtÞ ¼12
X2ðtÞdW1ðtÞ
X1ð0Þ ¼ 0
X2ð0Þ ¼ 1
8>>>>><>>>>>:
ð4:2Þ
Table 1. Error reduction possible by extrapolating Euler’s method.
Number of
intervals in t Euler’s method Extrapolation
800 11.0511
1600 5.5051 0.0409
3200 2.7452 0.0146
6400 1.3700 0.0052
12800 0.6811 0.0078
25600 0.3398 0.0015
51200 0.1728 0.0059
102400 0.0876 0.0023
204800 0.0195 0.0485
409600 0.0724 0.1643
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It is desired to estimate EFðX1ð1ÞÞ ¼ EX21ð1Þ: By application of Ito’s’s
formula, it can be shown that
EX21ðtÞ ¼
2
3ðe3t=2 2 1Þ þ t:
Hence,
EX21ð1Þ ¼
1
3ð2e3=2 þ 1Þ < 3:321126:
Table 2 presents the absolute errors in Euler’s method and those obtained in
extrapolating Euler’s method. The approximate values are based on 1,000,000
independent trials. Notice that the error in Richardson extrapolation becomes
small before rounding error becomes significant, whereas the error in Euler’s
method is eventually dominated by accumulation of rounding error for suf-
ficiently small step size h. Of course, the extrapolated Euler method also suffers
from rounding error as the step size decreases. However, the extrapolated Euler
method can obtain accurate results before rounding error dominates.
Figure 2. Illustration of the error reduction possible by extrapolating Euler’s method.
Rounding Error in Numerical Solution 297
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The next example illustrates that, although rounding error is present in all
second-order schemes, the accuracy generally is much better than that of
Euler’s method for large step sizes before rounding error dominates. The
calculational results of a weak second-order Runge–Kutta method described
by Abukhaled and Allen[1] and the results of Euler method are compared for
Table 2. Error reduction possible by applying Richardson
extrapolation
Number of
intervals in t Euler’s method
Extrapolated Euler
method
10 0.2567
20 0.1582 0.0597
40 0.0775 0.0032
80 0.0251 0.0272
160 0.0082 0.0087
320 0.0124 0.0166
Figure 3. Illustration of the errors in Euler and second-order methods.
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the following stochastic initial value problem
dXðtÞ ¼ ð1 þ XðtÞÞdt þ ð1 þ XðtÞÞdWðtÞ; 0 # t # 1
Xð0Þ ¼ 1:
(ð4:3Þ
The expectation of the solution to this problem is
EXðtÞ ¼ 2e t 2 1:
Thus,
EXð1Þ ¼ 2e 2 1 < 4:4366:
Figure 3 indicates that the errors of both second-order methods are much
smaller than the errors in Euler’s method before rounding error dominates for
small h. The approximate values are based on 1,000,000 independent trials.
CONCLUSION
Statistical rounding error analyses in numerical solution of stochastic
differential equations have been performed. Rounding error in Euler’s method
for stochastic differential equations has been analyzed and computationally
tested. It was found that rounding error is inversely proportional to the square
root of the step size and proportional to b 2t where b is the base and t is the
number of digits in the floating-point system. Richardson extrapolation was
applied to Euler’s method to alleviate the rounding error with regard to
approximation of functional expectation. Calculational results indicate that
higher-order methods may sometimes be used to obtain accurate results before
rounding error dominates as the step size is decreased.
ACKNOWLEDGMENTS
The research was supported by the Texas Advanced Research Program
Grants ARP 0202-44-6283, ARP 0212-44-1582, and the National Science
Foundation Grant NSF 1316-44-1775.
REFERENCES
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Rounding Error in Numerical Solution 299
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2. Arnold, L. Stochastic Differential Equations: Theory and Applications;
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3. Gard, T.C. Introduction to Stochastic Differential Equations; Marcel
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