B SERIES ANALYSIS OF STOCHASTIC RUNGE KUTTA METHODS THAT USE AN ITERATIVE SCHEME TO COMPUTE THEIR INTERNAL STAGE VALUES

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1 SIAM J. NUMER. ANAL. Vol. 47, No. 1, pp c 008 Society for Industrial and Applied Mathematics B SERIES ANALYSIS OF STOCHASTIC RUNGE KUTTA METHODS THAT USE AN ITERATIVE SCHEME TO COMPUTE THEIR INTERNAL STAGE VALUES KRISTIAN DEBRABANT AND ANNE KVÆRNØ Abstract. In recent years, implicit stochastic Runge Kutta SRK methods have been developed both for strong and wea approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for Itô- and Stratonovich-stochastic differential equations SDEs and applicable both for wea and strong convergence to analyze the order of the iterated Runge Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts. Key words. stochastic Runge Kutta method, composite method, stochastic differential equation, iterative scheme, order, internal stage values, Newton s method, wea approximation, strong approximation, growth functions, stochastic B-series AMS subject classifications. 65C30, 60H35, 65C0, 68U0 DOI / Introduction. In many applications, e.g., in epidemiology and financial mathematics, taing stochastic effects into account when modeling dynamical systems often leads to stochastic differential equations SDEs. Therefore, in recent years, the development of numerical methods for the approximation of SDEs has become a field of increasing interest; see, e.g., [16, ] and references therein. Many stochastic schemes fall into the class of stochastic Runge Kutta SRK methods. SRK methods have been studied both for strong approximation [1, 10, 11, 16], where one is interested in obtaining good pathwise solutions, and for wea approximation [8, 9, 16, 19, 1, 3], which focuses on the expectation of functionals of solutions. Order conditions for these methods are found by comparing series of the exact and the numerical solutions. In this paper, we will concentrate on the use of B-series and rooted trees. Such series are surprisingly general; as formal series they are independent of the choice of the stochastic integral, Itô or Stratonovich, or whether wea or strong convergence is considered. This is demonstrated in section. For solving SDEs which are stiff, implicit SRK methods have to be considered, which also has been done both for strong [4, 11, 1] and wea [7, 1, 17] approximation. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes lie simple iteration or some ind of Newton iterations. For the numerical solution of ODEs such iterative schemes have been studied [13, 14], and it was shown that certain growth functions defined on trees give a precise description of the development of the iterations. Exactly the same growth functions apply to SRKs, as we prove in section 3. Only when these results Received by the editors October, 007; accepted for publication in revised form June 9, 008; published electronically October 9, Fachbereich Mathemati, Technische Universität Darmstadt, Schloßgartenstr.7, D-6489 Darmstadt, Germany debrabant@mathemati.tu-darmstadt.de. Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway anne@math.ntnu.no. 181

2 18 KRISTIAN DEBRABANT AND ANNE KVÆRNØ are interpreted in terms of the order of the overall scheme, we distinguish Itô from Stratonovich SDEs, wea from strong convergence. This is discussed in sections 4 and 5. When considering strong convergence, it is difficult to implement fully implicit SRK methods in combination with Newton iterations due to the possible singularity of the numerical procedure. Therefore, various techniques have been developed to circumvent this problem [4]. One possibility is the use of so-called truncated random variables, which have finite distribution and can approximate the increment of Wiener processes to a chosen order [4, 3]. As the concrete choice of random variables in the numerical methods is not specified in this paper, all considerations are without any change also valid for SRK methods with such modified random variables. Another possibility is to use composite methods [31], which are combinations of a semi-implicit SRK and an implicit SRK. Based on the results for conventional SRK methods, convergence results for iterated composite methods are given in section 6. Finally, in section 7 we present two numerical examples. Let Ω, A, P be a probability space. We denote by Xt t I the stochastic process which is the solution of either a d-dimensional Itô or Stratonovich SDE defined by t t 1.1 Xt =x 0 + g 0 Xsds + g l Xs dw l s t 0 l=1 t 0 with an m-dimensional Wiener process W t t 0 and I =[t 0,T]. The integral w.r.t. the Wiener process has to be interpreted either as Itô integralwith dw l s =dw l s or as Stratonovich integral with dw l s = dw l s. We assume that the Borelmeasurable coefficients g l : R d R d are sufficiently differentiable and satisfy a Lipschitz and a linear growth condition. For Stratonovich SDEs, we require in addition that the g l are differentiable and that also the vectors g l g l satisfy a Lipschitz and a linear growth condition. Then the existence and uniqueness theorem [15] applies. To simplify the presentation, we define W 0 s =s, so that 1.1 can be written as t 1. Xt =x 0 + g l Xs dw l s. l=0 t 0 In the following we denote by Ξ a set of families of measurable mappings, Ξ:= ϕh} h 0 : ϕh : Ω R is A B-measurable h 0 }, and by Ξ 0 the subset of Ξ defined by Ξ 0 := ϕh} h 0 Ξ: ϕ0 0 }. Let a discretization I h = t 0,t 1,...,t N } with t 0 <t 1 <...<t N = T of the time interval I with step sizes h n = t n+1 t n for n =0, 1,...,N 1begiven. Now,we consider the general class of s-stage SRK methods given by Y 0 = x 0 and 1.3a Y n+1 = Y n + l=0 ν=0 z l,ν I d g l H l,ν for n =0, 1,...,N 1withY n = Y t n, t n I h, I d R d,d the identity matrix, and 1.3b H l,ν = 1 s Y n + r=0 Z l,νr,μ I d g r H r,μ

3 CONVERGENCE OF ITERATED SRK METHODS 183 for l =0,...,m and ν =0,...,M with 1 s =1,...,1 R s, g l H l,ν = g l,...,gl and z l,ν Ξ s 0, H l,ν 1 H l,ν s Zl,νr,μ Ξ s,s 0 for l, r =0,...,m, μ, ν =0,...,M. The formulation 1.3 is a slight generalization of the class considered in [7] and includes the classes of SRK methods considered in [4, 11, 18, 0, 8, 9, 30] as well as the SRK methods considered in [1, 16, 5]. It defines a d-dimensional approximation process Y with Y n = Y t n. Application of an iterative scheme yields 1.4a 1.4b H l,ν +1 = 1 s Y n + + Y n+1, = Y n + r=0 r=0 Z l,νr,μ I d g r H r,μ Z l,νr,μ I d J r,μ H r,μ +1 Hr,μ, l=0 ν=0 z l,ν I d g l H l,ν with some approximation J r,μ to the Jacobian of g r H r,μ and a predictor H l,ν 0. In the following we assume that 1.4a can be solved uniquely at least for small enough h n. We consider simple iterations with J r,μ = 0 i.e., predictor-corrector methods, modified Newton iterations with J r,μ = I s g r x 0, and full Newton iterations.. Some notation, definitions, and preliminary results. In this section we introduce some necessary notation and provide a few definitions and preliminary results that will be used later..1. Convergence and consistency. Here we will give the definitions for both wea and strong convergence and results which relate convergence to consistency. Let CP l Rd, R ˆd denote the space of all g C l R d, R ˆd fulfilling a polynomial growth condition [16]. Definition 1. A time discrete approximation Y =Y t t I h converges wealy with order p to X as h 0 at time t I h if for each f C p+1 P R d, R there exist a constant C f and a finite δ 0 > 0 such that EfY t EfXt C f h p holds for each h ]0,δ 0 [. Now, let le f h; t, x be the wea local error of the method starting at the point t, x with respect to the functional f andstepsizeh, i.e., le f h; t, x =E fy t + h fxt + h Y t =Xt =x. The following theorem due to Milstein [], which holds also in the case of general one-step methods, shows that, as in the deterministic case, consistency implies convergence.

4 184 KRISTIAN DEBRABANT AND ANNE KVÆRNØ Theorem 1. Suppose the following conditions hold: The coefficients g l are continuous, satisfy a Lipschitz condition, and belong to C p+1 P R d, R d for l =0,...,m. For Stratonovich SDEs, we require in addition that the g l are differentiable and that also the vectors g l g l satisfy a Lipschitz condition and belong to C p+1 P R d, R d for l =0,...,m. For sufficiently large r see, e.g., [] for details the moments E Y t n r exist for t n I h and are uniformly bounded with respect to N and n = 0, 1,...,N. Assume that for all f C p+1 P R d, R there exists a K CP 0 Rd, R such that le f h; t, x Kx h p+1 is valid for x R d and t, t + h I h, i.e., the approximation is wea consistent of order p. Then the method 1.3 is convergent of order p in the sense of wea approximation. Whereas wea approximation methods are used to estimate the expectation of functionals of the solution, strong approximation methods approach the solution pathwise. Definition. A time discrete approximation Y =Y t t I h converges strongly, respectively, in the mean square with order p to X as h 0 at time t I h if there exists a constant C and a finite δ 0 > 0 such that E Y t Xt Ch p, respectively, E Y t Xt Ch p holds for each h ]0,δ 0 [. In this article we will consider convergence in the mean square sense. But as by Jensen s inequality we have E Y t Xt E Y t Xt, mean square convergence implies strong convergence of the same order. Now, let le m h; t, x andle ms h; t, x, respectively, be the mean and mean square local error, respectively, of the method starting at the point t, x with respect to the step size h; i.e., le m h; t, x =E Y t + h Xt + h Y t =Xt =x, le ms h; t, x = E Y t + h Xt + h Y t =Xt =x. The following theorem due to Milstein [] which holds also in the case of general one step methods shows that in the mean square convergence case we obtain order p if the mean local error is consistent of order p and the mean square local error is consistent of order p 1. Theorem. Suppose the following conditions hold: The coefficients g l are continuous and satisfy a Lipschitz condition for l = 0,...,m,andE Xt 0 <. For Stratonovich SDEs, we require in addition that the g l are differentiable and that also the vectors g l g l satisfy a Lipschitz condition. There exists a constant K independent of h such that le m h; t, x K 1+ x h p1, le ms h; t, x K 1+ x h p+ 1

5 CONVERGENCE OF ITERATED SRK METHODS 185 with p 0, p 1 p +1 is valid for x R d, and t, t + h I h ; i.e., the approximation is consistent in the mean of order p 1 1 p and in the mean square of order p 1. Then the SRK method 1.3 is convergent of order p in the sense of mean square approximation. For Stratonovich SDEs, this result is also obtained by Burrage and Burrage []... Stochastic B-series. In this section we will develop stochastic B-series for the solution of 1. as well as for the numerical solution given by 1.3. B-series for deterministic ODEs were introduced by Butcher [6]. Today such series appear as a fundamental tool to do local error analysis on a wide range of problems. B-series for SDEs have been developed by Burrage and Burrage [1, ] to study strong convergence in the Stratonovich case, by Komori, Mitsui, and Sugiura [0] and Komori [18] to study wea convergence in the Stratonovich case, and by Rößler [6, 7] to study wea convergence in both the Itô and the Stratonovich case. However, the distinction between the Itô and the Stratonovich integrals depends only on the definition of the integrals, not on how the B-series are constructed. Similarly, the distinction between wea and strong convergence depends only on the definition of the local error. Thus, we find it convenient to present a uniform and self-contained theory for the construction of stochastic B-series. We will present results and proofs in a certain detail, since some intermediate results will be used in later sections. Following the idea of Burrage and Burrage, we introduce the set of colored, rooted trees related to the SDE 1.1, as well as the elementary differentials associated with each of these trees. Definition 3 trees. The set of m +1-colored, rooted trees T = } T 0 T 1 T m is recursively defined as follows: a The graph l =[ ] l with only one vertex of color l belongs to T l. Let τ =[τ 1,τ,...,τ κ ] l bethetreeformedbyjoiningthesubtreesτ 1,τ,...,τ κ each by asinglebranchtoacommonrootofcolorl. b If τ 1,τ,...,τ κ T,thenτ =[τ 1,τ,...,τ κ ] l T l. Thus, T l is the set of trees with an l-colored root, and T is the union of these sets. Definition 4 elementary differentials. For a tree τ T the elementary differential is a mapping F τ :R d R d defined recursively by a F x 0 =x 0, b F l x 0 =g l x 0, c If τ =[τ 1,τ,...,τ κ ] l T l,then F τx 0 =g κ l x 0 Fτ 1 x 0,Fτ x 0,...,Fτ κ x 0. As will be shown in the following, both the exact and the numerical solutions, including the iterated solutions as we will see later, can formally be written in terms of B-series. Definition 5 B-series. Given a mapping φ : T Ξ satisfying φ h 1 and φτ0 0, τ T \ }. A stochastic B-series is then a formal series of the form Bφ, x 0 ; h = τ T ατ φτh F τx 0,

6 186 KRISTIAN DEBRABANT AND ANNE KVÆRNØ where α : T Q is given by α =1, α l =1, ατ =[τ 1,...,τ κ ] l = 1 r 1!r! r q! κ ατ j, where r 1,r,...,r q count equal trees among τ 1,τ,...,τ κ. If φ : T Ξ s,thenbφ, x 0 ; h =[Bφ 1,x 0 ; h,...,bφ s,x 0 ; h]. The next lemma proves that if Y h can be written as a B-series, then fy h can be written as a similar series, where the sum is taen over trees with a root of color f and subtrees in T. The lemma is fundamental for deriving B-series for the exact and the numerical solution. It will also be used for deriving wea convergence results. Lemma 3. If Y h =Bφ, x 0 ; h is some B-series and f C R d, R ˆd, then fy h can be written as a formal series of the form.1 fy h = βu ψ φ uh Gux 0, u U f where U f is a set of trees derived from T,by a [ ] f U f,andifτ 1,τ,...,τ κ T,then[τ 1,τ,...,τ κ ] f U f. b G[ ] f x 0 =fx 0 and Gu =[τ 1,...,τ κ ] f x 0 =f κ x 0 F τ 1 x 0,...,Fτ κ x 0. c β[ ] f =1and βu =[τ 1,...,τ κ ] f = 1 r 1!r!...r q! κ ατ j, wherer 1,r,..., r q count equal trees among τ 1,τ,...,τ κ. d ψ φ [ ] f h 1 and ψ φ u =[τ 1,...,τ κ ] f h = κ φτ jh. Proof. WritingY h asab-series,wehave fy h =f ατ φτh F τx 0 = κ=0 τ T 1 κ! f κ x 0 τ T \ } ατ φτh F τx 0 κ 1 κ! =fx 0 + κ! r κ=1 1!r! r q! τ 1,τ,...,τ κ} T \ } κ ατ j φτ j h f κ x 0 F τ 1 x 0,...,Fτ κ x 0, where the last sum is taen over all possible unordered combinations of κ trees in T. For each set of trees τ 1,τ,...,τ κ T we assign a u =[τ 1,τ,...,τ κ ] f U f. The theorem is now proved by comparing term by term with.1. Remar 1. For example, in the definition of wea convergence, just f C p+1 P R d, R is required. Thus fy h can be written only as a truncated B-series, with a remainder term. However, to simplify the presentation in the following we assume that all derivatives of f,g 0,...,g l exist. We will also need the following result. Lemma 4. If Y h =Bφ Y,x 0 ; h and Zh =Bφ Z,x 0 ; h and f C R d, R ˆd, then f Y hzh = βu Υuh Gux 0 u U f

7 with CONVERGENCE OF ITERATED SRK METHODS 187 Υ[ ] f h 0, Υ[u =[τ 1,...,τ κ ] f h = κ κ φ Y τ j h φ Z τ i h with βu and Gux 0 given by Lemma 3. The proof is similar to the deterministic case see [4]. When Lemma 3 is applied to the functions g l on the right-hand side of 1. we get the following result: If Y h =Bφ, x 0 ; h, then i=1 j i. g l Y h = τ T l ατ φ l τh F τx 0 in which 1 if τ = l, φ lτh = κ φτ j h if τ =[τ 1,...,τ κ ] l T l. Theorem 5. Bϕ, x 0 ; h with The solution Xt 0 + h of 1. can be written as a B-series ϕ h 1, ϕ l h =W l h, ϕτ =[τ 1,...,τ κ ] l h = h 0 κ ϕτ j s dw l s. Proof. Write the exact solution as some B-series Xt 0 + h =Bϕ, x 0 ; h. By. the SDE 1. can be written as ατ ϕτh F τx 0 =x 0 + ατ τ T τ T l Comparing term by term we get ϕ h 1, and ϕτh = h 0 l=0 h 0 ϕ lτs dw l s F τx 0. ϕ τs dw l s for τ T l, l =0, 1,...,m. The proof is completed by induction on τ. The same result is given for the Stratonovich case in [, 18], but it clearly also applies to the Itô case. The definition of the order of the tree, ρτ, is motivated by the fact that E W l h = h for l 1andW 0 h =h. Definition 6 order. The order of a tree τ T is defined by and ρ =0, ρ[τ 1,...,τ κ ] f = ρτ =[τ 1,...,τ κ ] l = κ ρτ i + i=1 κ ρτ i i=1 1 for l =0, 1 otherwise.

8 188 KRISTIAN DEBRABANT AND ANNE KVÆRNØ Table.1 Examples of trees and corresponding functions ρτ, ατ, and ϕτ. The integrals ϕτ are also expressed in terms of multiple integrals J... for the Stratonovich S and I... for the Itô I cases; see [16] for their definition. In bracet notation, the trees will be written as l, [[ ] 0 ] 1, [ 1, 1 ] 0,and[ 1, [, ] 1 ] 0, respectively. τ ρτ ατ ϕτh 1 if l =0 1 if l 0 1 W l h = 1 1 h 0 s1 0 W s ds dw 1 s 1 = h 0 W 1s ds = h if l =0 J l S I l I J,0,1 I,0,1 S I J 1,1,0 I 1,1,0 + I 0,0 S I 3 1 h 0 W 1s 1 s 1 0 W s dw 1 s ds 1 4J,,1,1,0 +J,1,,1,0 +J 1,,,1,0 S = 4I,,1,1,0 +I,1,,1,0 +I 1,,,1,0 +I 0,1,1,0 +I,,0,0 + I 1,0,1,0 + I 0,0,0 I In Table.1 some trees and the corresponding values for the functions ρ, α, andϕ are presented. The following result is similar to results given in [1]. Theorem 6. If the coefficients Z l,ν,r,μ Ξ s,s 0 and z l,ν Ξ s 0, then the numerical solution Y 1 as well as the stage values can be written in terms of B-series H l,ν = B,x 0 ; h, Y 1 = BΦ,x 0 ; h for all l, ν, with.3a.3b and.4a.4b h 1 s, r h = τ =[τ 1,...,τ κ ] r h = Φ h 1, Φ l h = Φτ =[τ 1,...,τ κ ] l h = ν=0 Z l,νr,μ 1 s, Z l,νr,μ z l,ν κ z l,ν 1 s, ν=0 κ Φ r,μ τ j h τ j h. Proof. WriteH l,ν as a B-series, that is H l,ν = ατ h I d 1 s F τx 0. τ T

9 CONVERGENCE OF ITERATED SRK METHODS 189 Use the definition of the method 1.3 together with. to obtain H l,ν = 1 s x 0 + r=0 ατ τ T r Z l,νr,μ Φ r,μ τh I d 1 s F τx 0 r with Φ r,μ r τh = Φr,μ 1 r τh,...,φr,μ s r τh. Comparing term-byterm gives the relations.3. The proof of.4 is similar. To decide the wea order we will also need the B-series of the function f, evaluated at the exact and the numerical solution. From Theorem 5, Theorem 6, and Lemma 3 we obtain fxt 0 + h = βu ψ ϕ uh Gux 0, u U f fy 1 = u U f βu ψ Φ uh Gux 0, with and κ ψ ϕ [ ] f h 1, ψ ϕ u =[τ 1,...,τ κ ] f h = ϕτ j h κ ψ Φ [ ] f h 1, ψ Φ u =[τ 1,...,τ κ ] f h = Φτ j h. So, for the wea local error it follows le f h; t, x = u U f βu E[ψ Φ uh ψ ϕ uh] Gux. For the mean and mean square local error we obtain from Theorem 5 and Theorem 6, le ms h; t, x = E ατ Φτh ϕτh F τx, τ T le m h; t, x = τ T ατ E Φτh ϕτh F τx. With all the B-series in place, we can now present the order conditions for the wea and strong convergence for both the Itô and the Stratonovich case. 1 We have wea 1 As usual we assume that method 1.3 is constructed such that Eψ ϕuh =Oh ρu u U f and ϕτh =Oh ρτ τ T, respectively, where especially in the latter expression the O - notation refers to the L Ω-norm and h 0. These conditions are fulfilled if for i, j =1,...,s, N = 0, 1,...} it holds that z l,ν i = Oh l =0 Oh 1 l>0, Zl,νr,μ ij = Oh l =0 Oh 1 l>0.

10 190 KRISTIAN DEBRABANT AND ANNE KVÆRNØ consistency of order p if and only if.5 E ψ Φ uh =Eψ ϕ uh+oh p+1 u U f with ρu p + 1, where ρu =[τ 1,...,τ κ ] f = κ ρτ j.5 slightly weaens conditions given in [7], and mean square global order p if [4].6 Φτh =ϕτh+o h p+ 1 τ T with ρτ p,.7 EΦτh =Eϕτh+O h p+1 τ T with ρτ p + 1 and all elementary differentials F τ fulfill a linear growth condition. Instead of the last requirement it is also enough to claim that there exists a constant C such that g j y C y Rm, j =0,...,M which implies the global Lipschitz condition and all necessary partial derivatives exist []. 3. B-series of the iterated solution and growth functions. In this section wewilldiscusshowtheiteratedsolutiondefinedin1.4canbewrittenintermsof B-series, that is, = B,x 0 ; h and Y 1, = BΦ,x 0 ; h. H l,ν For notational convenience, in the following the h-dependency of the weight functions will be suppressed, so Φτh will be written as Φτ. Further, all results are valid for all l =0,...,m and ν =0,...,M. Assume that the predictor can be written as a B-series, H l,ν 0 = B 0,x 0 ; h satisfying 0 =1 s and 0 τ =Oh ρτ τ T. The most common situation is the use of the trivial predictor H l,ν = 1 s x 0,forwhich 0 = 1 s and 0 τ =0otherwise. The iteration schemes we discuss here differ only in the choice of J r,μ in 1.4. For all schemes, the following lemma applies. Lemma 3. Lemma 7. If H l,ν, The proof follows directly from = B,x 0 ; h, theny 1, = BΦ,x 0 ; h with Φ 1, Φ l = z l,ν 1 s, Φ τ =[τ 1,...,τ κ ] l = ν=0 ν=0 z l,ν κ τ j. Further, with fy 1, = βu ψ Φ u Gux 0 u U f κ ψ Φ [ ] f =1, ψ Φ u =[τ 1,...,τ κ ] f = Φ τ j, where βu and Gux 0 are given in Lemma 3.

11 CONVERGENCE OF ITERATED SRK METHODS 191 Fig Examples of trees and their growth functions for simple h, modified Newtonr, and full Newton d iterations. We are now ready to study each of the iteration schemes. In each case, we will first find the recurrence formula for τ. From this we define a growth function gτ. Definition 7 growth function. Agrowthfunctiong : T N is a function satisfying τ = τ, τ T, gτ τ =Φl,ν τ, τ T, gτ +1, for all 0. This result should be sharp in the sense that in general there exists τ with 0 τ τ and τ τ when<gτ. FromLemma7wealso have Φ τ =Φτ τ =[τ 1,...,τ κ ] l T, g τ = max κ gτ i, 3. ψ Φ u =ψ Φτ u =[τ 1,...,τ κ ] f U f, g u = max κ g τ i. The growth functions give a precise description of the development of the iterations. As we will see, the growth functions are exactly the same as in the deterministic case see [13, 14]. However, to get applicable results, we will at the end need the relation between the growth functions and the order. Further, we will also tae advantage of the fact that E Φτ =0andEψ Φ u = 0 for some trees. These aspects are discussed in the next sections. Examples of trees and the values of the growth functions for the three iteration schemes are given in Figure 3.1. The simple iteration. Simple iterations are described by 1.4a with J r,μ =0, that is, 3.3 H l,ν +1 = 1 s x 0 + r=0 Z l,νr,μ I d g r H r,μ By. and Theorem 6 we easily get the next two results. Lemma 8. If H l,ν 0 = B 0,x 0 ; h, thenh l,ν = B +1 1 s, +1 τ =[τ 1,...,τ κ ] r = Z l,νr,μ The corresponding growth function is given by h =0, h[τ 1,...,τ κ ] l =1+ max κ hτ j..,x 0 ; h, where κ Φ r,μ τ j. The function hτ is the height of τ, that is, the maximum number of nodes along one branch. The functions h τ andh u are defined by 3., with g replaced by h.

12 19 KRISTIAN DEBRABANT AND ANNE KVÆRNØ The modified Newton iteration. In this subsection, we consider the modified Newton iteration 1.4a with J r,μ = I s g r x 0. The B-series for H l,ν and the corresponding growth function can now be described by the following lemma. Lemma 9. If H l,ν 0 = B 0,x 0 ; h, thenh l,ν = B,x 0 ; h with s, +1 τ = Z l,νr,μ κ Φ r,μ τ j if τ =[τ 1,...,τ κ ] r T and κ, Z l,νr,μ Φ r,μ +1 τ 1 if τ =[τ 1 ] r T. The corresponding growth function is given by rτ 1 if κ =1, r =0, r l =1, rτ =[τ 1,...,τ κ ] l = 1+ max κ rτ j if κ. The function rτ is one plus the maximum number of ramifications along any branch of the tree. Proof. The iteration scheme 1.4a can be rewritten in B-series notation as 3.5 ατ +1 τ F τx 0=1 x 0 τ T M + ατ r=0 τ T r M + ατ 1 r=0 τ 1 T r Z l,νr,μ Φ r,μ τ F τx 0 r Z l,νr,μ Φ r,μ +1 τ 1 Φ r,μ τ 1 g r x 0F τ 1 x 0, wherewehaveused..clearly, +1 1 s for all 0and +1 r= Z l,νr,μ 1 s, proving the lemma for τ = r =[ ] r. Next, let τ =[τ 1 ] r,whereτ 1. Then F τx 0 =g rx 0 F τ 1. Comparing equal terms on both sides of the equation, using ατ =ατ 1, we get +1 τ = M Z l,νr,μ Φ r,μ r τ+φr,μ +1 τ 1 Φ r,μ τ 1. Since Φ r,μ r τ = Φr,μ τ 1 the lemma is proved for all τ = [τ 1 ] r. For τ = [τ 1,...,τ κ ] r with κ the last sum of 3.5 contributes nothing, thus +1 τ = M which concludes the proof of 3.4. Z l,νr,μ Φ r,μ τ, r

13 CONVERGENCE OF ITERATED SRK METHODS 193 The second statement of the lemma is obviously true for τ =. Letτ be any tree satisfying rτ + 1. Then either τ =[τ 1 ] l with rτ 1 +1orτ =[τ 1,...,τ κ ] l with κ andrτ i. In the latter case, we have by the hypothesis, by 3.4 and Theorem 6, that +1 τ = M Z l,νr,μ κ Φ r,μ τ j = τ. In the first case, it follows easily by induction on τ that +1 τ =Φl,ν τ since +1 τ = M Zl,νr,μ Φ r,μ +1 τ 1. The full Newton iteration. In this subsection, we consider the full Newton iteration 1.4a with = g r. J r,μ It follows that the B-series for H l,ν 3.6 H r,μ Lemma 10. If H l,ν 0 = B 0,x 0 ; h, thenh l,ν +1 1 s, M +1 τ = + Z l,νr,μ Z l,νr,μ κ i=1 and the corresponding growth function satisfy. = B,x 0 ; h with Φ r,μ τ j κ κ j i τ j Φ r,μ +1 τ i Φ r,μ τ i, Φ r,μ where τ =[τ 1,...,τ κ ] r and the rightmost is taen to be 1 s if κ =1. The corresponding growth function is given by d =0, d l =1, max κ dτ =[τ 1,...,τ κ ] l = dτ j if γ =1, max κ dτ j+1 if γ, where γ is the number of subtrees in τ satisfying dτ i =max κ dτ j. The function d is called the doubling index of τ. Proof. Using. and Lemma 4 the scheme 1.4a can be written as ατ +1 τ F τx 0=1 x where τ T M + ατ Z l,νr,μ Φ r,μ τ F τx 0 r r=0 τ T r M + βu Z l,νr,μ Υ r,μ u Gux 0, r=0 u U gr Υ r,μ u =[τ 1,...,τ κ ] gr = κ κ i=1 j i τ j Φ r,μ +1 τ i Φ r,μ τ i. Φ r,μ

14 194 KRISTIAN DEBRABANT AND ANNE KVÆRNØ From the definition of F τ, Gu =[τ 1,...,τ κ ] gr x 0 =Fτ =[τ 1,...,τ κ ] r x 0. The sum over all u U gr can be replaced by the sum over all τ T r, and the result is proved. Next, we will prove that dτ satisfies the implication 3.1 given in Definition 7. We will do so by induction on nτ, the number of nodes in τ. Since is the only tree satisfying nτ = 0, and Φ r,μ +1 =Φr,μ 1 s, the conclusion of 3.1 is true for all τ T with nτ =0. Let n 1 and assume by the induction hypothesis that the conclusion of 3.1 holds for any tree satisfying dτ +1and nτ < n. We will show that Φ r,μ +1 τ =Φr,μ τ for all τ satisfying d τ +1and n τ n. Let τ =[τ 1,...,τ κ ] l where nτ j < n for j =1,...,κ. Since d τ +1 there is at most one subtree τ j satisfying dτ j = + 1, let us for simplicity assume this to be the last one. Thus dτ j for j = 1,...,κ 1 anddτ κ +1. Consequently, Φ r,μ τ j =Φ r,μ τ j, j =1,...,κ 1 by the hypothesis of 3.1, and Φ r,μ +1 τ j=φ r,μ τ j, j =1,...,κ by the induction hypothesis. By 3.6 and Theorem 6, +1 τ = M = + Z l,νr,μ κ i=1 + Z l,νr,μ = τ, κ Z l,νr,μ Z l,νr,μ completing the induction proof. κ 1 Φ r,μ τ j κ j i Φ r,μ τ j Φ r,μ +1 τ i Φ r,μ τ i Φ r,μ τ j Φ r,μ τ κ κ 1 Φ r,μ τ j Φ r,μ τ κ Φ r,μ τ κ 4. General convergence results for iterated methods. Now we will relate the results of the previous section to the order of the overall scheme. In the following, we assume that the predictors satisfy the conditions τ = τ τ T with gτ G 0, } 0 τ τ, 0 τ T with gτ Ĝ0, where G 0 and Ĝ0 are chosen as large as possible. In particular, the trivial predictor satisfies G 0 = 0 while Ĝ0 =. We assume further that in analogy to 3.1 we have 4. } τ τ, 0, τ T, gτ +1 τ τ, 0 }, τ T, gτ +1, for all 0. By Lemmas 8, 9, and 10 this is guaranteed for the iteration schemes considered here.

15 CONVERGENCE OF ITERATED SRK METHODS 195 It follows from 3.1, 3., and 4. that Φ τ =Φτ τ T with g τ G 0 +, 4.3 Φ τ Φτ, 0} τ T with g τ Ĝ0 + as well as ψ Φ u =ψ Φ u u U f with g u G 0 +, 4.4 ψ Φ τ ψ Φ u, 0} u U f with g u Ĝ0 +. The next step is to establish the relation between the order and the growth function of a tree. We have chosen to do so by some maximum height functions, given by 4.5 G T q=max τ T g τ :ρτ q}, G T,ϕ q=max τ T g τ :Eϕτ 0,ρτ q}, G Uf q=maxg u :ρu q}, G Uf,ψ ϕ q=maxg u :Eψ ϕ u 0,ρu q}. u U f u U f Note that the definition relates to the weights of the exact, not the numerical, solution. We are now ready to establish results on wea and strong convergence for the iterated solution. Wea convergence. Let p be the wea order of the underlying scheme. The wea order of the iterated solution after iterations is minq,pif E ψ Φ u =Eψ Φ u u U f, ρu q + 1. If q p we can tae advantage of the fact that 0 = E ψ ϕ u =Eψ Φ u+oh p+1 for some u, and thereby relax the conditions to 4.6 ψ Φ u =ψ Φ u u U f with E ψ ϕ u 0, ψ Φ u ψ Φ u, 0} u U f with E ψ ϕ u =0. By 4.4, this is fulfilled for all u of order ρu min q,pif G Uf,Ψ ϕ q + 1 G 0 + and G Uf q + 1 Ĝ0 +. The results can then be summarized in the following theorem. Theorem 11. The iterated method is of wea order q p after max G Uf,ψϕ q + 1 G 0, G Uf q + 1 } Ĝ0 iterations. Strong convergence. The strong convergence case can be treated similarly. Let p now be the mean square order of the underlying method. The iterated solution is of order minp, q if Φ τ =Φτ τ T with ρτ q, 4.7 Φ τ =Φτ τ T with ρτ =q + 1, E φτ 0, Φ τ Φτ, 0} τ T with ρτ =q + 1, E φτ =0.

16 196 KRISTIAN DEBRABANT AND ANNE KVÆRNØ According to 4.3 these are satisfied if all the conditions G T q G 0 +, G T q + 1 Ĝ0 +, and G T,ϕ q + 1 G 0 + are satisfied. We can summarize this by the following theorem. Theorem 1. The iterated method is of mean square order q p after max max G T q, G T,ϕ q + 1 } G 0, G T q + 1 } Ĝ0 iterations. 5. Growth functions and order. In this section we will discuss the relation between the order of trees and the growth functions defined in section 3. Let us start with the following lemma. Lemma 13. For 1, h τ = ρτ + 1, r τ = ρτ, d τ = ρτ 1. The same result is valid for h u, r u, andg u. Proof. LetT h,, T r,,andt d, be sets of trees of minimal order satisfying hτ = τ T h,, rτ = τ T r,,anddτ = τ T d, see Figure 5.1, and denote this minimal order by ρ h,, ρ r, and ρ d,. Minimal order trees are built up only by stochastic nodes. It follows immediately that T h,1 = T r,1 = T d,1 = l : l 1}. Since ρ l =1/ forl 1, the results are proved for = 1. It is easy to show by induction on that T h, = [τ] l : τ T h, 1, l 1}, ρ h, = ρ h, =, 5.1 T r, = [ l1,τ] l : τ T r, 1,l 1,l 1}, ρ r, = ρ r, 1 +1= 1, T d, = [τ 1,τ ] l : τ 1,τ T d, 1, l 1}, ρ d, =ρ d, = 1 1. For each g being either h, r, ord, the minimal order trees satisfying g τ g, =, g u g, = are τ g, =[τ g,] l with τ g, T g, and l 1, and u g, =[τ g, ] f.bothare of order ρτ g, +1/. Let G T q andg Uf q be defined by 4.5. Then the following corollary holds. Corollary 14. For q 1 we have q 1 for simple iterations, G T q =G Uf q = q for modified Newton iterations, log q +1 for full Newton iterations. Fig Minimal order trees with gτ =3. The sets T g,3 consist of all such trees with only stochastic nodes.

17 CONVERGENCE OF ITERATED SRK METHODS 197 Proof. The minimal order trees are also the maximum height/ramification number/doubling index trees, in the sense that as long as ρτ g, q<ρτ g,+1 there are no trees of order q for which the growth function can exceed. Let T S T and Uf S U f be the set of trees with an even number of each ind of stochastic nodes. Further, let T I T 0 and Uf I U f be the set of trees constructed from the root 0 or f, by a finite number of steps of the form: i add one deterministic node, or ii add two equal stochastic nodes, neither of them being a father of the other. Clearly T I T S and Uf I U f S.From[5,6]wehave 5. E ϕτ =0 if τ E ψ ϕ u =0 if u T S T I U S f U I f in the Stratonovich case, in the Itô case, in the Stratonovich case, in the Itô case. Considering only trees for which E ϕ or E ψ ϕ are different from zero, we get the following lemma. Lemma 15. For 1, h +1 τ = ρτ if τ T S, +1 if τ T I, r if τ T S, τ = ρτ +1 if τ T I, d 1 if τ T S, τ = ρτ 1 +1 if τ T I. This result is also valid for h u, r u, andg u, witht replaced by U f. Proof. In the Stratonovich case, we consider only trees of integer order, which immediately gives the results. In the Itô case,letτ g,, τ g, be the minimal order trees used in the proof of Lemma 13. Unfortunately τ g, has a stochastic root, so τ g, T I, and there are no trees τ T I of order ρτ g, +1/ satisfying g τ =. When g is either r or d then the tree [τ g, l ] 0 T I if all the stochastic nodes are of color l 1. The order of this tree is ρτ g +3/, proving the result for r τ andd τ. Let ˆτ h, T I be a tree of minimal order satisfying h ˆτ h, =. Clearly, ˆτ h,1 can be either [ 0 ] 0 or [ l, l ] 0 with l 1, both of order. From the construction of trees in T I it is clear that the height of the tree can be increased only by one for each order, thus ρˆτ h, = +1. TheresultforUI f follows immediately. Let G T,ϕ q andg Uf,ψ ϕ q be given by 4.5. Then the analogue of Corollary 14 is as follows. Corollary 16. For q 1 we have in the Stratonovich case max0, q 1} G T,ϕ q =G Uf,ψ ϕ q = q log q +1 for simple iterations, for modified Newton iterations, for full Newton iterations,

18 198 KRISTIAN DEBRABANT AND ANNE KVÆRNØ Table 5.1 Number of iterations needed to achieve order p when using the simple, modified, or full Newton iteration scheme in the Itô or Stratonovich case for strong or wea approximation. Stratonovich Strong/wea appr. Wea appr. Strong appr. p simple mod. full simple mod. full simple mod. full Itô and in the Itô case max0, q 1} G T,ϕ q =G Uf,ψ ϕ q = max0, q 1} max0, log q } for simple iterations, for modified Newton iterations, for full Newton iterations. For the trivial predictor, Table 5.1 gives the number of iterations needed to achieve a certain order of convergence. The results concerning the Stratonovich case when considering strong approximation and using the simple iteration scheme where already obtained by Burrage and Tian [3] analyzing predictor corrector methods. 6. Convergence results for composite methods. Composite methods have been introduced by Tian and Burrage [31]. At each step either a semi-implicit Runge Kutta method or an implicit Runge Kutta method is used in order to obtain better stability properties, which results in the method 6.1a Y n+1 = Y n + λ n m +1 λ n l=0 ν=0 z 1,l,ν I d g l H 1,l,ν l=0 ν=0 for n =0, 1,...,N 1, t n I h, λ n 0, 1} and r=0 z,l,ν I d g l H,l,ν 6.1b H j,l,ν = 1 s Y n + Z j,l,νr,μ I d g r H j,r,μ, j =1,. Here, the method coefficients with superscripts 1 are those of the implicit SRK and the method coefficients with superscripts are those of the semi-implicit SRK. Let Φ 1,Φ be the corresponding weight-functions and Φ 1,Φ be the corresponding weight-functions of the iterated methods. Then the weight-function Φ of the composite method is given by Φ = λ 1 Φ 1 +1 λ 1 Φ, and similarly we have for the iterated method Φ = λ 1 Φ 1 +1 λ 1 Φ. It follows that the convergence conditions 4.6 and 4.7, respectively, are satisfied if and only if they are satisfied as well for the underlying implicit SRK as for the semi-implicit SRK. Thus, an iterated composite method has the same order as the original composite method, if in each step the number of iterations is chosen according to Theorem 11 and Theorem 1, respectively. For the trivial predictor, the number of iterations needed to achieve a certain order of convergence is again given by Table 5.1.

19 CONVERGENCE OF ITERATED SRK METHODS log 10 error p=0.51 for IPS Iter0 p=1.00 for IPS SimpleIter1 5 p=1.50 for IPS SimpleIter p=1.51 for IPS ModIter log h Fig Error of IPS applied to 7.1 without iteration, with one or two simple iterations, and with one modified Newton iteration the last two results nearly coincide. 7. Numerical examples. In the following, we analyze numerically the order of convergence of three SRK methods in dependence on the ind and number of iterations. In each example, the solution is approximated with step sizes 8,..., 1 and the sample average of M = 0,000 independent simulated realizations of the absolute error is calculated in order to estimate the expectation. As a first example, we apply the drift implicit strong order 1.5 scheme due to Kloeden and Platen [16], implemented as a stiffly accurate SRK scheme with six stages and denoted by IPS; i.e., for one-dimensional Wiener processes Y n+1 = Y n + 1 l=0 z l I d g l H, H = 1 6 Y n + 1 l=0 Z l I d g l H, h h Z 0 = h h , Z 1 = h h h , h h h h h a a 0 0 I 1 b + c b c d d 0 h z 0 =,a, a, 0, 0, h, z 1 = I 1,b+ c, b c, d, d, 0, a = I 1,0 1 I 1h, b = I 0,1 h h, c =I 1,1 h d, d = I 1,1,1 h, to the nonlinear SDE [16, 1] 7.1 dxt = 1 Xt+ Xt +1 dt + Xt +1dW t, X0 = 0, onthetimeintervali =[0, 1] with the solution Xt = sinht + W t. The results at time t = 1 are presented in Figure 7.1, where the orders of convergence correspond to the slope of the regression lines. As predicted by Table 5.1 we observe strong order 0.5 without iteration, strong order 1.0 for one simple iteration, and strong order 1.5 for two simple or one modified Newton iteration.

20 00 KRISTIAN DEBRABANT AND ANNE KVÆRNØ As second example, we apply the diagonal implicit strong order 1.5 method DIRK4 which for one-dimensional Wiener processes is given by Y n+1 = Y n + with coefficients 1 l=0 z l I d g l H, H = 1 3 Y n + z 0 = hα, z 1 = J 1 γ 1 + J 1,0 h γ, Z 0 = ha, Z 1 = J 1 B 1 + J 1,0 h B + hb 3, α = , , , , 1 l=0 Z l I d g l H γ 1 = , , , , γ = , , , , A = , B 1 = , B = , B 3 = to the corresponding Stratonovich version of 7.1. This method is constructed such that the regularity of the linear system which has to be solved in each modified Newton iteration step does not depend directly on J 1 and J 1,0. The results at time t = 1 are presented in Figure 7.. As predicted by Table 5.1 we observe no convergence without iteration, strong order 1.0 for one or two simple iterations or one modified Newton iteration, and strong order 1.5 in the case of three simple iterations or two modified Newton iterations. For typographical reasons, we restrict ourselves to an accuracy of A 16-digits version of the coefficients can be obtained on request from the authors.

21 CONVERGENCE OF ITERATED SRK METHODS 01 log 10 error p=0.01 for DIRK4 Iter0 p=1.00 for DIRK4 SimpleIter1 p=1.00 for DIRK4 SimpleIter p=1.50 for DIRK4 SimpleIter3 p=1.06 for DIRK4 ModIter1 p=1.50 for DIRK4 ModIter log h Fig. 7.. Error of DIRK4 applied to the Stratonovich version of 7.1 without iteration, with one, two, or three simple iterations, and with one or two modified Newton iterations the results for three simple iterations and two modified Newton iterations nearly coincide. Finally, we apply the drift implicit strong order 1.0 scheme due to Kloeden and Platen [16], implemented as a stiffly accurate SRK scheme in the form Y n+1 = Y n + l=0 ν=0 H ν = 1 Y n Z 00,0 = h h 0 0 Z 0l,μ = I μ,l h z l,ν I d g l H ν, l=0 Z νl,μ I d g l H μ,, Z 00,μ = I μ,l h ν =0,...,m, 0 0, Z 0 0 0l,0 0 0 =, I l 0, Z j0, =, Z h 0 jl,l =, h 0 Z jl,μ 0 0 = for l μ, z 0,0 = h 0 0, h, z 0,μ = 0, 0, z l,0 = I l, 0, z l,μ = I μ,l h, I μ,l h, j,l,μ=1,...,m, and denoted by IPS10 to the following nonlinear problem of dimension two driven by two Wiener processes in which there is no commutativity between the driving terms, 1 dx 1 t = X 1t+ X 1 t + X t +1 dt + X t +1dW 1 t 7.a +cosx 1 t dw t, 7.b dx t = 7.c X0 =0. 1 X 1t+ X t +1 dt + X 1 t +1dW 1 t+sinx t dw t, As here we don t now the exact solution, to approximate it we use IPS10 with two modified Newton iterations and a step size ten times smaller than the actual step size. The multiple Itô integrals are approximated as described in [16]. The results at time

22 0 KRISTIAN DEBRABANT AND ANNE KVÆRNØ log 10 error p=0.49 for IPS10 Iter0 p=0.98 for IPS10 SimpleIter1 p=0.98 for IPS10 ModIter log h Fig Error of IPS10 applied to 7. without iteration, with one simple iteration and with one modified Newton iteration the last two results nearly coincide. t = 1 are presented in Figure 7.3. As predicted by Table 5.1 we observe strong order 0.5 without iteration and strong order 1.0 for one simple iteration or one modified Newton iteration. 8. Conclusion. For stochastic Runge Kutta methods that use an iterative scheme to compute their internal stage values, we derived convergence results based on the order of the underlying Runge Kutta method, the choice of the iteration method, the predictor, and the number of iterations. This was done by developing a unifying approach for the construction of stochastic B-series, which is valid both for Itôand Stratonovich-SDEs and can be used both for wea and strong convergence. We expect this to be useful also for the further development and analysis of stochastic Runge Kutta type methods. Acnowledgement. We than the anonymous referees for their valuable comments and suggestions, which helped to improve this paper. REFERENCES [1] K. Burrage and P. M. Burrage, High strong order explicit Runge Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 1996, pp Special issue celebrating the centenary of Runge Kutta methods. [] K. Burrage and P. M. Burrage, Order conditions of stochastic Runge Kutta methods by B-series, SIAM J. Numer. Anal., , pp [3] K. Burrage and T. H. Tian, Predictor-corrector methods of Runge Kutta type for stochastic differential equations, SIAM J. Numer. Anal., 40 00, pp [4] K. Burrage and T. H. Tian, Implicit stochastic Runge Kutta methods for stochastic differential equations, BIT, , pp [5] P. M. Burrage, Runge Kutta methods for stochastic differential equations, Ph.D.thesis,The University of Queensland, Brisbane, [6] J. C. Butcher, Coefficients for the study of Runge Kutta integration processes, J.Austral. Math. Soc., , pp [7] K. Debrabant and A. Rößler, Diagonally drift-implicit Runge Kutta methods of wea order one and two for Itô SDEs and stability analysis, Appl. Numer. Math., 008, in press, doi: /j.apnum [8] K. Debrabant and A. Rößler, Families of efficient second order Runge Kutta methods for the wea approximation of Itô stochastic differential equations, Appl. Numer. Math., 008, in press, doi: /j.apnum [9] K. Debrabant and A. Rößler, Classification of stochastic Runge Kutta methods for the wea approximation of stochastic differential equations, Math. Comput. Simulation, , pp

23 CONVERGENCE OF ITERATED SRK METHODS 03 [10] T. C. Gard, Introduction to stochastic differential equations, Monographs and Textboos in Pure Appl. Math. 114, Marcel Deer Inc., New Yor, [11] D. B. Hernández and R. Spigler, Convergence and stability of implicit Runge Kutta methods for systems with multiplicative noise, BIT, , pp [1] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAMJ. Numer. Anal., , pp [13] K. R. Jacson, A. Kværnø, and S. P. Nørsett, The use of Butcher series in the analysis of Newton-lie iterations in Runge Kutta formulas, Appl. Numer. Math., , pp International Conference on Scientific Computation and Differential Equations Aucland, [14] K. R. Jacson, A. Kværnø, and S. P. Nørsett, An analysis of the order of Runge Kutta methods that use an iterative scheme to compute their internal stage values, BIT, , pp [15] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics 113, nd ed., Springer-Verlag, New Yor, [16] P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics 1, nd ed., Springer-Verlag, Berlin, [17] Y. Komori, Wea first- or second-order implicit Runge Kutta methods for stochastic differential equations with a scalar Wiener process, J. Comput. Appl. Math., , pp [18] Y. Komori, Multi-colored rooted tree analysis of the wea order conditions of a stochastic Runge Kutta family, Appl. Numer. Math., , pp [19] Y. Komori, Wea second-order stochastic Runge Kutta methods for non-commutative stochastic differential equations, J. Comput. Appl. Math., , pp [0] Y. Komori, T. Mitsui, and H. Sugiura, Rooted tree analysis of the order conditions of ROWtype scheme for stochastic differential equations, BIT, , pp [1] V. Macevičius and J. Navias, Second order wea Runge Kutta type methods of Itô equations, Math. Comput. Simulation, , pp [] G. N. Milstein, Numerical integration of stochastic differential equations, Mathematics and its Applications 313, Kluwer Academic Publishers Group, Dordrecht, Translated and revised from the 1988 Russian original. [3] G. N. Milstein, Y. M. Repin, and M. V. Tretyaov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., , pp [4] S. P. Nørsett and A. Wolfbrandt, Order conditions for Rosenbroc type methods, Numer. Math., , pp [5] W. P. Petersen, A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations, SIAM J. Numer. Anal., , pp [6] A. Rößler, Stochastic Taylor expansions for the expectation of functionals of diffusion processes, Stoch. Anal. Appl., 004, pp [7] A. Rößler, Rooted tree analysis for order conditions of stochastic Runge Kutta methods for the wea approximation of stochastic differential equations, Stoch. Anal. Appl., 4 006, pp [8] A. Rößler, Runge Kutta methods for Itô stochastic differential equations with scalar noise, BIT, , pp [9] A. Rößler, Second order Runge Kutta methods for Itô stochastic differential equations, Technical report Preprint 479, TU Darmstadt, 006. [30] A. Rößler, Second order Runge Kutta methods for Stratonovich stochastic differential equations, BIT, , pp [31] T. H. Tian and K. Burrage, Two-stage stochastic Runge Kutta methods for stochastic differential equations, BIT, 4 00, pp [3] A. Tocino and J. Vigo-Aguiar, Wea second order conditions for stochastic Runge Kutta methods, SIAM J. Sci. Comput., 4 00, pp