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

код для вставкиСкачатьAppl. 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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