Truncated Methods for Stochastic Equations:A Review

LIU Wei, MAO Xuerong

(1.Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;2.Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, UK)

Abstract：The truncated Euler-Maruyama(EM)method was originally proposed by Mao(2015, J.Comput.Appl.Math.).After that, plenty of works employed the idea to construct numerical approximations to different types of stochastic equations.Due to the bloom of the research in this direction, we give a thorough review of the recent development in this paper, along which we also point out some potential and challenging research.

Key words：stochastic different equations; super-linear coefficients; truncated methods; explicit methods.

1 Introduction

In recent decades, stochastic differential equations(SDEs)have been broadly used to model uncertain phenomena in different areas, such as finance, biology, chemistry, and physics[1, 12, 34-35].However, the explicit closed form of the true solution is rarely found.Even for linear SDEs, their explicit expressions of true solutions involve stochastic integrals and hence the statistical properties of the true solutions in most cases can only be seen from numerical simulations.Therefore, numerical methods for SDEs become extremely important in applications.

Since the solutions to SDEs can be understood as stochastic processes, the errors of numerical solutions could be investigated in difference sense, such as in distribution, in probability, inLpetc.The study of numerical methods of SDEs needs the knowledge about stochastic processes and numerical analysis, which makes it a hard but interesting topic.We refer the readers to the monographs[2, 36-37]for the introductions of fruitful methods and their properties.

Among different numerical methods, the Euler-Maruyama(EM)method is the simplest one, which is an explicit approach and can be implemented easily[3].The EM method is a straightforward extension of the forward Euler method for ordinary differential equations(ODEs).Although, the forward Euler method is quite inefficient for ODEs[4], the EM method has been a popular method for SDEs for a long time.Two main reasons are as follows.

(1)The application of SDEs in practices often leads to the simulation of a large number of the sample paths.Explicit methods, such as the EM method, are much faster than implicit methods, such as the semi-implicit Euler method(also refereed as the backward EM method)[5], which require to solve some non-linear systems in each iteration.

(2)Compared with other explicit methods with the higher convergence rate, such as the Runge-Kutta methods[6], the EM method has much simpler structure.This is because that the higher convergence rate for the case of SDEs needs quite a lot of effort to deal with the multiple stochastic integrals, which is more difficult than the case of deterministic integrals.

However, it is proved that the moment of the EM method is divergent from that of the true solution to an SDE with super-linear coefficients[7].From the theoretical point of view, this indicates that it is improper to use the EM method to approximate SDEs with super-linear coefficients any more, if one considers the strong convergence.But the EM type methods are still very attractive to many researchers due to reasons mentioned above.Therefore, plenty of modified explicit Euler type methods have been proposed.

Among all the different interesting modifications of the EM method, the tamed Euler method would be the first and a nice attempt[8].After that, different approaches are proposed[9, 38-41], we just mention some of them here and refer the readers to the references therein.

In this paper, we focus on the truncated EM method, which is a modified explicit Euler type method.The method was originally proposed by Mao in[10]for SDEs with both drift and diffusion coefficients growing super-linearly.The strong convergence was proved in[10]and the convergence rate was derived in[11].After these two initial works, plenty of interesting papers adapt the idea to construct different numerical methods for different types of stochastic equations.

The main purpose of this paper is to review the recent developments in the truncated methods for stochastic equations.We try to introduce what have been done and what have not without too much mathematics, and refer the readers to each individual paper for the detailed and rigorous analysis.

The paper is constructed as follows.The original truncated EM method is briefed in Section 1.The recent developments of this type of method for SDEs are discussed in Section 2.Section 3 sees the works for SDEs with different delay terms.Section 4 reviews the relevant papers for SDEs driven by other noises besides the Brownian motions.Section 5 concludes this paper by mentioning some potential future research.

2 Review of the original works

Let us present a brief review on the original ideas in[10-11].We refer the readers to these two papers for more detailed terminologies.

Consider ad-dimensional SDE

dx(t)=f(x(t))dt+g(x(t))dB(t),

(1)

andB(t)is anm-dimensional Brownian motion.

Following assumptions are imposed on the coefficients to guarantee the existence and uniqueness of the strong solution and to indicate that the super-liner terms are allowed.

Assumption1 There are constantsL1>0 andγ≥0 such that

|f(x)-f(y)|∨|g(x)-g(y)|≤L1(1+|x|γ+|y|γ)|x-y|,

(2)

(3)

(4)

Remark1 It is not hard to check that these three assumptions can cover SDEs like

dx(t)=-(x(t)+x5(t))dt+x2(t)dB(t),t≥0,

where the super-linear terms appear in both drift and diffusion coefficients.According to[7], the EM method fails to convergence in the strong sense.

In[10-11], to handle with the super-linearity the author proposed to construct a truncation mapping in the following way.

(5)

Step 2 One may choose a numberΔ*∈(0, 1]and a strictly decreasing functionh:(0,Δ*]→(0, ∞)such that

(6)

Step 4 The truncated EM method for(1)is defined by

Xi+1=Xi+f(πΔ(Xi))Δ+g(πΔ(Xi))ΔBi,i=0, 1, 2, …,

(7)

whereX0=x0andΔBi=B((i+1)Δ)-B(iΔ).

Remark2 It is not hard to check that by using the truncation idea bothf(πΔ(x))andg(πΔ(x))are bounded byh(Δ).Since the step sizeΔis prescribed,f(πΔ(x))andg(πΔ(x))in this method actually are bounded by some pre-decided constant.

As bothf(πΔ(x))andg(πΔ(x))are globally Lipschitz continuous and bounded, the truncated method(7)is well defined and performs nicely in the sense of strong convergence.That is, if we define its continuous-time extension

then this numerical solutionXΔ(t)should be a good approximation in strong sense to the true solution of the following truncated SDE

dyΔ(t)=f(πΔ(yΔ(t)))dt+g(πΔ(yΔ(t)))dB(t),

(8)

as long asΔis sufficiently small.It is known that under the Khasminskii-type condition(4),yΔ(t)converges to the true solutionx(t)of the underlying SDE(1)in probability[12].It is hence obvious to see thatXΔ(t)converges to the true solutionx(t)in probability.However, additional Assumptions 1 and Assumptions 2 guarantee the convergence is in fact in strong sense as described in the following theorem[11].

(9)

for all sufficiently smallΔ∈(0,Δ*), then, for every such smallΔ,

(10)

for eachT>0, whereCTis a positive constant dependent onTbut independent ofΔ.

(11)

This shows that the convergence order of the truncated EM could be arbitrarily close to half.

3 Recent works on SDEs without time-delays

After the original works, plenty of papers are devoted to improve and modify the intial setting up.

Hu, Li and Mao in[13]pointed out the requirements on the functionhmay sometime too strong to force the step size to be too small.They showed the following conditions are efficient

(12)

Guo, Liu, Mao and Yue in[14]proposed the partially truncated EM method by observing that it is of necessity to truncated the super-linear terms while leaving the linear term as it is.To be more specific, the authors separate both thefandgin(1)into two parts that

f(x)=F1(x)+F2(x)andg(x)=G1(x)+G2(x),

whereF1andG1satisfy the Global Lipschitz conditions, and some super-linear terms appear inF2andG2.This setting of the partial truncation makes the investigation of the asymptotic behaviours more easily.Hence, apart from the strong convergence was discussed, the asymptotic stability and boundedness in the mean square sense of the partially truncated EM method were discussed.Guo, Liu and Mao in[15]adapted the idea in[13]to release the requirements on the step size for the partially truncated EM method.

Lan and Xia in[16]constructed a modified truncated EM method by using different the truncation mappingπΔ(x)and requirements on the functionhin(6).The strong convergence at the terminal timeTas well as over a time interval[0,T]were investigated.

Wen in[17]proposed a fully implicit truncated EM method, which allows the larger step size and the less computational cost compared with the original works[10-11].

Guo, Liu, Mao and Zhan in[18]investigated the combination of the truncated EM method with the multi-level Monte Carlo method[19]and discussed the computational cost and theconvergence of such a combination.

Yang, Wu, Kloeden and Mao in[20]investigated the truncated EM method for the SDEs with the H?lder diffusion coefficient and the super-linear drift coefficient.

All the works mentioned above looked at the Euler-type method, whose convergence rate is at most a half.For methods with the higher rate, Guo, Liu, Mao and Yue in[21]developed the truncated Milstein method for SDEs with the commutative noise, whose coefficients are allowed to grow super-linearly.The strong convergence rate was proved to be arbitrarily closed to one.

More recently, Li, Mao and Yin in[22]pointed out that the truncation mappingπΔ(x)in the original works[10-11]is sometimes too strong.To be more precise,πΔ(x)forcesf(πΔ(Xi))andg(πΔ(Xi))to be bounded by some constant as discussed in Remark 2.5, which is not necessary.

Since the idea is to apply the EM method for the SDEs with the truncated coefficients and the EM is able to handle SDEs with linear growing coefficients, it seems that truncating the coefficients into trunction functions satisfying the linear growth condition is enough.By such an observation, the authors in[22]proposed a brand new truncated EM method.To be more precise, this new truncated EM scheme is constructed in the following way.

Step 2 A constantΔ*∈(0, 1)is chosen and a strictly decreasingh:(0,Δ*]→(0, ∞)such that for ?Δ∈(0,Δ*]

whereKis a constant independent ofΔ.

Step 4 Finally, the new scheme is constructed as

and define

(13)

Remark3 It is clear that

f(πΔ(x))≤h(Δ)(1+|πΔ(x)|)andg(πΔ(x))≤h(Δ)(1+|πΔ(x)|),

where the coefficients are truncated to be of linear growth.This is significantly different from the original papers by Mao[10-11].

It was due to this new truncation function that Li, Mao and Yin[22]were be able to replace Assumption 3 by a much weaker assumption as follows.

Assumption4 There exists a pair of positive constantspandλsuch that

To see that the family of the drift and diffusion coefficients satisfying Assumption 4 is much larger than that satisfying Assumption 3, we list a number of examples(theCbelow denotes a positive constant):

(1)If there are positive constants a,ε, andλsuch that |xTg(x)|2≤a|x|4-ε+Candthat2xTf(x)+|g(x)|2≤λ|x|2+C, then Assumption 4 holds for anyp>0.

(2)If there are positive constants a,ε, andλsuch that |xTg(x)|2≥λ|x|4+Cand that 2xTf(x)+|g(x)|2≤a|x|2-ε+C, then Assumption 4 holds for any 0

(3)If there exists a positive constantλsuch that 2xTf(x)+|g(x)|2≤λ|x|2+C, then Assumption 4 holdsforp=2.

(4)If there are positive constants a,λ, andu>v+2 such that |xTg(x)|2≥λ|x|u+Cand that 2xTf(x)+|g(x)|2≤a|x|v+C, then Assumption 4 holds for 0

(5)If there are positive constants a,ε, andusuch that |xTg(x)|2≥a|x|u+2+Cand that 2xTf(x)+|g(x)|2≤(2a-ε)|x|u+C, then Assumption 4 holds for 0

One of the key results in Li, Mao and Yin[22]which shows the order of strong convergence of the new truncated EM is half.

(14)

for eachT>0, whereCTis a positive constant dependent onTbut independent ofΔ.

The authors in[22]also pointed out that the truncation strategy should vary according to the desired properties to be studied.This is another important contribution of the work.For instance, to study the stability in distribution of the underlying SDE, the authors modified the construction of the truncated EM in the following approach.

Step 2 A constantΔ*∈(0, 1)is chosen and a strictly decreasingκ:(0,Δ*]→(0, ∞)such that for ?Δ∈(0,Δ*]

Step 4 Now, the truncated EM scheme is defined as

and define

(15)

The following assumption guarantees the stability in distribution of the underlying SDE

(1)(see, e.g.,[22]).

Assumption5 There are positive constantsp,λ,ρandνsuch that

(16)

and

|x-y|2[2(x-y)T(f(x)-f(y))+|g(x)-g(y)|2]-

(2-ρ)|(x-y)T(g(x)-g(y))|2≤ -ν|x-y|4,

(17)

The following theorem established by[22]shows that the truncated EM(15)reproduces this stability in distribution very well.

It should also be mentioned that all the proofs in[22]work on the discrete version of the numerical scheme.This is quite different from most existing papers, where the continuous version of the numerical scheme that needs the whole path of the Brownian moition is used as an intermediary.Yang and Li in[23]extended the results in[22]to the non-linear switching diffusion systems.

4 Recent works on SDEs with delay terms

The truncating techniques have also been adapted to handle the super-linearity in SDEs with different types of delays.

The paper by Guo, Mao and Yue[24]was the first to extend the original works[10-11]to SDDEs with the constant delay.More precisely, they investigated the following equation Consider a nonlinear SDDE

dx(t)=f(x(t),x(t-τ))dt+g(x(t),x(t-τ))dB(t),t≥0,

(18)

with the initial data given by

(19)

Here

whilex(t-τ)represents the delay effect with the constant delay timeτ>0.We assume that the coefficientsfandgobey the well known Local Lipschitz condition.To prove the strong convergence of the truncated EM, they imposed the generalized Khasminskii-type condition.

Assumption6 There are constantsK1>0,K2≥0 andβ>2 such that

(20)

It should be pointed out that the high order terms -K2|x|β+K2|y|βmake this assumption cover many highly nonlinear SDDEs.To have a feeling about what type of highly nonlinear SDDEs to which their theory may apply, please consider, for example, the scalar SDDE

dx(t)=[a1+a2|x(t-τ)|4/3-a3x3(t)]dt+[a4|x(t)|3/2+a5x(t-τ)]dB(t),t≥0,

(21)

Steps 2 and 3 are the same as in Section 2.Step 4 is to define the discrete-time truncated EM solution by settingtk=kΔfork=-M, -(M-1), …, 0, 1, 2, … andXΔ(tk)=ξ(tk)fork=-M, -(M-1), …, 0 and then forming

XΔ(tk+1)=XΔ(tk)+f(πΔ(XΔ(tk)),πΔ(XΔ(tk-M)))Δ+g(πΔ(XΔ(tk)),πΔ(XΔ(tk-M)))ΔBk,

(22)

fork=0, 1, 2, …, whereΔBk=B(tk+1)-B(tk).

The continuous-time truncated EM solution is defined by

(23)

Different from the SDEs, the continuity of the initial dataξplays its role in the convergence of the truncated EM.Accordingly, the following additional condition was imposed in[24].

Assumption7 There is a pair of constantsK3>0 andγ∈(0, 1]such that the initial dataξsatisfies

|ξ(u)-ξ(v)|≤K3|u-v|γ, -τ≤v

On of the key results in[24]shows the strong convergence of the truncated EM inqth moment forq∈[1, 2).

Theorem4 Let Assumptions 6 and 7 hold.Then, for anyq∈[1, 2),

(24)

To get the strong convergence in qth moment forq≥2, they need to impose stronger assumption than Assumption 4.1.

(25)

They could then prove the following result in[24].

(26)

Two theorems above does not give the order of strong convergence.Under additional conditions, Guo, Mao and Yue in[24]were able to show the order is close to half.

Zhang, Song and Liu in[25]investigated the extension of the partially truncated EM method to the stochastic differential delay equations of the form(18)and the strong convergence was proved.The same group of authors also studied the truncated EM method for a more general dely equations in[26], i.e.stochastic functional differential equations, wherex(t-τ)in(18)was replaced byxt={x(t+u):-τ≤u≤0}.More recently, Zhang in[27]extended the truncated EM methods to a class of stochastic Volterra integro-differential equations, where the coefficients do not satisfy the global Lipschitz condition.

When the neutral term was taken into consideration in(18), i.e.thex(t)on the left hand side is replaced byx(t)-D(x(t-τ)), Lan and Wang in[28]and Lan in[29]studied the finite time convergence and asymptotic exponential stability of the modified truncated EM method, which could be an extension of[16].

Zhan, Gao, Guo and Yao in[30]extended the partially truncated EM method to pantograph stochastic differential equations, where the delay termx(qt)for someq∈(0, 1)replacesx(t-τ)in(18).In addition, Cong, Zhan and Guo in[31]extended the partially truncated EM method to hybrid SDE with the variable but bounded delays, i.e.x(t-τ)in(18)is replaced byx(t-δ(t))forδ(t):[0, ∞)→[0,τ].

For higher rate methods, Zhang, Yin, Song and Liu in[32]extended the truncated Milstein method to(18)and the strong convergence rate was proved to be close to one.

5 Recent works on SDEs driven by noises other than the Brownian motions

If the driven noise in(1)is replaced by other stochastic processes, not many works have been done.One of the main reasons for it, we believe, is that there is not too much technical difficulty if the driven noise has the similar properties as the Brownian motions that

(1)independent and identically distributed increment,

(2)all the moments for the increment exist.

Bearing the first bullet point in mind, the extensions of the truncated methods for SDEs driven by Lévy processes with the finite Lévy measure are quite straightforward, if the coefficient for the noise term is global Lipschitz.However, difficulties may appear if some super-linear terms are allowed in the coefficients for the noise part.Deng, Fei, Liu and Mao in[33]tackled such a problem by looking at the truncated EM method for SDEs driven by the Poisson jump process and the coefficient of the jump term is allowed to grow super-linearly.

The second bullet point is corresponding to those processes like the α-stable process, whosepth moment only exists forp<α.In this case, one needs to be much more careful to handle the tradeoff between the super-linear terms and driven processes.However, no result has been known in this direction so far.

6 Conclusion and future research

Since the original works by Mao in[10-11], the developments of the truncated methods for different types of stochastic equations have bloomed.Most of existing papers focused on the equations driven by the Brownian motions and the Euler-type methods that have the convergence rate of arbitrarily closed to a half.However, only a few works have been devoted to equations driven by other types of noises and methods with the higher convergence rate.

It is also not hard to see that few work has taken the variable step size into consideration.In addition, no work has looked at stochastic partial differential equations(SPDEs).Taking all the mentioned observations, we believe that the following four directions worth to be investigated.

(1)Investigation of truncating techniques for stochastic equations driven by noises other than the Brownian motions, such as fractional noises and Lévy processes with the infinite Lévy measure.

(2)Construction of methods with the convergence rate higher than one, for example Runge-Kutta methods.

(3)Developments of strategies to change the step size in each iteration to release the constrain on step size.(4)Employment of the truncating ideas to handle the temporal variable in the numerical methods for SPDEs with super-linear terms.

Acknowledgements

Wei Liu would like to thank the Natural Science Foundation of China(11701378, 11871343, 11971316), Chenguang Program supported by both Shanghai Education Development Foundation and Shanghai Municipal Education Commission(16CG50), and Shanghai Gaofeng & Gaoyuan Project for University Academic Program Development for their financial support.

Xuerong Mao would like to thank the Royal Society(WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund(NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC(EP/K503174/1)for their financial support.