Successive approximation of solutions to doubly perturbed stochastic differential equations with jumps

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Successive approximation of solutions to doubly perturbed stochastic differential equations with jumps

Wei Mao

^1,2

, Liangjian Hu

^B³

, Surong You

³

and Xuerong Mao

⁴

1College of Information Sciences and Technology, Donghua University, Shanghai, 201620, China

2School of mathematics and information technology, Jiangsu Second Normal University Nanjing 210013, China

3Department of Applied Mathematics, Donghua Univerisity, Shanghai, 201620, China

4Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, U.K.

Received 27 May 2017, appeared 5 December 2017 Communicated by John A. D. Appleby

Abstract. In this paper, we study the existence and uniqueness of solutions to doubly perturbed stochastic differential equations with jumps under the local Lipschitz conditions, and give the p-th exponential estimates of solutions. Finally, we give an example to illustrate our results.

Keywords: doubly perturbed stochastic differential equations, Lévy jumps, local non- Lipschitz condition, p-th exponential estimates.

2010 Mathematics Subject Classification: 60H10, 34F05.

1 Introduction

As the limit process from a weak polymers model, the following doubly perturbed Brownian motion

x_t= B_t+αmax

0≤s≤tx_s+β min

0≤s≤tx_s, (1.1)

was studied by P. Carmona [4] and J. R. Norris [11]. Because of its important application, many people have devoted their investigation to this model and obtained a lot of results, for example, see [2,3,5–7,12,14]. Motivated by above mentioned works, R. A. Doney and T. Zhang [8] studied the singly perturbed Skorohod equations

x_t= x₀+

Z _t

0 σ(x_s)dB_s+

Z _t

0 b(x_s)ds+αmax

0≤s≤tx_s. (1.2) they proved the existence and uniqueness of the solution to equation(1.2) where the coefficients b,σ satisfy the global Lipschitz conditions; Hu and Ren [9] and Luo [10] extended the global Lipschitz conditions of [8] to the case of non-Lipschitz conditions which are imposed by

BCorresponding author. Email: ljhu@dhu.edu.cn

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[13,17,18], they proved the existence and uniqueness of solutions to doubly perturbed neutral stochastic functional equations and doubly perturbed jump-diffusion processes, respectively.

However, for many practical situations, the nonlinear terms do not obey the global Lips- chitz and linear growth condition, even the non-Lipschitz condition. For example, consider the singly perturbed semi-linear stochastic differential equations

dx(t) =ax(t)dt+σ(x(t))b(t,x(t))dBt+βmax

0≤s≤tx(s), t ∈[0,T]. (1.3) where a ∈ R, β ∈ (_{0, 1})_{, and} σ(x) satisfies the local Lipschitz condition: For any integer N >0, there exists a positive constantk_N such that for all x,y∈ Rwith|x|,|y| ≤N, it follows that

|σ(x)−σ(y)| ≤k_N|x−y|_. _(1.4) Let us take

ρ(u) =











0, u=0,

u[log(u⁻¹)]^r, u∈(0,δ], δ[log(δ⁻¹)]^r, u∈[δ,+_∞],

wherer ∈ [0,¹₂)andδ ∈ (0, 1)is sufficiently small, Assumeb(t,x)satisfies the non-Lipschitz condition

|b(t,x)−b(t,y)| ≤ρ(|x−y|). (1.5) From the analysis of Section 5, the coefficients of equation (1.3) do not satisfy the global Lip- schitz condition [8] or non-Lipschitz condition [9,10]. In other words, the main results of [8–10] do not apply to equation (1.3). Therefore, it is very important to establish the existence and uniqueness theory of perturbed stochastic differential equations under some weaker conditions. The purpose of this paper is to study the existence and uniqueness of solutions to equation (2.1) with the local non-Lipschitz coefficients. Meantime, we will give the pth exponential estimates and the pth moment continuity of solutions.

This paper is organized as follows. In Section 2, we first give some preliminaries and assumptions on equation (2.1). In Section 3, we state and prove our main results. While in Section 4, we show that the pth moment of solution will grow at most exponentially. As an application of the pth exponential estimates, we give the continuity of the pth moment of solutions. Finally, we give an example to illustrate the theory in Section 5.

2 Preliminaries

Let (_Ω,F,{F_t}_t_≥₀,P) be a complete probability space with a filtration {F_t}_t_≥₀ satisfying the usual conditions (i.e. it is increasing and right continuous while F₀ contains all P-null sets). Let{w(t)}_t_≥₀ be a one-dimensional Brownian motion defined on the probability space (_Ω,F,P). Let {p¯ = p¯(t),t ≥ 0}be a stationaryF_t-adapted andR-valued Poisson point process. Then, for A ∈ B(R− {0}), hereB(R− {0}) denotes the Borelσ-field on R− {0}and 06∈the closure of A, we define the Poisson counting measureNassociated with ¯pby

N((_0,t]×A)_:=_#{0< s≤t, ¯p(s)∈ A}=

∑

t0<s≤t

I_A(p¯(s))_,

where # denotes the cardinality of set{·}. It is known that there exists aσ-finite measureπ such that

E[N((0,t]×A)] =π(A)t, P(N((0,t]×A) =n) = ^exp(−tπ(A))(π(A)t)ⁿ

n! .

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This measureπis called the Lévy measure. Moreover, by Doob–Meyer’s decomposition theorem, there exists a unique{F_t}-adapted martingale ˜N((0,t]×A)and a unique{F_t}-adapted natural increasing process ˆN((0,t]×A)such that

N((0,t]×A) =N^˜((0,t]×A) +N^ˆ((0,t]×A), t>0.

Here ˜N((0,t]×A)is called the compensated Lévy jumps and ˆN((0,t]×A) =π(A)tis called the compensator.

Let p ≥ 2, L^p([a,b];R) denote the family of F_t-measurable, R-valued process f(t) = {f(t,ω)}, t ∈ [a,b] such thatRb

a |f(t)|^pdt < _{∞. For} Z ∈ B(R− {0}), consider the following doubly perturbed stochastic differential equations (SDEs) with Lévy jumps

x(t) =x(0) +

Z _t

0 f(s,x(s))ds+

Z _t

0 g(s,x(s))dw(s) +

Z _t

0

Z

h(s,x(s−),v)N(ds,dv) +αmax

0≤s≤tx(s) +β min

0≤s≤tx(s)_, _(2.1) where α,β ∈ (0, 1), the initial value x(0) = x₀ ∈ R and f : [0,T]×R → R, g : [0,T]×R → R, h : [0,T]×R×Z → R are both Borel-measurable functions. In this paper, we assume that Lévy jumps N is independent of Brownian motion w and the random variable x₀ is independent ofw, Nand satisfies E|x₀|^p<_∞.

To obtain the main results, we suppose R

Zπ(dv) = π(Z) < _∞ and give the following conditions.

Assumption 2.1. For anyx,y∈ Randt∈ [0,T], there exist two functionsk(.),ρ(.)such that

|f(t,x)− f(t,y)| ∨ |g(t,x)−g(t,y)| ≤λ(t)k(|x−y|), Z

Z

|h(t,x,v)−h(t,y,v)|²π(dv)≤ λ²(t)ρ²(|x−y|),

where λ(t)∈ L²([0,T],R), and k(u),ρ(u)are two concave nondecreasing functions such that k(0) =ρ(0) =0 andR

0⁺ u

k²(u)+ρ²(u)du= _∞.

Assumption 2.2. There exist two positive constantsK₁,K₂such that sup

0≤t≤T

{|f(t, 0)| ∨ |g(t, 0)|} ≤K₁, sup

0≤t≤T

Z

|h(t, 0,v)|²π(dv)≤K2. Assumption 2.3. The coefficients satisfy|α|+|β|<1.

Assumption 2.4. For any integer N > 0, there exist two positive constants k_N and ρ_N such that

|f(t,x)− f(t,y)| ∨ |g(t,x)−g(t,y)| ≤λ(t)k_N(|x−y|),

and Z

Z

|h(t,x,v)−h(t,y,v)|²π(dv)≤ λ²(t)ρ²_N(|x−y|),

for any x,y ∈ R with |x|,|y| ≤ N. Here k_N(u),ρ_N(u) are two concave and nondecreasing functions such thatk_N(0) =ρ_N(0) =0 andR

0⁺ u

k²_N(u)+ρ²_N(u)du= _∞.

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Remark 2.5. Clearly, Assumptions2.1and2.2imply the linear growth condition. Sincek(·)is concave andk(0) =0, we can find a pair of positive constantsaandbsuch that

k(u)≤a+bu, for u≥0.

Therefore, for anyx ∈Randt∈ [0,T],

|f(t,x)| ∨ |g(t,x)| ≤λ(t)k(|x|) +K₁,

≤ aλ(t) +K₁+bλ(t)|x|. Similarly, we can obtain

Z

|h(t,x,v)|²π(dv)≤_4λ²(t)a²+2K₂+_4λ²(t)b²|x|²_.

In the sequel, to prove our main results we recall the following two lemmas.

Lemma 2.6([1]). Let k : R+ → R+ be a continuous, non-decreasing function satisfying k(0) = 0 andR

0⁺ ds

k(s) = +_{∞. Let u}(·)be a Borel measurable bounded non-negative function defined on [0,T] satisfying

u(t)≤u₀+

Z _t

0 v(s)k(u(s))ds), t∈ [0,T]

where u₀ >0and v(·)is a non-negative integrable function on[0,T]. Then we have u(t)≤G⁻¹

G(u₀) +

Z _t

0 v(s)ds

, where G(t) =Rt

t₀ du

k(u) is well defined for some t₀ >0, and G⁻¹is the inverse function of G.

In particularly, if u₀=0, then u(t) =0for all t∈[0,T]. Lemma 2.7([15]). Letφ:R+×Z→Rⁿand assume that

Z _t

0

Z

|φ(s,v)|^pπ(dv)ds< _∞, p ≥2.

Then, there exists D_p>0such that E sup

0≤t≤u

Z _t

0

Z

φ(s,v)N^˜(ds,dv)

p!

≤Dp

( E

Z _u

0

Z

|φ(s,v)|²π(dv)ds ₂^p

+E Z _u

0

Z

|φ(s,v)|^pπ(dv)ds )

.

3 Existence and uniqueness theorem

In this section, we study the existence and uniqueness of solutions to doubly perturbed SDEs with Lévy jumps and the local non-Lipschitz coefficients.

Let us consider the following equation x(t) =x(0) +

Z _t

0 f(s)ds+

Z _t

0 g(s)dw(s) +

Z _t

0

Z

Zh(s−,v)N(ds,dv) +αmax

0≤s≤tx(s), (3.1)

with the initial datax(₀) =x₀and f ∈ L²([_0,T]_;R)_,g∈ L²([_0,T]_;R)_andh∈ L²([_0,T]×Z;R)_.

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Proposition 3.1. Under Assumptions2.2–2.3, Equation(3.1)has a unique solution x(t)on[0,T]. The proof of Proposition3.1is given in the Appendix.

Now, we construct a successive approximation sequence using a Picard type iteration. Let x⁰(t) =x₀,t ∈[0,T], define the following Picard sequence:

xⁿ(t) =x(0) +

Z _t

0 f(s,xⁿ⁻¹(s))ds+

Z _t

0 g(s,xⁿ⁻¹(s))dw(s) +

Z _t

0

Z

Zh(s,xⁿ⁻¹(s−),v)N(ds,dv)+αmax

0≤s≤txⁿ(s) +β min

0≤s≤txⁿ(s). (3.2) Obviously, according to proposition3.1, the solutionxⁿ(t)of equation (3.2) exists.

In what follows,C>0 is a constant which can change its value from line to line.

Lemma 3.2. Under Assumptions2.1–2.3, there exists a constant C₁ >0such that for any t∈ [0,T] E max

0≤t≤T|xⁿ(t)|²≤C₁. (3.3) Proof. For anys≥0, it follows that from (3.2)

|xⁿ(s)| ≤ |x(0)|+

Z _s

0 f(σ,xⁿ⁻¹(σ))dσ

+

Z _s

0 g(σ,xⁿ⁻¹(σ))dw(σ) +

Z _s

0

Z

Zh(σ,xⁿ⁻¹(σ−),v)N(dσ,dv)

+|α|

0max≤σ≤sxⁿ(σ)

+|β|

0min≤σ≤sxⁿ(σ)

≤ |x(0)|+

Z _s

+

Z _s

0 g(σ,xⁿ⁻¹(σ))dw(σ) +

Z _s

0

Z

+ (|α|+|β|) max

0≤σ≤s|xⁿ(σ)|. (3.4) Taking the maximal value on both sides of (3.4), by the Hölder inequality, the Doob’s martingale inequality and Assumption 2.3, we have

(1− |α| − |β|)²Emax

0≤s≤t|xⁿ(s)|²

≤E|x(0)|²+Emax

0≤s≤t

Z _s

2

+Emax

0≤s≤t

Z _s

0 g(σ,xⁿ⁻¹(σ))dw(σ)

2

+Emax

0≤s≤t

Z _s

0

Z

2

≤C

E|x(0)|²+E Z _t

0

|f(s,xⁿ⁻¹(s))|²ds+E Z _t

0

|g(s,xⁿ⁻¹(s))|²ds +[8+Tπ(Z)]E

Z _t

0

Z

|h(s,xⁿ⁻¹(s−),v)|²π(dv)ds

. Therefore, we get

Emax

0≤s≤t|xⁿ(s)|²

≤ ^C

(1− |α| − |β|)²[_E|x(₀)|²+_2E

Z _t

0

h

|f(_s,_xⁿ⁻¹(_s))− f(_{s, 0})|²+|g(_s,_xⁿ⁻¹(_s))−g(_{s, 0})|²ⁱds +2[8+Tπ(Z)]E

Z _t

0

Z

|h(s,xⁿ⁻¹(s−),v)−h(s, 0,v)|²π(dv)ds +2E

Z _t

0

|f(s, 0)|²+|g(s, 0)|²ds+₂[₈+Tπ(Z)]E Z _t

0

Z

|h(s, 0,v)|²π(dv)ds]_.

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By Assumptions2.1 and2.2, we have Emax

0≤s≤t|xⁿ(s)|²≤ ^C (1− |α| − |β|)²

E|x(0)|²+4K₁²T+2[8+Tπ(Z)]K₂T +4E

Z _t

0

|λ(s)|²k²(|xⁿ⁻¹(s)|)ds +2[8+Tπ(Z)]E

Z _t

0

|λ(s)|²ρ²(|xⁿ⁻¹(s−)|)ds

. Then the Jensen inequality implies that

Emax

0≤s≤t|xⁿ(s)|²≤ ^C (1− |α| − |β|)²

E|x(0)|²+4K₁²T+2[8+Tπ(Z)]K₂T +4

Z _t

0

|λ(s)|²k²((E|xⁿ⁻¹(s)|²)¹²)ds +2[8+Tπ(Z)]

Z _t

0

|λ(s)|²ρ²((E|xⁿ⁻¹(s)|²)¹²)ds

.

Lettingγ(x) =k²(x¹²) +ρ²(x¹²), it follows that Emax

0≤s≤t|xⁿ(s)|² ≤ ^C (1− |α| − |β|)²

E|x(0)|²+4K²₁T+2[8+Tπ(Z)]K₂T +2[8+Tπ(Z)]

Z _t

0

|λ(s)|²γ(E|xⁿ⁻¹(s)|²)ds

. (3.5) By Assumption2.1, we have that γ is a non-decreasing continuous function, γ(0) = 0 and R

0⁺ 1

γ(x)dx =_{∞. Since} ^k⁽_x^x⁾_, ^ρ⁽_x^x⁾_,_k⁰₊(_x)_andρ⁰₊(_x)are non-negative, non-increasing functions, we have that

γ⁰₊(x) =x⁻¹² h

k(x¹²)k⁰₊(x) +ρ(x¹²)ρ⁰₊(x)ⁱ

is a non-negative, non-increasing function, thus γis a non-negative, non-decreasing concave function. Sinceγ(·)is concave and γ(0) = 0, we can find a pair of positive constantsaandb such that

γ(u)≤a+bu, foru≥_0, we obtain

Emax

0≤s≤t|xⁿ(s)|²≤ ^C (1− |α| − |β|)²

1+E|x(0)|²+a Z _t

0

|λ(s)|²ds +b

Z _t

0

|λ(s)|²E max

0≤σ≤s|xⁿ⁻¹(σ)|²ds

. (3.6)

Set

r(t) =

C

(1− |α| − |β|)²(1+E|x(0)|²+a Z _t

0

|λ(s)|²ds

e^b^R⁰^t^|^λ⁽^s^)|²^ds, thenr(·)is the solution to the following ordinary differential equation:

r(t) = ^C (1− |α| − |β|)²

1+E|x(0)|²+a Z _t

0

|λ(s)|²ds+b Z _t

0

|λ(s)|²r(s)ds

.

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By recurrence, it is easy to verify that for eachn≥0, Emax

0≤s≤t|xⁿ(s)|²≤r(t). Sincer(t)is continuous and bounded on [0,T], we have

Emax

0≤s≤t|xⁿ(s)|² ≤C₁ <+_∞, for any n≥1. The proof is complete.

Lemma 3.3. Let Assumptions2.1–2.3hold, then{xⁿ(t)}_n_≥₁ defined by(3.2)is a Cauchy sequence.

Proof. For anyn,m≥1, we have

|Xⁿ(t)−X^m(t)|

≤

Z _t

0

h

f(s,xⁿ⁻¹(s))− f(s,x^m⁻¹(s))ⁱds

+

Z _t

0

h

g(s,xⁿ⁻¹(s))−g(s,x^m⁻¹(s))ⁱdw(s) +

Z _t

0

Z

h

h(s,xⁿ⁻¹(s−),v)−h(s,xⁿ⁻¹(s−),v)ⁱN(ds,dv) +|α|

0max≤s≤txⁿ(s)− max

0≤s≤tx^m(s)

+|β|

0min≤s≤txⁿ(s)− min

0≤s≤tx^m(s)

≤

Z _t

0

h

f(s,xⁿ⁻¹(s))− f(s,x^m⁻¹(s))ⁱds

+

Z _t

0

h

g(s,xⁿ⁻¹(s))−g(s,x^m⁻¹(s))ⁱdw(s) +

Z _t

0

Z

h

h(s,xⁿ⁻¹(s−),v)−h(s,xⁿ⁻¹(s−),v)ⁱN(ds,dv) + (|α|+|β|)max

0≤s≤t|xⁿ(s)−x^m(s)|. (3.7)

Taking the maximal value on both sides of (3.7), by the Hölder inequality, the Doob’s martingale inequality and Assumption 2.3, we have

(₁− |α| − |β|)²Emax

0≤s≤t|xⁿ(_s)−x^m(_s)|²

≤ Emax

0≤s≤t

Z _s

0

h

f(σ,xⁿ⁻¹(σ))− f(σ,x^m⁻¹(σ))ⁱdσ

2

+Emax

0≤s≤t

Z _s

0

h

g(σ,xⁿ⁻¹(σ))−g(σ,x^m⁻¹(σ))ⁱdw(σ)

2

+Emax

0≤s≤t

Z _s

0

Z

h

h(σ,xⁿ⁻¹(σ−),v)−h(σ,x^m⁻¹(σ−),v)ⁱN(dσ,dv)

2

≤ TE Z _t

0

f(s,xⁿ⁻¹(s))− f(s,x^m⁻¹(s))

2ds+4E Z _t

0

g(s,xⁿ⁻¹(s))−g(s,x^m⁻¹(s))

2ds

+ [₈+2Tπ(Z)]E Z _t

0

Z

h(s,xⁿ⁻¹(s−),v)−h(s,x^m⁻¹(s−),v)²π(dv)ds.

By Assumption2.1and Jensen’s inequality, we get Emax

0≤s≤t|xⁿ(s)−x^m(s)|²

≤ ¹

(1− |α| − |β|)²

(T+4)

Z _t

0

|λ(s)|²k²((E|xⁿ⁻¹(s)−x^m⁻¹(s)|²)¹²)ds + (8+2Tπ(Z))

Z _t

0

|λ(s)|²ρ²((E|xⁿ⁻¹(s)−x^m⁻¹(s)|²)¹²)ds

.

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Similar to (3.5), we obtain Emax

0≤s≤t|xⁿ(s)−x^m(s)|²≤ ¹²+T+2Tπ(Z) (₁− |α| − |β|)²

Z _t

0

|λ(s)|²γ(E|xⁿ⁻¹(s)−x^m⁻¹(s)|²)ds. (3.8) By the inequality (3.3) and Fatou’s lemma, it is easily seen that

lim sup

n,m→∞ E(max

0≤s≤t|xⁿ(s)−x^m(s)|²)

≤ ¹²+T+2Tπ(Z) (1− |α| − |β|)²

Z _t

0

|λ(s)|²γ lim sup

n,m→_∞

E max

0≤σ≤s|xⁿ(σ)−x^m(σ)|²

!

ds. (3.9) Owing to Lemma2.6, we immediately get that

lim sup

n,m→_∞

E(max

0≤s≤t|xⁿ(s)−x^m(s)|²) =0, for allt ∈[0,T], (3.10) Then{xⁿ(t)}_n_≥₁is a Cauchy sequence. The proof is complete.

Now, we state and prove our main results.

Theorem 3.4. Let Assumptions2.1–2.3hold, then equation(2.1)has a unique solution x(t)on[0,T]. Proof. According to (3.10), it follows that there existsx(t)∈ L²([_0,T]_;R)_{such that}

nlim→_∞E sup

0≤s≤t

|xⁿ(s)−x(s)|²=0.

Then the Borel–Cantelli lemma can be used to show thatxⁿ(t)converges tox(t)almost surely uniformly on [0,T] as n → ∞. Taking limits on both sides of (3.2) and letting n → _{∞, we} obtain thatx(t)is a solution of equation (2.1).

Now we devote to proving the uniqueness of equation (2.1). Suppose x(t) and y(t) are two solutions of equation (2.1) with initial value x₀andy₀, we have

|x(t)−y(t)|

≤ |x0−y0|+

Z _t

0

[f(s,x(s))− f(s,y(s))]ds

+

Z _t

0

[g(s,x(s))−g(s,y(s))]dw(s) +

Z _t

0

Z

[h(s,x(s−),v)−h(s,y(s−),v)]N(ds,dv)

+|α|

0max≤s≤tx(s)−max

0≤s≤ty(s) +|β|

0min≤s≤tx(s)− min

0≤s≤ty(s) .

Then, in the same way as the proof of (3.8) one can show that Emax

0≤s≤t|x(s)−y(s)|²≤C|x₀−y₀|²+C Z _t

0

|λ(s)|²γ

E

0max≤σ≤s|x(σ)−y(σ)|²

ds fort ∈[0,T]. By Lemma2.6, we get

Emax

0≤s≤t|x(_s)−y(_s)|²≤ G⁻¹

G(_C|x0−y0|²) +_C

Z _t

0

|λ(_s)|²ds

, whereG(t) =Rt

1 ds

γ(s). In particular, ifx₀ =y₀, then

G(C|x0−y0|²) =−_∞, G(C|x0−y0|²) +C Z _t

0

|λ(s)|²ds= −_∞.

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Obviously, Gis a strictly increasing function, then Ghas an inverse function which is strictly increasing, andG⁻¹(−_∞) =0. Finally, we obtain

Emax

0≤s≤t|x(s)−y(s)|²=0,

for any t∈[0,T]which implies the uniqueness. This completes the proof.

Example 3.5. We define the functionsk(·),ρ(·)byk(u) =√ uand

ρ(_u) =











0, u=0,

u q

log(u⁻¹), u∈(0,e⁻²], C

u+ ¹

3e²

, u∈(e⁻²,∞) whereC>0. Then k(·)andρ(·)satisfy Assumption2.1in Theorem3.4.

Theorem 3.6. Let Assumptions 2.2–2.4 hold. Then, there exists a unique solution {x(t)}₀_≤_t_≤_T to equation(2.1).

Proof. LetT₀ ∈(0,T), for eachN ≥1, we define the truncation function f_N(t,x)as follows:

f_N(t,x) =







f(t,x), |x| ≤N, f

t,N_|^x_x_|

, |x|>N,

and g_N(t,x), h_N(t,x,v)similarly. Then f_N, g_N andh_N satisfy Assumption2.1 due to that the following inequality about f_N, g_N andh_N hold:

|f_N(t,x)− f_N(t,y)| ∨ |g_N(t,x)−g_N(t,y)| ≤2λ(t)k_N(|x−y|), Z

Z

|h_N(t,x,v)−h_N(t,y,v)|²π(dv)≤2λ²(t)ρ²_N(|x−y|),

where x,y∈ Randt ∈[0,T₀]. Therefore, by Theorem3.4, there exists a unique solutionx_N(t) andx_N+1(t), respectively, to the following equations

xN(t) =x(0) +

Z _t

0 fN(s,xN(s))ds+

Z _t

0 gN(s,xN(s))dw(s) +

Z _t

0

Z

Zh_N(s,x_N(s−),v)N(ds,dv) +αmax

0≤s≤tx_N(s) +β min

0≤s≤tx_N(s), x_N+1(t) =x(0) +

Z _t

0 f_N+1(s,x_N+1(s))ds+

Z _t

0 g_N+1(s,x_N+1(s))dw(s) +

Z _t

0

Z

Zh_N+1(s,x_N+1(s−),v)N(ds,dv) +αmax

0≤s≤tx_N+1(s) +β min

0≤s≤tx_N+1(s). Define the stopping times

σ_N :=T₀∧_inf{t∈ [_0,T]_:|x_N(t)| ≥N}_, σ_N+1:=T₀∧inf{t∈ [0,T]:|x_N+1(t)| ≥N+1},

τ_N :=σ_N∧σ_N+1,

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where we set inf{φ}= _∞as usual. Similar to (3.7), we obtain

|x_N+1(t)−x_N(t)|

=

Z _t

0

[f_N+1(s,x_N+1(s))− f_N(s,x_N(s))]ds

+

Z _t

0

[g_N+1(s,x_N+1(s))−g_N(s,x_N(s))]dw(s) +

Z _t

0

Z

[h_N+1(s,x_N+1(s−),v)−h_N(s,x_N(s−),v)]N(ds,dv) + (|α|+|β|)max

0≤s≤t|x_N+1(s)−xN(s)|.

Again the Hölder inequality, the Doob’s martingale inequality imply that E max

0≤s≤t∧τN

|x_N+1(s)−x_N(s)|²

≤ ¹

(1− |α| − |β|)²

5TE Z _t∧τN

0

|f_N+1(s,x_N+1(s))− f_N(s,x_N(s))|²ds +20E

Z _t∧τN

0

|g_N+1(s,x_N+1(s))−g_N(s,x_N(s))|²ds +₂₀[₂+_Tπ(_Z)]_E

Z _t_∧_τ_N

0

Z

|hN+1(_s,_x_N₊₁(_s−),v)−hN(_s,_x_N(_s−),v)|²π(_dv)_ds

. (3.11) Noting that for any 0≤s≤ τN,

f_N+1(s,x_N(s)) = f_N(s,x_N(s)), g_N+1(s,x_N(s)) =g_N(s,x_N(s)), h_N+1(s,x_N(s−),v) =h_N(s,x_N(s−),v),

we derive that E max

0≤s≤t∧τN

|x_N+1(s)−x_N(s)|²

≤ ¹

(1− |α| − |β|)²

5TE Z _t∧τN

0

|f_N+1(s,x_N+1(s))− f_N+1(s,x_N(s))|²ds +20E

Z _t∧τN

0

|g_N+1(s,x_N+1(s))−g_N+1(s,x_N(s))|²ds +₂₀[₂+Tπ(Z)]E

Z _t_∧_τ_N

0

Z

|h_N+1(s,x_N+1(s−)_,v)−h_N+1(s,x_N(s−)_,v)|²π(dv)ds

. Then it follows from Assumption2.4that

Emax

0≤s≤t|x_N+1(s∧τ_N)−x_N(s∧τ_N)|²

≤ ^5T+20+20[2+Tπ(Z)]

(1− |α| − |β|)²

Z _t_∧_τ_N

0

|λ(s)|²γ_N+1(E|x_N+1(s)−x_N(s)|²)ds

≤ ^5T+20+20[2+Tπ(Z)]

(1− |α| − |β|)²

Z _t

0

|λ(s)|²γ_N+1

E max

0≤σ≤s|x_N+1(σ∧τ_N)−x_N(σ∧τ_N)|²

ds, where γ_N(·) = k²_N(·¹²) +ρ²_N(·¹²). Obviously, by Assumption 2.4, we have that γ_N(·) is a non-negative, non-decreasing concave function, γ_N(0) = 0 and R

0⁺ 1

γN(x)dx = ∞. By using Lemma2.6 again, it follows that

E sup

0≤s≤t

|x_N+1(s∧τN)−xN(s∧τN)|² =0.

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Therefore, we obtain that

x_N+1(t) =x_N(t), fort∈ [0,T₀∧τ_N].

For each ω ∈ Ω, there exists an N₀(ω) > 0 such that 0 < T₀ ≤ τ_N₀. Now define x(t) by x(_t) =_x_N₀(_t)_for_t∈[_0,_T₀]_{. Since}_x(_t∧τ_N) =_x_N(_t∧τ_N), it follows that

x(t∧τ_N) =x(0) +

Z _t_∧_τ_N

0 f_N(s,x_N(s))ds+

Z _t_∧_τ_N

0 g_N(s,x_N(s))dw(s) +

Z _t_∧_τ_N

0

Z

ZhN(s,xN(s−),v)N(ds,dv) +α max

0≤s≤t∧τN

xN(s) +β min

0≤s≤t∧τN

xN(s)

=x(0) +

Z _t_∧_τ_N

0 f(s,x(s))ds+

Z _t_∧_τ_N

0 g(s,x(s))dw(s) +

Z _t_∧_τ_N

0

Z

Zh(s,x(s−),v)N(ds,dv) +α max

0≤s≤t∧τN

x(s) +β min

0≤s≤t∧τN

x(s). Letting N→∞, then yields

x(t) =x(0) +

Z _t

0 f(s,x(s))ds+

Z _t

0 g(s,x(s))dw(s) +

Z _t

0

Z

Zh(s,x(s−),v)N(ds,dv) +αmax

0≤s≤tx(s), t∈[0,T0].

Since T₀is arbitrary, then we have x(t)is the solution of equation (2.1) on[0,T]. The proof is complete.

4 p-th moment exponential estimates

In this section, we will give the pth exponential estimates of solutions to equation (2.1).

Assumption 4.1. Assume λ(t) ∈ L^p([0,T],R), p > 2. For any x,y ∈ R andt ∈ [0,T], there exists a functionρ(·)and a constant K₃such that

Z

|h(t,x,v)−h(t,y,v)|^pπ(dv)≤λ^p(t)ρ^p(|x−y|), sup

0≤t≤T

Z

|h(t, 0,v)|^pπ(dv)≤K₃, whereρ(.)is defined as Assumption2.1.

Remark 4.2. In particular, we see clearly that if let ρ(u) = Ku, L > 0, then Assumption 4.1 reduces to the linear growth condition. That is, for anyx ∈Randt∈[0,T], we have

Z

|h(t,x,v)|^pπ(dv)≤2^p⁻¹λ^p(t)K^p|x|^p+2^p⁻¹K₃. Theorem 4.3. Let Assumptions2.1–2.3and4.1hold, for any p≥2

E max

0≤t≤T|x(t)|^p≤(₁+CE|x(₀)|^p+C₄T)e^C⁵^R⁰^T^|^λ⁽^t^)|^p^dt, (4.1) where C₄ and C₅are two positive constants of the inequality(4.11).

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Proof. For anyt≥0, it follows from (2.1) that

|x(t)| ≤ |x(0)|+

Z _t

0 f(s,x(s))ds

+

Z _t

0 g(s,x(s))dw(s) +

Z _t

0

Z

Zh(s,x(s−),v)N(ds,dv)

+ (|α|+|β|)max

0≤s≤t|x(s)|. (4.2) Taking the maximal value on both sides of (4.2), by Holder’s inequality, the Burkholder inequality and Assumption 2.3, we have

(1− |α| − |β|)^pEmax

0≤s≤t|x(s)|^p

≤4^p⁻¹

E|x(₀)|^p+_E_max

0≤s≤t

Z _s

0 f(_σ,_x(σ))_dσ

p

+_E_max

0≤s≤t

Z _s

0 g(_σ,_x(σ))_dw(σ)

p

+Emax

0≤s≤t

Z _s

0

Z

Zh(σ,x(σ−),v)N(dσ,dv)

p . Therefore, we get

Emax

0≤s≤t|x(s)|^p≤ C

E|x(0)|^p+Emax

0≤s≤t

Z _s

0 f(σ,x(σ))dσ

p

+Emax

0≤s≤t

Z _s

0 g(σ,x(σ))dw(σ)

p

+Emax

0≤s≤t

Z _s

0

Z

p

. (4.3)

whereC= ₍ ⁴^p⁻¹

1−|α|−|β|)^p. Using Hölder’s inequality, we get Emax

0≤s≤t

Z _s

0 f(σ,x(σ))dσ

p

≤T^p⁻¹E Z _t

0

|f(s,x(s))|^pds

≤T^p⁻¹E Z _t

0

|f(s,x(s))− f(s, 0)|^pds By the basic inequality

|a+b|^p ≤^h1+e

1 p−1

ip−1

|a|^p+ |b|^p e

, p>1, a,b∈Rⁿ, for anye>0, it follows that

Emax

0≤s≤t

Z _s

0 f(σ,x(σ))dσ

p

≤T^p⁻¹h 1+e

p−11

ip−1

E Z _t

0

|f(s,x(s))− f(s, 0)|^p+|f(s, 0)|^p e

ds.

By Assumptions2.1,2.2and lettinge=K₁^p⁻¹, we obtain Emax

0≤s≤t

Z _s

0 f(σ,x(σ))dσ

p

≤T^p⁻¹(1+K₁)^p⁻¹E Z _t

0

[λ^p(s)k^p(|x(s)|) +K₁]ds.

In fact, because the functionk(·)is concave and increasing, there must exist a positive number Lsuch that

k^p(|x|)≤L(1+|x|^p), for all p≥2. (4.4)

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Hence,

Emax

0≤s≤t

Z _s

0 f(σ,x(σ))dσ

p

≤ LT^p⁻¹(1+K₁)^p⁻¹

Z _t

0

|λ(s)|^p(1+E|x(s)|^p)ds

+T^pK₁(1+K₁)^p⁻¹. (4.5) By using the Burkholder–Davis–Gundy inequality and the Hölder inequality, we have a positive real number C_p such that the following inequality holds:

Emax

0≤s≤t

Z _s

0 g(σ,x(σ))dw(σ)

p

≤C_pE _Z _t

0

|g(s,x(s))|²ds ₂^p

≤C_pT²^p⁻¹E Z _t

0

|g(s,x(s))|^pds.

By the similar arguments, we have Emax

0≤s≤t

Z _s

0 g(σ,x(σ))dw(σ)

p

≤LCpT^p²⁻¹(1+K₁)^p⁻¹

Z _t

0

|λ(s)|^p(1+E|x(s)|^p)ds

+CpT²^pK₁(1+K₁)^p⁻¹. (4.6) Now, we will estimate the fourth term of (4.3). Using the basic inequality|a+b|^p ≤2^p⁻¹(|a|^p+

|b|^p), we have

Emax

0≤s≤t

Z _s

0

Z

p

≤2^p⁻¹E

0max≤s≤t

Z _s

0

Z

Zh(σ,x(σ−),v)N^˜(dσ,dv)

p

+2^p⁻¹E

0max≤s≤t

Z _s

0

Z

Zh(σ,x(σ−),v)π(dv)dσ

p

. (4.7)

By Lemma2.7and Hölder’s inequality, it follows that, E

0max≤s≤t

Z _s

0

Z

h(_σ,x(σ−)_,v)N^˜(dσ,dv)

p

≤Dp

( T^p²⁻¹E

Z _t

0

Z

|h(s,x(s−),v)|²π(dv) ^p₂

ds+E Z _t

0

Z

|h(s,x(s−),v)|^pπ(dv)ds )

(4.8) and

E

0max≤s≤t

Z _s

0

Z

Zh(σ,x(σ−),v)π(dv)dσ

p

≤T^p⁻¹[π(Z)]²^pE Z _t

0

_Z

Z

|h(s,x(s−),v)|²π(du) ^p₂

ds. (4.9)

Inserting (4.8) and (4.9) into (4.7), and by Assumption4.1, we have Emax

0≤s≤t

Z _s

0

Z

p

≤2^p⁻¹(DpT^p²⁻¹+T^p⁻¹[π(Z)]²^p)E Z _t

0

2|λ(s)|²ρ²(|x(s−)|) +2K₂^p₂ ds +₂^2p⁻²D_pE

Z _t

0

[|λ(s)|^pρ^p(|x(s−)|) +K₃]ds.