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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 750147, 8 pages
A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations
1Department of Maths and Statistics, Curtin University, Perth, WA 6845, Australia
2School of Statistics & Maths, Zhongnan University of Economics and Law, Wuhan 430073, China
3Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
Received 1 December 2012; Revised 13 March 2013; Accepted 14 March 2013
Academic Editor: Xinguang Zhang
Copyright © 2013 Yanli Zhou et al. 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.
Stochastic delay differential equations with jumps have a wide range of applications, particularly, in mathematical finance. Solution of the underlying initial value problems is important for the understanding and control of many phenomena and systems in the real world. In this paper, we construct a robust Taylor approximation scheme and then examine the convergence of the method in a weak sense. A convergence theorem for the scheme is established and proved. Our analysis and numerical examples show that the proposed scheme of high order is effective and efficient for Monte Carlo simulations for jump-diffusion stochastic delay differential equations.
Stochastic delay differential equation models play a very important role in the study of many fields such as economics and finance, chemistry, biology, microelectronics, and control theories. Models of this type can more accurately describe many phenomena in the real world by taking into account the effect of time delay and random noise. By time delay, it means a certain period of time is required for the effect of an action to be observed after the moment when the action takes place. This phenomenon exists in most systems in almost any area of science; for example, a patient shows symptoms of an illness days or even weeks after he/she was infected. Similarly, random noises appear in almost all real world phenomena and systems, for example, the motion of molecules and the price of assets in financial markets. Hence, study of SDDEs is an important undertaking in order to understand real world phenomena and systems precisely.
Over the last couple of decades, a lot of work has been carried out to study differential equations with delay and/or random noises, for example, [1–5]. The best-known and well-studied theory and systems include the delay differential equations (DDEs) presented by Kolmanvskii and Myshkis  and their stochastic generalizations and the stochastic delay differential equations (SDDEs) established by Mohammed [7, 8], Mao [9, 10], and Mohammed and Scheutzow . Other SDDEs theories of interest include, for instance, the so-called SDDEs with Markovian switching and Poisson jumps. These models have been investigated in the literature [12–14].
Analytical solutions of SDDEs can hardly be obtained. It is thus important to develop and study discrete-time approximation methods for solving SDDEs. Discrete-time approximations may be divided into two categories: weak approximations and strong approximations . Some implicit and explicit numerical approximation methods for SDDEs in strong approximation sense were derived by Küchler and Platen . Weak numerical methods for SDDEs have been studied by Küchler and Platen. The Monte Carlo simulation method has also been developed as a powerful simulation method for SDEs, while weak numerical approximations are required for Monte Carlo simulation [9, 17–20].
In this paper, we extend a fully weak approximation method for SDDEs to the type of jump-diffusion SDDEs: subject to the initial condition where , is the time delay which is assumed to be constant at all time, is an -adapted -dimensional Wiener process, and denotes the Poisson random measure. Also here we denote by the almost sure left-hand limit of . The coefficient and are -dimensional vectors of Borel measurable functions. Further, defined on is a -matrix of Borel measurable function.
The main contributions of this work include construction of a numerical approximation method for weak solutions of SDDEs with jump-diffusion, and establishment of a theoretical result for the convergence of the scheme.
The remaining part of this paper is organised as follows. In the following section, we give the conditions for the existence and uniqueness of the solution of the jump-diffusion SDDEs (1) which is applicable to any cases where solution exists, and present various lemmas to be used later for the proof of the convergence theorem. We then introduce, in Section 3, a general weak approximation scheme, where the simplified stochastic Taylor approximation scheme with order is constructed; followed by a convergence theorem and its proof. In Section 4, we give a numerical example to demonstrate the application and the convergence of the numerical scheme.
In this section, we present some basic concepts and definitions to be used in later sections, then establish the conditions for the existence and uniqueness of solution to the Jump-diffusion SDDEs (1), and then give some lemmas to be used later for the proof of the convergence theorem. In this work, we assume that the -dimensional vector valued function for the initial condition, , is right continuous and has left-hand limits.
From (1), we have the following integral form of the jump-diffusion equation SDDE: where , for , are a sequence of pairs of jump times and corresponding values generated by the Poisson random measure .
Definition 1. Given a filtered probability space , a stochastic process given by is known as a solution of (1) subject to the initial condition (2) if is -adapted, the integrals in the equation are well defined, and equalities (3) and (2) hold almost surely. Moreover, if any two solution processes are indistinguishable on with the same initial segment and the same path on , and where is the Euclidean norm, then if (1) has a solution, it is a unique solution for this initial value problem.
To guarantee the existence of a unique solution of the jump-diffusion SDDE (1), we assume that the coefficients of (1) satisfy the following Lipschitz conditions: for and , as well as the growth conditions for , .
We denote by the Banach space of all -dimensional continuous functions on equipped with the supremum norm . Furthermore, we suppose that the set of -valued continuous process is -measurable with
Now we present some lemmas to be used later for the proof of the convergence theorem. Consider a right continuous process . is called a discrete-time numerical approximation with maximum step size , if it is obtained by using a time discretization , and the random variable is -measurable for . Further, can be expressed as a function of and the discrete-time .
Because of dealing with the approximation of solutions of jump-diffusion SDDEs, we introduce a concept of weak order convergence due to Kloeden and Platen .
Definition 3. A discrete-time approximation converges weakly towards at time with order if for each there is a constant , independent of , such that where denotes the set of all polynomials .
We now give some auxiliary results to prepare for the proof of the weak convergence theorem to be presented.
Lemma 4. For and , we have for and .
The proof of the lemma for the case with no delay was established by Platen and Bruti-Liberati , and a similar procedure can be used for the proof of this lemma.
Lemma 5. Given , there is a bounded constant satisfying for and .
Lemma 6. Given , there is a finite constant satisfying for every .
Lemma 7. Given , there is and a bounded constant satisfying for each , , , and , where for all and .
Lemma 8. For , there exist and a finite constant satisfying for each , , and .
3. The Jump-Adapted Weak Taylor Approximation Scheme
In Monte Carlo simulations for functionals of jump-diffusion SDDEs, one uses numerical approximations evaluated only at discretization time. Here, we first give a jump adapted weak approximation Taylor scheme of order , and then study the basic properties of the discrete Taylor approximation in a weak order sense.
First we define the jump-adapted time discretization. Let . The jump adapted time discretization used throughout this paper is with the maximum step size . We choose the time discretization in such a way that all jump times are at the nodes of the time discretization. If the discretization node is not a jump time, then is -measurable. Otherwise, is -measurable. Also, throughout the paper, we denote the set of all multiindices by where the element is called a multiindex of length and has zero length. In the following, by a component of a multi-index we refer to the integration with respect to the th Wiener process in a multiple stochastic integral. A component with corresponds to integration with respect to time .
Now define the following operators for the coefficient functions:
A subset is the hierarchical set, and its corresponding remainder set is defined by . For each , we can then define the hierarchical set . The weak Taylor method of order is then constructed as follows: where
The multiple stochastic integral is then defined recursively as follows: where is obtained from by deleting its last component, while is obtained from by deleting its first component.
Now, we give the weak convergence theorem of the Taylor approximation with order .
Theorem 9. Given , let be the results obtained from the Taylor scheme (17) corresponding to with maximum step size . Suppose that for , and converges to weakly with order . Assume that the coefficients, , , , are in the space , for and , and the coefficient functions , with , satisfy the growth condition , with , for all , , and . Then for any there is a positive constant , which does not depend on , such that
Proof. For and , consider the Itô process below:
Then we can get .
Define also the process , by for , , and for .
By the definition of the functional and the terminal condition of the stochastic process , we have
Since converges towards weakly with order , one has
By Lemma 4, we can write
From the properties of stochastic integrals, we obtain
Therefore, we have where
In the following, we proceed to estimate and in Steps 1 and 2, respectively, and then complete the proof in Step 3.
Step 1. Let us assume that is so smooth that the deterministic Taylor expansion may be applied. Hence, by expanding in , we get where the remainder term is where is a diagonal matrix with for and and , respectively.
Therefore, according to the properties of expectation and absolute value, we get
By (31), the Hölder inequality, and Lemma 7, we get
Similarly, by Lemma 5 and the Cauchy-Schwarz inequality, we have
Now, from the Cauchy-Schwarz inequality, Lemmas 8 and 6, and inequalities (33) and (34), we obtain
Step 2. Now we estimate the term in inequality (27). By the jump coefficient and the smooth function , applying the Taylor expansion yields
Similarly, we can estimate the reminders as follows:
Then, by applying the Hölder inequality, Lemmas 8 and 6, inequalities (37) to estimate the inequality above, we get
Step 3. Finally, by inequalities (27) and (35) as well as (38), we have
4. A Numerical Example
Here we give an illustrative example to demonstrate the application and the convergence of the proposed numerical scheme. We consider the following linear SDDE with Poisson jumps: where , , and are, respectively, the drift coefficient, the diffusion coefficient and the jump coefficient; is the jump intensity; and and are the delay coefficients.
By the method for solving linear stochastic differential equations in , the analytical solution for is derived as follows: where
According to the weak Taylor approximation scheme (17)–(19) proposed in Section 3, we now expand it with weak order 1 (well known as Euler scheme): Here we have used the jump adapted time discretization, and is the maximum step size.
For higher accuracy and efficiency, one needs to construct higher order numerical schemes. We now give a Taylor scheme of weak order two below:
Next, we study the convergence of the two numerical schemes presented above by using the weak errors measured by and compare the results obtained from these two schemes to the explicit exact solution. We estimate the weak errors by running a very large number of simulations. The exact number depends on the implemented scheme. We use the following parameters: , , , , , and .
In Figure 1, we give the sample paths under the two approximation schemes and the numerical explicit solution of (40). We can see from the figure that the weak Taylor scheme path is closer to the analytical solution line than the Euler scheme.
Now we present the numerical errors generated by the two numerical schemes presented above. From Table 1, we notice that, for all the step sizes used in the numerical experiments, the weak Taylor method is more accurate. Moreover, the errors of the weak order two Taylor method decrease faster than the Euler scheme.
In this work, we have extended previous research on weak convergence to a more general class of stochastic differential equations involving both jumps and time delay. We proved that under the Poisson random measure and a fixed time delay, a simplified Taylor method gives weak convergence rate arbitrarily close to order . There is much scope for further work in the context of weak solution of jump-diffusion SDDEs. For example, it is clearly of great importance to extend the weak convergence theory to the case where coefficients in the equations are not globally Lipschitz, and to develop and analyze new methods that maintain good properties of convergence and stability.
The authors thank the anonymous referees and the editors for their constructive comments and suggestions. Yanli Zhou acknowledges financial support from Curtin International Postgraduate Research Scholarship (CIPRS) and Chinese Scholarship Council (CSC). Acknowledgement is also made to the China National Social Science Foundation (SSF) for the research support (10BJY104).
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