Scholarly article on topic 'Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients'

Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients Academic research paper on "Mathematics"

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Academic research paper on topic "Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients"

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 675781,17 pages doi:10.1155/2012/675781

Research Article

Numerical Solutions of Stochastic

Differential Equations Driven by Poisson Random

Measure with Non-Lipschitz Coefficients

Hui Yu and Minghui Song

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Correspondence should be addressed to Minghui Song, songmh@lsec.cc.ac.cn Received 15 January 2012; Revised 20 March 2012; Accepted 22 March 2012 Academic Editor: Said Abbasbandy

Copyright © 2012 H. Yu and M. Song. 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.

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.

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 j e (0.5,1,1.5,...} was presented. Moreover, N. Bruti-Liberati and 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 j e (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 Section 5 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 Section 2, 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. Let u1 V u2 = max{ui,u2} and ui A u2 = min(ui,u2}. Let | ■ | and (■, •) be the Euclidean norm and the inner product of vectors in Rd,d e 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| = y/trace(ATA). Let L|o(q; Rd) denote the family of Rd-valued Fo-measurable random variables £ with E|£|2 < oo. [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 following d-dimensional SDE driven by Poisson random measure is considered in our paper:

for t > 0 with initial condition x(0-) = x(0) = x0 e L|o (q; Rd), where x(t-) denotes lims^t-x(s) and pç(dv x dt) := pç(dv x dt) - Ç(dv)dt.

The drift coefficient a : Rd ^ Rd, the diffusion coefficient b : Rd ^ Rdxm, and the jump coefficient c : Rd x e ^ 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 (qw,FW, (FW)t>0,PW). A Poisson random measure

p,(w,dv x dt) is on qj x e x [0,to), where e c Rr \ {0} with r e N, and its deterministic compensated measure $(dv)dt = Xf (v)dvdt, that is, e(p^(dv x dt)) = $(dv)dt. f (v) is a probability density, and we require finite intensity X = $(e) < to. The Poisson random measure is defined on a filtered probability space (qj,FJ, (FJ)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 (q, F, (Ft)t>0, P), where q = qw x QJ, F = FW x FJ, (Ft)t>0 = (FW)t>0 x (FJ)t>0, P = PW x PJ and Fo contains all P-null sets.

Now, the condition of non-Lipschitz coefficients is given by the following assumptions.

Assumption 2.1. For each integer k > 1, there exists a positive constant Ck, dependent on k, such that

\a(x) - a(y)\2 V \b(x) - b(y) \2 < Ck|x - y\2, (2.2)

for x,y e Rd with |x|v|y|< k, k e N. And there exists a positive constant C such that

j" \c(x, v) - c(y, v)\2$(dv) < C\x - y\2, (2.3)

for x, y e Rd .

Assumption 2.2. There exists a positive constant L such that

2(x, a(x)> + |b(x)|2 + f |c(x, v)|2$(dv) < l( 1 + |x|2), (2.4)

•J e

for x e Rd .

A unique global solution of (2.1) exists under Assumptions 2.1 and 2.2, see [12]. Assumption 2.3. Consider

|a(0)|2 + |b(0)|2 + J" |c(0,v)|2$(dv) < L, L > 0. (2.5)

Actually, Assumptions 2.1 and 2.3 imply the linear growth conditions

|a(x)|2 V |b(x)|2 < Ck( 1 + |x|2), (2.6)

for x e rd with |x| < k and Ck > 0, and

[ |c(x,v)|2$(dv) < C( 1 + |x|2), (2.7)

for x e 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 Assumptions 2.1 and 2.2, for any pair of e e (0,1) and T > 0, there exists a sufficiently large integer k*, dependent on e and T, such that

P(Tfc < T) < e, Vk > k*, (2.8)

where Tk = inf{i > 0 : |x(t)| > k} for k > 1.

Proof. Using Ito's formula (see [1]) to |x(t)|2, for t > 0, we have

|x(t)|2 = |x0|2 + J ((2x(s-),a(x(s-))) + |b(x(s-))|2)ds

+ J | (|x(s-) + c(x(s-),v)|2 - |x(s-)|2 - (2x(s-),c(x(s-),v)))ç(dv)ds

+ J (2x(s-),b(x(s-)))dW(s) + ^ | (|x(s-)+ c(x(s-),v)|2 -|x(s-)|2)pç(dv x ds),

which gives

ft' k /

E|x(t A Tk)|2 = E|x0|2 + E j ((2x(s-),a(x(s-))) + |b(x(s-))|2jds

(tATk ( 2

+ E |c(x(s-),v)| Ç(dv)ds

= E|x0|2 + E J (2x(s-),a(x(s-))) + |b(x(s-))|2 + J ^(x^^rf^dv)^ds,

(2.10)

for t e [0, T]. By Assumption 2.2, we thus have

(tATk / x

E|x(t A Tk)|2 < E|x0|2 + E j L( 1 + |x(s-)|2jds

^ (2.11)

< E|x0|2 + LT + L E|x(s A Tk-)|2ds,

for t e [0, X]. Consequently by using the Gronwall inequality (see [13]), we obtain

E|x(t A Tk)|2 < (E|x0|2 + LX^eLT, (2.12)

for t e [0, X]. We therefore get

(E|x0|2 + LX^eLT > E|x(X A Tk)|2 > e(|x(Tk)|2/{Tk<x}) > k2P(Tk < X), (2.13) which means

p(Tk < X) < e^(E|x0|2 + LX). (2.14) So for any e e (0,1), we can choose

eLT E|xo|2 + LTe

(2.15)

such that

P(Tk < T) < e, Vk > k*.

Hence, we have the result (2.8).

(2.16) □

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, X], where X is a positive number.

Given a step size at e (0,1), the Euler method applied to (2.1) computes approximation Xn « x(tn), where tn = nAt, n = 0,1,..., by setting X0 = x0 and forming

rtn+1 {

Xn+1 = Xn + a(Xn)At + b(Xn)AWn + c(Xn,v)p^(dv x dt), (3.1)

J t„ J e

where aWn = W(tn+1) - W(tn).

The continuous-time Euler method is then defined by

a(Z(s))ds + Jo J(

X(t) := Xo + a(Z(s))ds ^ b(Z(s))dW(s) ^ c(Z(s),v)p^(dv x ds),

0 J £

where Z(t) = Xn for t e [tn, tn+1), n = 0,1,

Actually, we can see in [8], = p(t) := p^(s x [0,t])} is a stochastic process that counts the number of jumps until some given time. The Poisson random measure p,(dv x dt) generates a sequence of pairs {(k,£i), i e {1,2,...,p$(T)}} for a given finite positive constant T if X < to. Here {ii : q ^ R+, i e {1,2,...,p^(T)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity X, and {& : q ^ s, i e {1,2,.. .,p$(T)} is a sequence of independent identically distributed random variables, where ¿,i is distributed according to $(dv)/$(s). Then (3.1) can equivalently be the following form:

(,- \ Pi(in+l)

fl(X„) - c(Xn,v)$(dv)j At + b(Xn)AWn + £ c(X„,&). (3.3)

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 Assumptions 2.1 and 2.3, for any T > 0, there exists a positive constant K1(k), dependent on k and independent of At, such that for all At e (0,1) the continuous-time Euler method (3.2) satisfies

i|X(t) - Z(t)|2 < Ki(k)At,

for 0 < t < T A Tk A pk, where pk = inf{t > 0 : |X(t)| > k} for k > 1 and Tk is defined in Lemma 2.4.

Proof. For 0 < t < T A Tk A pk, there is an integer n such that t e [tn, tn+1). So it follows from (3.2) that

X(t) - Z(t) = Xn + J a(Z(s))ds + J b(Z(s))dW(s) + J" j c(Z(s),v)p$(dv x ds) - Xn.

Thus, by taking expectations and using the Cauchy-Schwarz inequality and the martingale properties of dW(t) and (dv x dt), we have

i|X(t) - Z(t)|

f t 2 ft |2

a(Z(s))ds + 3E b(Z(s))dW(s)

| tn tn |

c(Z(s),v)p$(dv x ds)

Jtn Je

< 3e([ 12ds [ |a(Z(s))|2ds ) + 3E [ \b(Z(s))\2ds

\Jt„ Jtn J Jtr.

C C |c(Z(s),v)|20(dv)ds

J tn Je

■ 3E

' tn J e

< 3atE j" |a(Z(s))|2ds + 3eJ |b(Z(s))|2ds + 3eJ J |c(Z(s),v)|20(dv)ds,

where the inequality \u1 + u2 + u3\2 < 3|mi|2 + 3|w212 + 3|w312 for u1,u2,u3 e Rd is used. Therefore, by applying Assumptions 2.1 and 2.3, we get

tt E |a(Z(s))|2ds < CkE (l + |Z(s)|2)ds < CkAt + Ckk2At,

e|" |b(Z(s))|2ds < Ck At + (Ckk2at, (3.7)

E I" I" |c(Z(s),v)|2^(dv)ds < CAt + Ck2 At,

which lead to

E|x(f) - Z(i)|2 < at(3Cfcat + 3k2CkAt + 3Cfc + 3k2Ck + 3C + 3k2C), (3.8)

for t e [0,T A Tk A pk]. Therefore, we obtain the result (3.4) by choosing

Ki (k) = 6Ck + 6k2Ck + 3CC + 3k2C. (3.9)

In accord with Lemma 2.4, 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 Assumptions 2.1,2.2, and 2.3, for any pair of e e (0,1) and T > 0, there exist a sufficiently large k* and a sufficiently small At\ such that

P(pk* < T) < e, Vat < at1, (3.10)

where pk» is defined in Lemma 3.1.

Proof. Applying generalized Ito's formula (see [1]) to |X(t)| , for t > 0, yields

|x(t)|2 = |X0|2 + (2X(s),a(Z(s))) + |b(Z(s))|2)ds

|X(s) + c(Z(s),v)|2 - |X(s)|2 - (lX(s),c(Z(s),v)^\$(dv)ds

| (2X(s),b(Z(s))}dW(s) + j | (|X(s) + c(Z(s),v)|2 - ^(s)Q(dv x ds). 0 0 s (3.11)

By taking expectations, we thus have

2 f iKPk

E|X(t A pk) |2 = E|X012 + E J^ ^^(lX(s), a(Z(s))) + |b(Z(s))|2 + J" |c(Z(s),v)|20(dv))ds = E|X012 + E J^ Pk((2X(s),a(X(s)) ) + |b(X(s)) |2 + J |c(X(s),v)|2$(dv)^ds

+ ej (lX(s),a(Z(s)) - a(X(s)) )ds + ej"^|b(Z(s))|2 - |b(X(s))Qds + e| ^|c(Z(s),v)|2 -^(X(s),^Q$(dv)ds.

(3.12)

For t e [0,T]. Now, by using the inequalities (ui,u2) < |ui||u2| for ui,u2 e R d, (2.2) in Assumption 2.1, Fubini's theorem, Cauchy-Schwarz's inequality, and Lemma 3.1, we get

J (2X(s),a(Z(s)) - a(X(s)) )ds < 2eJ |X(s)| |a(Z(s)) - a(X(s))d

< 2kVck f e|z(s a pk) - X(s A pk) |ds

ft / I - 12 X1/2

< ^E|Z(s A pk) - X(s A pk) | j ds

< 2kTyjCkKi(k)At.

Journal of Applied Mathematics And, similarly as above, we have

j Pk(i\b(Z(s))\2 - |b(x(s))Qds < E^ Pk(\b(Z(s))\ + |b(x(s))|)

x(\b(Z(s))\-|b(x(s))|) ds

< 2\]Ck(1 + k2)E j" Pk|b(Z(s)) - bixs)^

< 2\]CkCk(1 + k2) J" E^(s a pk) - X(s A pk^ds

< 2^CkCkK1(k)(1 + k2)at. (3.14)

Moreover, in the same way, we obtain

Ej P j (\c(Z(s),v)\2 -^(X(s),Vf)$(dv)ds

= E J Pk | ^c(Z(s),v) - c(X(s),v) + c(x(s),v)|2 - ^(X(s),^Q$(dv)ds < E J Pk j ^2|c(Z(s),v) - c(X(s),v)|2 + |c(x(s),v)Q$(dv)ds

ftAPk , _ ,2 _ ftAPk / ,_ ,2\

< 2CEJ |Z(s) - X(s)| ds + CEj (^1 + |x(s)| j ds

< 2C j E|Z(s a pk) - X(s A pk)|2ds + Ce j 1 + |X(s)Qds

_ _ ftApk ,_ ,2 < 2CTK1(k)at + CT + CE j |x(s)| ds,

(3.15)

where the inequality \u1 + u2\2 < 2\u1 \2 + 2\u2\2 for u1,u2 e 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

|x(t A pk)|2 < E\X0\2 + LE j" ^ 1 + |x(s)Q ds + Ce j |x(s)|2ds

+ 2kTyjCkK1(k)At + 2^CkCfcK1(k)(1 + k2) at + 2CTK1(k) at + Ct

10 Journal of Applied Mathematics

< (l + C) | E|X(s A Pk) |2ds + E|X0|2 + LT + Ct

f2kT^jCkK\(k) + 2^CkCfcKx(k)(1 + k2^ Vai + 2CTK\ (k) at,

(3.16)

for 0 < t < T. Therefore, by the Gronwall inequality (see [13]), for 0 < t < T, we get

E|X(t A pk)| < «1 a4 + a4a2(k)VKt + a4a3(k)At, (3.17)

«1 = e|x0|2 + lt + ct,

«2 (k) = 2kT\jCkK\(k) + 2^CkCkKx(k)(1 + k2), Os(k) = 2CTK\(k), a4 = exp ^ LT + (C^.

(3.18)

We thus obtain that

k2P(pk < T) < E^X(pk)| !{pk<t}j < e| X(T a< «1«4 + a4a2(k)VA~t + (k) at.

(3.19)

So for any e e (0,1), we can choose sufficiently large integer k = k* such that

aa e "fc^ < 2,

and choose sufficiently small at^ e (0,1) such that

a4a2(k*)\/At1 + a4a3(k*)At'

(3.20)

(3.21)

Hence, we have

p(pk* < T) < e, Vat < At*v (3.22)

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, X]. At the beginning, we give a lemma based on Lemma 3.1.

Lemma 4.1. Under Assumptions 2.1 and 2.3, for any X > 0, there exists a positive constant K2(k), dependent on k and independent of At, such that for all At e (0,1) the solution of (2.1) and the continuous-time Euler method (3.2) satisfy

E( sup lx(t A Tk A pk) - X(t A Tk A pk \0<t<T'

< Kz(k)At,

where Tk and pk are defined in Lemmas 2.4 and 3.1, respectively. Proof. From (2.1) and (3.2), for any 0 < tt < X, we have

E( sup lx(t A Tk A pk) - X(t A Tk A pk

\0<t<t'1

\0<t<t' Ji

■3E ( sup

0<t<t'

¡•tATkApk

(a(x(s-)) - a(Z(s)))ds

tATkApk 0

■3E ( sup

0 t t'

(b(x(s-)) - b(Z(s)))dW(s)

*tATkApk r

J J (c(x(s-),v) - c(Z(s),v))p$(dv x ds)

where the inequality \u\ + u2 + u3|2 < 3|mi|2 + 3|m2|2 + 3|m3|2 for u\,u2,u3 e 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

EI sup

0<t<t'

r tATk Apk

(a(x(s-)) - a(Z(s)))ds

/ r tATk Apk r tATk Apk

J su^ 12d^ |a(x(s-)) - a(Z(s))|2ds

0<t<t' 0 0

ft'ATkApk

|a(x(s-)) - a(Z(s))|2ds

ft' ATk Apk .__.2 \ / ft ATk Apk . _ .2

< 2TCkE

ft ATk Afik .__.2 \ / ft ATk Afik . _ 12

J |X(s) - Z(s)| ds ) + 2TCkeM |x(s-) - X(s)| ds

< 2TCkj" e|x(s a Tk A pk) - Z(s ATk Apk)| ds

+ 2TCk j E|x(s a Tk A pk-) - X(s ATk Apk)| ds

< 2T2CkK1(k)at + 2TCk i E( sup | x (u A Tk A pk-) - X(u A Tk A Pk)| ) ds. (4.3)

JQ \0<u<s' 1 /

Moreover, by using the martingale properties of dW(t) and p$(dv x dt), Assumption 2.1, Lemma 3.1, and Fubini's theorem, we have

EI sup

\ 0<t<t'

tATk Apk

(b(x(s-)) - b(Z(s)))dW(s)

M'ATkApk

< 4E |b(x(s-)) - b(Z(s))|2ds

f t ATk Apk __2 f tATk Apk _ 2

< 8Ckej |X(s) - Z(s)| ds + 8Ckej |x(s-) - X(s)| ds

< 8Ckj" e|x(s a Tk A pk) - Z(s ATk Apk)| ds

+ 8Ck j" E|x(s a Tk A pk-) - X(s A Tk A pk)| ds

< 8TCkKi(k)At + 8Ck | E( sup |x(u ATk Apk-) - X(u ATk Apk)| )ds

JQ \0<u<s' 1 /

/ f tATk Apk f 2\

E| sup (c(x(s-),v) - c(Z(s-),v))p$(dv x ds) !

\0<f<f J Q Js J

At ATkApk A

J J (c(x(s-),v) - c(Z(s-),v))p$(dv x ds)

ftt ATkApk f

|c(x(s-),v) - c(Z(s-), v) |2$(dv)ds

< 8TCK1(k)at + 8C i e( sup |x(u A Tk A pk-) - X(u A Tk A pk)f)ds. (4.4)

Jo \Q<u<s' 1 '

Hence, by substituting (4.3) and (4.4) into (4.2), we get

Ei sup |x(t A Tk A pk) - X(t A Tk A pk) I

\0<t<f1 1

< at (6X2CkK1 (k) + 24XCK (k) + 24XCK1 (k)) + (6XCk + 24Ck + 24C) (4.5) xf e( sup |x(u A Tk A pk-) - X(u A Tk A pk)| )ds.

J0 \0<u<s' 1 /

So using the Gronwall inequality (see [13]), we have the result (4.1) by choosing K2(k) = (6X2CkK1(k) + 24XCkK1(k) + 24XCKx(k^ exp(6X2Ck + 24XCk + 24XC). (4.6)

Now, let's state our theorem which demonstrates the convergence in probability of the continuous-time Euler method (3.2).

Theorem 4.2. Under Assumptions 2.1,2.2, and 2.3, for sufficiently small e,g e (0,1), there is a At* such that for all At < At*

Pi sup | x(t) - X(t)|2 > g) < e, (4.7)

0< KX1 1 '

for any X > 0.

Proof. For sufficiently small e,g e (0,1), we define

w : sup |x(t) - X(t)|2 > g}-. (4.8)

According to Lemmas 2.4 and 3.2, there exists a pair of k* and At^ such that

ptc < x ) < I, p(pk* < X) < 3, Vat < at1.

We thus have

p(q) < p(q n {tk* A pk* > t}) + P(Tfc. A pk* < t) < p(qn {tk*apk*> t^ + P(Tk* < t) + P

(q n T* A pk*> T}) + P(Tk* < T) + P(pk* < T) (4.10)

< p(q n {Tk* A pk*>T}) + y, for at < at1. Moreover, according to Lemma 4.1, we have

gP(qn {Tk* apk*>T}) < E I{Tk

X ' \ 0<t<T1 1

/ | — A (4.11)

< E( sup |x(t A Tk* A pk*) - ^t A Tk* A P&) | 1

\0<f<r 1 /

< K2(k*)At,

which leads to

p(Q n {Tk* A pk*>T}) < 3, (4.12) for at < at|. Therefore, from the inequalities above, we obtain

P(q) < e, (4.13)

for at < at*, where At* = min{ a* at^}. □

We remark that the continuous-time Euler solution X(t) (3.2) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths, not just only their at-increments. Therefore, the last theorem shows the convergence in probability of the discrete Euler solution (3.1).

Theorem 4.3. Under Assumptions 2.1,2.2, and 2.3, for sufficiently small e,g e (0,1), there is a At* such that for all At < At*

P(|x(t) - Z(t)|2 > g, 0 < t < T) < e, (4.14)

for any T > 0.

Proof. For sufficiently small e,g e (0,1), we define

q = [w : |x(t) - Z(t)|2 > g, 0 < t < tJ. (4.15)

Journal of Applied Mathematics A similar analysis as Theorem 4.2 gives

P(fl) < p(q n {Tk* A pk* > T}) + (4-16)

Recalling that

çP(¿2 n {Tk* A pk* > T}) < e(|x(T) - Z(T)|2I{Tk*APk*>T})

< E|x(T A Tk* A pk*) - Z(T A Tk* A pk*) |2

< sup |x(i A Tk* A pk*) - X(t A Tk* A pk*^^ (4.17)

\q<î<t'

+ 2e|x(t A Tk* A pk*) - Z(T A Tk* A pk*)|2 < 2Ki(k*)At + 2K2(k*)At,

and using Lemmas 3.1 and 4.1, we get that

P(¿2 n {Tk* A pk*>T^ < 3, (4.18)

for sufficiently small at. Consequently, the inequalities above show that

P(2) < e, (4.19)

for all sufficiently small at.

So we complete the result (4.14). □

5. Numerical Example

In this section, a numerical example is analyzed under Assumptions 2.1, 2.2, and 2.3 which cover more classes of SDEs driven by Poisson random measure. Now, we consider the following equation:

dx(t) = a(x(t-))dt + b(x(t-))dW(t) + j" c(x(t-),v)p$(dv x dt), t > Q, (5.1)

with x(0) = x(Q-) = Q, where d = m = r = 1. The coefficients of this equation have the form

a(x) = 1 (x - x3), b(x) = x2, c(x,v) = xv. (5.2)

The compensated measure of the Poisson random measure p$(dv x dt) is given by $(dv)dt = Xf (v)dvdt, where X = 5 and

f (v) = —L- exp(-, 0 < v< cc (5.3)

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

2(x, a(x)) + \b(x)\2 + J \c(x,v)\2$(dv) = x(x - x3) + x4 + J x2v2X exp^-dv

< (l + 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, c). Given the stepsize at, we can have the Euler method

X„+1 = Xn + 1 (Xn - X3) at + X2nAWn + Xnj"1 j vp$(dv x dt), (5.5)

with X0 = 0.

And in Matlab experiment, each discretized trajectory is actually given in detail by the following.

Algorithm

Simulate Xn+1 := Xn + (1/2)(Xn - X3 - 10—Xn)at + X2naWn;

Simulate variable p^(tn+1) - p$(tn), where (tn) is from Poisson distribution with parameter Xtn;

Simulate (tn+1) - (tn) independent random variables ii uniformly distributed on the interval p(tn),p^(tn+1));

Simulate p^(tn+1) - (tn) independent random variables li with law f (v);

obtain Xn+1 = Xn+1 + ^£^"¿+1 Itn<k<tn+1 &.

Subsequently, we can get the results in Theorems 4.2 and 4.3.

Acknowledgment

This work is supported by the NSF of China (no. 11071050).

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