An analytic approximation of solutions of stochastic differential delay equations with Markovian switching

Mathematical and Computer Modelling 50 (2009) 1379–1384

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An analytic approximation of solutions of stochastic differential delay equations with Markovian switchingI Jianhai Bao ∗ , Zhenting Hou School of Mathematics, Central South University, Changsha, Hunan 410075, PR China

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Article history: Received 3 September 2008 Received in revised form 3 July 2009 Accepted 7 July 2009 Keywords: Taylor approximation Strong convergence Markovian switching Stochastic differential delay equation

abstract In this paper, we are concerned with the stochastic differential delay equations with Markovian switching (SDDEwMSs). As stochastic differential equations with Markovian switching (SDEwMSs), most SDDEwMSs cannot be solved explicitly. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEwMSs in the sense of the Lp -norm when the drift and diffusion coefficients are Taylor approximations. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction The theory of stochastic differential equations (SDEs) has attracted much attention since it is not only academically challenging but also of practical importance, and has played an important role in many fields such as insurance, finance, population dynamics and control [1–3]. Recently, much work has been done on SDEwMSs, one of the important classes of hybrid systems, for example, see Mao and Yuan [2], Basak et al. [4], Luo [5], Luo et al. [6], Mao [7], Mao et al. [8], Mariton [9], Yuan et al. [10], Shaikhet [11] and Yuan and Mao [12]. As SDEwMSs, most SDDEwMSs cannot be solved explicitly. Even when such a solution can be found, it may be only in implicit form or too complicated to visualize and evaluate numerically [13]. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMS. For the numerical methods of SDEwMSs and SDDEwMSs, here, we highlight the great contribution of Mao and Yuan [2]. However, the rate of convergence to the true solution by the numerical solution is different for different numerical schemes [13]. Recently, Jankovic et al. [14] have investigated the rate of convergence between the true solution and numerical solution of the following SDE dx(t ) = a(t , x(t ))dt + b(t , x(t ))dB(t ), in the sense of the Lp -norm when the drift and diffusion coefficients are Taylor approximations of the functions a and b, up to arbitrary fixed derivatives. Since the rate of convergence for such a numerical method is faster than the result derived in [13], in this paper, we intend to generalize the above method to the SDDEwMS case, and consider the strong convergence between the true solution and numerical solution to SDDEwMS if the drift and diffusion coefficients are Taylor approximations, up to arbitrary fixed derivatives. To the best of our knowledge, so far there seem to be no existing results. Therefore, the aim of this paper is to close this gap.

I This work was partially supported by NNSF of China (Grant No. 10671212).

∗

Corresponding author. E-mail address: [email protected] (J. Bao).

0895-7177/$ – see front matter Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.07.006

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2. Approximation scheme and hypotheses Let {Ω , F , {Ft }t ≥0 , P } be a probability space with a filtration satisfying the usual conditions, i.e., the filtration is continuous on the right and F0 contains all P-zero sets. Let B(t ), t ≥ 0, be a one-dimensional Brownian motion defined on the probability space. For τ > 0, let C ([−τ , 0]; R) denote the family of continuous functions ϕ from [−τ , 0] to R equipped with the norm kϕk = sup−τ ≤s≤0 |ϕ(s)|. Denote by CFb 0 ([−τ , 0]; R) the family of all bounded, F0 -measurable, C ([−τ , 0]; R)p

valued random variables. For p > 0, denote by LFt ([−τ , 0]; R) the family of all Ft -measurable, C ([−τ , 0]; R)-valued random variables ϕ = {ϕ(θ ) : −τ ≤ θ ≤ 0} satisfying sup−τ ≤θ≤0 E |ϕ(θ )|p < ∞. Let {r (t ), t ∈ R+ = [0, ∞)} be a right continuous Markov chain on the probability space {Ω , F , {Ft }t ≥0 , P } taking values in a finite state space S = {1, 2, . . . , M }, where M is some positive integer, with generator Γ = (γij )M ×M given by P (r (t + ∆) = j|r (t ) = i) =



γij ∆ + o(∆), 1 + γii ∆ + o(∆),

if i 6= j, if i = j,

where ∆ > 0. Here γij ≥ 0 is the transition rate from i to j, if i 6= j; while γii = − j6=i γij . We further assume that Markov chain r (·) is independent of the Brownian motion B(·). It is known that almost every sample path of r (t ) is a right continuous step function with a finite number of simple jumps in any finite subinterval of R+ . In the present paper, we consider the following Itô SDDEwMS:

P

dx(t ) = f (t , x(t ), x(t − τ ), r (t ))dt + g (t , x(t ), x(t − τ ), r (t ))dB(t ),

(2.1)

on the time interval [0, T ] and the initial condition x0 = ξ , where ξ ∈ CF0 ([−τ , 0]; R) is independent of B(·). Assume that b

f : [0, ∞) × R × R × S → R,

g : [0, ∞) × R × R × S → R.

For the existence and uniqueness of the solution to (2.1), we make the following rather general assumptions [2, Theorem 7.10, p. 277]: (H1) f and g satisfy Lipschitz and linear growth condition, that is, for any t ≥ 0, there exists a constant L > 0 such that

|f (t , x, y, i) − f (t , x, y, i)| ∨ |g (t , x, y, i) − g (t , x, y, i)| ≤ L(|x − x| + |y − y|), |f (t , x, y, i)|2 ∨ |g (t , x, y, i)|2 ≤ L2 (1 + |x|2 + |y|2 ), for those x, y, x, y ∈ R and i ∈ S. In addition to the assumptions above, we further assume that (H2) There exist constants K1 > 0 and γ ∈ (0, 1] such that, for all −τ ≤ s < t ≤ 0 and r ≥ 2, E |ξ (t ) − ξ (s)|r ≤ K1 (t − s)γ . (H3) f and g have Taylor approximations in the second argument, up to m1 th and m2 th derivatives, respectively. (m +1) ( m +1 ) (H4) fx 1 (t , x, y, i) and gx 2 (t , x, y, i) are uniformly bounded, i.e., there exist positive constants L1 and L2 obeying sup

[0,T ]×R×R×S

|fx(m1 +1) (t , x, y, i)| ≤ L1 ,

sup

[0,T ]×R×R×S

|gx(m2 +1) (t , x, y, i)| ≤ L2 .

For some sufficiently large integer N, we define the time step by

1=

τ N

,

where 0 < 1  1. Then, the approximation solution to (2.1) is computed by y(t ) = ξ (t ) on −τ ≤ t ≤ 0, and, for any t ≥ 0, y(t ) = ξ (0) +

Z tX m1 (i) fx (s, z1 (s), z2 (s), r¯ (s)) (y(s) − z1 (s))i ds i! 0 i=0

Z tX m2 (i) gx (s, z1 (s), z2 (s), r¯ (s)) + (y(s) − z1 (s))i dB(s), i! 0 i =0

k = 0, 1, 2, . . . ,

(2.2)

where, for tk = k1 with integer k ≥ 0, z1 (t ) =

∞ X k=0

I[tk ,tk+1 ) y(tk ),

z2 (t ) =

∞ X k=0

I[tk ,tk+1 ) y(tk − τ ),

r¯ (t ) =

∞ X

I[tk ,tk+1 ) r (tk ).

(2.3)

k=0

Remark 2.1. If we let m1 = m2 = 0, then the approximate scheme (2.2) is reduced to EM method for SDDEwMS, which has been discussed in [2,15].

J. Bao, Z. Hou / Mathematical and Computer Modelling 50 (2009) 1379–1384

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In order to show the strong convergence between the numerical solutions and the exact solutions to (2.1), we will need another assumption: (H5) There exists a positive constant Q such that

 E

sup |x(t )|p



t ∈[0,T ]

 ∨E

sup |y(t )|(M +1)

2p

t ∈[0,T ]

 ≤Q

for any positive number p, where M = max{m1 , m2 }. Remark 2.2. Under the Lipschitz and linear growth condition, assumption (H1), [2, Theorem 7.3, p. 274] shows that the exact solutions to (2.1) admit finite moments. That is, for any p ≥ 0,

 E

sup |x(t )|p



t ∈[0,T ]

< ∞.

3. Necessary lemmas and the main result Since the proof of the main result is very technical, to begin with, we present several lemmas which will play an important role in the subsequent section. Lemma 3.1. Let the hypotheses (H1), (H3), (H4) as well as (H5) hold. Then, for 2 ≤ r ≤ (M + 1)p, E |y(t ) − z1 (t )|r ≤ C 1r /2 ,

t ≥ 0,

(3.1)

where C is a constant independent of 1. Proof. For simplicity, denote A(t , y(t ), z1 (t ), z2 (t ), r¯ (t )) =

m1 (i) X fx (t , z1 (t ), z2 (t ), r¯ (t ))

i!

i=0

B(t , y(t ), z1 (t ), z2 (t ), r¯ (t )) =

m2 (i) X gx (t , z1 (t ), z2 (t ), r¯ (t ))

i!

i=0

(y(t ) − z1 (t ))i , (y(t ) − z1 (t ))i .

Obviously, for any t ≥ 0, there exists an integer k ≥ 0 such that t ∈ [tk , tk+1 ). Then, by (2.3) we have y(t ) − z1 (t ) = y(t ) − y(tk )

Z

t

=

A(s, y(s), z1 (s), z2 (s), r¯ (s))ds +

Z

t

B(s, y(s), z1 (s), z2 (s), r¯ (s))dB(s).

(3.2)

tk

tk

Taking into account the Burkholder–Davis–Gundy inequality [2, Theorem 2.13, p. 70] and the Hölder inequality, we derive that, for some positive constant Cr ,

 Z

E |y(t ) − z1 (t )|r ≤ 2r −1 E

r

t tk

" ≤ 2r −1 (t − tk )r −1

r 

t

Z

A(s, y(s), z1 (s), z2 (s), r¯ (s))ds + E

tk

B(s, y(s), z1 (s), z2 (s), r¯ (s))dB(s)

t

Z

E |A(s, y(s), z1 (s), z2 (s), r¯ (s))|r ds tk

+ Cr (t − tk )

r /2−1

Z

#

t

E |B(s, y(s), z1 (s), z2 (s), r¯ (s))| ds r

tk

  = 2r −1 (t − tk )r /2−1 (t − tk )r /2 J1 (t ) + Cr J2 (t ) . Now, by the mean value theorem there is a θ ∈ (0, 1) such that J1 (t ) =

Z

t

E |f (s, y(s), z2 (s), r¯ (s)) − [f (s, y(s), z2 (s), r¯ (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))]|r ds tk

Z

t

= tk

r (m +1) fx 1 (s, z1 (s) + θ (y(s) − z1 (s)), z2 (s), r¯ (s)) m1 +1 E f (s, y(s), z2 (s), r¯ (s)) − (y(s) − z1 (s)) ds. (m1 + 1)!

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This, together with (H1), (H3)–(H5), yields that, for some constant C1 , J1 (t ) ≤ 2r −1

Z t

E (|f (s, y(s), z2 (s), r¯ (s))|2 )r /2 +

tk r −1

Z t"

≤2

r /2 r

3

Lr1

[(m1 + 1)!]

L (1 + E |y(s)| + E |z2 (s)| ) + r

r

tk



E |y(s) − z1 (s)|(m1 +1)r ds r

2(m1 +1)r Lr1

[(m1 + 1)!]r

# (m1 +1)r

(E |y(s)|

(m1 +1)r

+ E |z1 (s)|

) ds

= C1 (t − tk ). In the same way as J1 (t ) was done, we can also show that there exists a positive constant C2 for which J2 (t ) ≤ C2 (t − tk ). Next, recalling the boundedness of J1 (t ) and J2 (t ), we then have some positive constant C , independent of 1, such that E |y(t ) − z1 (t )|r ≤ C 1r /2 . The desired assertion is therefore complete.



Lemma 3.2. Under the hypotheses (H1)–(H5), for 2 ≤ r ≤ (M + 1)p, E |y(t − τ ) − z2 (t )|r ≤ C¯ 1γ ,

t ≥ 0,

where C¯ is a positive constant independent of 1. Proof. Clearly, for any t ≥ 0, there exists an integer k ≥ 0 such that t ∈ [tk , tk+1 ). In what follows, we split the following three cases to complete the proof. Case one. If −τ ≤ tk − τ ≤ t − τ ≤ 0, we then derive from (H2) that E |y(t − τ ) − z2 (t )|r = E |y(t − τ ) − y(tk − τ )|r = E |ξ (t − τ ) − ξ (tk − τ )|r ≤ K1 1γ . Case two. Let 0 ≤ tk − τ ≤ t − τ . For such case, following the proof of Lemma 3.1, E |y(t − τ ) − z2 (t )|r ≤ C3 1r /2 , where C3 is a positive constant. Case three. Let −τ ≤ tk − τ ≤ 0 ≤ t − τ . Note that E |y(t − τ ) − z2 (t )|r ≤ 2r −1 (E |y(t − τ ) − ξ (0)|r + E |y(tk − τ ) − ξ (0)|r ). Then, by Case one and Case two, it follows easily that, for some positive constant C4 , E |y(t − τ ) − z2 (t )|r ≤ C4 (1r /2 + 1γ ). Now, combining the three cases altogether, for 2 ≤ r ≤ (M + 1)p and γ ∈ (0, 1], E |y(t − τ ) − z2 (t )|r ≤ C 1γ ,

t ≥ 0,

where C is a constant independent of 1.



In view of the proceeding assertion, we can expect the sequence of the numerical solutions y(t ) converges to the true solution x(t ) to (2.1) as 1 → 0, in the sense of the Lp -norm. This can be done by the following theorem, which is the main result of the paper. Theorem 3.1. Under the hypotheses (H1)–(H5), for any p ≥ 2 and T ≥ 0,

 lim E

1→0

sup |x(t ) − y(t )|

p



t ∈[0,T ]

= 0.

Proof. Using the same notations in Lemma 3.1, it is easy to see from (2.1) and (2.2) that

Z t sup |x(t ) − y(t )| = sup (f (s, x(s), x(s − τ ), r (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s)))ds t ∈[0,T ] t ∈[0,T ] 0 p Z t + (g (s, x(s), x(s − τ ), r (s)) − B(s, y(s), z1 (s), z2 (s), r¯ (s)))dB(s) . p

0

J. Bao, Z. Hou / Mathematical and Computer Modelling 50 (2009) 1379–1384

1383

By the Burkholder–Davis–Gundy inequality [2, Theorem 2.13, p. 70] and the Hölder inequality, for some positive constant cr , compute

 E

sup |x(t ) − y(t )|p



t ∈[0,T ]

Z t p  ≤ 2p−1 E sup f (s, x(s), x(s − τ ), r (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))ds t ∈[0,T ] 0 Z t p  + E sup g (s, x(s), x(s − τ ), r (s)) − B(s, y(s), z1 (s), z2 (s), r¯ (s))dB(s) t ∈[0,T ]

0



≤ 2p−1 T p−1

T

Z

E |f (s, x(s), x(s − τ ), r (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))|p ds

0

+ cr T p/2−1

T

Z



E |g (s, x(s), x(s − τ ), r (s)) − B(s, y(s), z1 (s), z2 (s), r¯ (s))|p ds .

(3.3)

0

For convenience, denote that T

Z

J3 (t ) =

E |f (s, x(s), x(s − τ ), r (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))|p ds, 0 T

Z

J4 (t ) =

E |g (s, x(s), x(s − τ ), r (s)) − B(s, y(s), z1 (s), z2 (s), r¯ (s))|p ds. 0

However, J3 (t ) =

T

Z

E |f (s, x(s), x(s − τ ), r (s)) − f (s, y(s), z2 (s), r (s)) + f (s, y(s), z2 (s), r (s)) 0

− f (s, y(s), z2 (s), r¯ (s)) + f (s, y(s), z2 (s), r¯ (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))|p ds "Z T p ≤3 E |f (s, x(s), x(s − τ ), r (s)) − f (s, y(s), z2 (s), r (s))|p ds 0 T

Z

E |f (s, y(s), z2 (s), r (s)) − f (s, y(s), z2 (s), r¯ (s))|p ds

+ 0

#

T

Z

E |f (s, y(s), z2 (s), r¯ (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))| ds . p

+

(3.4)

0

Now, by virtue of the hypotheses (H1), we obtain that T

Z

E |f (s, x(s), x(s − τ ), r (s)) − f (s, y(s), z2 (s), r (s))| ds ≤ L p

p

0

T

Z

E (|x(s) − y(s)| + |x(s − τ ) − z2 (s)|)p ds 0

p p−1

T

Z

(E |x(s) − y(s)|p + E |x(s − τ ) − y(s − τ ) + y(s − τ ) − z2 (s)|p )ds

≤L 2

0

≤ Lp 22p−2

T

Z

(E |x(s) − y(s)|p + E |x(s − τ ) − y(s − τ )|p + E |y(s − τ ) − z2 (s)|p )ds.

(3.5)

0

On the other hand, following from [15], there is a positive constant C5 which is independent of 1 such that T

Z

E |f (s, y(s), z2 (s), r (s)) − f (s, y(s), z2 (s), r¯ (s))|p ds ≤ C5 1.

(3.6)

0

Furthermore, by Lemma 3.1 and (H4), there exists a θ1 ∈ (0, 1) such that T

Z

E |f (s, y(s), z2 (s), r¯ (s)) − A(s, y(s), z1 (s), z2 (s), r¯ (s))|p ds 0

(m1 +1)

T

Z =

E

|fx

0 p

≤

L1 TC

[(m1 + 1)!]p

(s, z1 (s) + θ1 (y(s) − z1 (s)), z2 (s), r¯ (s))|p |y(s) − z1 (s)|(m1 +1)p ds [(m1 + 1)!]p

1(m1 +1)p/2 .

(3.7)

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J. Bao, Z. Hou / Mathematical and Computer Modelling 50 (2009) 1379–1384

Substituting (3.5)–(3.7) into (3.4), together with Lemma 3.2, we then yield that

"

p

J3 (t ) ≤ 3p C5 1 +

p 2p−2

L1

[(m1 + 1)!]p

TC 1(m1 +1)p/2 + Lp 22p−2 C¯ T 1γ

#

T

Z

(E |x(s) − y(s)| + E |x(s − τ ) − y(s − τ )| )ds . p

+L 2

p

(3.8)

0

Similarly, there is a positive constant C6 satisfying

" J4 ( t ) ≤ 3

p

p

C6 1 +

p 2p−2

L2 TC

[(m2 + 1)!]p

1(m2 +1)p/2 + Lp 22p−2 C¯ T 1γ

#

T

Z

(E |x(s) − y(s)| + E |x(s − τ ) − y(s − τ )| )ds . p

+L 2

p

(3.9)

0

Hence, putting (3.8) and (3.9) into (3.3) gives immediately that

 E

sup |x(s) − y(s)|p



≤ C7 1 + C8 1(M +1)p/2 + C9 1γ + C10

t

Z

0≤s≤t

(E |x(s) − y(s)|p + E |x(s − τ ) − y(s − τ )|p )ds 0

≤ C7 1 + C8 1(M +1)p/2 + C9 1γ + 2C10

t

Z

 E

0



sup |x(r ) − y(r )|p ds, 0≤r ≤s

where C7 , C8 , C9 , C10 are some positive constants. Now, an application of the Gronwall inequality gives that

 E

sup |x(s) − y(s)|p




≤ e2C10 T C7 1 + C8 1(M +1)p/2 + C9 1γ .

(3.10)

0≤s≤t

The desired result is then complete.



Remark 3.1. If we remove the Markov chain r (t ) and (2.1) discussed without time lag, then the parameters C7 , C9 in (3.10) are equal to zero. Therefore, our result is reduced to the result derived in [14]. In other words, our result is the generalization of paper [14]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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