Runge Kutta methods for numerical solution of stochastic dierential equations

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1 Journal of Computational and Applied Mathematics Runge Kutta methods for numerical solution of stochastic dierential equations A.Tocino a; ; 1, R.Ardanuy b a Departamento de Matematicas, Universidad de Salamanca, Plaza de la Merced, 1, 7008-Salamanca, Spain b Departamento de Estadstica, Universidad de Salamanca, Plaza de los Cados, 7008-Salamanca, Spain Received September 000; received in revised form February 001 Abstract The way to obtain deterministic Runge Kutta methods from Taylor approximations is generalized for stochastic dierential equations, now by means of stochastic truncated expansions about a point for suciently smooth functions of an Itô process.a class of explicit Runge Kutta schemes of second order in the weak sense for systems of stochastic dierential equations with multiplicative noise is developed.also two Runge Kutta schemes of third order have been obtained for scalar equations with constant diusion coecients.numerical examples that compare the proposed schemes to standard ones are presented. c 00 Elsevier Science B.V. All rights reserved. MSC: 65U05; 60H10; 4F05 Keywords: Stochastic dierential equations; Weak approximation; Runge Kutta methods; Weak numerical schemes; Explicit schemes 1. Introduction Stochastic dierential equations are becoming increasingly important due to its application for modelling stochastic phenomena in dierent elds, e.g. physics or economics see [7,]. Unfortunately, in many cases analytic solutions of these equations are not available and we are forced to use numerical methods to approximate them.roughly speaking, there are two basic ways to achieve these approximations.when sample paths of the solutions need to be approximated, mean-square convergence is used and the methods so obtained are called strong.when we are only interested in the moments or other functionals of the solution, which implies a much weaker form of convergence Corresponding author. addresses: bacon@gugu.usal.es A. Tocino, raa@gugu.usal.es R. Ardanuy. 1 This author was partially supported by CICYT under the project BFM /0/$ - see front matter c 00 Elsevier Science B.V. All rights reserved. PII: S

2 0 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics than that needed for pathwise approximations, the methods are called weak.in the present paper we will only refer to weak methods; for this reason words like scheme, order and so on mean weak scheme, weak order... In previous works various mean-square and weak numerical methods have been derived; extended presentations on this subject are given in Milstein [1] and Kloeden and Platen [7].Analogously with the deterministic case, stochastic Taylor schemes are obtained by truncating stochastic Taylor expansions.the practical diculty of employing Taylor approximations is that they require to determine many derivatives.in this investigation we are interested in methods of Runge Kutta RK type, i.e.one step methods which avoid the use of derivatives.rk schemes in the strong sense have been proposed, for instance, by Chang [], Hernandez and Spigler [4,5], Klauder and Petersen [6], Mauthner [8], McShane [9] and Rumelin [14]; see also Kloeden and Platen [7], Milstein [1] and the references in Saito and Mitsui [15].On the other hand, Milstein [10 1], Talay [16] and Kloeden and Platen [7] present RK schemes in the weak sense. In this paper a class of explicit second order and two explicit third order RK schemes are developed.the second order class contains a scheme proposed by Milstein [10].In analogy with the ordinary case, these RK schemes have been obtained by matching their truncated stochastic expansion about a point with the corresponding Taylor approximation.the required truncated expansions about a point for functions of the solution of a stochastic dierential equation have been derived from Itô-Taylor expansions in Tocino and Ardanuy [17].. Weak approximations In this paper we consider a ltered probability space ; F; F t ;P, an m-dimensional Wiener process {W t =Wt 1 ;:::;Wt m } t 0 and a d-dimensional stochastic dierential equation sde dx t = at; X t dt bt; X t dw t ; t 0 6 t 6 T; 1 where a=a 1 ;:::;a d denotes the d-dimensional drift vector and b=b ij the d m-diusion matrix. Let s 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 dened 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 [1].We shall suppose that all of the initial moments E[ X 0 r ], r =1; ;::: exist; so, the moments of every X t will exist see [1].Let X t;x 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 dened in [t 0 ;T] R d which have polynomial growth with respect to x and let C P the subspace of functions f C P for which all partial derivatives up to order =1; ;::: belong to C P. Next, to Eq.1 we consider the one-step approximation X t;x t h=x At; x; h; ; where A is some R d -valued function and a random vector.we shall say that the one-step approximation X = X t;x converges weakly to X = X t;x with order 1 if there exists a function Kx C P

3 such that E A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics l X ij x ij l X ij x ij 6 Kxh1 ; i j =1;:::;d; l=1;:::; 1; where z i denotes the ith component of the vector z.from it s obvious that the dierences between the moments from the rst up to th inclusively of the vector X and the corresponding moments of its approximation X have 1 order of smallness in h.the number 1 will be called the local order of the approximation. Let be given an equidistant discretization {t 0 ;t 1 ;:::;t N } of the time interval [t 0 ;T] with step size =T t 0 =N.From the one-step approximation we construct the discrete approximation also called scheme: X 0 = X 0 ; 4 X n1 = X n At n ; X n ;; n ; n=0;:::;n 1: We shall say that the discrete approximation X ={ X 0 ; X 1 ;:::; X N } based on a step size converges weakly to X with order if for each g C P there exists a constant K g 0 not depending on such that E[g X N gx T ] 6 K g : The number in the above denition is the order of the scheme on an interval.the following theorem see [11] establishes the relation between the order of a one-step approximation and the order of the scheme generated by such approximation: Theorem 1. Suppose that the coecients of Eq. 1 are continuous; satisfy a Lipschitz condition and a linear growth bound and belong to C P. Suppose that the one-step approximation has order 1 and for each 0 h 1 veries E[At k ;x;h; k ] 6 K1 x h; At k ;x;h; k 6 M k 1 x h 1= ; where M is a function of C P such that M k has moments of all orders. Then the scheme 4 has order. Based on the above result one can obtain see [7] for the details the weak Taylor schemes: next to the sde 1 consider the operators 5 L L k = b ik ; i ; j i; k =1;:::;m; 6

4 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics where c ij = m k=1 bik b jk, i; j =1;:::;d.Given N, we denote by the set of all multi-indices =j 1 ;:::;j l, j k {0; 1;:::;m}, of length l {1;:::;} and by v the multi-index of length zero. Given a function f :[t 0 ;T] R d R d, if we drop the remainder of the Itô Taylor expansion of ft; X t for the hierarchical set {v}, we obtain the order truncated Itô Taylor expansion ft; X t ft 0 ;X t0 L ft 0 ;X t0 I ; 7 where, if =j 1 ;:::;j l, we have denoted L = L j1 L jl, I = t 1d = t sl s dw j1 t t t t s 1 dw j l s l and dw 0 =dt.from 7, if ft; x=x one obtains the one-step approximation X t;x t =x L ft; x I : 8 It can be shown that if the coecients a and b are as in Theorem 1 then the one-step approximation 8 veries that for each g C P there exist constants K 0 and r N such that E[g X t;x t h gx t;x t h] 6 K1 x r h 1 9 and hence it has order 1.On the other hand, if the functions L f, where ft; x =x and, grow at most linearly with respect to x then one can see that the one-step approximation 8 veries conditions 5.We can conclude, by Theorem 1, that under the above conditions the scheme generated by 8 X n1 = X n L ft n ; X n I ;n 10 converges weakly with order.it s called the order weak Taylor scheme. The order Taylor scheme was rst proposed by Milstein [10].Talay [16] proved it.milstein [11] also proposed the third order Taylor scheme for systems with additive noises.weak Taylor schemes of any order were constructed by Platen; see Kloeden and Platen [7]. We shall say that two one-step approximations X t;x and X t;x are -equivalent, X t;x t h X t;x t h, if there exists a function Kx C P such that l l E X ij x ij X ij x ij 6 Kxh1 ; i j =1;:::;d; l=1;:::; 1: It s obvious that if the one-step approximations X t;x and X t;x are -equivalent then either both or none of them have order 1.For example, by 9, an Itô process a solution of a sde and its truncated Itô Taylor expansion of order are -equivalent approximations if the coecients of the equation are continuous, satisfy both Lipschitz and linear growth conditions and belong to C P. Once obtained the one-step Taylor approximation 10, which has local order 1, we can construct schemes of order by means of -equivalent approximations to it.an example see [7] of this technique is the simplied order weak Taylor scheme X n1 = X n L ft n ; X n Î ;n ; 11

5 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics where the variables Î ;n,, are such that there is a constant K 0 verifying [ l l E I k ;n Î k ;n] 6 K1 1 k=1 k=1 for all choices of multi-indices k {v} with k =1;:::;l and l =1;:::; 1. If =, d = m = 1 scalar case and Ŵ n is any variable satisfying the moment conditions E[Ŵ n ] E[Ŵ n ] E[Ŵ n ] E[Ŵ n 4 ] E[Ŵ n 5 ] 6 K 1 for some constant K 0, it s easy to prove that the variables Î 0;n =I 0;n =, Î 0;0;n =I 0;0;n = =, Î 1;n =Ŵ n, Î 0;1;n = Î 1;0;n = 1 Ŵ n, Î 1;1;n = 1 Ŵ n satisfy 1.Then we have the simplied order Taylor scheme X n1 = X n bŵ n a 1 bb 01Ŵ n 1 b 10 ab 01 1 b b 0 ba 01 Ŵ n 1 a 10 aa 01 1 b a 0 ; 14 where for a function g = gt; x with t; x R we have denoted g ij t n; X j n 15 and g = g 00 = gt n ; X n. Notice that a Gaussian random variable W n N0; or a three-point distributed random variable Ŵ n with PŴ n = = PŴ n = = 1 6, PŴ n =0= satisfy the moment conditions 1. In the multi-dimensional case, the kth component of the simplied order Taylor scheme is given by X k n1 = X k n b kj Ŵ j n a k 1 b Ŵ nŵ j l n V ij;n 1 b @t a i i; i; c ij l=1 i 1 c b kj Ŵ l n a k ; j where Ŵ 1 n;:::;ŵ m n are independent random variables satisfying the moment conditions 1 and V ij;n, i; j =1;:::;m, are independent two-point distributed random variables with PV ij;n = = 1 =PV ij;n = if j i; V ii;n = ; 17 V ij;n = V ji;n if j i:

6 4 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics If =, d = m = 1 and the diusion coecient b is a constant, from the third order weak Taylor scheme one can construct the simplied order weak Taylor scheme, given by X n1 = X n bŵ n a ba 01 Ẑ n 1 a 10 aa b a ba 01 ba 11 aba 0 b a 0 Ŵ n 1 6 b a 0 Ŵ n 1 6 a 10a aa 01 b a 01 a 0 a 0 aa 11 b a 1 ab a 0 a a b4 a 04 ; 18 where Ŵ n and Ẑ n are correlated Gaussian random variables with Ŵ n N0;; Ẑ n N0; =; E[Ŵ n Ẑ n ]= =: 19 In the sequel, for simplicity of notation we shall often abbreviate I ;n,ŵ n,ẑ n, etc.to I,Ŵ and Ẑ, respectively.obviously, we can obtain -equivalent schemes to the order scheme given in 11 by replacing the Î s by new variables Ĩ s satisfying [ l ] l E Î k Ĩ k 6 K 1 ; k {v}; l=1;:::; 1 0 k=1 k=1 for some constant K 0.For example, the new family can contain all the variables of the old one except one of them, say Î, replaced by Ĩ; if 0 holds in this case it reduces to compare the products which contain Î with the corresponding with Ĩ we shall write Î Ĩ.In general, we shall say that Î and Ĩ are -equivalent, in symbols Î Ĩ, if by replacing in an approximation the variable Î by Ĩ the new approximation is -equivalent to the old one.for example, in the scalar case, if Ŵ is as in 1 and the function At; x; ; Ŵ of the one-step approximation is a linear combination of products of the form i Ŵ j, i; j =0; 1;::: note that the simplied Taylor approximation 14 belongs to this class, by 0 it s clear that Ŵ Ŵ; Ŵ ; 1 Ŵ 4 : Each variable i Ŵ j with mean-square order 5= i.e. i j== 5 is -equivalent to zero because it has zero mean and its product with every variable of the family has at least mean-square order.obviously, the variables i Ŵ j with mean-square order ; 7 ; 4;::: are -equivalent to zero: i Ŵ j 0 if i j= 5 : In the multi-dimensional case, when Ŵ 1 ;:::;Ŵ m are independent random variables satisfying the moment conditions 1 and the function At; x; ; Ŵ is a linear combination of products of the form i Ŵ 1 j1 Ŵ m jm and variables V ij satisfying 17, we have the following -equivalences: { Ŵ i Ŵ j if i = j; 0 if i j;

7 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics Ŵ i if i = j = k; Ŵ i Ŵ j Ŵ k Ŵ i if j = k i; 0 if i j; i k;j k; 4 i Ŵ 1 j1 Ŵ m jm 0 if i j 1 j m 5 : 5 Similarly, in the scalar case, if Ŵ and Ẑ verify 19 and the function At; x; ; Ŵ which denes the approximation is a linear combination of products i Ŵ j, i; j =0; 1;::: and Ẑ, we have the following -equivalences: Ŵ Ŵ ; Ŵ ; i Ŵ j 0 if i j= 7 : 6. Runge--Kutta schemes An important disadvantage of simplied Taylor schemes is that they require to determine many derivatives.using the idea of the deterministic case we shall obtain RK schemes by replacing the derivatives in simplied Taylor schemes by new evaluations of the coecients of the equation.an explicit s-stage stochastic Runge Kutta scheme will be given by X n1 = X n where 1 =0, 1 = X n, s j at n j ; j Ŵ k n k=1 s j k b k t n j ; j R; 7 j 1 j = X n ji at n i ; i Ŵ k n k=1 j 1 k jib k t n i ; i ; j =1;:::;s; and R is a t term.the numerical constants j ; k j ; j ; ij ; k ij and the term R must be chosen so that the approximation 7 is -equivalent to the simplied order Taylor scheme.since the truncated expansion of order of a process is, under appropriate conditions, -equivalent to the process, it suces to choose the parameters and R so that a -equivalent approximation to the truncated expansion of order of 7 is equal to the simplied order Taylor scheme.note that the t term is free and then it s obvious that for every family of parameters it can be chosen so that the required equality is fullled.but our goal is to avoid the use of derivatives in the scheme.then, for eciency, the number of derivatives in R must be notoriously smaller than in the Taylor scheme. If for a family of parameters we have the equality with R = 0, the scheme does not involve any derivative; it s then a Runge Kutta scheme in strict sense.if R contains one or more derivatives we shall say that it s a Runge Kutta type scheme.for example, the second order scheme in Milstein [1, pp ], is a RK type scheme with s = 4 which contains one derivative.

8 6 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics Generalizing Butcher arrays, the coecients occurring in 7 can be displayed m s s1 s;s 1 1 s1 1 s;s 1 m s1 m s;s 1 R 1 s 1 s s 1 1 s m 1 m s 1 m s where the rst matrix contains the coecients corresponding to the deterministic part and each of the remaining ones contains the coecients corresponding to the stochastic part with respect to a Wiener component. As in the deterministic case, in order to match the truncated expansion of 7 with the simplied Taylor scheme we need an expression of the order truncated expansion of a process ft ; X t X in terms of and X = X t X t.this expression has been obtained by Tocino and Ardanuy [17] for = in the multi-dimensional case and for = in the scalar case.so, the formula ft ; X t X f c j 1 i;j;k=1 f i l j;k=1 c jk 1 k 4 f X k c ij c 4 f X i X j j 8 where all of 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 15, 8 reduces to 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 : 9 Similarly, in the scalar case, if bt; x=b is a constant, the formula ft ; X t X f 00 f 10 f 01 X i; f 0 b4 4 f 04 f 11X f 0 X f 0 b f 4 b4 f b6 f 06 6

9 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics f 1 b f 1 b4 4 f X 05 f 1 b X f X 04 f 0 : 0 6 shows the -equivalence between the process and its third order truncated expansion. 4. Two-stage Runge--Kutta schemes In the scalar case, for s = the RK scheme 7 takes the form X n1 = X n { 1 a at n ; } { 1 b bt n ; }Ŵ R; 1 where = X n a bŵ ; for simplicity, in the above expression and from now on, when a function in a scheme is evaluated at t n ; X n we shall omit such point; notice that we have also abbreviated Ŵ n to Ŵ.The coecients a and b are supposed to belong to CP 6.By 9 and the equivalences 1 and we shall obtain a -equivalent approximation to 1 and we shall match this approximation with the simplied order Taylor scheme 14. By 9 we get at n ; X n a bŵ a a 10 a 01 a bŵ 1 a 0 b a 1 b b 01 a 0 b4 4 a 04 a 11 b a 0 a bŵ and then, using 1 and, we have 1 a 0a bŵ at n ; X n a bŵ a 01 bŵ a Similarly, we get a 10 aa 01 1 a 0b : bt n ; X n a bŵ Ŵ bŵ bb 01 Ŵ b 10 Ŵ ab 01 Ŵ b b 0 Ŵ bb 11 1 b b 0 abb 0 :

10 8 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics From and we see that approximation 1 is -equivalent to X n1 = X n 1 bŵ 1 a bb 01 Ŵ a 01 b b 10 ab 01 b b 0 Ŵ a 10 aa 01 1 a 0 b bb 11 1 b b 0 abb 0 R: 4 Now, we compare the above approximation with the simplied order Taylor approximation 14. Firstly, let s suppose = k is a constant.since in this case b 11 = b 0 = b 0 = 0, 4 and 14 coincide if the constants verify 1 =1; 1 =1; = 1 ; = 1 ; = 1 ; = 1 ; = 1 ; = 1 ; = 1 ; and R= 1 bb 01.Notice that the rst column contains the equations which appear in the deterministic case b 0 and the second one contains the analogues for the stochastic part of the scheme.the equations in the third column contain both deterministic and stochactic parameters.the above system has the unique solution = = =1, 1 = = 1 = =1=, which leads to the scheme X n1 = X n 1 bŵ n 1 bt n ; X n a bŵ n Ŵ n 1 a 1 at n ; X n a bŵ n 1 b@b : By construction, this scheme is -equivalent to the simplied order Taylor scheme.on the other hand, since b is continuous and we have supposed is a constant, we have that b is a Lipschitz function which grows at most linearly and therefore it s immediate to show that if a also grows at most linearly then approximation 5 veries conditions 5.By Theorem 1, we deduce the following. Theorem. Suppose that the coecients of Eq.1 belong to CP 6 ; that the drift a veries both Lipschitz and linear growth conditions in x and is a constant. Then the scheme given in 5 is of second order in the weak sense. shall be a known constant, at each step, in scheme 5 we have to evaluate the drift a at two points, the diusion coecient b at two points, as well as generating the random variable Ŵ n ; remember that in the Taylor scheme we have to evaluate @a=@t a=@x at one point and to generate Ŵ n.notice also that this scheme coincides in the analysed case with the second order RK scheme proposed by Milstein [10]. The Butcher array of the scheme would be bb To match 4 with 14 in a more general case we can take the same parameters, but R must change because new partial derivatives could appear.for example, b=@x = 0 then 14 contains

11 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics the same terms that in the above case but in 4 it appears the term bb 11 = 1 bb 11 ;so we must take R = 1 bb 01 b 11.Note that now the scheme contains two derivatives.and it s clear that in more general cases we can continue in this way i.e. with the same parameters and by increasing the number of derivatives in R to derive second order schemes of the form 1. Notice that we have also derived that only is a constant there exists a second order Runge Kutta scheme in strict sense as 7 with s =. 5. A class of second order schemes In view of the above results we must take s in 7 in order to obtain for the general case a second order RK scheme that does not include most of the derivatives participating in 14.So, let us consider schemes of the form X n1 = X n { 1 a at n ; } where { 1 b bt n ; bt n ; }Ŵ R; 6 = X n a bŵ; = X n a bŵ; = X n a bŵ: Using 9, 1 and analogously as in the above section, we shall obtain that the approximation 6 is -equivalent to X n1 = X n 1 bŵ 1 a bb 01 Ŵ a 01 bŵ b 10 Ŵ ab01 Ŵ b b 0 Ŵ a 10 aa 01 1 a 0 b bb 11 1 b b 0 abb0 R: 7 This approximation will be equal to the simplied second order Taylor approximation 14 if the parameters satisfy the nonlinear system 1 =1; = 1 ; = 1 ; = 1 ; = 1 ; 1 =1; = 1 ; = 1 ; =0; = 1 6 ; =0 8

12 0 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics and if R = 1 bb 01Ŵ.The left column system has the unique solution 1 = = 1 ; = = =1: 9 On substituting 9 on the right column of 8, it s easy to see that the system obtained has the one-parameter solution 1 = 1 ; 1 = 6 ; = 6 ; = 1 =1; = ; 0: 40 Then, taking R = 1 bb 01Ŵ, for each 0, 9 and 40 give a scheme X n1 = X n 1 bŵ 1 n 6 bt n ; X n a bŵ n Ŵ n t 6 b n ; X n a 1 bŵ n Ŵ n 1 a 1 at n ; X n a bŵ n 1 Ŵ n ; 41 which, by construction, is -equivalent to the order Taylor scheme.on the other hand, it s easy to show that it veries 5 if a; b have at most linear growth, and so, by Theorem 1, we have proved the following. Theorem. Suppose that the coecients a; bofeq.1 satisfy a Lipschitz condition and belong to CP 6. If a; b have at most linear growth then the RK schemes of 41 are of second order in the weak sense. At each step in a scheme of the family 41 we have to evaluate the drift a at two points, the diusion coecient b at three points and the at one point, as well as generating a random variable Ŵ n satisfying 1. Notice that the second order RK scheme proposed by Milstein [10] belongs to the above class. 6. Multi-dimensional generalization The construction of a second order schemes family in the multi-dimensional case can be accomplished by the procedure of the previous section.now we shall consider schemes whose kth component can be written as X k n1 = X k n { 1 a k a k t n ; } { j 1 bkj j bkj t n ; j bkj t n ; }Ŵ j R; 4

13 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics where = X n a 1 b 1 Ŵ 1 m b m Ŵ m ; = X n a 1 b 1 Ŵ 1 m b m Ŵ m ; = X n a 1 b 1 Ŵ 1 m b m Ŵ m : Here the constants and the t term R must be chosen so that the above scheme is -equivalent to the simplied order Taylor scheme 16. By 8 and the equivalences 5 we obtain that a t k n ; X n a r b r Ŵ r and that a k i 1 b kj t n ; X n a b kj Ŵ kj i bij j r=1 i; a j bir b jr r 4 r b r Ŵ r Ŵ j i bir r Ŵ r Ŵ i ai Ŵ j b kj i s;l=1 c sl i;l=1 b l bir b ls r s Ŵ r Ŵ s Ŵ b kj b l b l ai b lj j 44 if j =1;:::;m; besides, a similar equivalence to 44 can be obtained for each b kj t n ; Ŵ j with and r replaced by and r, respectively.using these last three equivalences we get the -equivalent to 4 scheme X k n1 = X k n j 1 j j bkj Ŵ j 1 a k i; j i kj l bli Ŵ i Ŵ j l=1 i;l=1 i bij Ŵ j

14 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics Ŵ j i ai Ŵ j 1 j;r; s=1 j r s j @ b l bir b ls Ŵ r Ŵ s Ŵ j i;l=1 i 1 j j b kj i 1 i; r=1 s;l=1 r c a j bir b b kj b l j j b kj l ai b lj R: 45 i;l=1 By 4 we have j r s b kj r l bir b ls Ŵ r Ŵ s Ŵ j j;r; s=1 i;l=1 j j j b l bij b lj Ŵ j i;l=1 j r j b l bir b lr Ŵ j r=1 r j r=1 r j i;l=1 r r j b kr r l bij b lr Ŵ j ; i;l=1 with this substitution, scheme 45 shall be equal to the simplied order Taylor scheme 16 if the parameters verify the systems 1 =1; = 1 ; = 1 ; j = 1 ; j = 1 ; j 1 j j =1;

15 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics j j = 1 ; j j = 1 ; j j j j =0; j j j j = 1 6 ; j j j j =0; j =1;:::;m 46 and j i j i =0; i;j =1;:::;m; i j 47 and if R = 1 b Ŵ i Ŵ j l ij i; l=1 r=1 r j r=1 r j j r j r b l bir b lr Ŵ j r r j b kr r l bij b lr Ŵ j : i;l=1 For each j =1;:::;m system 46 is analogous to 8; so, we can deduce that 46 has the m-parametric family of solutions 1 = = 1 ; j = = =1; j 1 = 1 ; j = 1 6 ; j = j j 6 ; = =1; j = 1 ; j j where j 0;;:::;m.By using the above values with 47 we obtain that i = j, i; j =1;:::;m; if we denote = 1, we can write i = i, i = i =, where 1 =1; i {±1}; i=;:::;m, and then j = 1 6 ; j = 6 ; j =1;:::;m and R = 1 b Ŵ i Ŵ j l ij i; 1 6 l=1 r=1 i;l=1 r b l bir b lr r b l bij b lr Ŵ j : 48

16 4 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics This family of solutions leads to the class of schemes { X k n1 = X k 1 n b kj 1 1 bkj t n ; X n a } 1 bkj t n ; X n a 1 r b r Ŵ r Ŵ j {a 1 k a k t n ; X n a r=1 r b r Ŵ r r=1 } b r Ŵ r R; r=1 where 0; 1 =1; ;:::; m {±1}; Ris given by 48 and the variables Ŵ j and V ij satisfy 1 and 17, respectively.they are -equivalent to the simplied order Taylor scheme; with the same reasoning employed in the scalar case we can conclude that these schemes are of second order in the weak sense if a and b belong to CP 6, verify a Lipschitz condition and, together with the partial kj =@x i ;@ b kj =@x l ; i;l;k=1;:::;d; ;:::;m, have at most linear growth.in particular, if = = m = 1 we have the one-parameter class of schemes { X k n1 = X k 1 n b kj 1 1 bkj t n ; X n a bŵ 1 bkj t n ; X n a 1 } bŵ Ŵ j 1 {ak a k t n ; X n a bŵ } R; where 0 and R is given by 48 with j =1, j =1;:::;m. 7. Third order RK schemes By the same technique employed in previous sections we will obtain now two third order RK schemes for scalar stochastic dierential equations d = m = 1 with constant diusion coecient bt; x=b.in this case a three-stage RK scheme of the form 7 can be written as where X n1 = X n { 1 a at n ; at n ; } bŵ R; 49 = X n 1 a bŵ; = X n 1 a at n ; bŵ: Here we have to nd out the parameters and R in such a way that 49 is -equivalent to the simplied order Taylor scheme 18.And, as before, it suces to choose them after replacing 49 by its third order truncated expansion.

17 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics We begin by evaluating the third order truncated expansion of at n ;.By using 0 and equivalences 6 we get at n ; X n 1 a bŵ a a 01 b Ŵ a 10 aa a 0b Ŵ a 11 b Ŵ aa 0 b 1 Ŵ 1 a 0b Ŵ aa a a 0 1 a 1 1 b a 04b 1 aa 0b 1 1 a 0 b4 4 a 04 : 50 Now let s calculate the third order truncated expansion of at n ;.For simplicity, we denote M = X n = 1 a at n ; bŵ.by 6 we have that 4 0 M, and then, by using 0 we get at n ; X n M a a 10 a 01 M a 0 b4 4 a 04 a 11 M M a 0 a 1 b a M M 04 a 0 : 6 51 The third order truncated expansion of M can be obtained from 50; and from its value, using 6, it s easy to show the equivalences M bŵ 1 a a 01 b Ŵ a 10 aa a 0b ; 5 M b Ŵ 1 a ; 5 M b Ŵ ab 1 Ŵ 1 a a 01 b ; 54 M b ; 55 M b Ŵ ab 1 : 56 Substituting 5 56 into 51 and using the obtained expression together with 50 we have that the scheme 49 shall be -equivalent to X n1 = X n bŵ 1 a a 01 b Ŵ a 10 aa a 0b Ŵ ba 01 Ŵ

18 6 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics ba 11 Ŵ aa 0 b 1 1 Ŵ 1 a 0b Ŵ 1 a 1 1 a 04b b 1 aa 0b a 01a 0 b 1 a0 1 4 b4 a 04 aa a a a 01 a 10 aa 01 1 R: 57 Schemes 18 and 57 coincide if the constants satisfy the system 1 =1; = 1 ; 1 1 = 1 ; = 1 ; 1 1 = 1 ; 1 1 = 1 ; = 1 6 ; 1 = 1 6 ; =1; = 1 ; 1 1 = 1 ; = 1 ; = 1 ; 1 1 = 1 ; = 1 ; = 1 ; = 1 6 ; 58 and if R = ba 01 {Z W } 1 1 b a 0.The equations on the left column are the same which appear in the deterministic case.as it s known, see REA [1], they reduce to the system 1 =1; = 1 ; = 1 ; = 1 6 ; 1 = ; 1 = ; 59 Using 59, the equations on the right column of 58 reduce to =1; = ; = ;

19 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics = 1 ; = 60 and R = ba 01 Z 1 W 1 1 b a 0.The ordinary case system 59 has two one-parameter and one two-parameter families of solutions; see [].It s easy to see that none of the solutions of the one-parameter families one of the families corresponds to the case =0, ==; the other one to the case = = can be a solution of 60.On the other hand, since we have in the two-parameter family that 1 = 1 1 ; = 1 ; ; 0; 0; ; by imposing on a solution of this family to verify 60 we get =0; the roots of this equation are =1, =1= and =4=.If = 1 then = and, as we have said, these solutions do not verify 60.Each of the other roots leads to a solution of system If = 1 we have the solution == ; = 1 ; 1 = 1 1 ; = 1 6 ; = 8 9 ; 1 =; 1 = 1 ; = ; and if = 4 we have = 1 = ; = 4 ; 1 = 1 8 ; =1; = 1 8 ; 1 = 1 ; 1 = 8 ; =4: Each solution denes a scheme -equivalent to the simplied order Taylor scheme.the rst one is given by X n1 = X n bŵ n 1 1 a 1 6 at n ; X n bŵ n a 8 9 a t n ; X n 1 bŵ n 1 a at n ; X n bŵ n a ba 01 Z n 1 Ŵ n 1 1 b a 0 ; 61 and the second one by X n1 = X n bŵ n 1 8 a a t n 1 ; X n 1 bŵ n 1 a 1 8 a t n 4 ; X n 4 bŵ n 8 a 4a t n ; X n 1 bŵ n 1 a ba 01 Z n 1 Ŵ n 1 1 b a 0 : 6

20 8 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics Their Butcher arrays are, respectively, R R Since it s immediate to show that these two schemes verify conditions 5 if a=@x grow at most linearly, we have the following. Theorem 4. Suppose that the diusion coecient b in the sde 1 is a constant. Suppose also that the drift coecient a belongs to CP 8 ; veries a Lipschitz condition and; together with the partial a=@x ; have at most linear growth. Then the RK schemes 61 and 6; where Ŵ n ; Z n are variables satisfying 19; have order in the weak sense. Both schemes require at each step to evaluate a at three at one point a=@x at one point and to generate the variables Ŵ n,z n verifying 19 note that the simplied Taylor scheme also requires to evaluate a 10 ;a 11 ;a 0 ;a 0 ;a 1 ;a Numerical results In this section, numerical results from the implementation of the second and third order schemes proposed in the paper are compared to those from the implementation of well-known schemes of the same order; we have also used in each example a lower order scheme to contrast with them. The two-stage order Runge Kutta method proposed in 5 will be denoted by RK-st; we will denote by RK-st the particular three-stage order Runge Kutta method of the family 41 with =1=; the order Runge Kutta method proposed in 61 will be denoted by RK.In order to compare with them, we shall use the Euler method, the simplied order Taylor method 14, denoted by Taylor, a second order Runge Kutta method proposed by Platen see [7, p.485], denoted by RK-PL, and the simplied order Taylor scheme 18. As test problems we have taken linear and nonlinear one-dimensional stochastic dierential equations d = m = 1 for which the exact solution X t in terms of the Wiener process is known; our aim is to simulate the known value E[gX T ], where we have chosen gx=x or gx=x. For each example we have used N = 5000 simulations for step sizes = 1 ;:::; 5 to compute the approximated value of the known expectation.the mean and the standard deviation of the errors for each considered scheme are summarized in Tables 1 4. Example 1. Consider the nonautonomous linear equation dx t =t X t dt t dw t ; X 0 =1:

21 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics Table 1 Euler Taylor RK-st Error St.dev. Error St.dev. Error St.dev Table Euler RK-PL RK-st Error St.dev. Error St.dev. Error St.dev Table Euler RK-PL RK-st Error St.dev. Error St.dev. Error St.dev Table 4 Taylor Taylor RK Error St.dev. Error St.dev. Error St.dev

22 40 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics We estimate the exact value E[X T ]=e 1 T at the point T = 0 we can apply the second order RK-st scheme.in this case this scheme coincides with RK-st.We compare them with the Taylor scheme; the Euler scheme is also included.the values are summarized in Table 1.The results obtained using the RK-st scheme are similar to those from the second order Taylor scheme; the advantage of RK-st is that it is derivative free. Example. Consider the nonlinear stochastic dierential equation 1 dx t = X 1= t 6X = t dt X = t dw t ; X 0 =1: with solution X t =t 1W t =.Since the equation is autonomous, we can use the second order RK-PL scheme to estimate the exact value E[X 1 ] = 8 and compare the obtained values with those from the second order RK-st scheme.the results are summarized in Table. Example. Consider the nonlinear equation given in Example above.here we approximate the value E[X1 ] = using RK-PL and RK-st schemes; the results obtained are shown in 5 Table. In Examples and we observed practically no dierences between the proposed second order Runge Kutta scheme RK-st and the second order one proposed by Platen.RK-PL is completely derivative free; in opposition, RK-st has only one derivative, is simpler and valid for the nonautonomous case. Example 4. Consider the linear nonautonomous stochastic dierential equation dx t =tx t 10tdt b dw t ; X 0 =10; with constant diusion coecient b =0:1.The exact value E[X 1 ]=0e 1= 10 was estimated using Taylor and the RK schemes, both of third order.to simulate the Gaussian variables Ŵ n and Z n satisfying 19 we have taken Ŵ = U 1 and Z = 1 = U 1 1 U, where U 1 and U are two independent Gaussian variables with distribution N0; 1.Although the results, summarized in Table 4, are not extensive enough to justify general conclusions, they suggest that RK is more ecient than Taylor. References [1] L.Arnold, Stochastic Dierential Equations, Wiley, New York, [] J.C.Butcher, The Numerical Analysis of Ordinary Dierential Equations.Runge Kutta and General Linear Methods, Wiley, Chichester, [] C.C.Chang, Numerical solution of stochastic dierential equations with constant diusion coecients, Math.Comput [4] D.B. Hernandez, R. Spigler, A-stability of Runge Kutta methods for systems with additive noise, BIT

23 A. Tocino, R. Ardanuy / Journal of Computational and Applied Mathematics [5] D.B.Hernandez, R.Spigler, Convergence and stability of implicit Runge Kutta methods for systems with multiplicative noise, BIT [6] J.R. Klauder, W.P. Petersen, Numerical integration of multiplicative-noise stochastic dierential equations, SIAM J. Numer.Anal [7] P.E.Kloeden, E.Platen, Numerical Solution of Stochastic Dierential Equations, Springer, Berlin, 199. [8] S.Mauthner, Step size control in the numerical solution of stochastic dierential equations, J.Comput.Appl.Math [9] E.J. Mcshane, Stochastic Calculus and Stochastic Models, Academic Press, New York, [10] G.N. Milstein, A method of second order accuracy integration of stochastic dierential equations, Theory Probab. Appl [11] G.N.Milstein, Weak approximation of solutions of systems of stochastic dierential equations, Theory Probab.Appl [1] G.N. Milstein, Numerical Integration of Stochastic Dierential Equations, translated version of 1988 Russian work, Kluwer, Dordrecht, [1] REA Research, Education Association, Advanced Methods for Solving Dierential Equations, New York, 198. [14] W.Rumelin, Numerical treatment of stochastic dierential equations, SIAM J.Numer.Anal [15] Y.Saito, T.Mitsui, Stability analysis of numerical schemes for stochastic dierential equations, SIAM J.Numer. Anal [16] D.Talay, Ecient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications, in: H.Korezlioglu, G.Mazziotto, J.Szpirglas Eds., Lecture Notes in Control and Inform.Sci., Vol.61, Springer, Berlin, 1984, pp [17] A.Tocino, R.Ardanuy, Truncated Itô Taylor expansions, Stochastic Anal.Appl., to appear.