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On Stability of Weak Schemes for Stochastic Differential Systems With One Multiplicative Noise.

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Appl. Num. Anal. Comp. Math. 1, No. 2, 377 ? 385 (2004) / DOI 10.1002/anac.200410005
On Stability of Weak Schemes for Stochastic Differential Systems
With One Multiplicative Noise
Marwan I. Abukhaled1
1
Department of Mathematics and Statistics American University of Sharjah Sharjah, United Arab Emirates
Received 20 October 2004, revised 4 December 2004, accepted 4 December 2004
Published online 20 December 2004
Key words Stochastic differential systems, weak numerical schemes, mean square stability.
AMS 60H10, 65C20, 65C05.
The logarithmic matrix norm with respect to l2 matrix norm will be used to investigate mean square stability
for a class of second-order weak schemes when applied to 2-dimensional linear stochastic differential systems
with one multiplicative noise.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
An autonomous scalar Ito? stochastic differential equation has the form
dZ(t) = f (t, Z(t)) dt + g(t, Z(t)) dW (t), 0 ? t ? T
(1)
Z(t0 ) = Z0
where W (t) is a standard Wiener process whose Gaussian increments, W (t + ?t) ? W (t), are normally distributed random variables with mean zero and variance one.
The conditions that the drift and diffusion functions (f and g, respectively) must satisfy in order for equation
(1) to have exact solution in Ito? sense are almost impossible to hold in practical applications [5]. Numerical
methods where trajectories of solution are computed at discrete points are used to solve SDEs (see for example
[8]). While order of convergence and simplicity of implementation are desired features of a numerical scheme,
its stability is a required feature to avoid a possible explosion of the numerical solution. Recent years have marked
many advances in the investigation of stability of numerical schemes for SDEs (e.g.,[1, 6, 7, 9])
Definition 1.1 The equilibrium position, Z(t) ? 0, is said to be mean square stable if for every > 0, there
exists a ?1 > 0 such that
Z(t) < where Z(t) =
that
for all t ? 0 and |Z0 | < ?1
(2)
E|Z(t)|2 , in which E stands for the expected value. Moreover, if there exists a ?2 > 0 such
lim Z(t) = 0
t??
for all |Z0 | < ?2
then the equilibrium position is said to be asymptotically mean square stable.
Definition 1.2 Suppose that the equilibrium position of equation (1) is mean square stable. Then a numerical
scheme that produces the iterations {Zn } to approximate the solution Z(tn ) is said to be mean square stable if
lim Zn = 0
n??
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
378
Marwan Abukhaled: Stability of Weak Schemes for SD Systems
Saito and Mitsui established mean square stability criteria for Euler-Maruyama scheme when applied to certain
types of 2-dimensional SD systems with one multiplicative noise [10]. In this paper, mean square stability of a
class of Runge-Kutta schemes when applied to same types of SD systems will be investigated. In section 2, we
will have a brief look at the class of Runge-Kutta methods under investigation. In section 3, a closer look at the
meaning of mean square stability for 2-dimensional SD systems will be given. In section 4, we define numerical
mean square stability and describe the restrictions under which a class of Runge-Kutta schemes is mean square
stable with respect to the logarithmic matrix norm in l2 . Some numerical experiments will be provided in section
5 to illustrate the theoretical analysis. Conclusion remarks are summarized in section 6.
2 A Class of Weak Second-Order Methods
Definition 2.1 A discrete time approximation Zn converges weakly towards the exact solution Z(tn ) with
order k as h ? 0, if for every smooth function F , there exists a positive constant C, independent of h, such that
|E(F (Z(tn ))) ? E(F (Zn ))| = Chk
Consider the explicit class of Runge-Kutta-type methods given by
Zn+1 = Zn + ?1 hf (Zn ) + ?1 hg(Zn )?1
+?2 hf (K1 ) + ?2 hg(K1 )?2
+?4 hf (K2 ) + ?4 hg(K2 )?3
where
(3)
?
K1 = Zn + ?3 hf (Zn ) + ??3 hg(Zn )?1
K2 = Zn + ?5 hf (Zn ) + ?5 hg(Zn )?1
(4)
and ?1 , ?2 , and ?3 are independent, normally distributed random variables with mean zero and variance one. It
was shown in [3] that methods (3)-(4) are of second-order accuracy in the weak sense provided that ?s satisfy the
nonlinear system
?
?1 + ?2 + ?4
=
?2 ?3 + ?4 ?5
=
?
?
?1 (?2 ?3 + ?4 ?5 ) =
1
1
2
1
2
(5)
System (5) has infinite number of real and imaginary solutions [3].
3 Stability of 2-Dimensional SDEs
The 2-dimensional Ito? linear SD system with one multiplicative noise is given by
dZ(t) = M Z(t) + SZ(t)dW (t)
Z(0) = 1
where
Z(t) =
Z1 (t)
Z2 (t)
, M=
�0
0
�
(6)
, S=
?
?
?
?
, and 1 =
1
1
Let Z(t) be the solution vector of (6) and define P (t) = E(Z(t)Z(t)t ) where Z(t)t is the vector transpose.
Then the symmetric matrix P (t) given by
E(Z1 (t)Z2 (t))
E(Z1 (t))2
(7)
P (t) =
E(Z1 (t)Z2 (t))
E(Z2 (t))2
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 2 (2004) / www.anacm.org
379
satisfies the IVP [4]
dP
=
dt
P (0) =
M P + P M t + SP S t
(8)
Z0 Zt0
The symmetry property of P leads to the ODE system [10]
dX(t)
= ?X(t)
dt
where
and
(9)
? ?
?
E(Z1 (t))2
X 1 (t)
X(t) = ? X 2 (t) ? = ? E(Z2 (t))2 ?
E(Z1 (t)Z2 (t))
X 3 (t)
?
?
2�+ ?2
?
?2
?=
??
?2
2�+ ? 2
??
?
2??
?
2??
�+ �+ ?? + ??
(10)
Definition 3.1 The logarithmic matrix norm of A denoted by 祊 [A] is defined by
祊 [A] = lim
+
h?0
I + hAp ? 1
h
where �p denotes a matrix norm, and I is the A-size identity matrix.
In this paper, the logarithmic matrix where p = 2 will be used. It is rather easy to verify the following identity
�[A] =
1
?(A + A? ), where ? is the maximum eigenvalue.
2
The following mean square stability criterion for the equilibrium position of system (6) follows intuitively
[10]
Lemma 3.2 System (6) is asymptotically mean square stable with respect to the logarithmic norm 祊 if and
only if
祊 [?] < 0
Using lemma 1 and direct computation of �[?], it is possible to establish mean square stability criteria for
the equilibrium position of (6) for certain diffusion coefficients. Throughout the remainder of this manuscript, it
will be assumed that �< �< 0 (this is a natural assumption for asymptotic stability of ODEs.)
0 ?
Example 3.3 In (6), let S =
. The resulting system is called the singly anti-diagonal (SAD) [10].
? 0
In this case, the matrix given in (10) becomes
?
?
2�? 2
0
?
0
? = ? ? 2 2�2
0
0
�+ �+ ?
By direct computation, we obtain that
�[?] =
1
max{2�+ 2�+ 2? 2 , 2�+ 2�+ 2C, 2�+ 2�+ 2C}
2
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
380
where C =
Marwan Abukhaled: Stability of Weak Schemes for SD Systems
� + � ? 2��+ ? 4 . This implies that
�[?] = max{�+ �+ ? 2 , �+ �+ C}
But C ? �?�+? 2 , which leads to �+�+C ? 2�+? 2 ? 2�+? 2 . Moreover, �+�+? 2 ? 2�+? 2 .
Consequently �[?] ? 2�+ ? 2 .
? 0
. It is straightRemark 3.4 In (6), a simultaneously diagonalizable system (SD) is obtained if S =
0 ?
forward to verify that �[?] = � [?] = 2�+ ? 2 .
4 Stability of Runge-Kutta-Type Methods
When scheme (3)-(4) is applied to the 2-dimensional Ito? linear stochastic differential system given in (6), we
obtain the iteration
where
X?n+1 = ??X?n
(11)
? ?
?
E(Z?n1 )2
X?n1
X?n = ? X?n2 ? = ? E(Z?n2 )2 ?
X?n3
E(Z?n1 Z?n2 )
(12)
?
The matrix ?? in (11), called the stability matrix, will be given in Theorem 2. Under the pth matrix norm �p , it
is evident that lim X?n = 0 if ?? < 1.
n??
p
Definition 4.1 A numerical scheme is said to be stable in the mean square sense with respect to a logarithmic
norm p provided that
(13)
?? < 1
p
Theorem 4.2 When method (3)-(4) is applied to the test equation (6), the stability matrix ??, is given by
?
?
??11 ??12 ??13
?? = ? ??21 ??22 ??23 ?
??31
where
??11
=
??12
=
??13
=
??21
=
??22
=
??32
??33
2
1 2
1
2
1 + (2�+ ? )h + 2
? + �+ ?? ? + ???
h2
2
2
1
3 2 2
+
�? + � h3 + � h4
4
4
1
3
? 2 h + 2�? 2 + (? + ?)2 ? 2 h2 + � ? 2 h3
2
4
3
2??h + (? + ?)(? 2 ? + ?2 ?) + 4??�h2 + � ??h3
2
1
3
? 2 h + 2�? 2 + (? + ?)2 ? 2 h2 + � ? 2 h3
2
4
2
1 2
1
2
2
1 + (2�+ ? )h + 2
? + �+ ?? ? + ??
h2
2
2
2
3
1
+( � ? 2 + � )h3 + � h4
4
4
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 2 (2004) / www.anacm.org
??23
=
??31
=
??32
=
??33
=
381
3
2??h + [(? + ?)(? 2 ? + ? 2 ?) + 4??�]h2 + � ??h3
2
1
3
2
2
??h + (? + ?)(? ? + ? ?) + ??(�+ �) h2 + �?�?h3
2
4
1
3
??h + (? + ?)(? 2 ? + ? 2 ?) + ??(�+ �) h2 + �?�?h3
2
4
1
2
2 (�+ �) + (�+ �)(?? + ??)
h2
1 + (�+ �+ ?? + ??)h +
+ 12 ?2 ? 2 + ??( 12 ?? + ? 2 + ?? + ?2 )
3
1
1
3
1
+ �?�? + �� + � �+ �?�? h3 + � � h4
4
2
2
4
4
Criterion 1 For the singly anti-digonal case (SAD) in which S =
0
?
?
0
, scheme (3)-(4) is mean square
stability if
1
??11 + ??22 + C
<1
?? = max ??33 ,
2
2
where C =
(14)
(??11 ? ??22 )2 + 4(??12 + ??21 )2
P r o o f. Substitute ? = ? = 0 and ? = ? = ? in (6), and then by direct computation obtain that
1
?? = max{2??33 , ??11 + ??22 + C, ??11 + ??22 ? C}}
2
2
where C = (??11 ? ??22 )2 + 4(??12 + ??21 )2 . Obviously C is nonnegative, hence the result.
Criterion 2 For the simultaneously diagonalizable case (SD) in which S =
0
?
?
0
, scheme (3)-(4) is
mean square stable if
?? = max{??11 , ??22 , ??33 } < 1
(15)
2
P r o o f. Let ? = ? = ? and ? = ? = 0 in (6) and then directly compute ?? to obtain the result.
2
Criterion 3 For the singly diagonal and anti-diagonal case (SDAD) in which S =
restrictions �= �= m, and ?? > 0, scheme (3)-(4) is mean square stable if
??
2
=
3?
2
2?? ? m + F
2
3
?
2
2 2
+(F + 4? ? + 3 2?? F )h2
?
+(3 2?? + 2F )h + 2 < 1
1 4 4
m h +
2
?
?
?
?
and under the
3 2 3
m h
2
?
?
provided that ?2 2 > F = ? 2 + ? 2 + 2m > ? 2.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
382
Marwan Abukhaled: Stability of Weak Schemes for SD Systems
P r o o f. For S =
?
?
?
?
, and under the assumption �= �= m, the entries of ?? can be computed.
??11
=
??12
=
??13
=
??21
??22
=
=
2
1 2
1 2
2
2
h2
1 + (2m + ? )h + 2
? +m +? ? + ?
2
2
3 2 2
1
3
+
m ? + m h3 + m4 h4
4
4
1
3
2
2
2 2
? h + 2m? + (2?) ? h2 + m2 ? 2 h3
2
4
3
2?? h + 2?(? 3 + ? 2 ? ) + 4?? m h2 + m2 ?? h3
2
??12
??11
??23
=
??13
??31
=
??32
??33
=
=
2
3
?? h + ?(? 3 + ? 2 ? ) + 2?? m h2 + m2 ?? h3
4
??31
1 + (2m + ? 2 + ? 2 )h
1 4
2
2
2
2 1 2
2
+ 2m + 2m(? + ? ) + ? + ? ( ? + 3? ) h2
2
2
3
3
1
+ m2 ? 2 + m3 + m2 ? 2 h3 + m4 h4
4
4
4
Hence
?
?
3 2
3 2
S, F1 ?
S, F2 }
?? = max{F1 +
4
4
2
where
F1
=
S
=
F2
=
1 4 4
3
m h + (F ? m) m2 h3 + (F 2 + 4? 2 ? 2 )h2 + 2F h + 2
2
2
?? h(4 + 4F h + 3m2 h2
3 2 3
1 4 4
3
m h + F ? m ? ?2
m h
2
2
2
+(F 2 ? 4? 2 ? 2 ? 8m? 2 )h2 + (2F ? 4? 2 )h + 2
?
?
Under the restriction ?2 2 > F = ? 2 + ? 2 + 2m > ? 2, it is evident
that S > 0. Now comparing the
?
2
3
coefficients of hn , n = 0, 1, .., 4 for both F1 and F2 shows that F1 +
S = ?? . Hence the result.
4
2
Remark 4.3 Assuming that �< �< 0 it is not possible to reach a closed form for the mean square stability
criterion with respect to �2 . However, it was established in [2], that the mean square stability with respect to
�? is given by
(16)
?? max {A, B} < 1
?
where
A
=
??11 + ??12 + ??13
B
=
??22 + ??21 + ??23
in which ??ij are as given in theorem 1.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 2 (2004) / www.anacm.org
383
5 Numerical Examples
In the examples below, we applied methods (3)-(4) on the three types of SD systems discussed in section 4
(SD, SAD, SDAD). We will investigate mean square stability of these systems based on the criteria that were
established in section 4 and then verify the results with the actual computations of 25,000 sample paths of RungeKutta methods. We plot the sample paths against time t for different step size h.
Example 5.1 Consider the 2-dimensional SD system given by
?6 0
2 0
dZ(t) =
Z dt +
Z dW (t)
0 ?4
0 2
1
Z(0) =
1
For h = 0.25, ?? = 0.9375 < 1, so the Runge-Kutta method is stable. For h = 0.5, ?? = 11.750 > 1
2
2
and hence the scheme is not stable. Numerical results are shown in Fig. 1.
10
10
10
10
Xn�
Xn�
Xn�
0
10
0
10
?10
10
?20
10
Xn�
Xn�
Xn�
?10
10
?20
0
5
10
15
20
10
0
t
5
10
15
20
t
Fig. 1 (left) h=0.25, (right) h=0.25
Example 5.2 Consider the 2-dimensional SAD system given by
?6 0
0 1
dZ(t) =
Z dt +
Z dW (t)
0 ?3
1 0
1
Z(0) =
1
For h = 0.25, ?? = 0.441 < 1, so the Runge-Kutta method is stable. For h = 0.5, ?? = 6.397 and
2
2
hence the scheme is not stable. Numerical results are shown in Fig. 2.
Example 5.3 Consider the 2-dimensional SDAD system given by
?4 0
1 0.1
dZ(t) =
Zdt +
ZdW (t)
0 ?4
0.1 1
1
Z(0) =
1
For h = 0.25, ?? = 0.440, so the Runge-Kutta method is stable and for h = 0.5, ?? = 2.372 and hence
2
2
the scheme is not stable. Numerical results are shown in Fig. 3.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
384
Marwan Abukhaled: Stability of Weak Schemes for SD Systems
10
10
10
10
X �
n
Xn�
Xn�
0
10
0
10
?10
?10
10
10
?20
10
X �
n
Xn�
Xn�
?20
0
5
10
15
20
10
0
5
10
15
20
Fig. 2 (left) h=0.25, (right) h=0.25
5
5
10
10
Xn�
Xn�
Xn�
0
0
10
10
Xn�
Xn�
Xn�
0
1
2
3
4
5
0
1
2
3
4
5
Fig. 3 (left) h=0.25, (right) h=0.25
6 Conclusions and Future Work
In this paper, we have established mean square stability criteria for a class of Runge-Kutta methods when applied
to certain types of 2-dimensional linear stochastic differential systems with one multiplicative noise. As the class
of Runge-Kutta schemes discussed in this paper are known to be of second order accuracy in the weak sense,
it is left for future work to investigate if other weak second-order schemes have similar mean square stability
criteria. It is also left for future work to discuss the more general complex form of M and S in (6). Another
interesting discussion topic will be the investigation of mean square stability of numerical methods when applied
to stochastic differential systems of dimension n > 2.
References
[1] M.I. Abukhaled, Mean Square Stability of Second-Order Weak Numerical Methods for Stochastic Differential Equations,
Applied Numerical Mathematics, 48 (2004), 127-134.
[2] M.I. Abukhaled, Mean Square Stability of a Class of Runge-Kutta Methods for 2-Dimensional Stochastic Differential
Systems, Applied Num. Anal. and Comp. Math, 1(2004), 77-89.
[3] M.I. Abukhaled and E.J. Allen, A Class of Second-Order Runge-Kutta Methods for Numerical Solution of Stochastic
Differential Equations, Stochastic Analysis and Applications, 16(1998), 977-991.
[4] L. Arnold, Stochastic Differential Equations, Wiley, New York, 1974.
[5] T.C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988.
[6] D.J. Higham, Mean Square and Asymptotic Stability of Numerical Methods for Stochastic Differential Equations, University of Strathclyde Mathematics Research Report 39 (1998).
[7] N. Hoffman, Stability of Weak Numerical Schemes for Stochastic Differential Equations, Computers Math. Applic.,28(1994), 45-75.
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Appl. Num. Anal. Comp. Math. 1, No. 2 (2004) / www.anacm.org
385
[8] P. Kloeden and E. Platen, Numerical Solutions for Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
[9] Y. Saito and T. Mitsui, Stability Analysis of Numerical Schemes for Stochastic Differential Equations, SIAM J. Numer.
Anal., 33(1996), 2254-2267.
[10] Y. Saito and T. Mitsui, Mean Square Stability of Numerical Schemes for Stochastic Differential Systems, Preprint
c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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