Rooted Tree Analysis for Order Conditions of Stochastic RungeKutta Methods for the Weak Approximation of Stochastic Differential Equations
Abstract
A general class of stochastic RungeKutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multidimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic RungeKutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic RungeKutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic RungeKutta methods converging with an arbitrarily high order.
tochastic RungeKutta method \sepstochastic differential
equation \sepweak approximation \seprooted tree analysis \seporder condition
MSC 2000: 65C30 \sep60H35 \sep65L05 \sep60H10 \sep34F05
1 Introduction
In recent years many numerical methods have been proposed for the
approximation of stochastic differential equations (SDEs), see
e.g. [7], [9], [10], [11],
[14], [19] and [20]. Mainly, numerical
methods for strong and for weak approximations can be
distinguished. While strong approximations focus on a good
approximation of the path of a solution, weak approximations are
applied if a good distributional approximation is needed. In
Section 2 of the
present paper, a class of stochastic RungeKutta (SRK) methods for
the weak approximation of Itô and Stratonovich SDEs is
introduced. As in the deterministic setting, order conditions for
SRK methods are calculated by comparing the numerical solution
with the exact solution over one step assuming exact initial
values. Therefore, the actual solution of the SDE and the
numerical approximation process have to be expanded by a
stochastic Taylor series. However, even for low orders such
expansions become much more complex than in the deterministic
setting where it is already a lengthy task. In order to handle
this task in an easy way, a rooted tree theory based on three
different kinds of colored nodes is established in
Section 3, which is a generalization of
the rooted tree theory due to Butcher [3]. Thus,
colored trees are applied in
Section 4 and
5 to give a representation of the
solution and the approximation process calculated with the SRK
method in order to allow a rooted tree analysis of order
conditions. A similar approach with two different kinds of nodes
has been introduced by Burrage & Burrage [1],
[2] for a SRK method converging in the strong sense as
well as in Komori et al. [8] for ROWtype schemes for
Stratonovich SDEs. Finally, the main
Theorem 6.4 presented in
Section 6 immediately yields all
order conditions for the coefficients and the random variables of
the introduced SRK method such that it converges with an
arbitrarily given order in the weak sense. As a result of this
theorem, the lengthy calculation and comparison of Taylor
expansions can be
avoided.
Let be a probability space with a
filtration and let
for some . We
consider the solution of either a
dimensional Itô stochastic differential equation system
(1) 
or a dimensional Stratonovich stochastic differential equation system
(2) 
Let be the measurable initial condition such that for some holds where denotes the Euclidean norm if not stated otherwise. Here, is an dimensional Wiener process w.r.t. . SDE (1) and (2) can be written in integral form
(3) 
for , where the th column of the matrix function is denoted by for . Here, the second integral w.r.t. the Wiener process
has to be interpreted either as an Itô integral in case of SDE
(1) or as a Stratonovich integral in case of
SDE (2), which is indicated by the
asterisk.
The solution of a Stratonovich SDE
with drift and diffusion is also a solution of an Itô
SDE as in (1) and therefore also a diffusion
process, however with the modified drift
(4) 
for and provided that is sufficiently differentiable, i.e.
(5) 
The solution of the stochastic differential equation (3) is sometimes denoted by in order to emphasize the initial condition. We suppose that the drift and the diffusion are measurable functions satisfying a linear growth and a Lipschitz condition
(6)  
(7) 
for all and all with
some constant . Then the conditions of the Existence and
Uniqueness Theorem are fulfilled for the Itô
SDE (1) (see, e.g., [6]). If the
conditions also hold with replaced by the modified drift
in the Itô SDE, then the Existence and Uniqueness
Theorem also applies to the Stratonovich
SDE (2).
In the following, let denote the
space of times continuously differentiable functions for which all partial derivatives
up to order have polynomial growth. That is, for which there
exist constants and depending on ,
such that holds
for all and any partial derivative
of order .
Let be a
discretization of the time interval such
that
(8) 
and define for with the maximum step size
In the following, we consider a class of approximation processes of the type where is a random variable or in general a vector of random variables, with moments of sufficiently high order, and is a vector valued function of dimension . We write and we construct the sequence
(9) 
where is independent of , while for is independent of and . Then we can define weak convergence with some order of an approximation process.
Definition 1.1
Since we are interested in calculating a global approximation converging in the weak sense with some desired order , we make use of the following theorem due to Milstein (1986) [13] which is stated with an appropriate notation.
Theorem 1.2
Let be the solution of SDE (1) or of SDE (2). Suppose the following conditions hold:

for sufficiently large (specified below) the moments do exist and are uniformly bounded with respect to and ,

assume that for all the following local error estimation
(11) is valid for , any with and .
Then for all and all the following global error estimation
(12) 
holds for all , where is a constant and where is the maximum step size of the corresponding discretization , i.e. the method (9) has order of accuracy in the sense of weak approximation.
A proof of Theorem 1.2 can be found in [11], [12], [13] and [17]. Lemma 1.3 gives sufficient conditions such that condition (ii) of Theorem 1.2 (see also [11], [12], [13]) holds.
Lemma 1.3
Suppose that for given by (9) and the conditions
(13)  
(14) 
hold where has moments of all orders, i.e. , , with constants and independent of . Then for every even number the expectations exist and are uniformly bounded with respect to and , if only exists.
2 A Class of Stochastic RungeKutta Methods
In the following a class of stochastic RungeKutta methods is introduced for the approximation of both Itô and Stratonovich stochastic differential equation systems w.r.t. an dimensional Wiener process. In order to preserve the most possible generality, the considered class of stochastic RungeKutta methods is of type (9) and has the following structure
(15) 
where is an arbitrary finite set of multiindices with elements and , , are some suitable random variables. For the weak approximation of the solution of the dimensional SDE system (3), considered either with respect to Itô or Stratonovich calculus, the general class of stage stochastic RungeKutta methods is given by
(16)  
for with
for , and , where
for . Here are the coefficients of the SRK method and as usual the weights can be defined by
(17) 
with . If for and , , then (2) is called an explicit SRK method, otherwise it is called implicit. The class of SRK methods introduced above can be characterized by an extended Butcher array

(18) 
for and for . We assume that the random variables satisfy the moment condition
(19) 
for all and , . The moment condition ensures a contribution
of each random variable having an order of magnitude
. This condition is in accordance with the order of
magnitude of the increments of the Wiener process. Further, the
moment condition is necessary for the estimates of the reminder
terms of the Taylor expansion of the SRK approximation presented
in Section 6.
Some SRK schemes which belong to the introduced general class of
SRK methods can be found in [15], [16] and
[17]. Further, many RungeKutta type schemes proposed in
recent literature like in [7], [8],
[10] or [21] are covered. Usually, the set
may consist of some multiindices
with for and the random
variables may be chosen as multiple Itô or Stratonovich
integrals of type or , depending on the calculus that is
used.
For example, the SRK scheme RI1WM due to
Rößler [17] for the Itô
SDE (1) in the case of and with is defined by (2) with
for . Further, we define in the case of , in the case of or and in the case of for . Here, , , are independent random variables defined by and . The , , are defined by with independent random variables such that for , and for and . Thus, we can characterize the SRK method (2) by the following Butcher array for with :

The coefficients of the order SRK scheme RI1WM are given in
Table 1. For detailed calculations of the
order conditions and the corresponding coefficients we refer to
[17].
As an example for a SRK scheme due to Rößler applicable to
the Stratonovich SDE (2) with
and fulfilling a commutativity condition (see
[16], [17] for details) we choose now
and
for and . The coefficients of such a method can be represented by the Butcher array taking for the form

For detailed calculations of the order conditions we refer to [16] and [17]. The coefficients of the order 2.0 SRK scheme RS1WM are presented in Table 2.
3 Stochastic Rooted Tree Theory
The SDE system (3) can be represented by an autonomous SDE system
(20) 
with one additional equation representing time. Hence, it is sufficient to treat autonomous SDE systems in the following. First of all, we give a definition of colored graphs which will be suitable in the rooted tree theory for SDEs w.r.t. a multidimensional Wiener process (see [18]).
Definition 3.1
Let be a positive integer.

A monotonically labelled Stree (stochastic tree) t with nodes is a pair of maps with
so that for . Unless otherwise noted, we choose the set where is a variable index with .

denotes the set of all monotonically labelled Strees w.r.t. . Here two trees and just differing by their colors and are considered to be identical if there exists a bijective map with and so that holds for .
So defines a father son relation between the nodes, i.e. is the father of the son . Furthermore the color , which consists of one element of the set , is added to the node for . Here, \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0cm, radius=1.6mm, treefit=loose] \TC* [tnpos=r] is a deterministic node and \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0cm, radius=1.6mm, treefit=loose] \TC [tnpos=r] is a stochastic node with a variable index . In the case of the node of type \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0cm, radius=1.6mm, treefit=loose] \Tdot [tnpos=r] is denoted as the root and always sketched as the lowest node of the graph. However, in the case of , the nodes and may also serve as the root of the tree. The variable index is associated with the th component of the corresponding dimensional Wiener process of the considered SDE. In case of a onedimensional Wiener process one can omit the variable indices since we have for all (see also [17]). As an example Figure 1 presents two elements of .
For the labelled Stree in Figure 1 we have and . The color of the nodes is given by , , and .
Definition 3.2
Let . We denote by the number of deterministic nodes, by the number of stochastic nodes and by the number of pairs of stochastic nodes with the same variable index. The order of the tree t is defined as with .
The order of the trees and presented in Figure 1 can be calculated as . Every labelled Stree can be written as a combination of three different brackets defined as follows.
Definition 3.3
If are colored trees then we denote by
the tree in which are each joined by a single branch to \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0cm, radius=1.6mm, treefit=loose] \Tdot , \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0cm, radius=1.6mm, treefit=loose] \TC* and \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0cm, radius=1.6mm, treefit=loose] \TC [tnpos=r] , respectively (see Figure 2).
\pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=1cm, radius=1.6mm, treefit=loose, nodesepB=1mm] \Tdot \Tr \Tr \Tr[edge=none] \Tr  \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=1cm, radius=1.6mm, treefit=loose, nodesepB=1mm] \TC* \Tr \Tr \Tr[edge=none] \Tr  \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=1cm, radius=1.6mm, treefit=loose, nodesepB=1mm] \TC [tnpos=r] \Tr \Tr \Tr[edge=none] \Tr 
Therefore proceeding recursively, for the two examples
and in
Figure 1 we obtain
and
.
Due to the fact that we are interested in calculating weak
approximations, it will turn out that we can concentrate our
considerations to one representative tree of each equivalence
class.
Definition 3.4
Let and be elements of . Then the trees t and u are equivalent, i.e. , if the following hold:


There exist two bijective maps
so that the following diagram commutes
The set of all equivalence classes under the relation is denoted by . We denote by the cardinality of t, i.e. the number of possibilities of monotonically labelling the nodes of t with numbers .
Thus, a monotonically labelled Stree u is equivalent to t, if each label is replaced by and if each stochastic node with variable index is replaced by an other stochastic node . Thus, all trees in Figure 3 belong to the same equivalence class as in the example above, since the indices and are just renamed either by and or and , respectively. Finally the graphs differ only in the labelling of their number indices.
\pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=1cm, radius=1.6mm, treefit=loose] \Tdot [tnpos=l]1 \pstree\TC* [tnpos=l]2\TC [tnpos=l]3 [tnpos=r] \TC [tnpos=l]4 [tnpos=r]  \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=1cm, radius=1.6mm, treefit=loose] \Tdot [tnpos=l]1 \pstree\TC* [tnpos=l]2\TC [tnpos=l]4 [tnpos=r] \TC [tnpos=l]3 [tnpos=r]  \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=0.1cm, radius=1.6mm, treefit=loose] \Tn \pstree[treemode=U, dotstyle=otimes, dotsize=3.2mm, levelsep=1cm, radius=1.6mm, treefit=loose] \Tdot [tnpos=l]1 \TC [tnpos=l]2 [tnpos=r] \pstree\TC* [tnpos=l]3 \TC [tnpos=l]4 [tnpos=r] 
For every rooted tree , there exists a corresponding elementary differential which is a direct generalization of the differential in the deterministic case (see, e.g., [3]). For , the elementary differential is defined recursively by
for single nodes and by