MAOWei(毛偉)，HU Liang-jian(胡良劍)

1 School of Mathematics and Information Technology，Jiangsu Second Normal University，Nanjing 210013，China

2 College of Science，Donghua University，Shanghai201620，China

Stochastic Approximate Solutions of Stochastic Differential Equationswith Random Jump Magnitudes and Non-Lipschitz Coefficients

MAOWei(毛偉)1，HU Liang-jian(胡良劍)2*

1 School of Mathematics and Information Technology，Jiangsu Second Normal University，Nanjing 210013，China

2 College of Science，Donghua University，Shanghai201620，China

A class of stochastic differential equations with random jump magnitudes(SDEwRJM s)is investigated．Under non-Lipschitz conditions，the convergence of sem i-imp licit Euler method for SDEwRJM s is studied．The main purpose is to prove that the sem i-implicit Euler solutions converge to the true solutions in the mean-square sense．An exam p le is given for illustration．

stochastic differential equations(SDEs);random jump magnitudes;numerical analysis;non-Lipschitz coefficients

Introduction

Stochastic differential equations(SDEs)have been found many applications in economics，biology，finance，and ecology，etc．Qualitative theory of SDEs have been studied intensively for many scholars．Here，we refer to Mao［1］，Higham et al．［23］and references therein．Recently there is an increasing interest in the study of stochastic differential equationswith jumps(SDEw Js)(see Ref．［4］)．There is an evidence that the dynam ics of prices of financial instruments exhibit jumpswhich cannotbe adequately described by diffusion processes(see Ref．［5］)．Since only a lim ited class of SDEw Js admits explicit solutions，there is a need for the developmentof approximatemethods．Some of the results in this area can be found in Refs．［6- 13］where the convergence and stability of numerical schemes are considered．In particularly，Chalmers and Higham［9］studied a class of SDEs with random jump magnitudes(SDEwRJMs)which is a generation of SDEs with deterministic jump magnitudes［68，10，11，13］．In Ref．［9］，they presented the semi-implicit Euler solutions of SDEwRJMs and discussed the convergence and stability of the semi-implicit Euler solutions where the coefficients satisfying the Lipschitz conditions．

In the papers mentioned above，most of the convergence theory for numerical methods requires the coefficients of SDEw Js to be Lipschitz．However，the Lipschitz condition is often notmet by many systems in practice．For example，the follow ing semi-linear stochastic differential equations

1 Prelim inaries and Sem i-im plicit Euler Approximation

Let(Ω，F，P)be a complete probability space with a filtration(Ft)t≥0satisfying the usual conditions．Let{w(t)，t≥0}be an m-dimensional Wiener process defined on the probability space(Ω，F，P)adapted to the filtration(Ft)t≥0．Let T＞0，L1(［0，T］;n)denote the family of alln-valued measurable(Ft)-adapted processes f={f(t)}0≤t≤Tsuch thatWe also denote by L2(［0，T］;n×m)the family of alln×m-matrix-valued measurable(Ft)-adapted processes f={f(t)}0≤t≤Tsuch thatbe an F0-measurable Rn-valued random variable such that

In this paper，we consider a class of SDEwRJMs

for all0≤t≤T．Here x(0)=x0，f:n→n，g:n→n×mand h:n×n→n;w(t)is an m-dimensional W iener process;N(t)is a Poisson processwithmeanλt andγi，i=1，2，…are independent，identically distributed random variables representingmagnitudes for each jump．For some P≥2，there is a constant B such that

For system(2)，the semi-implicit Euler approximation on t∈{0，h，2h，…}is given by the iterative scheme

where 0≤θ≤1．Here yn≈x(tn)，w(tn)andΔNn=N(tn+1)－N(tn)，n=0，1，2，…，N are the Wiener and Poisson increments，respectively．

Let z1(t)=yn，z2(t)=yn+1，and=γN(tn)+1，t∈［tn，tn+1)，and then the continuous-time approximation is defined bywhich interpolates the discrete numerical approximation(3)．

To establish the strong convergence theorem，we need the follow ing hypotheses．

(H1)There exists a positive constant k0such that

(H2)For all x1，y1，x2，y2∈n，there exist two positive constants L，η≥0 such that

whereρ(·)is a concave nondecreasing function from R+to R+such that

and

Remark 1Let us give some concrete functionsρ(·)．Let k＞0 andδ∈(0，1)be sufficiently small．Defineρ1(u)=Lu，u∈R+，

They are all concave nondecreasing functions satisfying

Remark 2Similar to the proof of Theorem 3．1 and Lemma 4．2 in Ref．［14］，we can prove that Eq．(2)has a unique solutions on［0，T］under(H1)-(H2)and show the existence of the semi-implicit Euler approximate solutions(3)under (H2)．

2 Main Results

In this section，wewill show the strong convergence of the semi-implicit Euler solutions to the exact solutions under the non-Lipschitz condition．

First，let us quote the Bihari lemma［15］which is necessary for the proof of our result．

Lemma 1Let T＞0 and c＞0．Letρ∶R+→R+be a continuous non-decreasing function such thatρ(t)＞0 for all t＞0．Let u(·)＞0 be a Borelmeasurable bounded non-negative function on［0，T］，and let v(·)be a non-negative integrable function ond s for allholds for all t∈［0，T］such thatand G－1is the inverse function of G．

Lemma 2Under conditions(H1)and(H2)，there exists a positive constant c such that

Proo fApplying the Itformula towe obtain that

By using the Burkholder Davis Gundy(BDG)inequality［1］and the Young inequality，we get

and

Inserting Formulas(10)and(11)into Formula(9)，it follows that

where M=2－2θ+2θ2+k1+2λ+k2．So，(H1)and(H2)imply that

Given thatρ(·)is concave andρ(0)=0，we can find a pair of positive constants a and b such that k(u)≤au+b for all u＞0．So we have

From the Gronwall inequality，we derive that

Hence the required assertionmust hold．

Lemma 3Under conditions(H1)and(H2)，there exist two positive constants ci，i=1，2 such that

ProofFor any t∈［0，T］，choose n such that t∈［nh，(n +1)h)．Then

By using the basic inequality and the Holder inequality，we have

Again themartingale isometries，conditions(H1)and(H2)imply that

Similarly，we obtain that E(sup0≤t≤T≤c2h． The proof is completed．

Now，we can state ourmain result of this paper．

Theorem 1Let conditions(H1)and(H2)hold，then the semi-implicit Euler solutions(4)will converge to the true solutions of Eq．(2);that is，for any T＞0，

ProofLettingε(t)=x(t)－y(t)，from Eqs．(2)and (4)，we derive that，for 0≤t≤T，

Applying the It∧oformula to|ε(t)|2，it follows that

Taking expectation on both sides of Eq．(18)，one gets

where H1- H6stand for the successive terms．Let us estimate Hi，i=1，2．By(H2)andLemma 3，we have

Again the Jensen's inequality，Lemma 3and(H2)imply that

FortheestimateofH4，by(H2)andLemma3，weget

By applying the Holder inequality and E≤B，we obtain that

Inserting Formula ( 23) into Formula ( 22) ，

Now，estimate the following two martingale terms． By the BDG inequality and Lemma 3，it follows that

and

Combining Formulas ( 19) ( 21) and Formulas ( 24) ( 26) together，we have

where

Sinceρ(·)is aconcave function andρ(0)=0，wehave ρ(u)≥ρ(1)u，for0≤u≤1．So we obtain that

ByLemma1，

E［sup0≤s≤tNote that whenh→0，then M2Recalling the condition+M1t→－∞，→0．Soit follows that

The proof of Theorem 1 is now completed．

Remark 3 If ρ( u) = Lu，u≥0，then the condition ( H2 )implies a global Lipschitz condition． Our result of this paper isTheorem 3．4 of Ref．［9］ and the results of Ref．［9］aregeneralized and improved．

3 An Example

Let w ( t) be a scalar Brownian motion and N ( t) be ascalar Poisson processes． Assume that w( t) and N( t) areindependent． Consider a semi-linear SDEwRJMs of the form

where Eq and a(t)，b(t)are two square-integrable functions in［0，T］．Here x(0)=x0，

From Eq．(4)，the sem i-implicit Euler solution of Eq．(28)is defined by

Let z1(t)=yn，z2(t)=yn+1，andt∈［tn， tn+1)．Then we have the continuous semi-implicit Euler solution

Clearly，the coefficientsα(·)andβ(·)do not satisfy the Lipschitz condition．We have thatα(·)is a nondecreasing，positive and concave function on［0，∞］withα(0)=0 and

Sim ilarly，we also obtain thatβ(0)=0 isa nondecreasing，positive and concave function on［0，∞］withβ(0)=0 andTherefore，it follows that condition(H2)is satisfied．Consequently，the approximate solutions(29)will converge to the true solutionsof Eq．(28)for any t∈［0，T］in the sense of Theorem 1．

4 Conclusions

In this paper，the sem i-implicit Eulermethod is developed for a class of SDEwRJMs．Different from the Lipschitz conditions of Refs．［6- 11］，we propose the non-Lipschitz conditionswhich the coefficients of Eq．(2)satisfy．The main purpose is to prove that the sem i-implicit Euler approximate solutions converges to the exact solutions in the mean-square sense under non-Lipschitz condition．

［1］Mao X R．Stochastic Differential Equations and Applications［M］．2nd ed．Chichester，UK:Horwood Publishing，2007．

［2］Higham D J，Mao X R，Stuart A M．Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations［J］．SIAM Journal on Numerical Analysis，2002，40 (3):1041-1063．

［3］Higham D J，Mao X R，Stuart A M．Exponential Mean-Square Stability of Numerical Solutions to Stohcastic Differential Equations［J］．LMS Journal of Computation and Mathematics，2003，6:297-313．

［4］Oksendal B，Sulem A．Applied Stochastic Control of Jump Diffusions［M］．2nd ed．Berlin，Germany:Springer，2007．

［5］Jorion P．On Jump Processes in the Foreign Exchange and Stock Markets［J］．Review of Financial Studies，1988，1(4):427-445．

［6］Gardon A．The Order of Approximation for Solutions of It o∧-Type Stochastic Diffrential Equations with Jumps［J］．Stochastic Analysis and Application，2004，22(3):679-699．

［7］Higham D J，Kloeden P E．Numerical Methods for Nonlinear Stochastic Differential Equations with Jumps［J］．Numerische Mathematik，2005，101(1):101-119．

［8］Higham D J，K loeden P E．Convergence and Stability of Implicit Methods for Jump-Diffusion Systems［J］．International Journal of Numerical Analysis and Modeling，2006，3(2):125-140．

［9］Chalmers G，Higham D J．Convergence and Stability Analysis for Implicit Simulations of Stochastic Differential Equations with Random Jump Magnitudes［J］．Discrete Continuous Dynamical Systems B，2008，9(1):47-64．

［10］Wang X，Gan S Q．Compensated Stochastic Theta Methods for Stochastic Differential Equations with Jumps［J］．Applied Numerical Mathematics，2010，60(9):877-887．

［11］Buckwar E，Riedler M G．Runge-Kutta Methods for Jump-Diffusion Differential Equations［J］．Journal of Computational and Applied Mathematics，2011，236(6):1155-1182．

［12］Song M H，Yu H．Convergence and Stability of Implicit Compensated Euler Method for Stochastic Differential Equations with Poisson Random Measure［J］．Advances in Difference Equations，2012:214．

［13］Hu L，Gan S Q，Wang X J．Asymptotic Stability of Balanced Methods for Stochastic Jump-Diffusion Differential Equations［J］．Journal of Computational and Applied Mathematic s，2013，238 (1):126-143．

［14］Mao W，Mao X R．On the Approximations of Solutions to Neutral SDEs with Markovian Sw itching and Jumps under Non-Lipschitz Conditions［J］．Applied Mathematics and Computation，2014，230(1):104-119．

［15］Bihari I．A Generalization of a Lemma of Bellman and Its Application to Uniqueness Problem of Differential Equations［J］．Acta Mathematica Academiae Scientiarum Hungaricae，1956，7 (1):81-94．

O211．63;O241．5

A

1672-5220(2015)04-0642-06

date:2014-02-17

s:National Natural Science Foundations of China(Nos．11401261，11471071);Qing Lan Project of Jiangsu Province，China (No．2012);Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No．13KJB110005);the Grant of Jiangsu Second Normal University(No．JSNU-ZY-02);the Jiangsu Government Overseas Study Scholarship，China

*Correspondence should be addressed to HU Liang-jian，E-mail:ljhu@dhu．edu．cn