# Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients

Authors: Hui Yu; Minghui Song
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

The numerical methods in the current known literature require the stochastic differential equations (SDEs) driven by Poisson random measure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler's method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. The main aim is to investigate the convergence of the Euler method in probability to such equations with non-Lipschitz coefficients. Numerical example is given to demonstrate our results.


---

## Body

## 1. Introduction

In finance market and other areas, it is meaningful and significant to model the impact of event-driven uncertainty. Events such as corporate defaults, operational failures, market crashes, or central bank announcements require the introduction of stochastic differential equations (SDEs) driven by Poisson random measure (see [1, 2]), since such equations were initiated in [3].Actually, we can only obtain the explicit solutions of a small class of SDEs driven by Poisson random measure and so numerical methods are necessary. In general, numerical methods can be divided into strong approximations and weak approximations. Strong approximations focus on pathwise approximations while weak approximations (see [4, 5]) are fit for problems such as derivative pricing.We give an overview of the results on the strong approximations of stochastic differential equations (SDEs) driven by Poisson random measure in the existing literature. In [6], a convergence result for strong approximations of any given order γ∈{0.5,1,1.5,…} was presented. Moreover, N. Bruti-Liberatiand E. Platen (see [7]) obtain the jump-adapted order 1.5 scheme, and they also give the derivative-free or implicit jump-adapted schemes with desired order of strong convergence. And for the specific case of pure jump SDEs, they (see [8]) establish the strong convergence of Taylor’s methods under weaker conditions than the currently known. In [5, 7], the drift-implicit schemes which achieve strong order γ∈{0.5,1} are given. Recently, Mordecki et al. [9] improved adaptive time stepping algorithms based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision.  M. Wei [10] demonstrates the convergence of numerical solutions for variable delay differential equations driven by Poisson random measure. In [11], the developed Runge-Kutta methods are presented to improve the accuracy behaviour of problems with small noise to SDEs with Poisson random measure.Clearly, the results above require that the SDEs driven by Poisson random measure satisfy the global Lipschitz condition and the linear growth condition. However, there are many equations which do not satisfy above conditions, and we can see such equations in Section5 in our paper. Our main contribution is to present Euler’s method for these equations with non-Lipschitz coefficients. Here non-Lipschitz coefficients are interpreted in [12], that is to say, the drift coefficients and the diffusion coefficients satisfy the local Lipschitz conditions, the jump coefficients satisfy the global Lipschitz conditions, and the one-sided linear growth condition is considered. Our work is motivated by [12] in which the existence of global solutions for these equations with non-Lipschitz coefficients is proved, while there is no numerical method is presented in our known literature. And our aim in this paper is to close this gap.Our work is organized as follows. In Section2, the property of SDEs driven by Poisson random measure with non-Lipschitz coefficients is given. In Section 3, Euler method is analyzed for such equations. In Section 4, we present the convergence in probability of the Euler method. In Section 5, an example is presented.


## 2. The SDEs Driven by Poisson Random Measure with Non-Lipschitz Coefficients

Throughout this paper, unless specified, we use the following notations. Letu1∨u2=max{u1,u2} and u1∧u2=min{u1,u2}. Let |·| and 〈·,·〉 be the Euclidean norm and the inner product of vectors in Rd,d∈N. If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by |A|=trace(ATA). Let Lℱ02(Ω;Rd) denote the family of Rd-valued ℱ0-measurable random variables ξ with E|ξ|2<∞. [z] denotes the largest integer which is less than or equal to z in R. I𝒜 denotes the indicator function of a set 𝒜.The followingd-dimensional SDE driven by Poisson random measure is considered in our paper:
(2.1)dx(t)=a(x(t-))dt+b(x(t-))dW(t)+∫ɛ‍c(x(t-),v)p~ϕ(dv×dt),
for t>0 with initial condition x(0-)=x(0)=x0∈Lℱ02(Ω;Rd), where x(t-) denotes lims→t-x(s) and p~ϕ(dv×dt):=pϕ(dv×dt)-ϕ(dv)dt.The drift coefficienta:Rd→Rd, the diffusion coefficient b:Rd→Rd×m, and the jump coefficient c:Rd×ɛ→Rd are assumed to be Borel measurable functions.The randomness of (2.1) is generated by the following (see [9]). An m-dimensional Wiener process W={W(t)=(W1(t),…,Wm(t))T} with independent scalar components is defined on a filtered probability space (ΩW,ℱW,(ℱtW)t≥0,PW). A Poisson random measure pϕ(ω,dv×dt) is on ΩJ×ɛ×[0,∞), where ɛ⊆Rr∖{0} with r∈N, and its deterministic compensated measure ϕ(dv)dt=λf(v)dvdt, that is, E(pϕ(dv×dt))=ϕ(dv)dt. f(v) is a probability density, and we require finite intensity λ=ϕ(ɛ)<∞. The Poisson random measure is defined on a filtered probability space (ΩJ,ℱJ,(ℱtJ)t≥0,PJ). The Wiener process and the Poisson random measure are mutually independent. The process x(t) is thus defined on a product space (Ω,ℱ,(ℱt)t≥0,P), where Ω=ΩW×ΩJ,ℱ=ℱW×ℱJ,(ℱt)t≥0=(ℱtW)t≥0×(ℱtJ)t≥0,P=PW×PJ and ℱ0 contains all P-null sets.Now, the condition of non-Lipschitz coefficients is given by the following assumptions.Assumption 2.1.
For each integerk≥1, there exists a positive constant Ck, dependent on k, such that
(2.2)|a(x)-a(y)|2∨|b(x)-b(y)|2≤Ck|x-y|2,
for x,y∈Rd with |x|∨|y|≤k,k∈N. And there exists a positive constant C such that
(2.3)∫ɛ‍|c(x,v)-c(y,v)|2ϕ(dv)≤C|x-y|2,
for x,y∈Rd.Assumption 2.2.
There exists a positive constantL such that
(2.4)2〈x,a(x)〉+|b(x)|2+∫ɛ‍|c(x,v)|2ϕ(dv)≤L(1+|x|2),
for x∈Rd.A unique global solution of (2.1) exists under Assumptions 2.1 and 2.2, see [12].Assumption 2.3.
Consider(2.5)|a(0)|2+|b(0)|2+∫ɛ‍|c(0,v)|2ϕ(dv)≤L~,L~>0.Actually, Assumptions2.1 and 2.3 imply the linear growth conditions
(2.6)|a(x)|2∨|b(x)|2≤Ck~(1+|x|2),
for x∈Rd with |x|≤k and Ck~>0, and
(2.7)∫ɛ‍|c(x,v)|2ϕ(dv)≤C~(1+|x|2),
for x∈Rd and C~>0.The following result shows that the solution of (2.1) keeps in a compact set with a large probability.Lemma 2.4.
Under Assumptions2.1 and 2.2, for any pair of ϵ∈(0,1) and T>0, there exists a sufficiently large integer k*, dependent on ϵ and T, such that
(2.8)P(τk≤T)≤ϵ,∀k≥k*,
where τk=inf{t≥0:|x(t)|≥k} for k≥1.Proof.
Using Itô's formula (see [1]) to |x(t)|2, for t≥0, we have
(2.9)|x(t)|2=|x0|2+∫0t‍(〈2x(s-),a(x(s-))〉+|b(x(s-))|2)ds+∫0t∫ɛ‍(|x(s-)+c(x(s-),v)|2-|x(s-)|2-〈2x(s-),c(x(s-),v)〉)ϕ(dv)ds+∫0t‍〈2x(s-),b(x(s-))〉dW(s)+∫0t∫ɛ‍(|x(s-)+c(x(s-),v)|2-|x(s-)|2)p~ϕ(dv×ds),
which gives
(2.10)E|x(t∧τk)|2=E|x0|2+E∫0t∧τk‍(〈2x(s-),a(x(s-))〉+|b(x(s-))|2)ds+E∫0t∧τk∫ɛ‍|c(x(s-),v)|2ϕ(dv)ds=E|x0|2+E∫0t∧τk(〈2x(s-),a(x(s-))〉+|b(x(s-))|2+∫ɛ‍|c(x(s-),v)|2ϕ(dv))ds,
for t∈[0,T]. By Assumption 2.2, we thus have
(2.11)E|x(t∧τk)|2≤E|x0|2+E∫0t∧τk‍L(1+|x(s-)|2)ds≤E|x0|2+LT+L∫0t‍E|x(s∧τk-)|2ds,
for t∈[0,T]. Consequently by using the Gronwall inequality (see [13]), we obtain
(2.12)E|x(t∧τk)|2≤(E|x0|2+LT)eLT,
for t∈[0,T]. We therefore get
(2.13)(E|x0|2+LT)eLT≥E|x(T∧τk)|2≥E(|x(τk)|2I{τk≤T})≥k2P(τk≤T),
which means
(2.14)P(τk≤T)≤eLTk2(E|x0|2+LT).
So for any ϵ∈(0,1), we can choose
(2.15)k*=[eLTE|x0|2+LTeLTϵ]+1,
such that
(2.16)P(τk≤T)≤ϵ,∀k≥k*.
Hence, we have the result (2.8).


## 3. The Euler Method

In this section, we introduce the Euler method to (2.1) under Assumptions 2.1, 2.2, and 2.3. Subsequently, we give two lemmas to analyze the Euler method over a finite time interval [0,T], where T is a positive number.Given a step sizeΔt∈(0,1), the Euler method applied to (2.1) computes approximation Xn≈x(tn), where tn=nΔt,n=0,1,…, by setting X0=x0 and forming
(3.1)Xn+1=Xn+a(Xn)Δt+b(Xn)ΔWn+∫tntn+1∫ɛc(Xn,v)p~ϕ(dv×dt),
where ΔWn=W(tn+1)-W(tn).The continuous-time Euler method is then defined by(3.2)X¯(t):=X0+∫0t‍a(Z(s))ds+∫0t‍b(Z(s))dW(s)+∫0t‍∫ɛ‍c(Z(s),v)p~ϕ(dv×ds),
where Z(t)=Xn for t∈[tn,tn+1),n=0,1,….Actually, we can see in [8], pϕ={pϕ(t):=pϕ(ɛ×[0,t])} is a stochastic process that counts the number of jumps until some given time. The Poisson random measure pϕ(dv×dt) generates a sequence of pairs {(ιi,ξi),i∈{1,2,…,pϕ(T)}} for a given finite positive constant T if λ<∞. Here {ιi:Ω→R+,i∈{1,2,…,pϕ(T)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity λ, and {ξi:Ω→ɛ,i∈{1,2,…,pϕ(T)} is a sequence of independent identically distributed random variables, where ξi is distributed according to ϕ(dv)/ϕ(ɛ). Then (3.1) can equivalently be the following form:
(3.3)Xn+1=Xn+(a(Xn)-∫ɛ‍c(Xn,v)ϕ(dv))Δt+b(Xn)ΔWn+∑i=pϕ(tn)+1pϕ(tn+1)‍c(Xn,ξi).The following lemma shows the close relation between the continuous-time Euler method (3.2) and its step function Z(t).Lemma 3.1.
Under Assumptions2.1 and 2.3, for any T>0, there exists a positive constant K1(k), dependent on k and independent of Δt, such that for all Δt∈(0,1) the continuous-time Euler method (3.2) satisfies
(3.4)E|X¯(t)-Z(t)|2≤K1(k)Δt,
for 0≤t≤T∧τk∧ρk, where ρk=inf{t≥0:|X¯(t)|≥k} for k≥1 and τk is defined in Lemma 2.4.Proof.
For0≤t≤T∧τk∧ρk, there is an integer n such that t∈[tn,tn+1). So it follows from (3.2) that
(3.5)X¯(t)-Z(t)=Xn+∫tnt‍a(Z(s))ds+∫tnt‍b(Z(s))dW(s)+∫tnt∫ɛc(Z(s),v)p~ϕ(dv×ds)-Xn.
Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of dW(t) and p~ϕ(dv×dt), we have
(3.6)E|X¯(t)-Z(t)|2≤3E|∫tnt‍a(Z(s))ds|2+3E|∫tnt‍b(Z(s))dW(s)|2+3E|∫tnt∫ɛc(Z(s),v)p~ϕ(dv×ds)|2≤3E(∫tnt‍12ds∫tnt‍|a(Z(s))|2ds)+3E∫tnt‍|b(Z(s))|2ds+3E∫tnt∫ɛ|c(Z(s),v)|2ϕ(dv)ds≤3ΔtE∫tnt‍|a(Z(s))|2ds+3E∫tnt‍|b(Z(s))|2ds+3E∫tnt∫ɛ|c(Z(s),v)|2ϕ(dv)ds,
where the inequality |u1+u2+u3|2≤3|u1|2+3|u2|2+3|u3|2 for u1,u2,u3∈Rd is used. Therefore, by applying Assumptions 2.1 and 2.3, we get
(3.7)E∫tnt‍|a(Z(s))|2ds≤Ck~E∫tnt‍(1+|Z(s)|2)ds≤Ck~Δt+C~kk2Δt,E∫tnt‍|b(Z(s))|2ds≤Ck~Δt+C~kk2Δt,E∫tnt‍∫ɛ‍|c(Z(s),v)|2ϕ(dv)ds≤C~Δt+C~k2Δt,
which lead to
(3.8)E|X¯(t)-Z(t)|2≤Δt(3Ck~Δt+3k2Ck~Δt+3Ck~+3k2Ck~+3C~+3k2C~),
for t∈[0,T∧τk∧ρk]. Therefore, we obtain the result (3.4) by choosing
(3.9)K1(k)=6Ck~+6k2Ck~+3C~+3k2C~.In accord with Lemma2.4, we give the following lemma which demonstrates that the solution of continuous-time Euler method (3.2) remains in a compact set with a large probability.Lemma 3.2.
Under Assumptions2.1, 2.2, and 2.3, for any pair of ϵ∈(0,1) and T>0, there exist a sufficiently large k* and a sufficiently small Δt1* such that
(3.10)P(ρk*≤T)≤ϵ,∀Δt≤Δt1*,
where ρk* is defined in Lemma 3.1.Proof .
Applying generalized Itô's formula (see [1]) to |X¯(t)|2, for t≥0, yields
(3.11)|X¯(t)|2=|X0|2+∫0t‍(〈2X¯(s),a(Z(s))〉+|b(Z(s))|2)ds+∫0t‍∫ɛ‍(|X¯(s)+c(Z(s),v)|2-|X¯(s)|2-〈2X¯(s),c(Z(s),v)〉)ϕ(dv)ds+∫0t‍〈2X¯(s),b(Z(s))〉dW(s)+∫0t‍∫ɛ‍(|X¯(s)+c(Z(s),v)|2-|X¯(s)|2)p~ϕ(dv×ds).
By taking expectations, we thus have
(3.12)E|X¯(t∧ρk)|2=E|X0|2+E∫0t∧ρk‍(〈2X¯(s),a(Z(s))〉+|b(Z(s))|2+∫ɛ‍|c(Z(s),v)|2ϕ(dv))ds=E|X0|2+E∫0t∧ρk‍(〈2X¯(s),a(X¯(s))〉+|b(X¯(s))|2E|X0|2+|X0|2+E+∫ɛ‍|c(X¯(s),v)|2ϕ(dv))ds+E∫0t∧ρk‍〈2X¯(s),a(Z(s))-a(X¯(s))〉ds+E∫0t∧ρk‍(|b(Z(s))|2-|b(X¯(s))|2)ds+E∫0t∧ρk∫ɛ(|c(Z(s),v)|2-|c(X¯(s),v)|2)ϕ(dv)ds.
For t∈[0,T]. Now, by using the inequalities 〈u1,u2〉≤|u1||u2| for u1,u2∈Rd, (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarz’s inequality, and Lemma 3.1, we get
(3.13)E∫0t∧ρk‍〈2X¯(s),a(Z(s))-a(X¯(s))〉ds≤2E∫0t∧ρk|X¯(s)||a(Z(s))-a(X¯(s))|ds≤2kCk∫0t‍E|Z(s∧ρk)-X¯(s∧ρk)|ds≤2kCk∫0t‍(E|Z(s∧ρk)-X¯(s∧ρk)|2)1/2ds≤2kTCkK1(k)Δt.
And, similarly as above, we have
(3.14)E∫0t∧ρk‍(|b(Z(s))|2-|b(X¯(s))|2)ds≤E∫0t∧ρk‍(|b(Z(s))|+|b(X¯(s))|)×(|b(Z(s))|-|b(X¯(s))|)ds≤2Ck~(1+k2)E∫0t∧ρk‍|b(Z(s))-b(X¯(s))|ds≤2CkCk~(1+k2)∫0t‍E|Z(s∧ρk)-X¯(s∧ρk)|ds≤2TCkCk~K1(k)(1+k2)Δt.
Moreover, in the same way, we obtain
(3.15)E∫0t∧ρk‍∫ɛ‍(|c(Z(s),v)|2-|c(X¯(s),v)|2)ϕ(dv)ds=E∫0t∧ρk‍∫ɛ‍(|c(Z(s),v)-c(X¯(s),v)+c(X¯(s),v)|2-|c(X¯(s),v)|2)ϕ(dv)ds≤E∫0t∧ρk‍∫ɛ‍(2|c(Z(s),v)-c(X¯(s),v)|2+|c(X¯(s),v)|2)ϕ(dv)ds≤2CE∫0t∧ρk‍|Z(s)-X¯(s)|2ds+C~E∫0t∧ρk‍(1+|X¯(s)|2)ds≤2C∫0t‍E|Z(s∧ρk)-X¯(s∧ρk)|2ds+C~E∫0t∧ρk‍(1+|X¯(s)|2)ds≤2CTK1(k)Δt+C~T+C~E∫0t∧ρk‍|X¯(s)|2ds,
where the inequality |u1+u2|2≤2|u1|2+2|u2|2 for u1,u2∈Rd, (2.3) in Assumptions 2.1 and 2.3, Fubini's theorem, and Lemma 3.1 are used. Subsequently, substituting (3.13), (3.14), and (3.15) into (3.12) together with Assumption 2.2 leads to
(3.16)E|X¯(t∧ρk)|2≤E|X0|2+LE∫0t∧ρk‍(1+|X¯(s)|2)ds+C~E∫0t∧ρk‍|X¯(s)|2ds+2kTCkK1(k)Δt+2TCkCk~K1(k)(1+k2)Δt+2CTK1(k)Δt+C~T≤(L+C~)∫0t‍E|X¯(s∧ρk)|2ds+E|X0|2+LT+C~T+(2kTCkK1(k)+2TCkCk~K1(k)(1+k2))Δt+2CTK1(k)Δt,
for 0≤t≤T. Therefore, by the Gronwall inequality (see [13]), for 0≤t≤T, we get
(3.17)E|X¯(t∧ρk)|2≤α1α4+α4α2(k)Δt+α4α3(k)Δt,
where
(3.18)α1=E|X0|2+LT+C~T,α2(k)=2kTCkK1(k)+2TCkCk~K1(k)(1+k2),α3(k)=2CTK1(k),α4=exp⁡(LT+C~T).
We thus obtain that
(3.19)k2P(ρk≤T)≤E(|X¯(ρk)|2I{ρk≤T})≤E|X¯(T∧ρk)|2≤α1α4+α4α2(k)Δt+α4α3(k)Δt.
So for any ϵ∈(0,1), we can choose sufficiently large integer k=k* such that
(3.20)α1α4k*2≤ϵ2,
and choose sufficiently small Δt1*∈(0,1) such that
(3.21)α4α2(k*)Δt1*+α4α3(k*)Δt1*k*2≤ϵ2.
Hence, we have
(3.22)P(ρk*≤T)≤ϵ,∀Δt≤Δt1*.


## 4. Convergence in Probability

In this section, we present two convergence theorems of the Euler method to the SDE with Poisson random measure (2.1) over a finite time interval [0,T].At the beginning, we give a lemma based on Lemma3.1.Lemma 4.1.
Under Assumptions2.1 and 2.3, for any T>0, there exists a positive constant K2(k), dependent on k and independent of Δt, such that for all Δt∈(0,1) the solution of (2.1) and the continuous-time Euler method (3.2) satisfy
(4.1)E(sup⁡0≤t≤T|x(t∧τk∧ρk)-X¯(t∧τk∧ρk)|2)≤K2(k)Δt,
where τk and ρk are defined in Lemmas 2.4 and 3.1, respectively.Proof.
From (2.1) and (3.2), for any 0≤t′≤T, we have
(4.2)E(sup⁡0≤t≤t′|x(t∧τk∧ρk)-X¯(t∧τk∧ρk)|2)≤3E(sup⁡0≤t≤t′|∫0t∧τk∧ρk(a(x(s-))-a(Z(s)))ds|2)+3E(sup⁡0≤t≤t′|∫0t∧τk∧ρk(b(x(s-))-b(Z(s)))dW(s)|2)+3E(sup⁡0≤t≤t′|∫0t∧τk∧ρk∫ɛ(c(x(s-),v)-c(Z(s),v))p~ϕ(dv×ds)|2),
where the inequality |u1+u2+u3|2≤3|u1|2+3|u2|2+3|u3|2 for u1,u2,u3∈Rd is used. Therefore, by using the Cauchy-Schwarz inequality, (2.2) in Assumption 2.1, Lemma 3.1 and Fubini's theorem, we obtain
(4.3)E(sup⁡0≤t≤t′|∫0t∧τk∧ρk(a(x(s-))-a(Z(s)))ds|2)≤E(sup⁡0≤t≤t′∫0t∧τk∧ρk12ds∫0t∧τk∧ρk|a(x(s-))-a(Z(s))|2ds)≤TE(∫0t′∧τk∧ρk|a(x(s-))-a(Z(s))|2ds)≤2TCkE(∫0t′∧τk∧ρk|X¯(s)-Z(s)|2ds)+2TCkE(∫0t′∧τk∧ρk|x(s-)-X¯(s)|2ds)≤2TCk∫0t′E|X¯(s∧τk∧ρk)-Z(s∧τk∧ρk)|2ds+2TCk∫0t′E|x(s∧τk∧ρk-)-X¯(s∧τk∧ρk)|2ds≤2T2CkK1(k)Δt+2TCk∫0t′E(sup⁡0≤u≤s|x(u∧τk∧ρk-)-X¯(u∧τk∧ρk)|2)ds.
Moreover, by using the martingale properties of dW(t) and p~ϕ(dv×dt), Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have
(4.4)E(sup⁡0≤t≤t′|∫0t∧τk∧ρk‍(b(x(s-))-b(Z(s)))dW(s)|2)≤4E∫0t′∧τk∧ρk‍|b(x(s-))-b(Z(s))|2ds≤8CkE∫0t′∧τk∧ρk‍|X¯(s)-Z(s)|2ds+8CkE∫0t′∧τk∧ρk‍|x(s-)-X¯(s)|2ds≤8Ck∫0t′‍E|X¯(s∧τk∧ρk)-Z(s∧τk∧ρk)|2ds+8Ck∫0t′‍E|x(s∧τk∧ρk-)-X¯(s∧τk∧ρk)|2ds≤8TCkK1(k)Δt+8Ck∫0t′‍E(sup⁡0≤u≤s|x(u∧τk∧ρk-)-X¯(u∧τk∧ρk)|2)ds,E(sup⁡0≤t≤t′|∫0t∧τk∧ρk‍∫ɛ‍(c(x(s-),v)-c(Z(s-),v))p~ϕ(dv×ds)|2)≤4E|∫0t′∧τk∧ρk‍∫ɛ‍(c(x(s-),v)-c(Z(s-),v))p~ϕ(dv×ds)|2=4E∫0t′∧τk∧ρk‍∫ɛ‍|c(x(s-),v)-c(Z(s-),v)|2ϕ(dv)ds≤8TCK1(k)Δt+8C∫0t′‍E(sup⁡0≤u≤s|x(u∧τk∧ρk-)-X¯(u∧τk∧ρk)|2)ds.
Hence, by substituting (4.3) and (4.4) into (4.2), we get
(4.5)E(sup⁡0≤t≤t′|x(t∧τk∧ρk)-X¯(t∧τk∧ρk)|2)≤Δt(6T2CkK1(k)+24TCkK1(k)+24TCK1(k))+(6TCk+24Ck+24C)×∫0t′‍E(sup⁡0≤u≤s|x(u∧τk∧ρk-)-X¯(u∧τk∧ρk)|2)ds.
So using the Gronwall inequality (see [13]), we have the result (4.1) by choosing
(4.6)K2(k)=(6T2CkK1(k)+24TCkK1(k)+24TCK1(k))exp⁡(6T2Ck+24TCk+24TC).Now, let's state our theorem which demonstrates the convergence in probability of the continuous-time Euler method (3.2).Theorem 4.2.
Under Assumptions2.1, 2.2, and 2.3, for sufficiently small ϵ,ς∈(0,1), there is a Δt* such that for all Δt<Δt*(4.7)P(sup⁡0≤t≤T|x(t)-X¯(t)|2≥ς)≤ϵ,
for any T>0.Proof.
For sufficiently smallϵ,ς∈(0,1), we define
(4.8)Ω¯={ω:sup⁡0≤t≤T|x(t)-X¯(t)|2≥ς}.
According to Lemmas 2.4 and 3.2, there exists a pair of k⋆ and Δt1⋆ such that
(4.9)P(τk⋆≤T)≤ϵ3,P(ρk⋆≤T)≤ϵ3,∀Δt≤Δt1⋆.
We thus have
(4.10)P(Ω¯)≤P(Ω¯∩{τk⋆∧ρk⋆>T})+P(τk⋆∧ρk⋆≤T)≤P(Ω¯∩{τk⋆∧ρk⋆>T})+P(τk⋆≤T)+P(ρk⋆≤T)≤P(Ω¯∩{τk⋆∧ρk⋆>T})+2ϵ3,
for Δt≤Δt1⋆. Moreover, according to Lemma 4.1, we have
(4.11)ςP(Ω¯∩{τk⋆∧ρk⋆>T})≤E(I{τk⋆∧ρk⋆>T}sup⁡0≤t≤T|x(t)-X¯(t)|2)≤E(sup⁡0≤t≤T|x(t∧τk⋆∧ρk⋆)-X¯(t∧τk⋆∧ρk⋆)|2)≤K2(k⋆)Δt,
which leads to
(4.12)P(Ω¯∩{τk⋆∧ρk⋆>T})≤ϵ3,
for Δt≤Δt2⋆. Therefore, from the inequalities above, we obtain
(4.13)P(Ω¯)≤ϵ,
for Δt≤Δt⋆, where Δt⋆=min{Δt1⋆,Δt2⋆}.We remark that the continuous-time Euler solutionX¯(t) (3.2) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths, not just only their Δt-increments. Therefore, the last theorem shows the convergence in probability of the discrete Euler solution (3.1).Theorem 4.3.
Under Assumptions2.1, 2.2, and 2.3, for sufficiently small ϵ,ς∈(0,1), there is a Δt* such that for all Δt<Δt*(4.14)P(|x(t)-Z(t)|2≥ς,0≤t≤T)≤ϵ,
for any T>0.Proof.
For sufficiently smallϵ,ς∈(0,1), we define
(4.15)Ω~={ω:|x(t)-Z(t)|2≥ς,0≤t≤T}.
A similar analysis as Theorem 4.2 gives
(4.16)P(Ω~)≤P(Ω~∩{τk⋆∧ρk⋆>T})+2ϵ3.
Recalling that
(4.17)ςP(Ω~∩{τk⋆∧ρk⋆>T})≤E(|x(T)-Z(T)|2I{τk⋆∧ρk⋆>T})≤E|x(T∧τk⋆∧ρk⋆)-Z(T∧τk⋆∧ρk⋆)|2≤2E(sup⁡0≤t≤T|x(t∧τk⋆∧ρk⋆)-X¯(t∧τk⋆∧ρk⋆)|2)+2E|X¯(T∧τk⋆∧ρk⋆)-Z(T∧τk⋆∧ρk⋆)|2≤2K1(k⋆)Δt+2K2(k⋆)Δt,
and using Lemmas 3.1 and 4.1, we get that
(4.18)P(Ω~∩{τk⋆∧ρk⋆>T})≤ϵ3,
for sufficiently small Δt. Consequently, the inequalities above show that
(4.19)P(Ω~)≤ϵ,
for all sufficiently small Δt.
So we complete the result (4.14).


## 5. Numerical Example

In this section, a numerical example is analyzed under Assumptions2.1, 2.2, and 2.3 which cover more classes of SDEs driven by Poisson random measure.Now, we consider the following equation:(5.1)dx(t)=a(x(t-))dt+b(x(t-))dW(t)+∫ɛ‍c(x(t-),v)p~ϕ(dv×dt),t>0,
with x(0)=x(0-)=0, where d=m=r=1. The coefficients of this equation have the form
(5.2)a(x)=12(x-x3),b(x)=x2,c(x,v)=xv.
The compensated measure of the Poisson random measure pϕ(dv×dt) is given by ϕ(dv)dt=λf(v)dvdt, where λ=5 and
(5.3)f(v)=12πvexp⁡(-(ln⁡v)22),0≤v<∞
is the density function of a lognormal random variable.Clearly, the equation cannot satisfy the global Lipschitz conditions and the linear growth conditions. On the other hand, we have(5.4)2〈x,a(x)〉+|b(x)|2+∫ɛ‍|c(x,v)|2ϕ(dv)=x(x-x3)+x4+∫ɛ‍x2v2λ12πvexp⁡(-(ln⁡v)22)dv≤(1+5e2)(1+x2),
that is to say, Assumptions 2.1, 2.2, and 2.3 in Section 2 are satisfied. Therefore, Albeverio et al. [12] guarantee that (5.1) has a unique global solution on [0,∞).Given the stepsizeΔt, we can have the Euler method
(5.5)Xn+1=Xn+12(Xn-Xn3)Δt+Xn2ΔWn+Xn∫tntn+1∫ɛ‍vp~ϕ(dv×dt),
with X0=0.And in Matlab experiment, each discretized trajectory is actually given in detail by the following.Algorithm
SimulateXn+1-:=Xn+(1/2)(Xn-Xn3-10eXn)Δt+Xn2ΔWn;
Simulate variablepϕ(tn+1)-pϕ(tn), where pϕ(tn) is from Poisson distribution with parameter λtn;
Simulatepϕ(tn+1)-pϕ(tn) independent random variables ιi uniformly distributed on the interval [pϕ(tn),pϕ(tn+1));
Simulatepϕ(tn+1)-pϕ(tn) independent random variables ξi with law f(v);
obtainXn+1=Xn+1-+Xn∑i=pϕ(tn)+1pϕ(tn+1)‍Itn≤ιi<tn+1ξi.Subsequently, we can get the results in Theorems 4.2 and 4.3.



---

*Source: 675781-2012-07-10.xml*