Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 305945, 14 pages
doi:10.1155/2012/305945
Research Article
Analytic Approximation of the Solutions of
Stochastic Differential Delay Equations with
Poisson Jump and Markovian Switching
Hua Yang1 and Feng Jiang2
1
2
School of Mathematics and Computer, Wuhan Polytechnic University, Wuhan 430023, China
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
Correspondence should be addressed to Feng Jiang, fjiang78@gmail.com
Received 2 May 2011; Revised 2 July 2011; Accepted 5 July 2011
Academic Editor: Ying U. Hu
Copyright q 2012 H. Yang and F. Jiang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We are concerned with the stochastic differential delay equations with Poisson jump and
Markovian switching SDDEsPJMSs. Most SDDEsPJMSs cannot be solved explicitly as stochastic
differential equations. Therefore, numerical solutions have become an important issue in the study
of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence
between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and
diffusion coefficients are Taylor approximations.
1. Introduction
Recently there has been an increasing interest in the study of stochastic differential delay
equations with Poisson jump and Markovian switching SDDEsPJMSs. Svı̄shchuk and
Kazmerchuk 1 investigated stability of stochastic differential delay equations with linear
Poisson jumps and Markovian switchings. While Luo 2 discussed the comparison principle
and several kinds of stability of Itô stochastic differential delay equations with Poisson
jump and Markovian switching. Besides, Li and Chang 3 discussed the convergence of
the numerical solutions of stochastic differential delay equations with Poisson jump and
Markovian switching. In the present paper we will further research this topic and our focus
is on the convergence of numerical solution to stochastic differential delay equation with
Poisson jump and Markovian switching when the coefficients are Taylor approximations.
Stochastic differential delay equation with Poisson jump and Markovian switching
may be considered as extension of stochastic differential delay equation with Poisson
jump. Of course, it may also be regarded as an generalization of stochastic differential
delay equation with Markovian switching. Similar to stochastic differential delay equations
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with Poisson jump, explicit solutions can hardly be obtained for the stochastic differential
delay equations with Poisson jump and Markovian switching. Thus appropriate numerical
approximation schemes such as the Euler or Euler-Maruyama are needed if we apply
them in practice or to study their properties. There is an extensive literature concerning the
approximate schemes for either stochastic differential delay equations with Poisson jump or
stochastic differential delay equations with Markovian switching 4–12.
However, the rate of convergence to the true solution by the numerical solution
is different for different numerical schemes 13. Recently, Janković and Ilić 14 have
investigated the rate of convergence between the true solution and numerical solution of
the stochastic differential equations in the sense of the Lp -norm when the drift and diffusion
coefficients are Taylor approximations, up to arbitrary fixed derivatives. Moreover, Jiang et al.
15 generalized the Taylor method to stochastic differential delay equations with Poisson
jump. Since the rate of convergence for such a numerical method is faster than the result
obtained in 13, in this paper, we intend to generalize the above method to the SDDEsPJMSs
case and consider the strong convergence between the true solution and numerical; solution
to SDDEsPJMSs 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.
In Section 2, we introduce necessary notations and approximation scheme. Then comes
our main result that the Taylor approximate solutions will converge to the true solutions of
SDDEsPJMSs. The proof of this main result is rather technical so we present several lemmas
in Section 3 and then complete the proof in Section 4.
2. Approximation Scheme and Hypotheses
Throughout this paper, we let Ω, F, {Ft }t≥0 , P be a complete probability space with a
filtration {Ft }t≥0 satisfying the usual conditions i.e., it is increasing and right-continuous
while F0 contains all P -null sets. Let C−τ, 0; R be the family of continuous function ξ
from −τ, 0 to R with the norm ξ sup−τ≤t≤0 |ξt|. We also denote by CFb 0 −τ, 0; R the
family of all bounded, F0 -measurable, C−τ, 0; R-valued random variables and denote
p
by LFt −τ, 0; R the family of all Ft -measurable, C−τ, 0; R-valued random variables
ξ {ξt : −τ ≤ t ≤ 0} satisfying sup−τ≤t≤0 E|ξt|p < κ < ∞, where κ is a positive constant.
Let wt, t ≥ 0, be a one-dimensional Brownian motion defined on the probability
space and let Nt, t ≥ 0, be a scalar Poisson process with intensity λ which is independent
of wt. Let rt, t ≥ 0, be a right-continuous Markov chain on the probability space taking
values in a finite state space S {1, 2, . . . , N} with the generator Γ γij N×N given by
⎧
if i /
j,
⎨γij δ oδ,
P rt δ j | rt i ⎩1 γ δ oδ, if i j,
ij
2.1
j while γii − i / j γij . We
where δ > 0. Here γij ≥ 0 is the transition rate from i to j if i /
assume that the Markov chain r· is independent of the Brownian motion w· and Poisson
process Nt. It is well known that almost every sample path of r· is a right-continuous step
function with finite number of simple jumps in any finite subinterval of R 0, ∞.
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3
Consider stochastic differential delay equations with Poisson jump and Markovian
switching of the form
dxt ft, xt, xt − τ, rtdt gt, xt, xt − τ, rtdwt
ht, xt, xt − τ, rtdNt,
2.2
on the time interval 0, T with initial data ξt ∈ CFb 0 −τ, 0; R being independent of wt
and Nt, and r0 i0 ∈ S, where
f : 0, ∞ × R × R × S −→ R,
j
g : 0, ∞ × R × R × S −→ R,
j
h : 0, ∞ × R × R × S → R.
2.3
j
In the paper, fx , gx , and hx denote, respectively, jth-order partial derivatives of
f, g and h with respect to x. Further, to ensure the existence and uniqueness of the solution
to 2.2, we impose the following hypotheses.
H1 f, g, and h satisfy Lipschitz and linear growth condition; that is, there exists
a positive constant L > 0 such that
ft, x1 , x2 , i − f t, y1 , y2 , i ∨ gt, x1 , x2 , i − g t, y1 , y2 , i 2 2 ∨ ht, x1 , x2 , i − h t, y1 , y2 , i ≤ L x1 − y1 x2 − y2 ,
2.4
ft, x1 , x2 , i2 ∨ gt, x1 , x2 , i2 ∨ |ht, x1 , x2 , i|2 ≤ L2 1 |x1 |2 |x2 |2 ,
for x1 , x2 , y1 , y2 ∈ R and i ∈ S.
H2 There exist constants K1 > 0 and γ ∈ 0, 1 such that, for all −τ ≤ s < t ≤ 0
and ρ ≥ 2,
E|ξt − ξs|ρ ≤ K1 t − sγ .
2.5
H3 f, g, and h have Taylor approximations in the second argument, up to m1 th,
m2 th, and m3 th derivatives, respectively.
H4 Partial derivatives of the order m1 1, m2 1, and m3 1 of the functions f,
m1 1
m 1
m 1
g, and h, fx
t, x, y, i, gx 2 t, x, y, i, and hx 3 t, x, y, i, are uniformly bounded; that
is, there exist positive constants L1 , L2 , and L3 obeying
sup
m1 1 t, x, y, i ≤ L1 ,
fx
0,T ×R×R×S
sup
m2 1 t, x, y, i ≤ L2 ,
gx
0,T ×R×R×S
sup
m3 1 t, x, y, i ≤ L3 .
hx
0,T ×R×R×S
2.6
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For some sufficiently large integer M, we define the time step by h τ/M, where
0 < h 1. Then, the approximation solution to 2.2 is computed by yt ξt on −τ ≤ t ≤ 0,
and for any t ≥ 0,
yt ξ0 t m1
0
t m2
j
j
gx s, z1 s, z2 s, rs ys − z1 s dws
j!
j0
0
j
j
fx s, z1 s, z2 s, rs ys − z1 s ds
j!
j0
t m3
2.7
j
j
hx s, z1 s, z2 s, rs ys − z1 s dNs,
j!
j0
0
where, for tk kh with integer k ≥ 0,
z1 t ∞
z2 t ytk Itk ,tk1 t,
k0
∞
ytk − τItk ,tk1 t,
rt k0
∞
k0
rtk Itk ,tk1 t.
2.8
In order to show the strong convergence of the numerical solutions and the exact
solutions to 2.2, we will need the following assumption.
H5 There exists a positive constant K2 such that
E sup |xt|
p
m12 p
∨ E sup yt
≤ K2 ,
0≤t≤T
2.9
0≤t≤T
for any p > 0, where m max{m1 , m2 , m3 }.
Remark 2.1. H1 shows that the exact solutions to 2.2 admit finite moments; see 15. If
m1 m2 m3 0, then H1 shows also that the numerical solutions and the exact solutions
admit finite moments see, 1, 3.
We can now state our main result of this paper.
Theorem 2.2. Under assumptions H1 –H5 , for any p ≥ 2, then
p
0.
lim E sup xt − yt
h→0
2.10
0≤t≤T
The proof of this theorem is rather technical. We will present a number of useful
lemmas in Section 3 and then complete the proof in Section 4.
3. Lemmas
Throughout our analysis, Ci , i 1, 2, . . . denote generic constants, independent of h. In order
to prove the main theorem, the following lemmas are useful.
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5
Lemma 3.1. If assumptions H1 , H3 , H4 , and H5 hold, then, for 2 ≤ ρ ≤ m 1p,
ρ
Eyt − z1 t ≤ Chρ/2 ,
3.1
t ≥ 0,
where C is a positive constant independent of h.
Proof. For notation simplicity reason, let us denote that
A t, yt, z1 t, z2 t, rt m1 j
fx t, z1 t, z2 t, rt j0
j!
j
yt − z1 t ,
m2 j
j
gx t, z1 t, z2 t, rt B t, yt, z1 t, z2 t, rt yt − z1 t ,
j!
j0
3.2
m3 j
j
hx t, z1 t, z2 t, rt C t, yt, z1 t, z2 t, rt yt − z1 t .
j!
j0
Obviously, for any t ≥ 0, there exists an integer k ≥ 0 such that t ∈ tk , tk1 . Then, by 2.7
and 2.8 we obtain
yt − z1 t yt − ytk t
t
A s, ys, z1 s, z2 s, rs ds B s, ys, z1 s, z2 s, rs dws
tk
tk
t
C s, ys, z1 s, z2 s, rs dNs.
tk
3.3
Hence, we have
ρ
t
ρ
Eyt − z1 t ≤ 3ρ−1 E A s, ys, z1 s, z2 s, rs ds
tk
ρ
t E B s, ys, z1 s, z2 s, rs dws
tk
3.4
ρ t E C s, ys, z1 s, z2 s, rs dNs .
tk
By the Hölder inequality, we have
ρ
t
t ρ
ρ−1
E A s, ys, z1 s, z2 s, rs ds ≤ t − tk EA s, ys, z1 s, z2 s, rs ds.
tk
tk
3.5
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By the Burkholder-Davis-Gundy inequality and the Hölder inequality, we yield, for some
positive constant Cρ ,
ρ
t E B s, ys, z1 s, z2 s, rs dws
tk
t
ρ
≤ Cρ t − tk ρ/2−1
EB s, ys, z1 s, z2 s, rs ds.
3.6
tk
For the jump integral, we convert to the compensated Poisson process Nt
: Nt − λt,
which is a martingale with
2
t2
t2 2
EC s, ys, z1 s, z2 s, rs ds;
E C s, ys, z1 s, z2 s, rs dNs λ
t1
t1
3.7
see, for example, 1, 4, 12, 16. By the Burkholder-Davis-Gundy inequality and the Hölder
inequality, for some positive constant Cλ,ρ , we then obtain
ρ
t E C s, ys, z1 s, z2 s, rs dNs
tk
ρ
t
t λ
E C s, ys, z1 s, z2 s, rs dNs
C s, ys, z1 s, z2 s, rs ds
tk
tk
≤2
ρ−1
ρ
t E C s, ys, z1 s, z2 s, rs dNs
tk
ρ
t ρ−1 2 Eλ
C s, ys, z1 s, z2 s, rs ds
tk
≤ 2ρ−1 Cλ,ρ t − tk ρ/2−1
t
ρ
EC s, ys, z1 s, z2 s, rs ds
tk
2ρ−1 λρ t − tk ρ−1
t
tk
ρ
EC s, ys, z1 s, z2 s, rs ds.
3.8
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7
Now, by the mean value theorem there is a θ ∈ 0, 1 such that
J1 t t
ρ
Ef s, ys, z2 s, rs − f s, ys, z2 s, rs −A s, ys, z1 s, z2 s, rs ds
tk
m 1 s, z1 s θ ys − z1 s , z2 s, rs
fx 1
Ef s, ys, z2 s, rs −
m1 1!
tk t
ρ
m1 1 × ys − z1 s
ds.
3.9
This, together with H1 , H3 –H5 , yields that
J1 t ≤ 2
ρ−1
t 2 ρ/2
E f s, ys, z2 s, rs tk
≤2
ρ−1
t tk
≤2
ρ−1
t m1 1ρ
ds
E ys − z1 s
m1 1!ρ
ρ
L1
ρ
ρ
L 2m1 1ρ
3ρ/2 Lρ 1 Eys E|z2 s|ρ 1
m1 1!ρ
⎤
m1 1ρ
× Eys
E|z1 s|m1 1ρ ⎦ds
ρ
L 1 2K2 κ ρ/2 ρ
3
tk
L1 2m1 1ρ1
m1 1!ρ
K2 ds
C1 t − tk .
3.10
Similarly, we can also show that there are two positive constant C2 and C3 for which
J2 t t
ρ
EB s, ys, z1 s, z2 s, rs ds
tk
≤ C2 t − tk ,
t
ρ
EC s, ys, z1 s, z2 s, rs ds
J3 t 3.11
tk
≤ C3 t − tk .
Next, since the boundedness of J1 t, J2 t, and J3 t, we then have some positive constant C,
independent of h, such that
ρ
Eyt − z1 t ≤ Chρ/2 .
The desired assertion is complete.
3.12
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Lemma 3.2. If assumptions H1 –H5 hold, then, for 2 ≤ ρ ≤ m 1p,
ρ
Eyt − τ − z2 t ≤ Chγ ,
t ≥ 0,
3.13
where γ ∈ 0, 1 and C is a positive constant independent of h.
Proof. Clearly, for any t ≥ 0, there exists an integer k ≥ 0 such that t ∈ tk , tk1 . In what
follows, we split the following three cases to complete the proof.
Case 1. If −τ ≤ tk − τ ≤ t − τ ≤ 0, we then have, from H2 ,
ρ
ρ
Eyt − τ − z2 t Eyt − τ − ytk − τ E|ξt − τ − ξtk − τ|ρ ≤ K1 hγ .
3.14
Case 2. If 0 ≤ tk − τ ≤ t − τ, then, by Lemma 3.1, we have
ρ
Eyt − τ − z2 t ≤ C4 hρ/2 .
3.15
Case 3. If −τ ≤ tk − τ ≤ 0 ≤ t − τ, note that
ρ
ρ
ρ
Eyt − τ − z2 t ≤ 2ρ−1 Eyt − τ − ξ0 2ρ−1 Eytk − τ − ξ0 .
3.16
Then, by the above two cases, it follows easily that
ρ
Eyt − τ − z2 t ≤ C5 hρ/2 hγ .
3.17
Now, combining the three cases altogether, for 2 ≤ ρ ≤ m 1P and γ ∈ 0, 1,
ρ
Eyt − τ − z2 t ≤ Chγ ;
3.18
where C is a positive constant independent of h. The proof is complete.
Lemma 3.3. If assumptions H1 and H5 hold, then, for p ≥ 2,
T
p
1 h,
Ef s, ys, z2 s, rs − f s, ys, z2 s, rs ds ≤ C
3.19
p
2 h,
Eg s, ys, z2 s, rs − g s, ys, z2 s, rs ds ≤ C
3.20
p
3 h,
Eh s, ys, z2 s, rs − h s, ys, z2 s, rs ds ≤ C
3.21
0
T
0
T
0
1, C
2 , and C
3 are positive constants dependent on max0≤i≤N −γii , but independent of h.
where C
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9
Proof. Let n T/h be the integer part of T/h. Then
T
E
p
|fs, ys, z2 s, rs − f s, ys, z2 s, rs ds
0
3.22
tk1
n
f s, ys, z2 s, rs − f s, ys, z2 s, rtk p ds,
E
k0
tk
with tn1 being T . By H1 and H5 , we derive
tk1
E
ff s, ys, z2 s, rs − f f s, ys, z2 s, rtk p ds
tk
≤ 2p−1 E
tk1 f s, ys, z2 s, rs p f s, ys, z2 s, rtk p I{rs / rt
k }
ds
tk
≤ 2p Lp E
tk1 tk
≤ 2p Lp 3p/2−1
p/2
2
1 ys |z2 s|2
I{rs / rtk } ds
tk1
tk
≤ 2p Lp 3p/2−1
tk1
tk
3.23
p
E E 1 ys |z2 s|p I{rs / rtk } | rtk ds
E E1 2K2 κ | rtk E I{rs / rtk } | rtk ds.
Now, by the Markov property 6, we have
I{rtk i} P rs /
i | rtk i
E I{rs / rtk } | rtk i∈S
γij s − tk os − tk I{rtk i}
i∈S
≤
j
/i
I{rtk i} max −γij o .
i∈S
1≤i≤N
So, 3.19 is complete. Similarly, we can show 3.20 and 3.21.
3.24
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Journal of Applied Mathematics
4. Proof of Theorem 2.2
Let us now begin to prove our main result Theorem 2.2. Clearly,
p
E sup xt − yt
0≤t≤T
t
E sup fs, xs, xs − τ, rs − A s, ys, z1 s, z2 s, rs ds
0≤t≤T 0
t
gs, xs, xs − τ, rs − B s, ys, z1 s, z2 s, rs dws
0
p hs, xs, xs − τ, rs − Cs, ys, z1 s, z2 s, rsdNs
0
t
p t
E sup fs, xs, xs − τ, rs − A s, ys, z1 s, z2 s, rs ds
0≤t≤T 0
≤ 3p−1
p t
E sup gs, xs, xs − τ, rs − B s, ys, z1 s, z2 s, rs dws
0≤t≤T 0
p t
.
E sup hs, xs, xs − τ, rs − C s, ys, z1 s, z2 s, rs dNs
0≤t≤T 0
4.1
So, for any t1 ≤ T , by the Hölder inequality,
p t
E sup fs, xs, xs − τ, rs − A s, ys, z1 s, z2 s, rs ds
0≤t≤t1 0
≤T
p−1
t1
p
Efs, xs, xs − τ, rs − A s, ys, z1 s, z2 s, rs ds.
4.2
0
By the Burkholder-Davis-Gundy inequality, for some positive constant Cp , we have
p t
E sup gs, xs, xs − τ, rs − B s, ys, z1 s, z2 s, rs dws
0≤t≤t1 0
≤ Cp T p/2−1
t1
0
p
Egs, xs, xs − τ, rs − B s, ys, z1 s, z2 s, rs ds.
4.3
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11
By Doob’s martingale inequality and the compensated Poisson integral, for some positive
constant Cλ,p,T , we yield
p t
E sup hs, xs, xs − τ, rs − C s, ys, z1 s, z2 s, rs dNs
0≤t≤t1 0
≤
p
p−1
p
p−1
2
2p−1 λp T p−1
t1
Cλ,p,T
t1
p
Ehs, xs, xs − τ, rs − C s, ys, z1 s, z2 s, rs ds
0
p
Ehs, xs, xs − τ, rs − C s, ys, z1 s, z2 s, rs ds.
0
4.4
Now, by virtue of the assumption H1 , we obtain that
J4 t t1
p
Efs, xs, xs − τ, rs − A s, ys, z1 s, z2 s, rs ds
0
t1
Efs, xs, xs − τ, rs − f s, ys, z2 s, rs
0
f s, ys, z2 s, rs − f s, ys, z2 s, rs
p
f s, ys, z2 s, rs − A s, ys, z1 s, z2 s, rs ds
t
1
p
p−1
≤3
Efs, xs, xs − τ, rs − f s, ys, z2 s, rs ds
4.5
0
t1
p
Ef s, ys, z2 s, rs − f s, ys, z2 s, rs ds
0
t1
p
Ef s, ys, z2 s, rs − A s, ys, z1 s, z2 s, rs ds.
0
Moreover, by H1 and Lemma 3.2, we have
t1
p
Efs, xs, xs − τ, rs − f s, ys, z2 s, rs ds
0
≤ Lp
t1
p
E xs − ys |xs − τ − z2 s| ds
0
≤L 2
p 2p−2
t1
p
p
p Exs − ys Exs − τ − ys − τ Eys − τ − z2 s ds
0
≤ Lp 22p−2
t1
0
p p
Exs − ys Exs − τ − ys − τ ds Lp 22p−2 T Chγ ,
4.6
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Journal of Applied Mathematics
and by Lemma 3.1 and H4 , there exists a θ1 ∈ 0, 1 such that
t1
p
Ef s, ys, z2 s, rs − A s, ys, z1 s, z2 s, rs ds
0
t1
0
p
m1 1 s, z1 s θ1 ys − z1 s , z2 s, rs fx
ys − z1 sm1 1p ds
E
p
m1 1!
4.7
p
≤
L1 T C
m1 1!p
hm1 1p/2 .
Hence, by 4.6 and 4.7, together with Lemma 3.3, we yield
J4 t ≤ 3
p−1
1h C
p
L1 T C
m1 1!p
Lp 22p−2
t1
hm1 1p/2 Lp 22p−2 T Chγ
p p
E xs − ys E xs − τ − ys − τ ds .
4.8
0
Similarly, we have
J5 t t1
p
Egs, xs, xs − τ, rs − B s, ys, z1 s, z2 s, rs ds
0
2h C
≤3
p−1
p
L2 T C
m2 1!p
L 2
p 2p−2
t1
hm2 1p/2 Lp 22p−2 T Chγ
4.9
p p
E xs − ys E xs − τ − ys − τ ds ,
0
J6 t t1
p
Ehs, xs, xs − τ, rs − C s, ys, z1 s, z2 s, rs ds
0
≤3
p−1
3h C
L 2
p
L3 T C
m3 1!p
p 2p−2
t1 0
hm3 1p/2 Lp 22p−2 T Chγ
p
p E xs − ys E xs − τ − ys − τ ds .
4.10
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13
Now we use K to denote a generic positive constant that may change between occurrences.
Hence, by 4.2–4.4 and 4.8–4.10, we yield that
E
p
sup xt − yt
0≤t≤t1
m1p/2
≤ Kh Kh
Kh K
γ
t1
p p
Exs − ys Exs − τ − ys − τ ds
4.11
0
m1p/2
≤ Kh Kh
t1 p
Kh 2K
E sup xs − ys ds .
γ
0
0≤r≤s
The continuous Gronwall inequality then gives
E
p
sup xt − yt
≤ Kh Khm1p/2 Khγ e2KT .
4.12
0≤t≤T
This proof is therefore complete.
Remark 4.1. If m1 m2 m3 0, then the approximate scheme 2.7 is reduced to
Euler-Maruyama method for stochastic differential delay equations with Poisson Jump and
Markovian switching, which has been discussed in 3.
Remark 4.2. As is well known, Taylor approximation is effectively applicable in engineering
if the equations can be solved explicitly. If not, since polynomials are very useful analytic
functions, the approximation in the paper can be useful in other applications of stochastic
Taylor expansion, especially in the construction of various time discrete approximations of
Itô processes by using Itô-Taylor expansion such as Euler-Maruyama approximation and
Milstein approximation, which has order 1/2 and 1, respectively 3, 8, 13. All these show
that numerical methods based on Taylor expansions of higher degrees could be improved by
combining them with analytic approximations presented in this paper.
Acknowledgments
The project reported here was supported by the Foundation of Wuhan Polytechnic University
and Zhongnan University of Economics and Law for Zhenxing.
References
1 A. V. Svı̄shchuk and Y. I. Kazmerchuk, “Stability of stochastic Itô equations with delays, with jumps
and Markov switchings, and their application in finance,” Theory of Probability and Mathematical
Statistics, no. 64, pp. 141–151, 2002.
2 J. Luo, “Comparison principle and stability of Ito stochastic differential delay equations with Poisson
jump and Markovian switching,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 2, pp.
253–262, 2006.
3 R. Li and Z. Chang, “Convergence of numerical solution to stochastic delay differential equation with
Poisson jump and Markovian switching,” Applied Mathematics and Computation, vol. 184, no. 2, pp.
451–463, 2007.
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Journal of Applied Mathematics
4 D. J. Higham and P. E. Kloeden, “Strong convergence rates for backward Euler on a class of nonlinear
jump-diffusion problems,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 949–
956, 2007.
5 J. Bao and Z. Hou, “An analytic approximation of solutions of stochastic differential delay equations
with Markovian switching,” Mathematical and Computer Modelling, vol. 50, no. 9-10, pp. 1379–1384,
2009.
6 X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press,
London, UK, 2006.
7 X. Mao, C. Yuan, and G. Yin, “Approximations of Euler-Maruyama type for stochastic differential
equations with Markovian switching, under non-Lipschitz conditions,” Journal of Computational and
Applied Mathematics, vol. 205, no. 2, pp. 936–948, 2007.
8 Y. Saito and T. Mitsui, “Stability analysis of numerical schemes for stochastic differential equations,”
SIAM Journal on Numerical Analysis, vol. 33, no. 6, pp. 2254–2267, 1996.
9 B. Li and Y. Hou, “Convergence and stability of numerical solutions to SDDEs with Markovian
switching,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1080–1091, 2006.
10 L.-S. Wang, C. Mei, and H. Xue, “The semi-implicit Euler method for stochastic differential delay
equations with jumps,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 567–578, 2007.
11 C. Yuan and W. Glover, “Approximate solutions of stochastic differential delay equations with
Markovian switching,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 207–226,
2006.
12 F. Jiang, Y. Shen, and J. Hu, “Stability of the split-step backward Euler scheme for stochastic delay
integro-differential equations with Markovian switching,” Communications in Nonlinear Science and
Numerical Simulation, vol. 16, no. 2, pp. 814–821, 2011.
13 P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications
of Mathematics, Springer, Berlin, Germany, 1992.
14 S. Janković and D. Ilić, “An analytic approximation of solutions of stochastic differential equations,”
Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 903–912, 2004.
15 F. Jiang, Y. Shen, and L. Liu, “Taylor approximation of the solutions of stochastic differential delay
equations with Poisson jump,” Communications in Nonlinear Science and Numerical Simulation, vol. 16,
no. 2, pp. 798–804, 2011.
16 D. J. Higham and P. E. Kloeden, “Numerical methods for nonlinear stochastic differential equations
with jumps,” Numerische Mathematik, vol. 101, no. 1, pp. 101–119, 2005.
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