Periodic averaging method for impulsive stochastic differential equations with Lévy noise

Accepted Manuscript Periodic averaging method for impulsive stochastic differential equations with L´evy noise

Shuo Ma, Yanmei Kang

PII: DOI: Reference:

S0893-9659(19)30047-3 https://doi.org/10.1016/j.aml.2019.01.040 AML 5775

To appear in:

Applied Mathematics Letters

Received date : 28 November 2018 Revised date : 28 January 2019 Accepted date : 28 January 2019 Please cite this article as: S. Ma and Y. Kang, Periodic averaging method for impulsive stochastic differential equations with L´evy noise, Applied Mathematics Letters (2019), https://doi.org/10.1016/j.aml.2019.01.040 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript Click here to view linked References

Periodic averaging method for impulsive stochastic differential equations with L´ evy noise Shuo Ma

Yanmei Kang∗

School of Mathematics and Statistics, Xi’an JiaoTong University, Xi’an, Shaanxi, 710049, P. R. China Abstract: The purpose of this paper is to present a periodic averaging method for impulsive stochastic differential equations with L´ evy noise under non-Lipschitz condition. It is shown that the solutions of impulsive stochastic differential equations with L´ evy noise converge to the solutions of the corresponding averaged stochastic differential equations without impulses. Keywords: Impulsive stochastic differential equations; Averaging method; Periodic averaging; L´ evy noise;

1

Introduction

Impulse is a common phenomenon and the theory of differential equations involving impulses is developing as an active area of investigation due to the application in areas such as engineering, biology, physics and so on[1-3]. The systems with impulses are utilized for studying the dynamics of processes subject to abrupt changes at discrete moments. There have been massive literatures covering stability, existence and uniqueness and averaging method in deterministic case. Stochastic impulsive differential equations arising in very natural way as mathematical models are often applied to describe the case where deterministic changes with impulses are interwoven with noisy fluctuations[4-10]. Usually, these fluctuations are assumed to be Brownian motion[4,5,8,9,10]. As a generalisation of the pure-diffusion Brownian motion, L´ evy process allows not only the trajectory to change continuously most of the time but also jump discontinuities occurring at random times[11]. The stochastic differential equations (SDEs) with L´ evy noise have been widely employed in mathematical finance, quantum physics and mathematical biology. On the other hand, the averaging method provides a powerful tool in order to strike a balance between complex models that are more realistic and simpler models that are more amenable to analysis and simulation. The basic idea behind the averaging method is to approximate the original system by a simpler system. Starting with the work of Khasminskii[12], the averaging method has been developed for many types of SDEs[13-15]. In general, the results of averaging method in standard form mainly fall into two categories: approximating coupled slow component equation in two time-scales SDE by a non-coupled equation[13,14], and approximating a nonautonomous SDE by an autonomous SDE[15]. Unfortunately, up to now this method has not been considered to simplify a SDE with impulses. Motivated by the above discussion, our aim is to obtain the periodic averaging method for impulsive stochastic differential equations with L´ evy noise under non-Lipschitz condition. Different from previous work, this research is to approximate nonautonomous SDEs with impulses by autonomous SDEs without impulses. Especially, the so-called non-Lipschitz condition as a weaker condition than Lipschitz condition will be used to derive our main results. ∗ Corresponding

author. E-mail: [email protected]

1

2

Preliminaries

Throughout this paper, unless otherwise specified, we let (Ω, F , {Ft }t≥0 , P ) be a complete probability space with a filtration {Ft }t≥0 satisfying usual conditions (i.e., it is right continuous and increasing while F0 contains all P -null sets). Let t ≥ 0 and D([t0 , t]; Rn ) denote the family of functions ϕ from [t0 , t] → Rn that are right-continuous and have limits on the left. D([t0 , t]; Rn ) is equipped with the norm k ϕ k= sup | ϕ(θ) |, where | · | is the Euclidean norm in Rn . L2Ft ([t0 , t]; Rn ) denotes the family of all t0 ≤θ≤t

bounded, Ft -measurable, D([t0 , t]; Rn )-valued random variables. According to L´ evy-Itˆ o decomposition theorem and interlacing technique[11], a class of impulsive stochastic differential equations driven by L´ evy noise can be interpreted as R ˜ (dt, dz), t 6= ti , dx(t) = f (t, x(t))dt + g(t, x(t))dW (t) + Z h(t, x(t), z)N − − (2.1) ∆x(ti ) = x(t+ t = ti , i ∈ N, i ) − x(ti ) = Ji (x(ti )), x(0) = x0 , where x0 is a F0 -measurable, Rn -valued random variable and E | x0 |2 < ∞. x(t) denotes a n-dimensional state process. Let {t1 < t2 < · · · < ti < · · · } be a impulse time sequence. Functions f : [0, ∞]×Rn → Rn , g : [0, ∞] × Rn → Rn×n , h : [0, ∞] × Rn × Z → Rn and Ji : Rn → Rn . W (t) is a n-dimensional standard Brownian motion defined on the probability space (Ω, F , {Ft }t≥0 , P ) with {Ft }t≥0 satisfying the usual conditions, and N (dt, dz) is a real-valued Poisson counting measure with characteristic measure λ on a ˜ (dt, dz) = N (dt, dz) − λ(dz)dt. Throughout the paper, it is assumed measurable subset Z of (−∞, ∞), N that W and N are independent. Moreover, without loss of generality, we also assume that x(t) is right continuous at time ti , that is x(ti ) = x(t+ i ), where xt+ = lim x(t) and xt− = lim x(t). t→ti+

i

i

t→ti−

In order to derive the main results of this paper, we require the functions f (t, x), g(t, x), h(t, x, z) and Ji (x) to satisfy the following assumptions. A1 There exists a concave continuous nondecreasing function ψ(·), for ∀x, y ∈ Rn and t ∈ [0, ∞) such that | f (t, y) − f (t, x) |2

W

| g(t, y) − g(t, x) |2

WR

Z

| h(t, y, z) − h(t, x, z) |2 λ(dz) ≤ ψ(| y − x |2 ),

R du = ∞. Furthermore, since ψ(·) is a concave where ψ(·) is from R+ to R+ such that ψ(0) = 0, 0+ ψ(u) continuous nondecreasing function, there must exist positive constants α1 and α2 such that ψ(u) ≤ α1 u + α2 ,

f or

all

u ≥ 0.

A2 There exist positive constants L1 and L2 such that | Ji (x) |2 ≤ L1

and

| Ji (x) − Ji (y) |2 ≤ L2 | x − y |

for every x, y ∈ Rn . Remark 2.1.We see clearly that if ψ(u) = Lu, then Lipschitz condition is recovered. Assumption A1 is much weaker than the usual requirement of Lipschitz condition. Remark 2.2. Similar to Lemma 3.1 in [16], the solutions of impulsive stochastic differential equations (2.1) can be rewritten as Rt Rt RtR ˜ (ds, dz) + P Ji (x(ti )). x(t) = x0 + 0 f (s, x)ds + 0 g(s, x(s))dW (s) + 0 Z hσ (s, x(s), z)N (2.2) 0
2

3

Main results

In this section, we will study the periodic averaging method for impulsive SDEs with L´ evy noise. The following Theorem 3.1 shows that a nonautomatic SDE with impulses of the form (2.1) can be transformed to an automatic SDE without impulses. In order to derive a periodic averaging theorem, it is assumed that f, g, h are T -periodic in the first argument and the impulses are periodic in the sense that there exists a k ∈ N such that 0 ≤ t1 < t2 < · · · < tk < T and for every integer i > k, we have ti = ti−k + T , Ji = Ji−k . Let f¯(x) : Rn → Rn , ¯ z) : Rn × Z → Rn and J(x) ¯ g¯(x) : Rn → Rn×n , h(x, : Rn → Rn be measurable functions satisfying the following definitions. Z Z 1 T 1 T f¯(x) = f (s, x)ds, g¯(x) = g(s, x)ds, T 0 T 0 Z Z k 1 X 1 T ¯ ¯ Ji (x). h(s, x, z)λ(dz)ds, J(x) = h(x) = T 0 Z T i=1 Theorem 3.1. Let the assumptions A1 -A2 hold. Suppose that for every ε ∈ (0, ε0 ], the impulsive stochastic differential equation √ √ R ˜ (dt, dz), t 6= ti , dxε (t) = εf (t, xε (t))dt + εg(t, xε (t))dW (t) + ε Z h(t, xε (t), z)N ∆xε (ti ) = εJi (xε (ti )), t = ti , i ∈ N, xε (0) = x0 ,

and the stochastic differential equation √ ¯ ε (t)))dt + √ε¯ dyε (t) = ε(f¯(yε (t)) + J(y g (yε (t))dW (t) + ε yε (0) = y0 ,

R

Z

¯ ε (t), z)N ˜ (dt, dz), h(y

have solutions xε and yε , respectively, where ε ∈ (0, ε0 ] is a positive small parameter with ε0 a constant. Assume that functions f (t, x), g(t, x), h(t, x, z) are bounded in the first argument. Moreover, if there is a constant R > 0 such that | x0 − y0 |2 ≤ Rε, then there exist constants η1 > 0 and η2 > 0 such that E | xε (t) − yε (t) |2 ≤ η1 ε,

∀t ∈ [0, η2 ε−1 ],

for all ε ∈ (0, ε1 ]. Proof. There exists a constant M > 0 such that | f (t, x) |2 ≤ M , | g(t, x) |2 ≤ M , | h(t, x) |2 ≤ M , ¯ | f¯(x) |2 ≤ M , | g¯(x) |2 ≤ M and | h(x) |2 ≤ M for every t ∈ [0, ∞). By using the elementary inequality, for any t ∈ [0, ∞), it is easy to show that Rt | xε (t) − yε (t) |2 ≤ 5 | x0 − y0 |2 +5ε2 | 0 [f (s, xε (s)) − f¯(yε (s))]ds |2 Rt +5ε | 0 [g(s, xε (s)) − g¯(yε (s))]dW (s) |2 RtR ¯ ε (s), z)]N ˜ (ds, dz) |2 +5ε | 0 Z [h(s, xε (s), z) − h(y (3.1) ∞ R P t ¯ 2 2 Ji (xε (ti )) − 0 J(yε (s))ds | +5ε | i=1

= 5 | x0 − y0 |2 +I1 + I2 + I3 + I4 .

Now to deal with E sup I1 , we have 0≤s≤t

E sup I1 0≤s≤t

≤ 10ε2 E sup | 0≤s≤t

+10ε2 E sup | 0≤s≤t

Rs 0

Rs 0

[f (θ, xε (θ)) − f (θ, yε (θ))]dθ |2

[f (θ, yε (θ)) − f¯(yε (θ))]dθ |2 .

3

Due to H¨ older’s inequality and assumption A1 , we get Rs 10ε2 E sup | 0 [f (θ, xε (θ)) − f (θ, yε (θ))]dθ |2 0≤s≤t Rt 2 ≤ 10ε tα1 0 E sup | xε (θ) − yε (θ) |2 ds + 10ε2 α2 t2 . 0≤θ≤s

Let m be the largest integer such that mT ≤ t. Then for every j = {1, . . . , m}, we have Rs 10ε2 E sup | 0 [f (θ, yε (θ)) − f¯(yε (θ))]dθ |2 0≤s≤t

2

≤

20ε E sup | 0≤s≤t

m R P jT

(j−1)T [f (θ, yε (θ))

j=1

− f¯(yε (θ))]dθ |2

Rs +20ε2E sup | mT [f (θ, yε (θ)) − f¯(yε (θ))]dθ |2 0≤s≤t RT 2 120ε mT 0 E sup ψ(| yε (θ) − yε (jT ) |2 )dθ

≤

2

+60ε E sup | 0≤s≤t

+20ε2E sup | 0≤s≤t

0≤s≤t m R P

jT [f (θ, yε (jT )) (j−1)T

j=1 Rs mT [f (θ, yε (θ))

− f¯(yε (jT ))]dθ |2

− f¯(yε (θ))]dθ |2 .

Thanks to the H¨ older’s inequality, Burkholder-Davis-Gundy’s inequality and assumption A1 , it can be obtained that E sup | yε (s) − yε (jT ) |2 0≤s≤t Rs Rs ≤ 3ε2 tE sup jT | f¯(yε (θ)) |2 dθ + 3εCE jT | g¯(yε (θ)) |2 dθ 0≤s≤t Rs R ¯ ε (θ), z) |2 λ(dz)dθ +3εCE jT | Z h(y ≤ 3ε2 (m + 1)M T 2 + 3εCM T + 3εCM T, t ∈ [(j − 1)T, jT ].

By the definition of f¯, we can get

E sup | ≤

m

0≤s≤t m P j=1

m R P jT

j=1

(j−1)T

E sup | 0≤s≤t

RT 0

[f (θ, yε (jT )) − f¯(yε (jT ))]dθ |2

f (θ, yε (jT ))dθ − T f¯(yε (jT )) |2 = 0.

Consequently, we obtain the following estimate for E sup I1 , 0≤s≤t

E sup I1 0≤s≤t

≤ 10ε2 tα1

Rt

+120ε2mT

0

E sup | xε (θ) − yε (θ) |2 ds + 10ε2 α2 t2

RT

0≤θ≤s

2 2 0 E sup (α1 | yε (θ) − yε (jT ) | +α2 )dθ + 80ε tM T 0≤s≤t Rt ≤ 10ε2 (m + 1)T α1 0 E sup | xε (θ) − yε (θ) |2 ds + 10ε2 α2 (m + 1)2 T 2 0≤θ≤s

+80ε2(m + 1)M T 2 + 120ε2 m2 T 2 α1 (3ε2 (m + 1)M T 2 + 3εCM T + 3εCM T ) +120ε2mT 2 α2 Rt := εC1 0 E sup | xε (θ) − yε (θ) |2 ds + εO1 . 0≤θ≤s

Now to deal with E sup I2 , under the help of Burkholder-Davis-Gundy’s inequality, we have 0≤s≤t

E sup I2 0≤s≤t

Rs

| g(θ, xε (θ)) − g¯(yε (θ))] |2 dθ Rs ≤ 10εCE sup 0 | g(θ, xε (θ)) − g(θ, yε (θ)) |2 dθ 0≤s≤t Rs +10εCE sup 0 | g(θ, yε (θ)) − g¯(yε (θ)) |2 dθ. ≤ 5εCE

0

0≤s≤t

4

(3.2)

Similar to I1 , according to assumption A1 , we deduce Rs E sup 0 | g(θ, yε (θ)) − g¯(yε (θ)) |2 dθ 0≤s≤t

≤

m R P jT

2E sup

+2E sup ≤

Rs

0≤s≤t m P

6E sup

mT

| g(θ, yε (θ)) − g¯(yε (θ)) |2 dθ

| g(θ, yε (θ)) − g¯(yε (θ)) |2 dθ

R jT

0≤s≤t j=1 m P

(j−1)T

R jT

| g(θ, yε (θ)) − g(θ, yε (jT )) |2 dθ

¯(yε (jT )) |2 dθ (j−1)T | g(θ, yε (jT )) − g 0≤s≤t j=1 m R P jT ¯(yε (jT ) − g¯(yε (θ)) |2 dθ + 8M T +6E sup (j−1)T | g 0≤s≤t j=1 RT 12m 0 E sup ψ(| yε (θ) − yε (jT ) |2 )dθ + 24mT M + 8M T. 0≤s≤t +6E sup

≤

(j−1)T

0≤s≤t j=1

Then, we estimate E sup I2 as follows, 0≤s≤t

E sup I2 0≤s≤t

≤ 10εCα1

Rs 0

E sup | xε (θ) − yε (θ) |2 dθ + 10εCα2 (m + 1)T + 80εCM T 0≤s≤t

+240εCmT M + 120εCm2 T α1 (3ε2 (m + 1)M T 2 + 3εCM T + 3εCM T ) +120εCmα2T Rt := εC2 0 E sup | xε (θ) − yε (θ) |2 ds + εO2 .

(3.3)

0≤θ≤s

Again, by applying Burkholder-Davis-Gundy’s inequality, H¨ older’s inequality and assumption A1 , one obtains Rs E sup I3 ≤ 10εCα1 0 E sup | xε (θ) − yε (θ) |2 dθ + 10εCα2 (m + 1)T + 80εCM T 0≤s≤t

0≤s≤t

+240εCmT M + 120εCm2 T α1 (3ε2 M (m + 1)T 2 + 3εCM T + 3εCM T ) +120εCmα2T Rt := εC3 0 E sup | xε (θ) − yε (θ) |2 ds + εO3 .

(3.4)

0≤θ≤s

For E sup I4 , we use assumption A2 to derive that 0≤s≤t

E sup I4 0≤s≤t

≤ 10ε2 k(m + 1)E sup

k P

0≤s≤t i=1

+10ε2(m + 1) T12 ktE sup

| Ji (xε (ti )) |2 k R P s

0≤s≤t i=1 2

0

| Ji (yε (θ)) |2 dθ

≤ 10ε2 (m + 1)k 2 L1 + 10ε (m + 1)k 2 (m + 1)2 L1 := εO4 . Taking (3.2)-(3.5) into account, it can be deduced that E sup | xε (s) − yε (s) |2 0≤s≤t Rt ≤ ε(C1 + C2 + C3 ) 0 E sup | xε (θ) − yε (θ) |2 ds + ε(O1 + O2 + O3 + O4 ) 0≤θ≤s Rt := εC4 0 E sup | xε (θ) − yε (θ) |2 ds + εO5 . 0≤θ≤s

5

(3.5)

Finally, by utilizing Gronwall’s inequality, we get E sup | xε (s) − yε (s) |2 ≤ εO5 eεC4 t . 0≤s≤t

Consequently, choose η2 > 0 such that for every t ∈ [0, η2 ε−1 ] ⊆ [0, ∞), let η1 = O5 eC4 η2 , we can choose ε1 ∈ (0, ε0 ] such that for each ε ∈ (0, ε1 ] and t ∈ [0, η2 ε−1 ], E

sup t∈[0,η2 ε−1 ]

| xε (t) − yε (t) |2 ≤ η1 ε.



Remark 3.1. If the system (2.1) has no impulses, then the convergence in mean square of periodic averaging method for stochastic differential equations with L´ evy noise can be recovered. Moreover, compared with the existing literatures on the averaging method of impulsive differential equations[17, 18], it is for the first time that we consider it in the stochastic case.

Acknowledgment The work is financially supported by the National Natural Science Foundation of China (No. 11372233 and 11772241).

References [1]

A. J. Catll´ a, D. G. Schaeffer, T. P. Witelski, E. E. Monson, A. L. Lin. On Spiking Models for Synaptic Activity and Impulsive Differential Equations. Siam Review, 50(3) (2008) 553-569.

[2]

S. Girel, F. Crauste. Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation. Journal of Mathematical Biology, 76(7) 2018 1765-1795.

[3]

X. D. Li, M. Bohner, C. K. Wang. Impulsive differential equations: Periodic solutions and applications. Automatica, 52 (2015) 173-178.

[4]

D. H. He, Y. M. Huang. Ultimate boundedness theorems for impulsive stochastic differential systems with Markovian switching. Applied Mathematics Letters, 65 (2017) 40-47.

[5]

L. G. Xu, S. Z. S. Ge. The pth moment exponential ultimate boundedness of impulsive stochastic differential systems. Applied Mathematics Letters, 42 (2015) 22-29.

[6]

R. Sakthivel, J. Luo. Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. Journal of Mathematical Analysis and Applications, 356 (2009) 1-6.

[7]

A. Boudaoui, T. Caraballo, A. Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete and continuous dynamical systems series B, 22(7) (2017) 2521-2541.

[8]

L. G. Xu, Z. L. Dai, D. H. He. Exponential ultimate boundedness of impulsive stochastic delay differential equations. Applied Mathematics Letters, 85 (2018) 70-76.

[9]

R. Sakthivel, J. Luo. Asymptotic stability of nonlinear impulsive stochastic differential equations. Statistics and Probability Letters, 79 (2009) 1219-1223.

[10]

T. Caraballo, M. A. Hammami, L. Mchiri. Practical exponential stability of impulsive stochastic functional differential equations. Systems & Control Letters, 109 (2017) 43-48.

[11]

D. Applebaum. L´ evy Processes and Stochastic Calculus. 1st edition, Cambridge University Press, 2004, 2nd edition, 2009.

[12]

R. Z. Khasminskii. On the principle of averaging the Itˆ o stochastic differential equations. Kibernetika, 4 (1968) 260-279.

[13]

J. Xu. Lp (p > 2)-strong convergence of the averaging principle for slow-fast SPDEs with jumps. Journal of Mathematical Analysis and Applications, 445 (2017) 342-373.

[14]

S. Cerrai, M. Freidlin. Averaging principle for a class of stochastic reaction-diffusion equations. Probability Theory and Related Fields, 144 (2009) 137-177.

[15]

W. Mao, S. R. You, X. Q. Wu, X. R. Mao. On the averaging principle for stochastic delay differential equations with jumps. Advances in Difference Equations, (2015) 2015:70.

[16]

L. J. Shen, J. T. Sun. Global existence of solutions for stochastic impulsive differential equations. Acta Mathematica Sinica, English Series, 27(4) (2011) 773-780.

[17]

J. G. Mesquita, A. Slav´ık. Periodic averaging theorems for various types of equations. Journal of Mathematical Analysis and Applications, 387 (2012) 862-877.

[18]

M. Federson, J. G. Mesquita, A. Slav´ık. Basic results for functional differential and dynamic equations involving impulses. Mathematische Nachrichten, 286 (2013) 181-204.

6