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ABSTRACT
In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian.
1 Introduction
Determining numerical approximations to solutions of stochastic differential equations is of great interest, as easily evaluable analytical solutions to stochastic differential equations do rarely exist (Kloeden & Platen, Citation1999; Mao, Citation2008; Milstein & Tretyakov, Citation2004). The development of approximation algorithms has therefore been an active field of research during the last decades. Weak convergence of stochastic Runge–Kutta (SRK) methods is studied in literature (Buckwar et al., Citation2010; Debrabant & Rößler, Citation2008; Milstein, Citation1985; Tocino & Vigo-Aguiar, Citation2002). Efficient SRK methods that deliver second-order weak approximations for stochastic differential equations with multiple stochastic integrals have been introduced in Debrabant & Rößler, Citation2009; Rößler (Citation2009). Tang and Xiao (Citation2017, Citation2018, Citation2019) proposed and analyzed new classes of SRK methods to further reduce the computational effort.
Stiff stochastic differential equations are studied, for example, in Abdulle et al. (Citation2013), Higham (Citation2000) and Kloeden and Platen (Citation1999). To easily handle stiff stochastic differential equation systems, Milstein, Platen and Schurz introduced the balanced method, a class of quasi-implicit numerical schemes (Milstein et al., Citation1998). Alcock and Burrage (Citation2006) examined its stability and the choice of parameters. Balanced SRK methods are studied by Amiri and Hosseini (Citation2015). Further, Mardones and Mora (Citation2019) discussed first-order weak balanced schemes to stochastic differential equations. Finally, strong convergence of two classes of second-order balanced SRK methods applied to multidimensional Itô stochastic differential equations has been discussed in Rathinasamy et al. (Citation2020).
The main contribution of the current paper is to prove, generalizing a corresponding statement in Schurz (Citation2005) for the standard Euler method, that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods, and to apply this result to derive a corresponding compact result for Runge–Kutta methods. As a direct consequence, this proves the weak convergence order of the methods presented in Rathinasamy et al. (Citation2020) and Rathinasamy and Nair (Citation2018).
The paper is organized as follows: In Section 2, some basic results and definitions are gathered. In Section 3, the weak convergence of balanced one-step methods and balanced SRK methods is proven.
2 Preliminaries
Let be a complete probability space with a filtration
satisfying the usual conditions and
for some
. Consider the system of Itô stochastic differential equations of the form
where ,
is the drift vector,
is called diffusion matrix,
is an
-dimensional Wiener process adapted to the filtration, and
is assumed to be
-measurable.
Throughout this paper, we consider discrete time approximations with equidistant time discretization
with
of the time interval
with step sizes
for
and maximal step size
.
We consider SRK methods (Burrage & Burrage, Citation1996; Debrabant & Kværnø, Citation2008/09; Rößler, Citation2006, Citation2009)
Here, is the stage number of the SRK method,
multi-indices (e.g.,
), and the
and
are some random variables depending on the step size
that are independent of
,
.
Generalizing the ideas in Alcock and Burrage (Citation2006), Amiri and Hosseini (Citation2015), Kahl and Schurz (Citation2006); Mardones and Mora (Citation2019), Milstein et al. (Citation1998), Rathinasamy and Nair (Citation2018) and Schurz (Citation2005), a balanced version of EquationEquation (2)(2)
(2) is given as
where for an underlying SRK method of weak order using approximations
to the Brownian increments
the matrices
are chosen as
in which the -matrix valued functions
,
, satisfy the following assumption adapted from (Milstein et al., Citation1998). Here, we denote by
the set of possible values for
. For example, if
and
, then
, and if
and
, then
.
Assumption 1 For any sequence of real numbers with
,
,
and
for all step sizes
considered and
, the matrix
has an inverse fulfilling .
The constant relates to the individual bounds on the norm of the
matrices.
3 Weak convergence analysis for balanced Runge–Kutta methods
In this section, for , let
be the set of all
-times continuously differentiable functions
with uniformly bounded derivatives up to order
, such that for some constant
where denotes the Euclidean vector norm on
,
the Frobenius norm on
, and in general
To reduce the convergence analysis of balanced Runge–Kutta methods to the one of the underlying methods we will first establish the following theorem that generalizes a corresponding statement in Schurz (Citation2005) for the standard Euler method.
Theorem 1 Let be a numerical method with one-step representation
that is weakly consistent of order
, i. e. for
there exists a constant
depending only on
such that the local weak error fulfills
where denotes the solution of stochastic differential Equationequation (1)
(1)
(1) with initial value
. Let further
be a numerical method with one-step representation
, the sequence of approximations
defined by
and
and assume that for some
Assume finally that
Then is weakly convergent of order
, i.e. there exists
independent of
such that
Proof The proof follows the one of [22, Theorem 5.2], with a few changes. Suppose that and consider
Then we have for some that
where and
denotes the Euclidean scalar product in
. Applying the Cauchy–Schwarz inequality, we obtain that
with . Let
For the global weak error
it follows then
Applying this theorem to a balanced Runge–Kutta method and its underlying method, we obtain
Theorem 2 Let be given by the balanced SRK method (3), and assume that its underlying Runge–Kutta method (2) is weakly consistent of order
. Assume further that
a) the coefficients of stochastic differential Equationequation (1)
(1)
(1) fulfill a Lipschitz and linear growth condition, i. e. there exists
such that for all
and
b) that the coefficients of the balanced method fulfill that ,
, and that there exists
such that
and
for as well as
,
, and
.
Then is weakly convergent of order
.
Proof First, note that EquationEquation (3)(3)
(3) implies that
Using EquationEquations (5)(5)
(5) andEquation(6)
(6)
(6) , we obtain
which implies
for all , where
Here, denotes the number of elements in
.
Further, EquationEquation (3)(3)
(3) implies that
Let
where denotes the numerical approximation given by the underlying Runge–Kutta method (EquationEquation (2))
(2)
(2) of the balanced method at time point
when starting at time
with value
. It follows then from EquationEquations (2)
(2)
(2) and Equation(9)
(9)
(9) that
implying together with EquationEquation (4)(4)
(4) , Assumption 1, EquationEquations (6)
(6)
(6) ,Equation(5)
(5)
(5) and Equation(8)
(8)
(8) that
Similarly, we obtain from EquationEquation (9)(9)
(9) by using EquationEquation (4)
(4)
(4) , Assumption 1, EquationEquations (6
(6)
(6) ),Equation(5)
(5)
(5) and (Equation8
(8)
(8) )that
with , and in addition using EquationEquation (7)
(7)
(7) that
with
With EquationEquations (11)(11)
(11) andEquation(12)
(12,13)
(12,13) , it follows from [15, Lemma 9.1] that for each
, there exists a constant
such that for all
With EquationEquations (10)(10)
(10) and (Equation13
(14)
(14) ), the assertion follows now by Theorem 1.
As a direct consequence, Theorem 2 yields the weak convergence order of the balanced versions of W2Ito1, SRI1, W2Ito2 and SRK4 discussed in Rathinasamy et al. (Citation2020) and Rathinasamy and Nair (Citation2018).
Corollary 1 For sufficiently smooth drift and diffusion, the balanced versions of W2Ito1, SRI1, W2Ito2 and SRK4 are of weak convergence order 2.
Acknowledgements
The authors are grateful to two anonymous referees for their helpful hints which improved the presentation of the material.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Correction Statement
This article has been corrected with minor changes. These changes do not impact the academic content of the article.
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References
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