Shooting Methods for Numerical Solution of Stochastic Boundary-Value Problems

STOCHASTIC ANALYSIS AND APPLICATIONS Vol. 22, No. 5, pp. 1295 1314, 24 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 7949-142, USA; Fax: (6) 742-1112; E-mail: eallen@math.ttu.edu. 1295 DOI: 1.11/SAP-226465 Copyright # 24 by Marcel Dekker, Inc. 736-2994 (Print); 1532-9356 (Online) www.dekker.com

1296 Arciniega and Allen Mathematics Subject Classification: 65C3; 6H1. 1. INTRODUCTION Methods for numerically solving stochastic initial-value problems have been under much study (see, for example, Refs. [3 5,,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: >< d~uðtþ ¼ðA~uðtÞþ~aðtÞÞdt þ Xk ðb i ~uðtþþ~b i ðtþþ dw i ðtþ; t 1 F ~uðþþf 1 ~uð1þ ¼ ~ ; ð1:1þ where ~u, ~a, ~ b i 2 R n and A, F, F 1, B i 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 F F ¼ b and F 1 ¼ bf 1 where bf is a l n matrix of rank l and bf 1 is a ðn lþn matrix of rank n l. Also, < l < n. Ocone and Pardoux [6] and Zeitouni and Dembo [1] have established existence and uniqueness of solutions to Eq. (1.1). As defined in Ref. [1], ~u 2 R n is a solution to Eq. (1.1) if ~u is Stratonovich integrable and satisfies for all t 2½; 1Š: Z t Z t >< X k ~uðtþ ~uðþ¼ ða~uðsþþ~aðsþþds þ ðb i ~uðsþþ~b i ðsþþ dw i ðsþ; F ~uðþþf 1 ~uð1þ¼ ~ : 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,

Shooting Methods for Numerical Solution 1297 Ocone and Pardoux [6] and Zeitouni and Dembo [1] 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~u m ðtþ ¼ðA~u m ðtþþ~aðtþþdt þ Xk ðb i ~u m ðtþþ~b i ðtþþ dw i ðtþ; ð1:3þ for m ¼ 1; 2;...; n l þ 1, where ~u 1 ðþ ¼ ~, and ~u m ðþ for m ¼ 2; 3;...; n l þ 1 are chosen to be n l linearly independent vectors in the null space of F. 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 ~u m ðtþ obtained using (1.3) can be combined to form a solution to Eq. (1.1). To see this, let ~uðtþ ¼ X l m ~u m ðtþ; ð1:4þ where the l m,1 m n l þ 1, satisfy X l m ¼ 1 ð1:5þ and the linear system of rank ðn lþ: X l m ðf ~u m ðþþf 1 ~u m ð1þþ ¼ X l m F 1 ~u m ð1þ ¼ ~ : ð1:6þ In the next section, it is shown that the solution~uðtþ, of (1.1) can be written as given by (1.4) where the ~u m 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 l m can be calculated uniquely and the solution ~uðtþ can be approximated by combining the approximate

129 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 X l m d~u m ðtþ ¼ X l m ða~u m ðtþþ~aðtþþdt þ Xk ¼ðA~uðtÞþ~aðtÞÞdt þ Xk where (1.5) has been applied, that is, ðb i ~u m ðtþþ~b i ðtþþ dw i ðtþ ðb i ~uðtþþ ~ b i ðtþþ dw i ðtþ;! X l m ¼ 1: Also, the boundary conditions are satisfied as: F ~uðþþf 1 ~uð1þ ¼ X l m ðf ~u m ðþþf 1 ~u m ð1þþ ¼ X l m F 1 ~u m ð1þ ¼ ~ ; as ~u 1 ðþ ¼ ~ and~u m ðþ are linearly independent vectors in the null space of F for m ¼ 2; 3;...; n l þ 1, and P l m F 1 ~u m ð1þ ¼ ~. The next theorem shows that the l m,1m, 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 ~u m ðtþ of the initial-value problems (1.3).

Shooting Methods for Numerical Solution 1299 Theorem 2.1. The l m, 1 m n l þ 1 uniquely exist and satisfy Eqs. (1.5) and (1.6). Proof. For the original stochastic boundary-value problem: d~uðtþ ¼ðA~uðtÞþ~aðtÞÞdt >< þ Xk ðb i ~uðtþþ~b i ðtþþ dw i ðtþ; t 1 F ~uðþþf 1 ~uð1þ ¼ ~ ; ð2:1þ the solution ~uðtþ uniquely exists, which is proved in Refs. [6,1]. Consider the corresponding stochastic initial-value problem: d~u 1 ðtþ ¼ðA~u 1 ðtþþ~aðtþþdt >< þ Xk ðb i ~u 1 ðtþþ ~ b i ðtþþ dw i ðtþ; t 1 ð2:2þ ~u 1 ðþ ¼ ~ ; It is well-known that ~u 1 ðtþ is uniquely determined for this problem (see, for example, Refs. [2 4] ). Subtracting Eqs. (2.1) and (2.2) yields: >< d~wðtþ ¼A~wðtÞdt þ Xk ðb i ~wðtþþ dw i ðtþ; t 1 ð2:3þ F ~wðþþf 1 ~wð1þ ¼ F 1 ~u 1 ð1þ; where ~wðtþ ¼~uðtÞ ~u 1 ðtþ. As~uðtÞ and ~u 1 ðtþ uniquely exist, then ~wðtþ also uniquely exists. In particular, ~wðþ ¼ ~uðþ exists. Consider, next F F ~wðþ ¼ b " F ~wðþ ¼ b # ~wðþ ~ and F 1 ~wð1þ ¼ bf 1 ~wð1þ ¼ ~ : bf 1 ~wð1þ Then, F ~wðþþf 1 ~wð1þ ¼ " F b # ~wðþ ~ ¼ : bf 1 ~wð1þ bf 1 ~u 1 ð1þ Thus F ~wðþ ¼ ~ and F 1 ~wð1þ ¼ F 1 ~u 1 ð1þ. Hence ~wðþ is in the null space of F. However, the null space of F and the range of F 1 are identical,

13 Arciniega and Allen i.e., NullðF Þ¼RangeðF 1 Þ. Furthermore, dimðrangeðf 1 ÞÞ ¼ n l ¼ dimðnullðf ÞÞ. Therefore, one can let ~wðþ ¼ X l m ~v m ðþ; m¼2 where ~v m ðþ ¼~u m ðþ are linearly independent vectors in the range of F 1 and fl m g m¼2 are to be determined. However, as ~wðþ 2NullðF Þ, fl m g m¼2 uniquely exist. Consider now the stochastic initial-value problem: d~wðtþ ¼A~wðtÞdt þ Xk >< ðb i ~wðtþþ dw i ðtþ; t 1 ~wðþ ¼ X ð2:4þ l m ~v m ðþ: 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 ~v m ðtþ ¼~u m ðtþ ~u 1 ðtþ that solves the stochastic initial-value problem: >< d~v m ðtþ ¼A~v m ðtþdt þ Xk ðb i ~v m ðtþþ dw i ðtþ; t 1 ð2:5þ F ~v m ðþ ¼ ~ ; for m ¼ 2; 3;...; n l þ 1. Then, it is clear that ~wðtþ ¼ X l m ~v m ðtþ: m¼2 Thus, as F 1 ~wð1þ ¼ F 1 ~u 1 ð1þ, then l 2 ; l 3 ;...; l satisfy F 1 l m ~v m ð1þ ¼ F 1 ~u 1 ð1þ. Finally, notice that P m¼2 ~uðtþ ¼~wðtÞþ~u 1 ðtþ: Therefore, ~uðtþ ¼~u 1 ðtþþ X l m ~v m ðtþ ¼~u 1 ðtþþ X l m ð~u m ðtþ ~u 1 ðtþþ: m¼2 Thus, ~uðtþ ¼ X l m ~u m ðtþ; m¼2

Shooting Methods for Numerical Solution 131 where X F 1 l m ð~u m ð1þ ~u 1 ð1þþ¼ F 1 ~u 1 ð1þ; m¼2 X l m ¼ 1; and F ~v m ðþ ¼ ~ for each m ¼ 1; 2;...; n 1 þ l: Thus, l m, m 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] ): d~u m ðtþ ¼ ða~u m ðtþþ~aðtþþ þ 1 X k ðb 2 j 2 ~u m ðtþþb j ~ bj ðtþþ dt j¼1 þ Xk ðb i ~u m ðtþþ~b i ðtþþdw i ðtþ: ð3:1þ 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 ½; 1Š into ¼ t < t 1 < < t N ¼ 1; where t p ¼ ph for each p ¼ ; 1;...; N. It is assumed that the step size h is fixed, so that the common distance between the discrete times is h ¼ 1 N. For example, the Euler approximations to system (3.1) are stochastic

132 Arciniega and Allen processes satisfying the iterative scheme (see Refs. [3,4] ): ~u m;pþ1 ¼ ~u m;p þ h ða~u m;p þ ~aðt p ÞÞ þ 1 X k ðb 2 j 2 ~u m;p þ B j ~ bj ðt p ÞÞ j¼1 þ Xk pffiffiffi ðb i ~u m;p þ~b i ðt p ÞÞ h Zi ; ð3:2þ for each m ¼ 1; 2;...; n l þ 1. In the above scheme, ~u m;p denotes the approximation to the exact solution at the pth time step. That is ~u m;p ~u m ðt p Þ, for each p ¼ ; 1;...; N 1, m ¼ 1; 2;...; n l þ 1. Also, the random increments Z i are independent normal random variables with mean zero and variance unity, i.e., Z i 2 Nð; 1Þ (see Ref. [4] ). After (3.2) is solved numerically for each m ¼ 1; 2;...; n l þ 1, the approximate ^l m, for m ¼ 1; 2;...; n l þ 1, are calculated using X ^l m ðf ~u m;n þ F 1 ~u m;n Þ¼ X ^l m F 1 ~u m;n ¼ ~ with X ^l m ¼ 1 corresponding to (1.5) and (1.6). As a result, ~uðt p Þ X ^l m ~u m;p 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,] ). To be consistent with existing literature, the following notation is used in the present investigation. In particular, denote and ~fðt;~u m ðtþþ ¼ ða~u m ðtþþ~aðtþþ þ 1 2 Gðt;~u m ðtþþ ¼ Xk X k ðb i ~u m ðtþþ~b i ðtþþ: j¼1 ðb 2 j ~u m ðtþþb j ~b j ðtþþ

Shooting Methods for Numerical Solution 133 Then, the system d~u m ðtþ ¼~fðt;~u m ðtþþdt þ Gðt;~u m ðtþþd~wðtþ is equivalent to (3.1). In the above system, ~ f ¼ffi g is an n-vector-valued function, G ¼fg i;j g is an n k-matrix-valued function, ~W ¼fW i g is a k-dimensional Wiener process, and the solution ~u m is an n-dimensional process. The above system can be expressed as d~u m ðtþ ¼~fðt;~u m ðtþþ dt þ Xk ~g i ðt;~u m ðtþþ dw i ðtþ; where the ~g i are the columns of the matrix G and the W i are the independent scalar Wiener processes forming the components of ~W. Theorem 3.1. Consider the system of Ito stochastic differential equations, d~u m ðtþ ¼ ~ fðt;~u m ðtþþdt þ Gðt;~u m ðtþþd~wðtþ for m ¼ 1; 2;...; n l þ 1, where ~f 2 R n, G 2 R nk, and ~W 2 R k. Suppose ~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 m > for each m ¼ 1;...; n l þ 1 such that for all s; t 2½; 1Š, ~u m ;~v m 2 R n, jj~fðt;~u m ðtþþ ~fðt;~v m ðtþþjj þ jjgðt;~u m ðtþþ Gðt;~v m ðtþþjj Kjj~u m ~v m jj ð3:3þ jj ~ fðt;~u m ðtþþjj 2 þjjgðt;~u m ðtþþjj 2 K 2 ð1 þjj~u m jj 2 Þ ð3:4þ jj~fðs;~u m ðtþþ ~fðt;~u m ðtþþjj þ jjgðs;~u m ðtþþ Gðt;~u m ðtþþjj Kjjs tjj 1 2 : ð3:5þ

134 Arciniega and Allen Then, there exists a positive constant C m such that Ejj~u m ðt p Þ ~u m;p jj 2 C m 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] ): ~u m;pþ1 ¼ ~u m;p þ h ða~u m;p þ ~aðt p ÞÞ þ 1 X k ðb 2 j 2 ~u m;p þ B j ~b j ðt p ÞÞ j¼1 þ Xk pffiffiffi ðb i ~u m;p þ~b i ðt p ÞÞ h Zi þ Xk j 1 ;j 2 I ðj1 ;j 2 Þ Bj2 B j1 ~u m;p þ B j2 ~b j1 ; ð3:6þ where >< I ðj1 ;j 2 Þ ¼ Z tnþ1 Z s1 t n I ðj1 ;j 1 Þ ¼ 1 ðdwj1 Þ 2 h : 2 t n dw j 1 s 2 dw j 2 s 1 ; j 1 6¼ j 2 ð3:7þ 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 I ðj1 ;j 2 Þ.If~b j ðtþ ¼ ~ for j ¼ 1; 2;...; k and B j2 B j1 ¼ B j1 B j2 for j 1 ; j 2 ¼ 1; 2;...; k, then Milstein s method becomes: ~u m;pþ1 ¼ ~u m;p þ h ða~u m;p þ ~aðt p ÞÞ þ Xk þ 1 2 X k j 1 X k B i ~u m;p p Zi j 2 ðdw j1 ÞðDW j2 ÞB j2 B j1 ~u m;p ; ð3:þ and the evaluation of I ðj1 ;j 2 Þ is not required. The following theorem states that Milstein s method has second-order strong convergence in the mean ffiffiffi h

Shooting Methods for Numerical Solution 135 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~u m ðt p Þ ~u m;p jj 2 C m h 2 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 X l m F 1 ~u m ð1þ ¼ ~ ; and, l 1 ¼ 1 X l m : m¼2 ð3:9þ ð3:1þ Substituting Eq. (3.1) into Eq. (3.9) and rearranging terms yields X m¼2 l m F 1 ð~u m ð1þ ~u 1 ð1þþ ¼ F 1 ~u 1 ð1þ: ð3:11þ But F 1 ¼ bf 1 where bf 1 is a ðn lþn matrix of rank n l. Thus, the equation in (3.11) is a linear system of the form A ~ l ¼ ~b; ð3:12þ where A is an ðn lþðn lþ matrix, ~ l and ~b are vectors of length n l. In particular, ~b ¼ bf 1 ~u 1 ð1þ and A ¼½~a 1 ;...;~a n l Š, where

136 Arciniega and Allen ~a i ¼ bf 1 ½~u iþ1 ð1þ ~u 1 ð1þš. Then, ~ l ¼ A 1 ~b: ð3:13þ In numerical solution of the initial-value problem (3.1) the ~u m ð1þ, for m ¼ 2;...; n l þ 1 are being approximated by ^~u~u m ð1þ. Therefore, the linear system to be solved is an approximation of (3.12). Call this linear system ba^~ l ~ l ¼ ^~b~b; ð3:14þ where ba¼½^~a~a 1 ;...; ^~a~a n l Š, ^~a~a i ¼ bf 1 ½ ^~u~u iþ1 ^~u~u 1 ð1þš, and ~ ^~ b¼ bf 1 ^~u~u 1 ð1þ. However, notice by Theorems 3.1 and 3.2, that ba and ~ ^~ b are perturbations to A and ~ b for small time step h. It is well-known (see, e.g., Ref. [7] ) that " #" jj ~ l ~ ^~ ljj jj ~ jjajjjja 1 jj jja bajj ljj þ jj ~b ^~b~bjj # : 1 jja 1 jjjja bajj jjajj jj~bjj Hence, " Ejj ~ l ~ ^~ jj ljj E ~ # 1 2 1=2 ljjjja 1 jj 1=2 @ A EjjA bajj 2 1 jja 1 jjjja bajj " jj þ E ~ # 1 2 1=2 ljjjjajjjja 1 jj @ jj ~ A Ejj~b ^~b~bjj 1=2: 2 bjjð1 jja 1 jjjja bajjþ Thus, the error Ejj ~ l ~ ^~ ljj is proportional to the error obtained in estimating ~u m 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~uðtþ the approximate solution obtained using the initial-value system (3.1). Then, Ejj~uðt p Þ ^~u~uðt p Þjj 1=2 X Ejl m j þ Ejj ~ l ^~ l ~ ljj 1=2! 1=2 X Ejj~u m ðt p Þ ~u m;p jj X Ejj~u m;p jj! 1=2 :! 1=2

Shooting Methods for Numerical Solution 137 Proof. Consider Then, ^~u~uðt p Þ¼ X ^l m ~u m;p ~uðt p Þ: ~uðt p Þ ^~u~uðt p Þ¼ X lm ~u m ðt p Þ ^l m ~u m;p ¼ X l m ð~u m ðt p Þ ~u m;p Þþ X ðl m ^l m Þ~u m;p : ð3:15þ As a result, Ejj~uðt p Þ ^~u~uðt p Þjj 1=2 E 2 E4 X þ E4 X 2 jl m jjj~u m ðt p Þ ~u m;p jj jl m j 2! 1=4 X X jl m ^l m j 2! 1=4 E X! 1 1=2 @ jl m j 2 A þ Ejj ~ l ~ ^~ 1=2 ljj @ E@ 1=2! 1=2 þ E X jl m ^l m jjj~u m;p jj! 3 1=4 jj~u m ðt p Þ ~u m;p jj 2 5 X @ E@ X! 3 1=4 jj~u m;p jj 2 5 X! 1 1=2 jj~u m;p jj 2 A! 1=2! 1 1=2 jj~u m ðt p Þ ~u m;p jj 2 A 1=2 1=2 X Ejl m j! 1=2 X Ejj~u m ðt p Þ ~u m;p jj! 1=2 þ Ejj ~ l ~ ^~ 1=2 1=2 X ljj Ejj~u m;p jj! ; ð3:16þ

13 Arciniega and Allen using the Cauchy-Schwarz inequality and the inequality: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux t N ja i j 2 XN ja i j: 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: j ðxþ ¼ð 1þjðxÞÞdx þ jðxþdwðxþ >< jðþ ¼ ð4:1þ jð1þ ¼: Letting y 1 ðxþ ¼jðxÞ and y 2 ðxþ ¼j ðxþ, Eq. (4.1) becomes: dy 1 ðxþ ¼y 2 ðxþdx >< dy 2 ðxþ ¼ð 1 þ y 1 ðxþþdx þ y 1 ðxþdwðxþ y 1 ðþ ¼ y 1 ð1þ ¼: ð4:2þ Now, letting u 1 ðtþ ¼y 1 ðxþ, u 2 ðtþ ¼y 2 ðxþ, with t ¼ x, Eq. (4.2) becomes: d~uðtþ ¼ðA~uðtÞþ~aðtÞÞdt >< ðb i ~uðtþþ~b i ðtþþ dw i ðtþ; t 1 þ X2 F ~uðþþf 1 ~uð1þ ¼ ~ ; ð4:3þ

Shooting Methods for Numerical Solution 139 where ~aðtþ ¼ ; A ¼ 1 1 1 F 1 ¼ 1 ; B 1 ¼ 1 ; ~b 1 ¼ ~ ; ~b 2 ¼ ~ ; and B 2 ¼ : Thus, problem (4.3) has the form ; F ¼ 1 " # " # " #" #! " #" # u 1 ðtþ 1 u1 ðtþ u1 ðtþ d ¼ þ dt þ dw 1 ðtþ >< u 2 ðtþ 1 1 u 2 ðtþ 1 u 2 ðtþ " #" # " #" # " # 1 u1 ðþ u1 ð1þ þ ¼ u 2 ðþ 1 u 2 ð1þ ð4:4þ To solve this problem, consider two different solutions to a corresponding stochastic initial-value problem. Consider ~u m ðtþ, m ¼ 1; 2 where ~u 1 ðþ ¼ and ~u 2 ðþ ¼ : 1 The ~u m ðtþ for m ¼ 1; 2 solve the stochastic initial-value problem: d~u m ðtþ ¼ þ 1 ~u m ðtþ dt þ >< ~u m ðtþdwðtþ 1 1 1 for m ¼ 1; 2 where ~u m ðþ is given above : ð4:5þ Notice that Eq. (4.5) is solved for both ~u 1 ðtþ and ~u 2 ðtþ using different initial conditions but the same Wiener process. Euler and Milstein methods are used for comparison in numerically solving the corresponding stochastic initial-value problems. After ~u 1 ðtþ and ~u 2 ðtþ are numerically solved to time 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 Eðu 1 ð1=2þþ and Eðu 2 1ð1=2ÞÞ using both, the Euler and Milstein methods. The approximate values are based on 1, independent trials. Figure 4.1 illustrates the average of the approximate solution with 1, independent trials, using both, the Euler and Milstein ;

131 Arciniega and Allen Table 4.1. Approximate values of Eðu 1 ð1=2þþ and Eðu 2 1 ð1=2þþ. Euler shooting Milstein shooting Number of intervals in t (h) Eu Eu 2 Eu Eu 2 2.1251.156.1251.156 4.1169.13.1169.13.1155.135.1155.135 16.1153.135.1153.135 Methods. Two particular trajectories of the solutions are also shown. Absolute errors of the numerical solution at time t ¼ :5 areshownin Figure 4.2 for Euler and Milstein methods. As a second example, consider the following two-point stochastic boundary-value problem: >< d~uðtþ ¼ðA~uðtÞþ~aðtÞÞdt þ X2 ðb i ~uðtþþ~b i ðtþþ dw i ðtþ; t 1 F ~uðþþf 1 ~uð1þ ¼ ~ ð4:6þ Figure 4.1. Illustration of the average and two trajectories of the solution.

Shooting Methods for Numerical Solution 1311 Figure 4.2. Illustration of the absolute errors for the first example. where 2 3 2 3 2 3 1 2 1 6 7 6 7 ~aðtþ ¼4 5; A ¼ 4 1 3 15; B 1 ¼ 1 1 6 7 4 1 5; 1 1 1 4 1 2 3 2 3 2 3 B 2 ¼ 1 2 1 1 6 7 6 7 6 7 4 1 2 15; F ¼ 4 5; F 1 ¼ 4 1 1 5; 1 1 2 1 1 ~b 1 ¼ ~ ; ~b 2 ¼ ~ : As in the first example, the corresponding initial-value problem is solved numerically using three different initial conditions: 2 3 2 3 2 3 ~u 1 ðþ ¼4 5; ~u 2 ðþ ¼4 1 5; and ~u 3 ðþ ¼4 5: 1 The three numerical solutions are then combined to approximate the solution to the original boundary-value problem (4.6). Table 4.2 presents

1312 Arciniega and Allen Table 4.2. Approximate values of Eðu 1 ð1=2þþ and Eðu 2 1 ð1=2þþ. Euler shooting Milstein shooting Number of intervals in t (h) Eu Eu 2 Eu Eu 2 2.4493.219.4491.217 4.5256.2771.5262.2777.5729.335.5741.3317 16.59.362.62.3637 32.611.376.6134.36 64.613.372.62.393 approximations of Eðu 1 ð1=2þþ and Eðu 2 1ð1=2ÞÞ using both the Euler and Milstein methods. The approximate values are based on 1, 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 Eðu 1 ð1=2þþ and Eðu 2 1ð1=2ÞÞ are, respectively, :624 and :396. Notice that Milstein s method appears to converge Figure 4.3. Illustration of the average and two trajectories of the solution.

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 1, 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 1, 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.

1314 Arciniega and Allen ACKNOWLEDGMENTS The research was supported by the Texas Advanced Research Program Grant ARP 212-44-152 and the National Science Foundation Grant DMS-2115. REFERENCES 1. Allen, E.J.; Nunn, C.J. Difference methods for numerical solution of stochastic two-point boundary-value problems. In Proceedings of the First International Conference on Difference Equations; Elydi, S.N., Greef, J.R., Ladas, G, Peterson, A.C., Eds.; Gordan and Breach Publishers: Amsterdam, 1995. 2. Arnold, L. Stochastic Differential Equations: Theory and Applications; John Wiley & Sons: New York, 1974. 3. Gard, T.C. Introduction to Stochastic Differential Equations; Marcel Dekker: New York, 19. 4. Kloeden, P.E.; Platen, E. Numerical Solution of Stochastic Differential Equations; Springer-Verlag: New York, 1992. 5. Kloeden, P.E.; Platen, E.; Schurz, H. Numerical Solution of SDE Through Computer Experiments; Springer-Verlag: Berlin, 1994. 6. Ocone, D.; Pardoux, E. Linear stochastic differential equations with boundary conditions. Probab. Theory Rel. Fields 199, 2, 49 526. 7. Ortega, J.M. Numerical Analysis: A Second Course; Academic Press Inc.: London, 1972.. Talay, D. Simulation and numerical analysis of stochastic differential systems: A review. In Probabilistic Methods in Applied Physics; Lecture Notes in Physics; Springer-Verlag: New York, 1995; Vol. 451, 63 16. 9. Talay, D.; Turbano, L. Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 199, (4), 43 59. 1. Zeitouni, O.; Dembo, A. A Change of variables formula for Stratonovich integrals and existence of solutions for two-point stochastic boundary value problems. Probab. Theory Rel. Fields 199, 4, 411 425.