Rounding Error in Numerical Solution of Stochastic Differential Equations

Transcription

1 STOCHASTIC ANALYSIS AND APPLICATIONS Vol. 21, No. 2, pp , 2003 Rounding Error in Numerical Solution of Stochastic Differential 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 EjXðt n Þ 2 ^Y n j 2 and E½Fð ^Y n Þ 2 FðXðt n ÞÞŠ is investigated, where X(t n ) is the exact solution at the nth time step and Ŷ n 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 computational results are given. *Correspondence: Armando Arciniega, Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas , USA; aarcinie@ math.ttu.edu. 281 DOI: /SAP Copyright q 2003 by Marcel Dekker, Inc (Print); (Online)

2 282 Arciniega and Allen 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 8 >< >: dx dt 22 1:07 2ð1:07Þ ¼ xðtþ 2 2 tþ0:07 2: xð0þ ¼e 21 þ 1:07 0:07 ðtþ0:07þ 3 ; 0 # t # 1 ð1:1þ The exact solution to this problem is xðtþ ¼e t21 1:07 2 þ : t þ 0:07 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

3 Rounding Error in Numerical Solution 283 Figure 1. Illustration of the error reduction possible by extrapolating Euler s method. 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 Itô process X ¼ {XðtÞ :0# t # T} satisfying the stochastic differential equation ( dxðtþ ¼f ðt; XðtÞÞdt þ gðt; XðtÞÞdWðtÞ; Xð0Þ ¼X 0 ; 0 # t # T ð2:1þ

4 284 Arciniega and Allen where X(t) satisfies the equivalent Itô stochastic integral equation XðtÞ ¼X 0 þ Z t f ðs; XðsÞÞds þ Z t 0 0 gðs; XðsÞÞdWðsÞ; 0 # t # T: ð2:2þ Select a positive integer N $ 2 and partition the interval [0,T ] into 0 ¼ t 0, t 1,, t N ¼ T; where t n 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 T N : An Euler approximation to (2.1) is a stochastic process satisfying the iterative scheme ( Y 0 ¼ X 0 ; Y n ¼ Y n21 þ f ð ;Y n21 Þðt n 2 Þþgð ;Y n21 ÞðWðt n Þ 2 Wð ÞÞ ð2:3þ for each n ¼ 1;...;N; where Y n denotes the approximation to the exact solution at the nth time step. That is Y n < Xðt n Þ: Denote the random increments of the Wiener process W ¼ {WðtÞ : t $ 0} by DW n ¼ Wðt n Þ 2 Wð Þ: It is well known that these increments are independent normal random variables with mean zero and variance t n 2 ; for each n ¼ 1;...;N (see, for example, Ref. [2] ). If h ¼ t n 2 ; equation (2.3) takes the form ( Y 0 ¼ X 0 ; Y n ¼ Y n21 þ hf ð ;Y n21 Þþgð ;Y n21 ÞðDW n Þ 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 Hölder continuous of order 1 2 in the first variable. Specifically, there exists a constant K. 0 such

5 Rounding Error in Numerical Solution 285 that for all s; t [ ½0; TŠ; x; y [ R; j f ðt; xþ 2 f ðt; yþj þ jgðt; xþ 2 gðt; yþj # Kjx 2 yj ð2:5þ j f ðt; xþj 2 þjgðt; xþj 2 # K 2 ð1 þjxj 2 Þ ð2:6þ j f ðs; xþ 2 f ðt; xþj þ jgðs; xþ 2 gðt; xþj # Kjs 2 tj 1 2 : ð2:7þ Then, there exists a positive constant C 1 ¼ C 1 ðtþ such that EjXðt n Þ 2 Y n j 2 # C 1 h: 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 ( Y 0 ¼ X 0 ; ð2:8þ Y n ¼ Y n21 þ hf ð ; Y n21 Þþgð ; Y n21 ÞðDW n Þ for each n ¼ 1;...; N; is not used to calculate the approximation to the solution X(t n ) at the point t n. Instead, the following equation is used 8 < : ^Y 0 ¼ Y 0 þ e~ 0 ; ^Y n ¼ ^Y n21 þ hf ð ; ^Y n21 Þþgð ; ^Y n21 ÞðDW n Þþ e~ n ð2:9þ for each n ¼ 1;...; N; where e n 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 e n are normally distributed with mean zero and that e n is independent of e m for n m: In particular, it is assumed that Eð e~ n Þ¼0and Eð e~ 2 n Þ # ~Cd 2 for each n, where d is proportional to the unit roundoff error. That is, d / b 2t ; 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 Ŷ n in

6 286 (2.9) can be written in the form ( ^Y 0 ¼ Y 0 þ e 0 ; ^Y n ¼ Y n þ e n Arciniega and Allen ð2:10þ where e n is the accumulated error due to rounding. To see this, first notice that ^Y 0 ¼ Y 0 þ e 0 ¼ Y 0 þ e~ 0 : Therefore, where ^Y 1 ¼ ^Y 0 þ hf ðt 0 ; ^Y 0 Þþgðt 0 ; ^Y 0 ÞðDW 0 Þþ e~ 1 ¼ Y 0 þ e~ 0 þ hf ðt 0 ; Y 0 þ e~ 0 Þþgðt 0 ; Y 0 þ e~ 0 ÞðDW 0 Þþ e~ 1 ¼ Y 0 þ hf ðt 0 ; Y 0 Þþgðt 0 ; Y 0 ÞðDW 0 Þþe^ 1 þ e~ 1 ¼ Y 1 þ e^ 1 þ e~ 1 ¼ Y 1 þ e 1 ; Y 1 ¼ Y 0 þ hf ðt 0 ; Y 0 Þþgðt 0 ; Y 0 ÞðDW 0 Þ; e 1 ¼ e^ 1 þ e~ 1 ; and e^ 1 ¼ e~ 0 þ h½ f ðt 0 ; Y 0 þ e~ 0 Þ 2 f ðt 0 ; Y 0 ÞŠ þðdw 0 Þ½gðt 0 ; Y 0 þ e~ 0 Þ 2 gðt 0 ; Y 0 ÞŠ: Considering eˆ1, it follows that e^ 2 1 ¼ e~ 0 2 þ h 2 ½ f ðt 0 ; Y 0 þ e~ 0 Þ 2 f ðt 0 ; Y 0 ÞŠ 2 þðdw 0 Þ 2 ½gðt 0 ; Y 0 þ e~ 0 Þ 2 gðt 0 ; Y 0 ÞŠ 2 þ 2 e~ 0 h½ f ðt 0 ; Y 0 þ e~ 0 Þ 2 f ðt 0 ; Y 0 ÞŠ þ 2 e~ 0 ðdw 0 Þ½gðt 0 ; Y 0 þ e~ 0 Þ 2 gðt 0 ; Y 0 ÞŠ þ 2hðDW 0 Þ½ f ðt 0 ; Y 0 þ e~ 0 Þ 2 f ðt 0 ; Y 0 ÞŠ½gðt 0 ; Y 0 þ e~ 0 Þ 2 gðt 0 ; Y 0 ÞŠ: However, the random increments DW n are normally distributed with mean zero and variance h, i.e., DW n [ Nð0; hþ for each n. Therefore,

7 Rounding Error in Numerical Solution 287 from the preceding equation, Eðe^ 2 1 Þ¼Eð e2 ~ 0 Þþh 2 E{½ f ðt 0 ; Y 0 þ e~ 0 Þ 2 f ðt 0 ; Y 0 ÞŠ 2 } þ he{½gðt 0 ; Y 0 þ e~ 0 Þ 2 gðt 0 ; Y 0 ÞŠ 2 } þ E{2 e~ 0 h½ f ðt 0 ; Y 0 þ e~ 0 Þ 2 f ðt 0 ; Y 0 ÞŠ}: 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 Eðe^ 2 1 Þ # Eð e2 ~ 0 ÞþK 2 ½h 2 Eð e~ 2 0 ÞþhEð e2 ~ 0ÞŠ þ 2hKEð e2 ~ 0 Þ # ~Cd 2 ½1 þ 2hK þ hðh þ 1ÞK 2 Š: For notational convenience, denote r 1 h by Then, r 1 h ¼½2K þðt þ 1ÞK 2 Šh: Eðe^ 2 1 Þ # ~Cd 2 ð1 þ r 1 hþ: Since Eð e~ 0 Þ¼0 and Eð e~ 2 1 Þ # ~Cd 2 ; then Eðe 2 1 Þ¼E½ð e~ 1 þ e^ 1 Þ 2 Š¼Eð e~ 2 1 ÞþEðe2 ^ 1 Þ # ~Cd 2 þ ~Cd 2 ð1 þ r 1 hþ ¼ ~Cd 2 ½1 þð1 þ r 1 hþš: Similarly, the second step leads to and ^Y 2 ¼ Y 2 þ e 2 Eðe^ 2 2 Þ # ð1 þ r 1hÞEðe 2 1 Þ # ð1 þ r 1hÞ½Eð e~ 2 1 ÞþEðe^2 1 ÞŠ # ð1 þ r 1 hþ½ ~Cd 2 þ ~Cd 2 ð1 þ r 1 hþš ¼ ~Cd 2 ½ð1 þ r 1 hþþð1 þ r 1 hþ 2 Š:

8 288 Arciniega and Allen Hence, Eðe 2 2 Þ¼E½ð e~ 2 þ e^ 2 Þ 2 Š¼Eð e~ 2 2 ÞþEðe^2 2 Þ # ~Cd 2 þ ~Cd 2 ½ð1 þ r 1 hþþð1 þ r 1 hþ 2 Š ¼ ~Cd 2 ½1 þð1 þ r 1 hþþð1 þ r 1 hþ 2 Š: Continuing in this manner, the nth step leads to and Thus, ^Y n ¼ Y n þ e n Eðe^ 2 n Þ # ð1 þ r 1hÞEðe 2 n21 Þ # ð1 þ r 1 hþ½eð e~ 2 n21 ÞþEðe^ 2 n21 ÞŠ # ð1 þ r 1 hþ½ ~Cd 2 þ Eðe^ 2 n21 ÞŠ # ~Cd 2 ½ð1 þ r 1 hþþð1 þ r 1 hþ 2 þ þð1 þ r 1 hþ n Š: Eðe 2 n Þ¼E½ð e~ n þ e^ n Þ 2 Š Ce where C ¼ ~ r 1 T r 1 result: ¼ Eð e~ 2 n ÞþEðe^ 2 n Þ # ~Cd 2 ½1 þð1 þ r 1 hþþð1 þ r 1 hþ 2 þ þð1 þ r 1 hþ n Š # ~Cd 2 ð1 þ r 1hÞ n r 1 h # ~ Cd 2 r 1 h e r 1T ¼ C d 2 h ; and r 1 ¼½2K þðt þ 1ÞK 2 Š: This proves the following Theorem 2.2. Let ^Y 0 ;...; ^Y N be the approximations obtained using (2.9) or (2.10).Ife n are independently distributed random variables with Eð e~ n Þ¼0and Eð e~ 2 n Þ # ~Cd 2 ; then the accumulated error e n for any n satisfies Eðe 2 n Þ # C d 2 h for some positive constant C, where d is proportional to the unit roundoff error.

9 Rounding Error in Numerical Solution 289 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 ^Y 0 ;...; ^Y N be the approximations obtained using (2.10). Let Eð e~ n Þ¼0; Eð e~ 2 n Þ, ~Cd 2 ; and suppose the hypotheses of Theorem 2.1 are satisfied. Then, there exist positive constants C 1,C 2, and C 3 such that EjXðt n Þ 2 ^Y n j 2 # C 1 h þ C 2 þ C 3 d 2 : h Proof. First, notice that conditions (2.5) (2.7) guarantee a unique solution X(t) to (2.1). Denote g n by g n ¼ EjXðt n Þ 2 ^Y n j 2 : Then, g 0 ¼ EjXðt 0 Þ 2 ^Y 0 j 2 ¼ EjX 0 þ e~ 0 2 X 0 j 2 ¼ Ej e~ 0 j 2 # ~Cd 2 : Next, define Ŷ(t) by ^YðtÞ ¼ ^Y n21 þ e~ n þ Notice that Z t f ð ; ^Y n21 Þds Z t þ gð ; ^Y n21 ÞdWðsÞ: ð2:11þ Now, ^Yð Þ¼ ^Y n21 þ e~ n : XðtÞ 2 ^YðtÞ ¼Xð Þ 2 ^Y n21 2 e~ n Z t þ ½ f ðs; XðsÞÞ 2 f ð ; ^Y n21 ÞŠds Z t þ ½gðs; XðsÞÞ 2 gð ; ^Y n21 ÞŠdWðsÞ:

10 290 Arciniega and Allen and dðxðtþ 2 ^YðtÞÞ ¼ ½ f ðt; XðtÞÞ 2 f ð ; ^Y n21 ÞŠdt þ½gðt; XðtÞÞ 2 gð ; ^Y n21 ÞŠdWðtÞ: Applying Itô s formula to the preceding equation, one obtains dððxðtþ 2 ^YðtÞÞ 2 Þ ¼ {2ðXðtÞ 2 ^YðtÞÞðf ðt; XðtÞÞ 2 f ð ; ^Y n21 ÞÞ þðgðt; XðtÞÞ 2 gð ; ^Y n21 ÞÞ 2 }dt þ 2ðXðtÞ 2 ^YðtÞÞðgðt; XðtÞÞ 2 gð ; ^Y n21 ÞÞdWðtÞ: Integrating the preceding equation from to t n and taking expectations where, as before, g n ¼ EjXðt n Þ 2 ^Y n j 2 ; and noticing that leads to EjXð Þ 2 ^Yð Þj 2 ¼ g n21 þ Eð e~ 2 n Þ Z tn g n ¼ g n21 þ Eð e~ 2 n Þþ E{2ðXðsÞ 2 ^YðsÞÞðf ðs; XðsÞÞ 2 f ð ; ^Y n21 ÞÞ þðgðs; XðsÞÞ 2 gð ; ^Y n21 ÞÞ 2 }ds # g n21 þ ~Cd 2 Z tn þ {EjXðsÞ 2 ^YðsÞj 2 þ Ej f ðs; XðsÞÞ 2 f ð ; ^Y n21 Þj 2 þ Ejgðs; XðsÞÞ 2 gð ; ^Y n21 Þj 2 }ds ð2:12þ as Eð e~ 2 n Þ # ~Cd 2 for each n. Invoking the Lipschitz (2.5) and Hölder (2.7)

11 Rounding Error in Numerical Solution 291 conditions, one has j f ðs; XðsÞÞ 2 f ð ; ^Y n21 Þj 2 #3{j f ðs; XðsÞÞ 2 f ðs; Xð ÞÞj 2 þjfðs; Xð ÞÞ 2 f ð ; Xð ÞÞj 2 þjfð ; Xð ÞÞ 2 f ð ; ^Y n21 Þj 2 } þðs2 ÞþjXð Þ 2 ^Y n21 j 2 } n # 3K 2 jkðsþÿxðt nÿ1 Þj 2 þðsÿt nÿ1 Þ þjxðt nÿ1 Þÿ ^Y n j 2o ð2:13þ Similarly, jgðs; XðsÞÞ 2 gð ; ^Y n21 Þj 2 # 3K 2 {jxðsþ 2 Xð Þj 2 Now, consider þðs 2 ÞþjXð Þ 2 ^Y n21 j 2 }: ð2:14þ EjXðsÞ 2 Xð Þj 2 Z s Z s 2 ¼ E f ðt; XðtÞÞdt þ gðt; XðtÞÞdWðtÞ " Z s 2 Z s # 2 # 2 E f ðt; XðtÞÞdt þe gðt; XðtÞÞdWðtÞ # 2{ðs 2 Þ Z s Ej f ðt; XðtÞÞj 2 dt þ Z s Ejgðt; XðtÞÞj 2 dt} Z s # 2{½ðs 2 Þþ1Š K 2 Eð1 þjxðtþj 2 Þdt}: However, from Theorem 3.8 of Ref., [3] EjXðtÞj 2 # ð1 þ EjX 0 j 2 Þe Lt for some constant L. Therefore, EjXðsÞ 2 Xð Þj 2 # K 1 ðs 2 Þ; ð2:15þ

12 292 Arciniega and Allen where K 1 depends only on K, T, and X 0. Substituting (2.13), (2.14), and (2.15) into (2.12) leads to Z tn g n # g n21 þ ~Cd 2 þ {EjXðsÞ 2 ^YðsÞj 2 þ 6K 2 ½ðK 1 þ 1Þðs 2 Þþg n21 Š}ds # g n21 ð1 þ 6K 2 hþþ3k 2 ðk 1 þ 1Þh 2 Z tn þ EjXðsÞ 2 ^YðsÞj 2 ds þ ~Cd 2 : ð2:16þ Applying the Bellman Gronwall inequality: if a(t) and b(t) are measurable bounded functions such that for some ~L. 0; then aðtþ # bðtþþ~l aðtþ # bðtþþ~l to (2.16) with and Z t 0 Z t 0 aðsþds; aðtþ ¼EjXðtÞ 2 ^YðtÞj 2 e ~ Lðt2sÞ bðsþds bðtþ ¼ðEjXð Þ 2 ^Y n21 j 2 Þð1 þ 6K 2 hþþ3k 2 ðk 1 þ 1Þh 2 þ ~Cd 2 on ½ ; t n Š leads to g n # g n21 ð1 þ 6K 2 hþ Z tn þ e tn2s ½g n21 ð1 þ 6K 2 hþþ3k 2 ðk 1 þ 1Þh 2 þ ~Cd 2 Šds þ 3K 2 ðk 1 þ 1Þh 2 þ ~Cd 2 # ½g n21 ð1 þ 6K 2 hþþ3k 2 ðk 1 þ 1Þh 2 þ ~Cd 2 Še h : ð2:17þ

13 Rounding Error in Numerical Solution 293 Iterating (2.17) with g 0 ¼ Ej e~ 0 j 2 leads to g n # r n 2 Ej e~ 0j 2 þð3k 2 ðk 1 þ 1Þh 2 þ ~Cd 2 Þe h 1 2 r n 2 ; ð2:18þ 1 2 r 2 where r 2 ¼ð1þ6K 2 hþe h : Now, Ej e~ 0 j 2 # ~Cd 2 implies that r n 2 Ej e~ 0j 2 # C 2 d 2 ; where C 2 ¼ ~Ce Tð1þ6K 2Þ : Therefore, g n # C 2 d 2 þ h½3k 2 he h ðk 1 þ 1ÞŠ r rn 2 þ ~Cd 2 e h r rn 2 : However, ð2:19þ r n 2 # rt h 2 ¼½ð1 þ 6K 2 hþe h Š T h # e Tð1þ6K 2Þ ; and Also, he h r ¼ h 1 þ 6K 2 h 2 e 2h # 1 6K 2 : ~Cd 2 e h r rn 2 ¼ ~ Cd 2 he h h r rn 2 : Substituting all this into (2.19) leads to g n # C 1 h þ C 2 þ C 3 d 2 ; h ð2:20þ where C 1 ¼ ðk 1þ1Þ 2 e Tð1þ6K 2Þ C and C 3 ¼ ~ 6K e Tð1þ6K 2Þ : This completes the proof 2 of the theorem. A 2.3. Rounding Error for Functional Expectation (Weak Convergence) Let Ŷ n and Y n 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 E½FðY n Þ 2 FðXðt n ÞÞŠ ¼ c 1 h þ Oðh 2 Þ; where X(t n ) denotes the exact solution at the nth step and c 1 is a constant independent of h (see Ref. [9] ). This expansion justifies the Richardson

14 294 Arciniega and Allen 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 Since E½Fð ^Y n Þ 2 FðXðt n ÞÞŠ ¼ E½Fð ^Y n Þ 2 FðY n ÞþFðY n Þ 2 FðXðt n ÞÞŠ ^Y n ¼ Y n þ e n ; ¼ E½F 0 ðy n Þð ^Y n 2 Y n ÞŠ þ c 1 h þ Oðh 2 Þ it follows from Theorem 2.2 that Eðe 2 n Þ 1=2¼ ðeð ^Y n 2 Y n Þ 2 Þ 1=2 # Cd 2 1=2 : h Substituting this into (2.21) leads to where # ðeðf 0 ðy n ÞÞÞ 1=2 ðeð ^Y n 2 Y n Þ 2 Þ 1=2 þ c 1 h þ Oðh 2 Þ: ð2:21þ E½Fð ^Y n Þ 2 FðXðt n ÞÞŠ # M Cd 2 1=2 þc 1 h þ Oðh 2 Þ; ð2:22þ h M ¼ maxðf 0 ðyþþ 1=2 : Y[R This proves the following result: Theorem 2.4. Let Ŷ n and Y n 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 E½FðY n Þ 2 FðXðt n ÞÞŠ ¼ c 1 h þ Oðh 2 Þ; where X(t n ) denotes the exact solution at the nth step and c 1 is a constant independent of h. Then, E½Fð ^Y n Þ 2 FðXðt n ÞÞŠ # M Cd 2 1=2 þc 1 h þ Oðh 2 Þ h for some positive constants C and M.

15 Rounding Error in Numerical Solution 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 E½Fð ^Y n Þ 2 FðXðt n ÞÞŠ ¼ M Cd 2 1=2 þc 1 h þ Oðh 2 Þ: h Next, Euler s method is used with 2n time steps of equal length h/2 so that E½Fð ^Y h=2 2n Þ 2 FðXðt nþþš ¼ M 2Cd 2 1=2 h þc 1 h 2 þ Oðh 2 Þ: A combination of the two preceding equations yields E½2Fð ^Y h=2 2n Þ 2 Fð ^Y n Þ 2 FðXðt n ÞÞŠ ¼ ~M Cd 2 1=2 þoðh 2 Þ; h p where ~M ¼ Mð2 ffiffi 2 2 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 8 22 dxðtþ¼ XðtÞ 2 1:07 2ð1:07Þ 2 p tþ0:07 dt þ ffiffi 2 >< ðtþ0:07þ 3 10 dwðtþ; 0 # t # 1 ð4:1þ >: 2: Xð0Þ¼e 21 þ 1:07 0:07 It is desired to estimate EFðXð1ÞÞ ¼ EXð1Þ: The expectation of the solution to

16 296 Arciniega and Allen this problem is EXðtÞ¼e t21 1:07 2 þ t þ 0:07 and is obtained by application of Itô s formula. Therefore, EXð1Þ¼2: 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 8 dx 1 ðtþ ¼2dW 1 ðtþþx 2 2 ðtþdw 2ðtÞ; 0 # t # 1 >< dx 2 ðtþ ¼ 1 2 X 2ðtÞdW 1 ðtþ ð4:2þ X 1 ð0þ ¼0 >: X 2 ð0þ ¼1 Table 1. Error reduction possible by extrapolating Euler s method. Number of intervals in t Euler s method Extrapolation

17 Rounding Error in Numerical Solution 297 Figure 2. Illustration of the error reduction possible by extrapolating Euler s method. It is desired to estimate EFðX 1 ð1þþ ¼ EX 2 1ð1Þ: By application of Itô s s formula, it can be shown that Hence, EX 2 1 ðtþ ¼2 3 ðe 3t=2 2 1Þþt: EX 2 1 ð1þ ¼1 3 ð2e 3=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 sufficiently 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.

18 298 Arciniega and Allen Table 2. Error reduction possible by applying Richardson extrapolation Number of intervals in t Euler s method Extrapolated Euler method 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 Figure 3. Illustration of the errors in Euler and second-order methods.

19 Rounding Error in Numerical Solution 299 the following stochastic initial value problem ( dxðtþ ¼ð1 þ XðtÞÞdt þð1þxðtþþdwðtþ; 0 # t # 1 Xð0Þ ¼1: The expectation of the solution to this problem is EXðtÞ ¼2e t 2 1: Thus, EXð1Þ ¼2e 2 1 < 4:4366: ð4:3þ 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 , ARP , and the National Science Foundation Grant NSF REFERENCES 1. Abukhaled, M.I.; Allen, E.J. A class of second-order Runge Kutta methods for numerical solution of stochastic differential equations. Stoch. Anal. Appl. 1998, 16,

20 300 Arciniega and Allen 2. Arnold, L. Stochastic Differential Equations: Theory and Applications; John Wiley & Sons: New York, Gard, T.C. Introduction to Stochastic Differential Equations; Marcel Dekker: New York, Henrici, P. Elements of Numerical Analysis; John Wiley & Sons: New York, Kloeden, P.E.; Platen, E. Numerical Solution of Stochastic Differential Equations; Springer-Verlag: New York, Marchuk, G.I.; Shaidurov, V.V. Difference Methods and Their Extrapolations; Springer-Verlag: New York, Stoer, J.; Bulirsch, R. Introduction to Numerical Analysis; Springer- Verlag: New York, Talay, D. Simulation and numerical analysis of stochastic differential systems: a review. In Probabilistic Methods in Applied Physics; Kree, P., Wedig, W., Eds.; Lecture Notes in Physics; Springer-Verlag: New York, 1995; Vol. 451, Talay, D.; Turbano, L. Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 1990, 8 (4),

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