Split-step backward balanced Milstein methods for stiff stochastic systems

Applied Numerical Mathematics 59 (2009) 1198–1213

www.elsevier.com/locate/apnum

Split-step backward balanced Milstein methodsfor stiff stochastic systems ✩

Peng Wang a,∗, Zhenxin Liu b

a Institute of Mathematics, Jilin University, Changchun 130012, PR Chinab College of Mathematics, Jilin University, Changchun 130012, PR China

Available online 26 June 2008

Abstract

In this paper we discuss split-step backward balanced Milstein methods for solving Itô stochastic differential equations (SDEs).Four families of methods, a family of drifting split-step backward balanced Milstein (DSSBBM) methods, a family of modifiedsplit-step backward balanced Milstein (MSSBBM) methods, a family of drifting split-step backward double balanced Milstein(DSSBDBM) methods and a family of modified split-step backward double balanced Milstein (MSSBDBM) methods, are con-structed in this paper. Their order of strong convergence is proved. The stability properties and numerical results show theeffectiveness of these methods in the pathwise approximation of stiff SDEs.© 2008 IMACS. Published by Elsevier B.V. All rights reserved.

MSC: 60H10; 60H35; 65L20

Keywords: Stochastic differential equations; Balanced Milstein method; Stochastic Taylor expansion; Mean-square stability; Stiff equations

1. Introduction

In this paper we consider numerical methods for the strong solutions of Itô stochastic differential equations

dy(t) = f(t, y(t)

)dt + g

(t, y(t)

)dW(t), y(t0) = y0, t ∈ [t0, T ], y ∈ R

m, (1)

where W(t) is a Wiener process, whose increment �W(t) = W(t + �t) − W(t) is a Gaussian random variableN(0,�t). For simplicity in this paper numerical methods on a given time interval [t0, T ] are fixed by schemes basedon equidistant time discretization points tn = t0 + nh, n = 0,1, . . . ,N with step size h = (T − t0)/N , N = 1,2, . . . .

Numerical schemes for SDEs are recursive methods where trajectories of the solution are computed at discrete timesteps. These schemes are now abundant and classified according to their type (strong or weak) and order of con-vergence [10]. In this paper, we focus our attention on schemes that converge in the strong sense. We say that adiscrete time approximation yn converges strongly to the exact solution y(tn) with order p > 0 if there exist constantsh0 ∈ (0,∞) and C ∈ (0,∞), independent of h, such that

✩ The first author is partially supported by the Young Fund of the College of Mathematics at Jilin University. The second author is partiallysupported by SRFDP (20070183053) and the Young Fund of the College of Mathematics at Jilin University.

* Corresponding author.E-mail addresses: [email protected] (P. Wang), [email protected] (Z. Liu).

0168-9274/$30.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2008.06.001

P. Wang, Z. Liu / Applied Numerical Mathematics 59 (2009) 1198–1213 1199

E(∣∣y(tn) − yn

∣∣) � Chp

for all h ∈ (0, h0). For SDE (1), the well-known Euler–Maruyama method is given by [12]

yn+1 = yn + f (tn, yn)h + g(tn, yn)�Wn, (2)

where �Wn = W(tn+1) − W(tn), n = 1,2, . . . ,N − 1, and y0 = y(t0). However, strong convergence order of Euler–Maruyama method is merely 0.5, and it is unsuitable for solving stiff stochastic systems, for example, see [4,15]. Byincluding from the Itô–Taylor expansion the additional term

g(tn, yn)∂g

∂y(tn, yn)

t∫t0

s∫t0

dWz dWs = 1

2g(tn, yn)

∂g

∂y(tn, yn)

[(�Wn)

2 − h],

Milstein [13] has presented an important (Milstein) method with strong order 1.0, namely

yn+1 = yn + f (tn, yn)h + g(tn, yn)�Wn + 1

2g(tn, yn)

∂g

∂y(tn, yn)

[(�Wn)

2 − h]. (3)

In recent years many efficient numerical methods are constructed for solving different types of SDEs with differentproperties (for example, see [6,10,14,17]). In particular, several authors have presented different efficient implicitmethods for stiff SDEs (see [3,4,15,19,22]). In order to improve the stability properties of the numerical methods forsolving SDEs which are stiff in both the deterministic and stochastic components, some attempts have been made topropose modified implicit methods. In this paper we discuss split-step backward balanced Milstein methods for solv-ing stiff SDEs. In Section 2, we give a brief review of stochastic Taylor expansions. Four families of methods, a familyof drifting split-step backward balanced Milstein (DSSBBM) methods, a family of modified split-step backward bal-anced Milstein (MSSBBM) methods, a family of drifting split-step backward double balanced Milstein (DSSBDBM)methods and a family of modified split-step backward double balanced Milstein (MSSBDBM) methods, are pre-sented in Section 3. The convergence properties of these methods are discussed in Section 4. The stability propertiesand numerical results of these methods are reported in Sections 5 and 6, respectively.

2. Stochastic Taylor expansions

In this section we give a brief review of stochastic Taylor expansions. As a version of stochastic Taylor expan-sions, the Itô–Taylor expansions are obtained by generalizing the deterministic Taylor expansions and the Itô formula[10,11]. As with the deterministic Taylor expansions in the numerical analysis of ordinary differential equations, thestochastic Taylor expansions are the key to the numerical analysis for SDEs. The following special form of Itô–Taylorexpansions (based on an SDE with one Wiener process) is used in the analysis of this paper

y(tn+1) = yn + I0f (tn, yn) + I1g(tn, yn) + I11L1g(tn, yn) + I10L

1f (tn, yn) + I01L0g(tn, yn)

+ I111L1L1g(tn, yn) + I00L

0f (tn, yn) + I110L1L1f (tn, yn)I101L

1L0g(tn, yn)

+ I011L0L1g(tn, yn) + I1111L

1L1L1g(tn, yn) + R, (4)

where the two operators L0, L1 are defined by

L0 = f (t, x)∂

∂x+ 1

2g2(t, x)

∂2

∂x2, L1 = g(t, x)

∂

∂x.

The stochastic integral Ij1j2···jk−1jkis defined recursively by

Ij1j2···jk−1jk=

tn+1∫tn

Ij1j2···jk−1(t) dt, jk = 0,

Ij1j2···jk−1jk=

tn+1∫Ij1j2···jk−1(t) dW(t), jk = 1,

tn

1200 P. Wang, Z. Liu / Applied Numerical Mathematics 59 (2009) 1198–1213

with I0 = h, I1 = �Wn. For low order stochastic integrals, we have the following expressions (see Kloeden andPlaten [10])

I11 = 1

2

[(�Wn)

2 − h], I00 = 1

2h2, I111 = 1

6

[(�Wn)

3 − 3�Wn

],

I10 = 1

2

[�Wn + �Wn1√

3

]h, I01 = 1

2

[�Wn − �Wn1√

3

]h,

where �Wn and �Wn1 are mutually independent N(0,√

h) random variables. The remainder R in the Itô–Taylorseries (4) satisfies

E(R) = O(h5/2), E

(R2) = O

(h5).

Using the Itô formula, we have the following Taylor expansions of the functions of y(tn+1), given by

f(tn+1, y(tn+1)

) = f (tn, yn) + I0L0f (tn, yn) + I1L

1f (tn, yn) + I11L1L1f (tn, yn) + R1,

g(tn+1, y(tn+1)

) = g(tn, yn) + I0L0g(tn, yn) + I1L

1g(tn, yn) + I11L1L1g(tn, yn)

+ I10L1L0g(tn, yn) + I01L

0L1g(tn, yn) + I111L1L1L1g(tn, yn) + R2,

∂g

∂y

(tn+1, y(tn+1)

) = ∂g

∂y(tn, yn) + I0L

0 ∂g

∂y(tn, yn) + I1L

1 ∂g

∂y(tn, yn) + I11L

1L1 ∂g

∂y(tn, yn) + R3.

The remainders in these Taylor expansions satisfy

E(R1) = E(R2) = E(R3) = O(h2), E

(R2

1

) = E(R2

3

) = O(h3), E

(R2

2

) = O(h4).

3. Split-step backward balanced Milstein methods

For SDE (1), Milstein, Platen and Schurz [15] presented a class of balanced implicit (BI) method for stiff SDEs,namely

yn+1 = yn + f (tn, yn)h + g(tn, yn)�Wn + Cn(yn − yn+1), (5)

where

Cn = c0(tn, yn)h + c2(tn, yn)|�Wn|.In this method the functions c0 and c2 are called control functions. The control functions must satisfy some conditions.

Assumption 1. The c0 and c2 represent bounded m × m-matrix-valued functions. For any real numbers α0 ∈ [0, α1],α2 ∈ [−α2, α2], where α1 � h, |α2| � |(�Wn)

2 − h| for all step sizes h considered and (t, x) ∈ [0,∞] × Rm, the

matrix

M(t, x) = I + α0c0(t, x) + α2c2(t, x)

has an inverse and satisfies the condition∣∣M(t, x)−1∣∣ � K < ∞. (6)

Here I is the unit matrix, K is a positive constant.

Using the idea of the BI method and combining it with the Milstein method, Kahl [8] presented a class of balancedMilstein (BM) methods (see also Kahl and Schurz [9]), namely

yn+1 = yn + f (tn, yn)h + g(tn, yn)�Wn + 1

2g(tn, yn)

∂g

∂y(tn, yn)

[(�Wn)

2 − h] + Cn(yn − yn+1), (7)

where

Cn = c0(tn, yn)h + c2(tn, yn)[(�Wn)

2 − h].

The control functions c0 and c2 must satisfy Assumption 1.Drift implicit Milstein (DIBM) methods are given by [20]


yn+1 = yn + f (tn+1, yn+1)h + g(tn, yn)�Wn + 1

2g(tn, yn)

∂g

∂y(tn, yn)

[(�Wn)

2 − h] + Cn(yn − yn+1), (8)

where


2 − h].

The control functions c0 and c2 must satisfy Assumption 1. Semi-implicit balanced Milstein (SIBM) methods aregiven by [20]

yn+1 = yn +[f (tn+1, yn+1) − 1

2g(tn+1, yn+1)

∂g

∂y(tn+1, yn+1)

]h

+ g(tn, yn)�Wn + 1

2g(tn, yn)

∂g

∂y(tn, yn)(�Wn)

2 + Cn(yn − yn+1), (9)

where


2 − h].

The control functions c0 and c2 must satisfy Assumption 1.Let us rewrite the SDE (1) in the following form:

dy(t) = f(y(t)

)dt + g

(y(t)

)dW(t), y(t0) = y0, t ∈ [t0, T ], y ∈ R

m. (10)

For SDE (10), Higham, Mao and Stuart [7] presented a split-step backward Euler method, namely

Yn = yn + hf (Yn),

yn+1 = Yn + �Wng(Yn). (11)

Using the same implicit splitting technique, Wang [21] presented drifting split-step backward Milstein (DSSBM)method, namely

Yn = yn + hf (Yn),

yn+1 = Yn + �Wng(Yn) + 1

2g(Yn)g

′(Yn)[(�Wn)

2 − h]. (12)

The other one modified split-step backward Milstein (MSSBM) method is given by [21]

Yn = yn + h

[f (Yn) − 1

2g(Yn)g

′(Yn)

],

yn+1 = Yn + �Wng(Yn) + 1

2g(Yn)g

′(Yn)(�Wn)2. (13)

The convergence of methods (12) and (13) is proved in [21].For SDE (1), similar to methods (12) and (13), using this a class of implicit splitting techniques for balanced Mil-

stein method (8)–(9), we obtain following two families of methods, a family of drifting split-step backward balancedMilstein (DSSBBM) methods

Yn = yn + hf (tn, Yn),

yn+1 = Yn + �Wng(tn, Yn) + 1

2g(tn, Yn)

∂g

∂y(tn, Yn)

[(�Wn)

2 − h] + Cn(Yn − yn+1), (14)

where


2 − h],

a family of modified split-step backward balanced Milstein (MSSBBM) methods

Yn = yn + h

[f (tn, Yn) − 1

2g(tn, Yn)

∂g

∂y(tn, Yn)

],

yn+1 = Yn + �Wng(tn, Yn) + 1g(tn, Yn)

∂g(tn, Yn)(�Wn)

2 + Cn(Yn − yn+1), (15)
2 ∂y


where


2 − h].

For methods (14) and (15), the control functions c0 and c2 must satisfy Assumption 1. In addition, adding balancedterms for split-step stage, we obtain again following two families of methods, a family of drifting split-step backwarddouble balanced Milstein (DSSBDBM) methods

Yn = yn + hf (tn, Yn) + Cn(yn − Yn),

yn+1 = Yn + �Wng(tn, Yn) + 1

2g(tn, Yn)

∂g

∂y(tn, Yn)

[(�Wn)

2 − h] + Cn(Yn − yn+1), (16)

where


2 − h],

a family of modified split-step backward double balanced Milstein (MSSBDBM) methods

Yn = yn + h

[f (tn, Yn) − 1

2g(tn, Yn)

∂g

∂y(tn, Yn)

]+ Cn(yn − Yn),

yn+1 = Yn + �Wng(tn, Yn) + 1

2g(tn, Yn)

∂g

∂y(tn, Yn)(�Wn)

2 + Cn(Yn − yn+1), (17)

where


2 − h].

For methods (16) and (17), the control functions c0 and c2 must also satisfy Assumption 1.Now we consider choice of the functions c0 and c2 for methods (14)–(17) when m = 1. Alcock and Burrage [1]

have presented the optimal parameter selection for BI method (5). In practical computation the control functions c0 andc2 are often selected as constants under Assumption 1. Here we just present three typical criterions for 1-dimensioncase.

Criterion 1. For solving scalar SDE (1) (m = 1), a simple criterion for selecting c0 and c2 in the methods (14)–(17)is given by c0 = | ∂f

∂y(tn, yn)| and c2 = 0.

By c0h+c2[(�Wn)2 −h] = (c0 −c2)h+c2(�Wn)

2, introducing the stochastic balanced component, we can obtainthe following criterion.

Criterion 2. The criterion for selecting c0 and c2 in the methods (14)–(17) for solving scalar SDE (1) is given byc2 = |g(tn, yn)

∂g∂y

(tn, yn)| and

c0 =⎧⎨⎩

| ∂f∂y

(tn, yn)|, if | ∂f∂y

(tn, yn)| > |g(tn, yn)∂g∂y

(tn, yn)|,| ∂f∂y

(tn, yn)| + c2, if | ∂f∂y

(tn, yn)| � |g(tn, yn)∂g∂y

(tn, yn)|.

In addition, a more general criterion is given by the following criterion.

Criterion 3. For solving scalar SDE (1), the criterion for selecting c0 and c2 in the methods (14)–(17) is given byc2 > 0 and c0 − c2 > 0.

In order to avoid that Cn be close to −1, we can choose Criterion 2 or Criterion 3. In order to avoid that the resultingsample path distribution is likely to be biased, we can choose Criterion 3 since the functions c0 and c2 are selected asconstants.


4. Convergence properties

In this section the strong convergence order of split-step backward balanced Milstein methods is discussed. Forlocal error analysis, it is assumed that yn = y(tn) and yn+1, in the implicit terms f (tn+1, yn+1), g(tn+1, yn+1) andg(tn+1, yn+1)

∂g∂y

(tn+1, yn+1), is the exact solution y(tn+1). A similar assumption can be found in Petersen [16] whenconsidering the convergence properties of a splitting scheme for weak solutions of Itô SDEs.

Our convergence result makes the following assumption on the SDE (1).

Assumption 2. The functions f , g and g∂g∂y

in (1) satisfy the Lipschitz condition for constant K > 0, i.e.

∣∣f (t, a) − f (t, b)∣∣+∣∣g(t, a) − g(t, b)

∣∣ +∣∣∣∣g(t, a)

∂g

∂y(t, a) − g(t, b)

∂g

∂y(t, b)

∣∣∣∣ � K|a − b|,∀a, b ∈ R

m, ∀t ∈ [t0, T ],and linear growth bound, i.e.

∣∣f (t, a)∣∣2 + ∣∣g(t, a)

∣∣2 +∣∣∣∣g(t, a)

∂g

∂y(t, a)

∣∣∣∣2

� K2(1 + a2), ∀a ∈ Rm, ∀t ∈ [t0, T ].

To measure the strong convergence order of split-step backward balanced Milstein methods derived in this paper,we introduce the following convergence theorem given by Milstein [14,15].

Theorem 1. Assume for a one-step discrete time approximation y that the local mean error and mean-square errorfor all N = 1,2, . . . , and n = 0,1, . . . ,N − 1 satisfy the estimates∣∣E[(

yn+1 − y(tn+1)) ∣∣ yn = y(tn)

]∣∣ � K(1 + |yn|2

)1/2hp1 (18)

and (E

[(yn+1 − y(tn+1)

)2 ∣∣ yn = y(tn)])1/2 � K

(1 + |yn|2

)1/2hp2 (19)

with p2 � 12 and p1 � p2 + 1

2 . Then(E

[(yk − y(tk)

)2 ∣∣ y0 = y(t0)])1/2 � K

(1 + |y0|2

)1/2hp2−1/2 (20)

holds for each k = 0,1,2, . . . ,N . Here K is independent of h but dependent on the length of the time interval T − t0.

We now show that under Assumptions 1 and 2 the strong convergence order of split-step backward balancedMilstein methods is 1.0 and give following theorem.

Theorem 2. Let yk be the numerical approximation to y(tk) at time T after k steps with step size h = T/N ,N = 1,2, . . . . If applying one of split-step backward balanced Milstein methods (14)–(17) to the SDE (1) underAssumptions 1 and 2, then for all k = 0,1, . . . ,N , we have(

E[(

yk − y(tk))2 ∣∣ y0 = y(t0)

])1/2 = O(h).

Proof. A similar proof can be found in [15]. For the completeness of the paper, we give the proof here with somemodifications. At first, we show that the estimate (18) holds for the split-step backward balanced Milstein methods(14)–(17) with p1 = 2. Denote the local balanced Milstein approximation step

yBk+1 = yk + hf (tk, yk) + �Wkg(tk, yk) + 1

2g(tk, yk)

∂g

∂y(tk, yk)

[(�Wk)

2 − h] + Cn(yk − yk+1),

k = 0,1, . . . ,N − 1,

where Cn = c0(tk, yk)h + c2(tk, yk)[(�Wk)2 − h]. By Itô–Taylor expansions, there exists some constant K > 0 such

that


H1 := ∣∣E[(y(tn+1) − yn+1

) ∣∣ yn = y(tn)]∣∣

= ∣∣E[(y(tn+1) − yB

n+1

) ∣∣ yn = y(tn)] + E

[(yB

n+1 − yn+1)∣∣ yn = y(tn)

]∣∣� K

(1 + |yn|2

)1/2h2 + H2

with

H2 := ∣∣E[(yBn+1 − yn+1

) ∣∣ yn = y(tn)]∣∣

=∣∣∣∣E

[(I + Cn)

−1(

h(f (tn, yn) − f (tn, Yn)

) + �Wn

(g(tn, yn) − g(tn, Yn)

) + 1

2

[(�Wn)

2 − h]

×(

g(tn, yn)∂g

∂y(tn, yn) − g(tn, Yn)

∂g

∂y(tn, Yn)

)+ Cn(yn − Yn)

) ∣∣∣ yn = y(tn)

]∣∣∣∣� K

(1 + |yn|2

)1/2h2. (21)

The term Cn(yn − Yn) disappears in inequality (21) when we use methods (16) or (17). Thus the estimate (18) withp1 = 2 in Theorem 1 is satisfied for split-step backward balanced Milstein methods (14)–(17).

Similarly, using the Itô–Taylor expansions, we check estimate (19) for the local mean-square error of split-stepbackward balanced Milstein methods (14)–(17) and obtain for n = 0,1, . . . ,N − 1 by standard arguments

H3 := (E

[(y(tn+1) − yn+1

)2 ∣∣ yn = y(tn)])1/2

= (E

[(y(tn+1) − yB

n+1

)2 ∣∣ yn = y(tn)])1/2 + (

E[(

yBn+1 − yn+1

)2 ∣∣ yn = y(tn)])1/2

� K(1 + |yn|2

)1/2h3/2.

Thus we can choose in Theorem 1 the exponent p2 = 1.5 together with p1 = 2 and apply it to finally prove the strongorder γ = 1 (= p2 − 1

2 ) of split-step backward balanced Milstein methods, as was claimed in Theorem 2. �5. Stability properties

In this section we discuss stability properties of split-step backward balanced Milstein methods. We apply one-stepscheme to the scalar linear test equation

dy(t) = ay(t) dt + by(t) dW(t), y(t0) = y0 (22)

with known solution y(t) = y0e(a−b2/2)t+bW(t), which is represented by

yn+1 = R(a, b,h, J )yn,

where J is the standard Gaussian random variable J = J1/√

h ∼ N(0,1). Saito and Mitsui [18] introduced the fol-lowing definition of mean-square (MS) stability.

Definition 1. The numerical method is said to be MS-stable for a, b, h if

R(a, b,h) = E(R2(a, b,h, J )

)< 1.

R(a, b,h) is called MS-stability function of the numerical method.

The mean-square (MS) stability is a stochastic version of absolute stability, and it is a very important concept innumerical simulation of SDEs. Applying DSSBBM method (14) with c0 = −a and c2 = 0 to linear test equation (22),we can obtain

yn+1 = R1(p, q, J )yn,

where p = ah, q = b√

h and

R1(p, q, J ) = 1 − p + qJ + 12q2J 2 − 1

2q2

2.

(1 − p)


The MS-stability function of DSSBBM method with c0 = −a and c2 = 0 is given by

R1(p, q) = 1 − 2p + p2 + q2 + 12q4

(1 − p)4.

The DSSBBM method with c0 = −a and c2 = 0 will be MS-stable if R1(p, q) < 1. Applying DSSBBM method (14)with c0 = −a and c2 = b2 to linear test equation (22), we can obtain

yn+1 = R′1(p, q, J )yn,

where

R′1(p, q, J ) = 1 − p + qJ + 3

2q2J 2 − 32q2

(1 − p)(1 − p + q2(J 2 − 1)).

The MS-stability function of DSSBBM method with c0 = −a and c2 = b2 is given by

R′1(p, q) =

∞∫−∞

R′1(p, q, x)

1√2π

e− x22 dx.

The DSSBBM method with c0 = −a and c2 = b2 will be MS-stable if R′1(p, q) < 1. Applying MSSBBM method

(15) with c0 = −a and c2 = 0 to linear test equation (22), we can obtain

yn+1 = R2(p, q, J )yn,

where

R2(p, q, J ) = 1 − p + qJ + 12q2J 2

(1 − p)(1 − p + 12q2)

.

The MS-stability function of MSSBBM method with c0 = −a and c2 = 0 is given by

R2(p, q) = 1 − 2p + p2 + 2q2 − pq2 + 34q4

(1 − p)2(1 − p + 12q2)2

.

The MSSBBM method with c0 = −a and c2 = 0 will be MS-stable if R2(p, q) < 1. Applying MSSBBM method (15)with c0 = −a and c2 = b2 to linear test equation (22), we can obtain

yn+1 = R′2(p, q, J )yn,

where

R′2(p, q, J ) = 1 − p + qJ + 3

2q2J 2 − q2

(1 − p + q2

2 )(1 − p + q2(J 2 − 1)).

The MS-stability function of MSSBBM method with c0 = −a and c2 = b2 is given by

R′2(p, q) =

∞∫−∞

R′2(p, q, x)

1√2π

e− x22 dx.

The MSSBBM method with c0 = −a and c2 = b2 will be MS-stable if R′2(p, q) < 1. Applying DSSBDBM method


yn+1 = R3(p, q, J )yn,

where

R3(p, q, J ) = 1 − p + qJ + 12q2J 2 − 1

2q2

.
(1 − 2p)(1 − p)


The MS-stability function of DSSBBM method with c0 = −a and c2 = 0 is given by

R3(p, q) = 1 − 2p + p2 + q2 + 12q4

(1 − 2p)2(1 − p)2.

The DSSBDBM method with c0 = −a and c2 = 0 will be MS-stable if R3(p, q) < 1. Applying DSSBDBM method(16) with c0 = −a and c2 = b2 to linear test equation (22), we can obtain

yn+1 = R′3(p, q, J )yn,

where

R′3(p, q, J ) = (1 − p + qJ + 3

2q2J 2 − 32q2)

(1 − 2p + q2(J 2 − 1)).

The MS-stability function of DSSBDBM method with c0 = −a and c2 = b2 is given by

R′3(p, q) =

∞∫−∞

R′3(p, q, x)

1√2π

e− x22 dx.

The DSSBDBM method with c0 = −a and c2 = b2 will be MS-stable if R′3(p, q) < 1. Applying MSSBDBM method


yn+1 = R4(p, q, J )yn,

where

R4(p, q, J ) = 1 − p + qJ + 12q2J 2

(1 − 2p + 12q2)(1 − p)

.

The MS-stability function of MSSBDBM method is given by

R4(p, q) = 1 − 2p + p2 + 2q2 − pq2 + 34q4

(1 − 2p + 12q2)2(1 − p)2

.

The MSSBDBM method will be MS-stable if R4(p, q) < 1. Applying MSSBDBM method (17) with c0 = −a andc2 = b2 to linear test equation (22), we can obtain

yn+1 = R′4(p, q, J )yn,

where

R′4(p, q, J ) = (1 − p + qJ + 3

2q2J 2 − q2)

(1 − 2p + q2J 2 − 12q2)

.

The MS-stability function of MSSBDBM method with c0 = −a and c2 = b2 is given by

R′4(p, q) =

∞∫−∞

R′4(p, q, x)

1√2π

e− x22 dx.

The MSSBDBM method with c0 = −a and c2 = b2 will be MS-stable if R′4(p, q) < 1. In order to compare the

stability properties, applying BM method (7) with c0 = −a and c2 = 0 to linear test equation (22), we can obtain

yn+1 = R5(p, q, J )yn,

where

R5(p, q, J ) = 1 + qJ + 12q2J 2 − 1

2q2

.
1 − p


Fig. 1. MS-stable regions of Milstein type methods with c2 = 0.

The MS-stability function of BM method with c0 = −a and c2 = 0 is given by

R5(p, q) = 1 + q2 + 12q4

(1 − p)2.

The BM method will be MS-stable with c0 = −a and c2 = 0 if R5(p, q) < 1. Applying BM method (7) with c0 = a

and c2 = b2 to linear test equation (22), we can obtain

yn+1 = R′5(p, q, J )yn,

where

R′5(p, q, J ) = 1 + qJ + 3

2q2J 2 − 32q2

2 2.

1 − p + q (J − 1)


Fig. 2. MS-stable regions of Milstein type methods with c2 = b2.

The MS-stability function of BM method with c0 = −a and c2 = 0 is given by

R′5(p, q) =

∞∫−∞

R′5(p, q, x)

1√2π

e− x22 dx.

The BM method will be MS-stable with c0 = −a and c2 = 0 if R′5(p, q) < 1.

Fig. 1 gives the MS-stable regions of the BM, DSSBM, MSSBM, DSSBBM, MSSBBM, DSSBDBM and MSSB-DBM methods with c0 = a and c2 = 0, respectively. Fig. 2 gives the MS-stable regions of the BM, DSSM, MSSM,DSSBBM, MSSBBM, DSSBDBM and MSSBDBM methods with c0 = a and c2 = b2, respectively. The MS-stableregions are the areas under the plotted curves and symmetric about the p-axis. The MS-stability properties of DSS-BBM, MSSBBM, DSSBDBM and MSSBDBM methods are better than that of BM method, and the MS-stabilityproperties of MSSBBM and MSSBDBM method are better than that of DSSBBM and DSSBDBM methods. The MS-stability properties of DSSBBM, MSSBBM, DSSBDBM and MSSBDBM methods with c0 = a and c2 = 0 are better


Table 1Means of absolute errors for (23) and (24) (u = 5, v = 10, c2 = 0)

h BM DSSBBM DSSBDBM MSSBBM MSSBDBM

2−1 3.00e − 3 2.90e − 3 1.50e − 3 3.59e − 4 1.10e − 32−2 3.10e − 3 3.00e − 3 1.90e − 3 5.96e − 4 1.20e − 32−3 2.90e − 3 2.90e − 3 2.10e − 3 8.23e − 4 1.30e − 32−4 2.20e − 3 2.20e − 3 1.70e − 3 6.75e − 4 9.33e − 42−5 1.20e − 3 1.20e − 3 9.70e − 4 2.57e − 4 3.80e − 42−6 5.21e − 4 5.07e − 4 4.09e − 4 4.94e − 5 7.01e − 52−7 1.97e − 4 1.91e − 4 1.48e − 4 3.95e − 5 2.72e − 52−8 7.04e − 5 6.81e − 5 4.88e − 5 2.69e − 5 2.26e − 52−9 2.46e − 5 2.38e − 5 1.50e − 5 1.28e − 5 1.17e − 52−10 8.55e − 6 8.24e − 6 4.26e − 6 5.27e − 6 5.00e − 6

than that of DSSBM and MSSBM methods. The MS-stability properties of DSSBBM and DSSBDBM methods withc0 = a and c2 = b2 are better than that of DSSBM method. The MS-stability properties of MSSBBM and MSSBDBMmethods with c0 = a and c2 = b2 are better than that of MSSBM method. The MS-stable regions of all these methodsare semi-infinite.

6. Numerical results

Numerical results are reported in this section to confirm the convergence properties and stability properties of split-step backward balanced Milstein methods. Denoting y

(i)N as the numerical approximation to y(i)(tN ) at step point tN

in the i-th simulation of all 5000 simulations, we use means of absolute errors M , strong order 1.0 convergence ratesR1.0, defined by [2,3]

M = 1

5000

5000∑i=1

∣∣y(i)N − y(i)(tN )

∣∣, R1.0 = M

h,

to measure the accuracy and convergence property of split-step backward balanced Milstein methods. In simulationwe use the double precision data.

The first test equation is a two-dimensional linear SDE system whose Itô form is given by

dy(t) = Uy(t) dt + Vy(t) dW(t), y(t0) = y0, t ∈ [0,1], y ∈ R2, (23)

where U and V are matrices

U =(−u u

u −u

), V =

(v 00 v

). (24)

The exact solution of this equation is given by [10]

y(t) = P

(exp(ρ+(t)) 0

0 exp(ρ−(t))

)P −1y0, P = 1√

2

(1 11 −1

),

where ρ±(t) = (−u − 12v2 ± u)t + vW(t) and P −1 = P .

This equation is stiff in the deterministic (or stochastic) component if u (or v) is large. The stiffness of this linearsystem increases quadratically in terms of v. Suitable numerical results can be obtained only with smaller stepsizeif this stochastic system is stiff (see [2,19]). Table 1 gives the errors of BM, DSSBDBM, DSSBBM, MSSBDBMand MSSBBM methods with c0 = 5 and c2 = 0 when solving (23) and (24) with u = 5, v = 10 and y0 = (1,2).Table 2 gives the strong convergence rates of these balanced methods with c2 = 0, where we use the MATLABfunction to generate double precision random numbers. By Tables 1 and 2 we know that if h � 2−2 or h � 2−6 thenthe accuracy of DSSBBM method is better than that of BM methods, else their accuracies are same. The accuracyof MSSBBM method is better than of MSSBDBM method when h � 2−6. The accuracy of DSSBDBM method isbest when h = 2−10. Table 3 gives the errors of BM, DSSBDBM, DSSBBM, MSSBDBM and MSSBBM methodswith c0 = 5 and c2 = 4 when solving (23) and (24) with u = 5, v = 10 and y0 = (1,2). Table 4 gives the strong


Table 2Strong convergence rate R1.0 for (23) and (24) (u = 5, v = 10, c2 = 0)



Table 3Means of absolute errors for (23) and (24) (u = 5, v = 10, c2 = 4)



Table 4Strong convergence rate R1.0 for (23) and (24) (u = 5, v = 10, c2 = 4)



convergence rates of these balanced methods with c2 = 4. By Tables 3 and 4 we know that if h � 2−4 or h � 2−6 thenthe accuracy of DSSBBM method is better than that of BM methods, else their accuracies are same. The accuracy ofMSSBBM method is better than of other balanced methods when h � 2−5 or h � 2−7.

The second test stiff equation is a stochastic rotating problem given by

dy1(t) = βy2(t) dt + 1

2σ(y1(t) + y2(t)

)dW(t),

dy2(t) = −βy1(t) dt + 1

2σ(y1(t) + y2(t)

)dW(t). (25)

For this equation, a version with two Wiener process can be found in [15], and a determinate version can be foundin [5].


Fig. 3. Numerical simulation of the SDE (25) by derived methods in this paper.


Fig. 3. (Continued).

Using DSSBBM, DSSBDM, MSSBBM and MSSBDBM methods with h = 0.02, in Fig. 3 we give the numericalsimulation of Eq. (25) with β = 5, σ = 4, 0 � t � 100 starting at (y1(0), y2(0)) = (1,0), where we use the MATLABfunction to generate double precision random numbers. We set the control functions c0 = 5 and c2 = 0 (or c2 = 2)for split-step backward balanced Milstein methods. We observe in Fig. 3 for split-step backward balanced Milsteinmethods the approximate trajectories stay close to the origin (0,0), which replicates the behavior of the exact solution.For this model the split-step backward balanced Milstein methods stabilize the numerical solutions. The stabilityproperty of MSSBDBM methods is better than that of DSSBBM, DSSBDBM and MSSBBM methods for this model.

7. Conclusions

In this paper we have constructed split-step backward balanced Milstein methods for solving stiff SDEs. We havederived a family of drifting split-step backward balanced Milstein methods, a family of modified split-step backwardbalanced Milstein methods, a family of drifting split-step backward double balanced Milstein methods and a familyof modified split-step backward double balanced Milstein methods based on implicit splitting techniques. The orderconvergence of these methods is proved. The stability properties and numerical results show that these methods aresuitable for solving stiff SDEs. We will consider constructing methods with higher strong global convergence ordersand better stability properties in future work.

Acknowledgement

The authors would like to take this opportunity to thank Professor Yong Li for his instruction and encouragement.The authors are grateful to the anonymous referees for careful reading of the manuscript and valuable comments.

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Split-step backward balanced Milstein methods for stiff stochastic systems