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1 SIAM J. SCI. COMPUT. Vol. 4 No. pp c 00 Society for Industrial and Applied Mathematics WEAK SECOND ORDER CONDITIONS FOR STOCHASTIC RUNGE KUTTA METHODS A. TOCINO AND J. VIGO-AGUIAR Abstract. A general procedure to construct weak methods for the numerical solution of stochastic differential systems is presented. As in the deterministic case the procedure consists of comparing the stochastic expansion of the approximation with the corresponding Taylor scheme. In this way the authors obtain the order conditions that a stochastic Runge Kutta method must satisfy to have weak order two. Explicit examples of generalizations of the classical family of second order two-stage explicit Runge Kutta methods are shown. Also numerical examples are presented. Key words. stochastic differential equations weak approximation Runge Kutta methods weak numerical schemes explicit schemes AMS subject classifications. 65U05 60H10 4F05 PII. S Introduction. In the classical theory of Runge Kutta methods the technique for deriving the order conditions consists of matching the expansion of the solution generated by the Runge Kutta method with the Taylor expansion of the exact solution. This matching is carried out over the increments in the independent variable. This procedure leads to the order conditions and for each order a family of methods is obtained. The simplest family obtained in this way is the one-parameter family of second order two-stage Runge Kutta methods. To construct a similar theory in the stochastic case the first obvious step is to obtain the generalization of this second order family of Runge Kutta methods. This task can be divided into two objectives: the first one is to obtain the generalization of the order conditions and the second one is to find examples of families satisfying the obtained conditions. The main difficulty in the first objective arises from the fact that the Itô expansion depends on multiple integrals of the unity and this complicates any kind of matching. In fact there are few Runge Kutta methods for solving stochastic differential equations SDEs all of them of low order; see Platen [6] and sometimes they involve derivatives see Milshtein [4] Talay [7] or Tocino and Ardanuy [9]. The key to avoiding this difficulty is using Itô Taylor expansions in terms of the increments of the independent variable and of the solution. With them matching order conditions can be obtained as in the nonstochastic case. The obtained order conditions are stochastic in the sense that they contain random variables. For each choice of these random variables we have a different generalization of the classical order two family. This fact is new in the literature since we propose is an infinite number of weak second order methods generalizing each classical two-stage second order Runge Kutta method. Received by the editors April ; accepted for publication in revised form April 11 00; published electronically October This work was supported by Spanish CICYT under project BMF Departamento de Matemáticas Universidad de Salamanca E7008 Salamanca Spain bacon@ usal.es. Departamento de Matemática Aplicada Universidad de Salamanca E7008 Salamanca Spain jvigo@usal.es. 507

2 508 A. TOCINO AND J. VIGO-AGUIAR The paper begins with the fundamental concepts needed to obtain a stochastic scheme and an adequate expansion in terms of the increments of the independent variable and of the solution. The order conditions for the scalar case and particular families of methods satisfying them can be found in section. Section 4 shows the extension to the multidimensional case. A section of numerical examples is also included.. Weakapproximations and Itô Taylor expansions. We consider a filtered probability space Ω F F t P an m-dimensional Wiener process {W t =Wt 1...Wt m } t 0 and a d-dimensional SDE.1 dx t = at X t dt + bt X t dw t t 0 t T where a =a 1...a d denotes the d-dimensional drift vector and b =b ij the d m diffusion matrix. Denote b j =b 1j...b dj j =1...m. The functions a = at x and b j = b j t x are assumed to be defined and measurable in [t 0 T] R d and to satisfy both Lipschitz and linear growth bound conditions in x. These assumptions ensure the existence of a unique solution of the SDE.1 with the initial condition X t0 = X 0 if X 0 is F t0 -measurable see Arnold [1]. We assume all the initial moments E[ X 0 r ] < r =1...exist; thus the moments of every X t exist see Arnold [1]. Let X tx denote the solution of.1 starting at time t [t 0 T]atx R d. Let C P denote the space of functions ft x defined in [t 0 T] R d which have polynomial growth with respect to x and let C β P denote the subspace of functions f C P for which all partial derivatives up to order β =1... belong to C P. In addition to.1 we consider the one-step approximation. X tx t + h =x + At x h ξ where A is some R d -valued function and ξ is a random vector. We say that the onestep approximation X = X tx converges weakly to X = X tx with order β + 1 if there exists a function Kx C P such that. l E X ij x ij j=1 l X ij x ij Kxhβ+1 i j =1...d l =1...β + j=1 where z i denotes the ith component of the vector z. From.4 it is obvious that the differences between the moments from the first up to the β + th inclusive of the vector X and the corresponding moments of its approximation X have β + 1 order of smallness in h. The number β + 1 will be called the local order of the approximation. We give an equidistant discretization {t 0 t 1...t N } of the time interval [t 0 T] with stepsize = T t 0 /N. From the one-step approximation. we construct the discrete approximation also called scheme.4 X 0 = X 0 X n+1 = X n + At n X n ξ n n =0...N 1. We say that the discrete approximation X = {X 0 X 1...X N } based on a stepsize converges weakly to X with order β if for each g C β+ P there exists a constant K g 0 not depending on such that [ E gxn gx T ] K g β.

3 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 509 The number β in the above definition is the order of the scheme on an interval. Based on a theorem due to Mil shtein [5] that establishes the relation between the order of a one-step approximation and the order of the scheme generated by such approximation one can obtain schemes of order β by means of one-step approximations of local order β + 1. For example the truncated Itô Taylor expansion of order β of the Itô process X tx i.e. the expression obtained from the Itô Taylor expansion removing the terms which contain multiple integrals of multiplicities equal to or greater than β + 1isan approximation of local order β + 1 if the coefficients of the equation are continuous satisfy both Lipschitz and linear growth conditions and belong to C β+ P. The scheme obtained with the truncated Itô Taylor expansion of order β will be called the order β weak Taylor scheme see Kloeden and Platen [] for details. We say that two one-step approximations X tx and X tx are β-equivalent X tx t+ h β X tx t + h if there exists a function Kx C P such that l l E X ij x ij X ij x ij Kxhβ+1 l =1...β +. j=1 j=1 It is obvious that if the one-step approximations X tx and X tx are β-equivalent then either both or none of them have order β +1. For the one-step approximation given by the order β truncated Itô Taylor expansion of the Itô process X tx wehave.5 X tx = x + At x h ξ =x + c 1 ξ c s ξ s where c i t x are constants and the random variables ξ 1...ξ s are given by multiple integrals of the unity. It can be shown [ p. 1] that we can obtain β-equivalent approximations to.5 by replacing the variables ξ i with new variables ˆξ i satisfying [ l l.6 E ξ j k ˆξ k] j Kβ+1 j k {1...s} l =1...β +1 k=1 k=1 for some constant K>0. With this procedure from the order two Taylor approximation we can obtain the simplified order two weak Taylor scheme see [] for details; for the nonautonomous case see [9] given by.7 m X k n+1 = X k n + b kj Ŵ n j + a k + 1 m li bkj b x l Ŵ nŵ i n j + V ijn j=1 ij=1 l=1 + 1 m ij ak b x i + bkj + a i bkj t x i + 1 m b ik b lk b kj x i x l Ŵ n j j=1 il=1 k=1 + 1 ak t + a i ak x i + 1 m b ik b jk a k x i x j ij=1 k=1 where Ŵ 1 n...ŵ m n are independent random Gaussian random variables N0 and V ijn i j =1...m are independent two-point distributed random variables

4 510 A. TOCINO AND J. VIGO-AGUIAR with.8 PV ijn == 1 =PV ijn = V iin = V ijn = V jin if j>i. if j<i From now on for a function g = gt x with t x R we denote g = g 00 = gt n X n and g ij = i+j g.9 t i x j t n X n. With this notation in the scalar case d = m = 1 the simplified order two weak Taylor scheme is given by.10 X n+1 = X n + bŵn + a+ 1 bb 01 Ŵn + 1 b 10 + ab b b 0 + ba 01 Ŵn + 1 a 10 + aa b a 0 where Ŵn is any Gaussian random variable N0. In the order β Taylor approximation.5 we can consider the function A as a linear combination of ξ 0 =0ξ 1...ξ s. If we replace only one variable say ξ with a new variable ˆξ then condition.6 reduces to a comparison of the products containing β ˆξ with the corresponding products containing ξ. If.6 holds then we write ˆξ ξ since the change leads to a β-equivalent approximation. If ˆζ 1 ˆζ β = ˆξ 0 then we write ˆζ β 1 ˆζ. In the multidimensional case when Ŵ 1...Ŵm are independent random variables with distribution N0 and V ij are as in.8 we have the following -equivalences: { Ŵ i Ŵ j if i = j 0 if i j Ŵ i Ŵ j Ŵ k i Ŵ 1 j1 Ŵ m jm 0 Ŵ i Ŵ j Ŵ i Ŵ j Ŵ i if i = j = k Ŵ i if j = k i 0 if i j i k j k if i + j j m 5. { if i = j 0 if i j { 15 if i = j 0 if i j Other useful equivalences are.16 Ŵ i 4 6Ŵ i

5 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS Ŵ i 4 6 Ŵ i Ŵ i 5 Ŵ i 6 15 Ŵ i 45 Ŵ i 0. An important disadvantage of simplified Taylor schemes is that they require us to determine many derivatives. Using the idea of the deterministic case we obtain Runge Kutta schemes by replacing the derivatives in simplified Taylor schemes with new evaluations of the coefficients of the equation. As in the deterministic case in order to match the truncated expansion of the Runge Kutta approximation with the simplified Taylor scheme we need an expression of the order β truncated expansion of a process ft +X t +X in terms of and X = X t+ X t. This expression can be found in Tocino and Ardanuy [8] for β =. In the multidimensional case the formula.0 { ft +X t +X f + f t + f f x i Xi + t + c ij f t x i x j ij=1 + 1 } kl cij f c x l x i x j x k + 1 c ij c kl 4 f 4 x i x j x k x l ijk=1 l=1 ijkl=1 + f t x i + 1 c jk f x i x j x k X i f X i X j + x i x j jk=1 ij=1 where c ij = m k=1 bik b jk and all the functions are evaluated at t X t expresses the -equivalence between the process and its second order truncated expansion. In the scalar case d = m = 1 using notation.9 the above expression reduces to.1 ft +X t +X f 00 + f 10 +f 01 X + f 0 + b f 1 + b b 01 f 0 + b4 4 f 04 + f 11 + b f X 0 X + f 0.. Stochastic second order Runge Kutta methods in the scalar case. The goal of this section is to generalize the classical second order two-stage Runge Kutta methods for ordinary differential equations ODEs; see [] to SDEs. When b = 0.1 is nonstochastic and explicit second order two-stage Runge Kutta methods are given by.1 X n+1 = X n +α 1 k 0 + α k 1 where. k 0 = at n X n k 1 = at n + µ 0 X n + λ 0 k 0

6 51 A. TOCINO AND J. VIGO-AGUIAR and the constants satisfy the system. α 1 + α =1 µ 0 α =1/ λ 0 α =1/. The evaluations of a appear multiplied by which has mean-square order 1. In the stochastic case we have to include evaluations of b; it is natural that these evaluations appear multiplied by Ŵ but in this way the generalization always contains evaluations of some derivatives see [9]. Since the mean-square order of Ŵ is 1/ the approach for avoiding derivatives in the stochastic case consists of multiplying each evaluation of the diffusion coefficient b by new random variables of mean-square order 1/. Thus the proposed form of an explicit stochastic Runge Kutta scheme is.4 X n+1 = X n +α 1 k 0 + α k 1 + s o Γ+s 1 Λ+s Ω with.5 k 0 = at n X n s 0 = bt n X n k 1 = a t n + µ 0 X n + λ 0 k 0 +s 0 L s 1 = b t n + µ 0 X n + λ 0 k 0 +s 0 M s = b t n + µ 0 X n + λ 0 k 0 +s 0 N where Γ Λ Ω L M and N are random variables of mean-square order 1/. We seek values for the constants and conditions on these random variables in order that the scheme has order two in the weak sense. When bt X t = 0 these methods reduce to the form of the classical Runge Kutta methods.1. for nonstochastic differential equations. Particular cases of this situation are the schemes presented in Platen [6] and Vigo-Aguiar and Tocino [10]..1. Stochastic second order conditions. As we have said the way to obtain the weak order two conditions is to compare approximation.4.5 with the simplified order two Taylor scheme.11. The coefficients and variables for which the Runge Kutta approximation is -equivalent to the second order Taylor approximation give a second order Runge Kutta scheme. Using the truncated expansion.1 and that 0 we have.6 α 1 k 0 + α k 1 = α 1 at n X n +α a t n + µ 0 X n + λ 0 k 0 +s 0 L α 1 + α a+α a 01 bl+α µ 0 a 10 + α λ 0 aa α a 0 b L + a 11 + b a 0 bα µ 0 L + aa 0 bα λ 0 L. On the other hand using again the truncated expansion.1 we have

7 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 51.7 s o Γ+s 1 Λ+s Ω = b Γ+bt n + µ 0 X n + λ 0 a+bmλ+bt n + µ 0 X n + λ 0 a+bnω bγ+λ+ω+bb 01 M Λ+N Ω + b 10 µ 0 Λ+Ω+ab 01 λ 0 Λ+Ω + 1 b b 0 M Λ+N Ω + b b b b 0 µ 0 M Λ+N Ω +abb 0 λ 0 M Λ+N Ω + 1 b 0 + b b 1 + b b 01 b 0 + b4 4 b 04 µ 0 Λ+Ω + 1 a b 0 λ 0 Λ+Ω+a b 11 + b b 0 λ 0 µ 0 Λ+Ω. Inserting.6 and.7 in.4 and using the equivalences.11 and.1 it is easy to see that the Runge Kutta approximation.4 and the simplified Taylor approximation.11 are -equivalent if α 1 + α =1.8 µ 0 α =1/ λ 0 α =1/.9 and.10 α L 1 Ŵn α L 1 Γ+Λ+Ω Ŵn M Λ+N Ω 1 Ŵn µ 0 Λ+Ω 1 Ŵn λ 0 Λ+Ω 1 Ŵn M Λ+N Ω 1 Ŵn. Thus we have proved the following theorem. Theorem 1. Suppose the coefficients a and b of the SDE.1 with d = m =1 are continuous satisfy both Lipschitz and linear growth bound conditions in x and belong to C β+ P. The Runge Kutta scheme.4.5 has order two in the weak sense if.8.9 and.10 hold. Since the three equations of system.8 are the order two conditions in the classical two-stage Runge Kutta methods the schemes satisfying the theorem conditions are the stochastic generalizations of second order classical Runge Kutta methods. For each α 0 we have a solution to.8. Thus in the ordinary case each α 0 determines a second order method. Once α 0 is fixed each solution of.9 and.10 determines a stochastic generalization of that ordinary Runge Kutta method. Notice that α does not appear in system.10. Thus the evaluations of b have no influence on what ordinary method will be generalized. Only the point of evaluation of a affects this.

8 514 A. TOCINO AND J. VIGO-AGUIAR.. Particular families of second order Runge Kutta methods. The most simple variables of mean-square order 1/ are Ŵn and. Other variables with the same order are the products k Ŵn Ŵn k =1... In the rest of this section we consider linear combinations of some of the above variables in order to obtain special solutions of systems.9 and.10. First consider system.9. It is clear that L = ν 1 cannot be a solution. Another approach is to take L = ν 1 Ŵn; using.11 the system becomes α ν 1 =1/ α ν1 =1/. Its solution is α =1/ ν 1 = 1. Therefore only the improved Euler method can be generalized with this kind of variable. To extend the generalization to other Runge Kutta methods next we consider variables of the form Ŵn L = ν 1 Ŵn + ν. Now using the equivalences the system becomes.11 α ν 1 +ν =1/ α ν 1 +6ν 1 ν +15ν =1/. The solutions of.8 and.11 are the two one-parameter families and α 1 =1 α µ 0 = λ 0 = 1 α ν = α 1 =1 α µ 0 = λ 0 = 1 α α 1 6α ν 1 = 1 α ν α 1 ν = ν 1 = 1 ν 6α α with α 1/. Now consider system.10. Here we use linear combinations of the variables Ŵn and Ŵn. To be specific we take Ŵn Γ=γ 1 Ŵn + γ +γ Λ=λ 1 Ŵn + λ +λ Ŵn Ω=µ 1 Ŵn + µ +µ Ŵn M = β 1 Ŵn + β +β Ŵn N = δ 1 Ŵn + δ +δ Ŵn.

9 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 515 Using equivalences.1 and system.10 becomes.1 γ 1 + λ 1 + µ 1 =1 γ =0 γ =0 β λ 1 + β 1 λ + δ µ 1 + δ 1 µ =0 β λ 1 + β 1 λ + δ µ 1 + δ 1 µ =0 β λ β λ + δ µ δ µ = 1 β 1 λ 1 + β λ + β λ +6β λ + δ 1 µ 1 + δ µ + δ µ +6δ µ = 1 λ 0 λ 1 + µ 1 = 1 µ 0 λ 1 + µ 1 = 1 λ + µ =0 λ + µ =0 β λ 1 +β 1 λ 1 +6β β λ 1 +15β λ 1 +β 1 β λ +6β 1 β λ +6 β 1 β λ +0β 1 β λ + δ µ 1 +δ 1 µ 1 +6δ δ µ δ µ 1 +δ 1 δ µ +6δ 1 δ µ +6δ 1 δ µ +0δ 1 δ µ = 1 6 β 1 β λ 1 + β λ β λ β 1 λ 6 β β λ 0 β λ 6 δ 1 δ µ 1 +δ µ δ µ δ 1 µ 6 δ δ µ 0 δ µ =0 β 1 β λ 1 +1β 1 β λ 1 + β 1 λ +β β λ +6β λ + β λ +6β 1 λ +1 β β λ +45β λ +δ 1 δ µ 1 +1δ 1 δ µ 1 + δ 1 µ +δ δ µ +6δ µ +δ µ +6δ 1 µ +1δ δ µ +45δ µ =0. Examples of solutions of this system are the following one-parameter families. Family A. β = δ =0 µ 0 = λ 0 =1 γ 1 = 1 γ = γ =0 µ 1 = λ 1 = 1 4 µ = λ = 1 48µ λ = µ β = δ = 8µ µ 1 48µ β 1 = δ 1 =1+ µ 1 48µ where µ 0µ 1 4. Family B. β 1 = β = δ 1 = δ =0 γ = γ =0 γ 1 =1 4µ 5 µ 1 = λ 1 = 1µ 5 µ 0 = λ 0 = 5 48µ δ = β = 1 λ = µ =µ λ = µ 1µ

10 516 A. TOCINO AND J. VIGO-AGUIAR where µ 0. In short we have considered Runge Kutta schemes of the form X n+1 = X n +α 1 k 0 + α k 1 + Ŵn γ 1 Ŵn + γ +γ s Ŵn λ 1 Ŵn + λ +λ s 1 where + Ŵn µ 1 Ŵn + µ +µ s.14 k 0 = at n X n s 0 = bt n X n k 1 = a t n + µ 0 X n + λ 0 k 0 + Ŵn ν 1 Ŵn + ν s 0 s 1 = b t n + µ 0 X n + λ 0 k 0 + Ŵn β 1 Ŵn + β +β s 0 Ŵn s = b t n + µ 0 X n + λ 0 k 0 + δ 1 Ŵn + δ +δ s 0. We have obtained that these schemes have order two if the constants satisfy conditions.8.11 and.1. We chose examples of families of solutions of 1 the system. Thus for each α 1/ and µ R {0 4 } we have two second order Runge Kutta schemes one of them with ν 0. This two-parameter family of Runge Kutta methods is a stochastic generalization of.1. The schemes of this family are derivative free and at each step we have to evaluate the drift coefficient a at two points the diffusion coefficient b at three points and to generate the Gaussian variable Ŵn. Observe also that none of them requires less evaluations of a and b. Analogous considerations are valid for the second family of parameters. The schemes proposed by Vigo-Aguiar and Tocino [10] as a generalization of the ordinary Runge Kutta methods with α 1/ are of the form The method proposed by Platen [6] also has the same form. 4. Multidimensional case. In the multidimensional case the proposed Runge Kutta scheme is given by 4.1 with X n+1 = X n +α 1 k 0 + α k 1 + m ij=1 s ij o Γ ij + s ij 1 Λij + s ij Ωij k 0 = at n X n s ij 0 = bj t n X n m k 1 = a t n + µ 0 X n + λ 0 k 0 + s rr 0 L r r=1

11 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 517 { s ij 1 = b j t n + µ j 0 X n + λ j 0 k 0 +s jj 0 M j if i = j b j t n X n + s ii 0 P i if i j { s ij = b j t n + µ j 0 X n + λ j 0 k 0 +s jj 0 N j if i = j b j t n X n + s ii 0 Q i if i j where Γ ij Λ ij Ω ij L j M j N j P j and Q j i j =1...m are random variables of mean-square order 1/. As in the one-dimensional case we seek values for the constants and conditions for these random variables in order to have weak order two for the scheme. Using the truncated expansion.0 and the equivalence.1 we have that the kth component of α 1 k 0 + α k 1 satisfies 4. m α 1 a k +α a t k n + µ 0 X n + λ 0 a + b r L r α 1 + α a k +α r=1 m r=1 a k +α λ 0 x i ai + 1 α m +α µ 0 a k t x i + 1 +α λ 0 m ij=1 r=1 r=1 a k x i bir L r a k + α µ 0 m ij=1 rs=1 jk=1 c jk a k x i x j ai b jr L r. t a k x i x j bir b js L r L s a k x i x j x k b ir L r On the other hand using again the truncated expansion.0 we have that the kth component of s jj 0 Γjj + s jj 1 Λjj + s jj Ωjj satisfies 4. b kj Γ jj + b kj t n + µ j 0 X n + λ j 0 a +b j M j Λ jj +b kj t n + µ j 0 X n + λ j 0 a +b j N j Ω jj b kj Γ jj +Λ jj +Ω jj b kj x i bij M j Λ jj + N j Ω jj + bkj t µ 0Λ jj +Ω jj b kj x i ai λ 0 Λ jj +Ω jj + 1 b kj x i x l bij b lj M j Λ jj +N j Ω jj il=1 b kj t x i + 1 b kj x i x r x s crs b ij µ 0 M j Λ jj + N j Ω jj rs=1 b kj + x i x l ai b lj λ 0 M j Λ jj + N j Ω jj il=1 { + 1 b kj t + c il b kj t x i x l + 1 il=1 irs=1 l=1 sl cir c x l b kj x i x r x s

12 518 A. TOCINO AND J. VIGO-AGUIAR irsl=1 } c ir c sl 4 b kj x i x r x s x l µ 0 Λ jj +Ω jj b kj x i x l ai a l λ 0 Λ jj +Ω jj il=1 b kj t x i + 1 b kj x i x r x s crs a i λ 0 µ 0 Λ jj +Ω jj. rs=1 Finally in the same way if i j i j =1...m then the kth component of s ij 0 Γij + s ij 1 Λij + s ij Ωij satisfies b kj Γ ij + b kj t n X n + b i P i Λ ij + b kj t n X n + b i Q i Ω ij 4.4 b kj Γ ij +Λ ij +Ω ij rl=1 l=1 b kj x l b li P i Λ ij + Q i Ω ij b kj x r x l bri b li P i Λ ij +Q i Ω ij. Inserting and 4.4 in 4.1 and using equivalences.11 and.1 it is easy to see that the Runge Kutta approximation and the simplified Taylor approximation.7 are -equivalent if 4.5 α 1 + α =1 µ 0 α =1/ λ 0 α =1/ and for each i j =1...m i j α L j 1 Ŵ j n 4.6 α L j 1 α L i L j 0 and Γ jj +Λ jj jj +Ω Ŵ j n M j Λ jj + N j jj Ω 1 Ŵ j n 4.7 µ j 0 Λjj +Ω jj 1 Ŵ j n λ j 0 Λ jj +Ω jj 1 Ŵ j n M j Λ jj +N j jj Ω 1 Ŵ n j and Γ ij +Λ ij ij +Ω 0

13 4.8 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 519 P i Λ ij + Q i ij Ω 1 Ŵ n i Ŵ n j + V ij P i Λ ij +Q i ij Ω 1 Ŵ n. j Thus we have the following theorem. Theorem. Let a and b be as in Theorem 1. The Runge Kutta scheme 4.1 has order two in the weak sense if and 4.8 hold. As in the scalar case once α 0 is fixed each solution of and 4.8 determines a stochastic generalization of the corresponding ordinary Runge Kutta method. Here we also can propose particular examples of mean-square order 1/ satisfying the theorem. If we consider L j = ν j 1 Ŵ n j + ν j Ŵ n j j =1...m then it is immediate using equivalences and.15 that L i L j 0 if i j. Thus 4.6 reduces to the first two equations which become analogous to.11 for each pair ν j 1 νj. Following the one-dimensional case if we take for each j =1...m Γ jj = γ j 1 Ŵ j n + γ j Λ jj = λ j 1 Ŵ j n + λ j Ω jj = µ j 1 Ŵ j n + µ j M j = β j 1 Ŵ j n + β j N j = δ j 1 Ŵ j n + δ j +γ j Ŵ n j +λ j Ŵ n j +µ j Ŵ n j +β j Ŵ n j +δ j Ŵ n j then using equivalences.1 and system 4.7 becomes similar to system.1 for each j =1...m. Finally taking P j = θ j and Q j = η j it is a straightforward computation to show that 4.8 holds if θ j 0η j 0θ j η j and Γ ij = 1 θ j η j Λ ij = Ω ij = 1 θ j θ j η j 1 η j θ j η j Ŵ n j θ j + η j Ŵ n i Ŵ n j + V ij Ŵ n j η j Ŵ n i Ŵ n j + V ij Ŵ n j + θ j Ŵ n i Ŵ n j + V ij. Generalizing the conclusions of the above section for each α 1/ each µ j R {0 1 4 } and each θj η j with θ j η j θ j η j 0 we have an example of a second

14 50 A. TOCINO AND J. VIGO-AGUIAR order multidimensional Runge Kutta scheme. This four-parameter family of Runge Kutta methods is a stochastic generalization of.1. The schemes of this family are derivative free and at each step we have to generate the Gaussian independent variables Ŵ n j and the variables V ij verifying.8. The multidimensional scheme proposed by Kloeden and Platen [ pp ] corresponds to the particular solution α 1 = 1 α = 1 λ 0 =1 µ 0 =1 ν j 1 =1 νj =0 β j = δj =0 µj 0 = λj 0 =1 γ j 1 = 1 γj = γj =0 µj 1 = λj 1 = 1 4 µ j = λj = 1 4 λj = µj = 1 4 βj = δj =1 βj 1 = δj 1 =0 θj = η j =1 for each j =1...m. 5. Numerical examples. In this section numerical examples are presented to illustrate some of the methods we have developed. The schemes selected are generalizations of two well-known ordinary Runge Kutta methods: the improved and the modified Euler methods. From Families A and B introduced in section. we have chosen a scheme generalizing each one of these methods. Method SIE-A. If we take µ = 1 4 in Family A then for each α we have a scheme generalizing the corresponding Runge Kutta method. For α =1/ wehave the extension of the improved Euler method X n+1 = X n + 1 k 0 + k s 0 + s 1 + s Ŵn s Ŵn s 1 where k 0 = at n X n s 0 = bt n X n k 1 = a t n +X n + k 0 +Ŵns 0 s 1 = b t n +X n + k 0 + s 0 s = b t n +X n + k 0 s 0. Method SME-A. If we take µ = 1 4 α = 1 in Family A then we obtain a stochastic generalization of the modified Euler method X n+1 = X n + k s 0 + s 1 + s Ŵn s Ŵn s 1 where k 0 = at n X n s 0 = bt n X n

15 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 51 k 1 = a t n + 1 X n + 1 k 0 + s 1 = b t n +X n + k 0 + s 0 s = b 6 4 t n +X n + k 0 s 0. Ŵn Ŵn s 0 Method SIE-B. If we take µ = 1 in Family B then for each α we have a scheme generalizing the corresponding Runge Kutta method. The method of this group that generalizes the improved Euler method α =1/ is X n+1 = X n + 1 k 0 + k s 1 + s s 0 Ŵn + 1 s 1 s Ŵn where k 0 = at n X n s 0 = bt n X n k 1 = a t n +X n + k 0 +Ŵns 0 s 1 = b t n X n k s = b t n X n k Ŵn s 0 Ŵn s 0. Method SME-B. If we take α =1 µ = 1 in Family B then we obtain a stochastic generalization of the modified Euler method X n+1 = X n + k s 1 + s s 0 Ŵn + 1 s 1 s Ŵn where k 0 = at n X n s 0 = bt n X n k 1 = a t n + 1 X n + 1 k 0 + s 1 = b t n X n k s = b t n X n k Ŵn + 4 Ŵn s 0 Ŵn s Ŵn s 0 The method denoted by SIE-A was proposed by Platen see [] or [6]. The others are new in the literature. To compare their accuracy we present two examples: one is a linear equation and the other is nonlinear. For each scheme we compute the approximations and compare the results with the exact value. The errors depend on

16 5 A. TOCINO AND J. VIGO-AGUIAR Table 1 Errors and standard deviations in the approximation of E[X ] in 5.1. SIE-A SME-A SIE-B SME-B error st. d. error st. d. error st. d. error st. d Table Errors and standard deviations in the approximation of E[X 1 ] in 5.. SIE-A SME-A SIE-B SME-B error st. d. error st. d. error st. d. error st. d the stepsize. All the computations were done on an IBM-PC with a Pentium-II processor using Mathematica and 5000 independent trials. To simulate the variable Ŵn we have taken a Gaussian variable with distribution N0. The first test problem we considered was the Black Scholes stochastic differential equation used in option pricing: 5.1 dx t = µx t dt + σx t dw t X 0 = x 0. Since the equation is linear it can be easily solved see []. The solution X t is given in terms of the initial value x 0 and the parameters µ and σ. It is easy to see that E[X t ]=x 0 e µt. This first moment was approximated at the point t = when x 0 =1 µ =1.5 and σ =0.5. The obtained results are showed in Table 1. It can be seen that there are practically no differences between the four proposed second order Runge Kutta schemes. In all the integrations the error decreases and the standard deviation increases with the stepsize. Our second test problem is the nonlinear SDE 5. dx t = X 0 =1 1 X1/ t +6X / t dt + X / t dw t with solution X t =t +1+ Wt. We approximated the first moment of the solution at point t = 1. The exact value is E[X 1 ] = 8. The results summarized in Table show that in this problem the errors obtained with the schemes which generalize the modified Euler method are lower for every step length than those of the improved Euler generalizations.

17 SECOND ORDER STOCHASTIC RUNGE KUTTA METHODS 5 6. Summary. In this paper we obtained the conditions that a stochastic Runge Kutta method must satisfy to have order two in the weak sense. We considered the family of classical second order two-stage explicit Runge Kutta methods for solving ODEs and using these conditions we obtained the first generalization of these methods for the stochastic case. This generalization is not unique but depends on the stochastic variables used. In view of the developed numerical examples no differences in the behavior of the proposed methods can be concluded. REFERENCES [1] L. Arnold Stochastic Differential Equations Wiley New York [] P.E. Kloeden and E. Platen Numerical Solution of Stochastic Differential Equations Springer Berlin 199. [] J.D. Lambert Numerical Methods for Ordinary Differential Systems Wiley New York [4] G.N. Milshtein A method of second-order accuracy integration of stochastic differential equations Theory Probab. Appl pp [5] G.N. Mil shtein Weak approximation of solutions of systems of stochastic differential equations Theory Probab. Appl pp [6] E. Platen An introduction to numerical methods for stochastic differential equations Acta Numer pp [7] D. Talay Efficient numerical schemes for the approximation of expectations of functionals of the solution of an SDE and applications in Filtering and Control of Random Processes Lecture Notes in Control and Inform. Sci. 61 Springer Berlin 1984 pp [8] A. Tocino and R. Ardanuy Truncated Itô -Taylor expansions Stochast. Anal. Appl pp [9] A. Tocino and R. Ardanuy Runge-Kutta methods for numerical solution of stochastic differential equations J. Comput. Appl. Math pp [10] J. Vigo-Aguiar and A. Tocino An infinite family of second order weak explicit Runge-Kutta methods J. Comput. Methods Sci. Engrg pp

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