Exact Finite-Difference Schemes for 𝑑-Dimensional Linear Stochastic Systems with Constant Coefficients

Abstract

The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

1. Introduction

Certainty barely exists as a natural phenomenon in isolation. On the contrary, the phenomenon of certainty is an integral part of a complex environment full of interrelations and interactions. Physics is the general analysis of nature, including elements of matter, motion, space, and time, all of which are relevant to the concept of certainty and uncertainty. In order to capture a clear understanding of physical movements in the natural world by using modeling techniques, it is inevitable to take the effects, as the result of the uncertainty upon any development course of any entities, into full consideration. The stochastic differential equations are one of the best known techniques to depict physical movement in precision. For instance, statistical physics is based on the ergodicity assumption of the development of a system and stochastic differential equations can give perfect solutions to such scenarios. Moreover, the stochastic differential equations and relevant applications have been applied in such a variety of scientific fields such as stochastic control, stochastic neural network, financial economic studies, species dynamics, and electronic engineering [1–5]. In this regard, it is of crucial importance to explore stochastic differential equations. Major research findings of the stochastic differential equations can be found in [4, 5].

Although Mao [4] and Kloeden and Platen [6] have managed to calculate the expression of the explicit solution to certain types of stochastic differential equations, it is not always possible to derive the explicit solution to equations of such kind. Strong nonlinearity and coupling characteristics of stochastic differential equations are the major reason and, in this sense, it is of significant importance to establish an effective numerical method [6–15] that is able to make rational judgment by using computer simulations. In saying so, a new branch of computational mathematics is emerging as the numerical solutions to stochastic differential equations. The fundamental theories in this area can be found in [6, 15].

In recent years, the exact finite-difference schemes of ordinary differential equations have captured more and more attention from the academia. Such exact finite-difference schemes are the same to the solutions to the original equations, whilst being able to retain the characteristics of the solutions. Further, such exact finite-difference schemes can be utilized to rationalize the construction of nonstandardized finite-difference schemes [16], for example, the exponentially fitted method [17–19]. As is well known, the linear ordinary differential equations with constant coefficients involve an exact finite-difference scheme (see [16, 20, 21]) and some scholars have discussed exact discretizations of other systems, for example, Vigo-Aguiar and Ferràndiz [22], Cieśliński [23, 24], Sakamoto et al. [25], Mickens and Washington [26], Roeger et al. [27–29], and so on. However, there are only a few published papers [30–32] discussing whether an exact finite-difference scheme exists for widely applied linear stochastic differential equations with constant coefficients. In order to bridge the gap, this paper aims to calculate the exact finite-difference schemes of general 𝑑-dimensional linear stochastic differential equations with constant coefficients and hence apply the results to the Kubo oscillator so as to prove the structure-preserving property of the exact finite-difference schemes, which is a stochastic sample from [24].

The following part of the paper is organized as follows. In Section 2, the authors will discuss the exact finite-difference schemes for Itô and Stratonovich type stochastic differential equations. In Section 3, the authors demonstrate how to apply the findings generalized from Section 2 to the Kubo oscillator. Finally, the authors will use numerical examples to illustrate the validity of the findings.

2. The Exact Finite-Difference Schemes for Linear Stochastic Differential Equations with Constant Coefficients

Let(Ω,𝐹,𝑃)be a complete probability space with a filtration{𝐹_𝑡}. The filtration{𝐹_𝑡}is increasing and right continuous, and𝐹₀contains all𝑃-null sets. Let𝐵(𝑡)=(𝐵₁(𝑡),𝐵₂(𝑡),…,𝐵_𝑚(𝑡))^𝑇be a standard𝑚-dimensional Brownian motion defined on(Ω,𝐹,𝑃), whose incrementΔ𝐵_𝑘=𝐵_𝑘(𝑡+ℎ)−𝐵_𝑘(𝑡)(𝑘=1,2,…,𝑚) is a Gaussian random variable𝑁(0,ℎ).

Then we discuss numerical methods for strong solutions to stochastic differential equations: (𝑡)(𝑥(𝑡+𝑑𝑥=𝑓))𝑑𝑡_𝑚𝑘=1𝑔_𝑘(𝑥(𝑡))𝑑𝐵_𝑘(𝑡)𝑥(0),𝑡≥0,=𝑥₀∈𝑅^𝑑, (1) where the deterministic term𝑓(𝑥)is the drift coefficient and the stochastic terms𝑔_𝑘(𝑥)(𝑘=1,2,…,𝑚) are diffusion coefficients. The solution of (1) can be written as 𝑥(𝑡)=𝑥₀+^𝑡₀𝑓(𝑥(𝑠+))𝑑𝑠_{𝑚𝑘=1𝑡𝑡0}𝑔_𝑘(𝑥(𝑠))𝑑𝐵_𝑘(𝑠). (2) Given an equidistant discretization of the interval[0,𝑡]with grid points𝑡_𝑛(𝑛=1,2,…,𝑁) and letting𝜉_𝑛=𝜃𝑡_𝑛+(1−𝜃)𝑡_𝑛−1(𝜃∈[0,1]), the stochastic integrals in (2) can be calculated by the limit of the approximating sums in the mean square sense, as𝑁→∞, ^𝑁_𝑛=1𝑔_𝑘𝑥𝜉_𝑛𝐵_𝑘𝑡_𝑛−𝐵_𝑘𝑡_𝑛−1(𝑘=1,2,…,𝑚). (3) The values of stochastic integrals depend on the choice of𝜃and the Itô integrals when𝜃=0. The corresponding Itô type stochastic differential equations are the equations using the usual notation (1) and the Stratonovich integrals when𝜃=1/2. The corresponding Stratonovich type stochastic differential equations are denoted by 𝑑𝑥(𝑡)=𝑓₁(𝑥(𝑡+))𝑑𝑡_𝑚𝑘=1𝑔_𝑘(𝑥(𝑡))∘𝑑𝐵_𝑘(𝑡)𝑥(0),𝑡≥0,=𝑥₀∈𝑅^𝑑. (4) The relationships between these two types of stochastic differential equations are given by 𝑓₁(𝑥(𝑡(𝑥(𝑡−1))=𝑓))2^𝑚_𝑘=1𝑔^′_𝑘(𝑥(𝑡))𝑔_𝑘(𝑥(𝑡)). (5)

The exact finite-difference schemes of𝑑-dimensional linear stochastic differential equations with constant coefficients of Itô type can be discussed as 𝑑𝑥(𝑡)=𝐹𝑥(𝑡)𝑑𝑡+^𝑚_𝑘=1𝐺_𝑘𝑥(𝑡)𝑑𝐵_𝑘(𝑡)𝑥(0),𝑡≥0,=𝑥₀∈𝑅^𝑑, (6) where𝐹and𝐺_𝑘(𝑘=1,2,…,𝑚) are𝑑×𝑑matrices and𝐵_𝑘(𝑡)(𝑘=1,2,…,𝑚) are independent one-dimensional Brownian motion. Suppose that the matrices𝐹,𝐺₁,𝐺₂,…,𝐺_𝑚are commutative; that is, 𝐹𝐺_𝑘=𝐺_𝑘𝐹,𝐺_𝑘𝐺_𝑗=𝐺_𝑗𝐺_𝑘,𝑘,𝑗=1,2,…,𝑚. (7) Letℎbe the time step and𝑡_𝑛=𝑛ℎ.Δ𝐵^𝑘_𝑛=𝐵_𝑘(𝑡_𝑛+1)−𝐵_𝑘(𝑡_𝑛)(𝑘=1,2,…,𝑚) denotes the increment of Brownian motion. Then the following theorem can be derived.

Theorem 1. Equation (6) admits the exact finite-difference schemes given by 𝑥_𝑛+11=exp𝐹−2^𝑚_𝑘=1𝐺²_𝑘ℎ+^𝑚_𝑘=1𝐺_𝑘Δ𝐵^𝑘_𝑛𝑥_𝑛, (8) where𝑥_𝑛is the exact value of solution𝑥(𝑡)to (6) at𝑡=𝑡_𝑛.

Proof. Using the theories in [4, Chapter 3], the general solution of (6) can be expressed in terms of𝐹,𝐺₁,𝐺₂,…,𝐺_𝑚as follows: 𝑥(𝑡)1=exp𝐹−2^𝑚_𝑘=1𝐺²_𝑘𝑡+^𝑚_𝑘=1𝐺_𝑘𝐵_𝑘(𝑡)𝑥₀. (9) Inserting𝑡_𝑛and𝑡_𝑛+1into (9), it can be deduced that 𝑥_𝑛=𝑥𝑡_𝑛1=exp𝐹−2^𝑚_𝑘=1𝐺²_𝑘𝑡_𝑛+^𝑚_𝑘=1𝐺_𝑘𝐵_𝑘𝑡_𝑛𝑥₀,𝑥_𝑛+1=𝑥𝑡_𝑛+11=exp𝐹−2^𝑚_𝑘=1𝐺²_𝑘𝑡_𝑛+1+^𝑚_𝑘=1𝐺_𝑘𝐵_𝑘𝑡_𝑛+1𝑥₀1=exp𝐹−2^𝑚_𝑘=1𝐺²_𝑘ℎ+𝑡_𝑛+^𝑚_𝑘=1𝐺_𝑘Δ𝐵^𝑘_𝑛+𝐵_𝑘𝑡_𝑛𝑥₀1=exp𝐹−2^𝑚_𝑘=1𝐺²_𝑘ℎ+^𝑚_𝑘=1𝐺_𝑘Δ𝐵^𝑘_𝑛𝑥_𝑛, (10) which ends the proof.

Consider a𝑑-dimensional linear stochastic differential equation with constant coefficient of Stratonovich type driven by𝑚independent one-dimensional Brownian motion: 𝑑𝑦(𝑡)=𝐴𝑦(𝑡)𝑑𝑡+^𝑚_𝑘=1𝐶_𝑘𝑦(𝑡)∘𝑑𝐵_𝑘(𝑡)𝑦(0),𝑡≥0,=𝑦₀∈𝑅^𝑑, (11) where𝐴and𝐶_𝑘(𝑘=1,2,…,𝑚) are𝑑×𝑑matrices. It is assumed that the matrices𝐴,𝐶₁,𝐶₂,…,𝐶_𝑚are commutative; that is, 𝐴𝐶_𝑘=𝐶_𝑘𝐴,𝐶_𝑘𝐶_𝑗=𝐶_𝑗𝐶_𝑘,𝑘,𝑗=1,2,…,𝑚. (12) The exact finite-difference schemes can be constructed by using the general solution of (11).

Theorem 2. Equation (11) admits the exact finite-difference schemes given by 𝑦_𝑛+1=exp𝐴ℎ+^𝑚_𝑘=1𝐶_𝑘Δ𝐵^𝑘_𝑛𝑦_𝑛, (13) where𝑦_𝑛is the exact value of solution𝑦(𝑡)to (11) at𝑡=𝑡_𝑛.

Proof. Firstly, we prove that the general solution to (11) is given by 𝑦(𝑡)=exp𝐴𝑡+^𝑚_𝑘=1𝐶_𝑘𝐵_𝑘(𝑡)𝑦₀. (14) Equation (14) follows (9) by a known relation (5) between Itô and Stratonovich type stochastic differential equations. Taking into account that𝑦_𝑛=𝑦(𝑡_𝑛)and, in particular,𝑦(0)=𝑦₀, it can be obtained that 𝑦_𝑛=𝑦𝑡_𝑛=exp𝐴𝑡_𝑛+^𝑚_𝑘=1𝐶_𝑘𝐵_𝑘𝑡_𝑛𝑦₀,𝑦_𝑛+1=𝑦𝑡_𝑛+1=exp𝐴𝑡_𝑛+1+^𝑚_𝑘=1𝐶_𝑘𝐵_𝑘𝑡_𝑛+1𝑦₀=exp𝐴ℎ+𝑡_𝑛+^𝑚_𝑘=1𝐶_𝑘Δ𝐵^𝑘_𝑛+𝐵_𝑘𝑡_𝑛𝑦₀=exp𝐴ℎ+^𝑚_𝑘=1𝐶_𝑘Δ𝐵^𝑘_𝑛𝑦_𝑛. (15) This completes the proof of the theorem.

Remark 3. Although the existence of the exact solution of (6) and (11) is known, it is impossible to compute the exact value of the solution at a fixed time, because of the randomness of Brownian motion. The author of [33] has used a Matlab program to simulate the exact solution of one-dimensional linear stochastic differential equations. The codes in the program produce a discretized Brownian path𝑊_𝑖(𝑖=1,2,…,𝑁) with the help of the computer. The value of Brownian motion𝐵(𝑡)at𝑡=𝑡_𝑘is obtained by𝐵(𝑡_𝑘)=∑^𝑘−1_𝑙=1𝑊_𝑙. Then it is possible to derive the value of exact solution𝑥(𝑡)at𝑡=𝑡_𝑘. Such sum must be calculated in every step of the iteration, increasing the amount of computation needed. However, the scheme proposed in this paper can overcome such a problem.

3. An Application to the Kubo Oscillator

Consider the Kubo oscillator [34]: 𝑑𝑝(𝑡)=−𝑎𝑞(𝑡)𝑑𝑡−𝑏𝑞(𝑡)∘𝑑𝐵(𝑡),𝑝(0)=𝑝₀(𝑡)(𝑡)(𝑡)(𝑡)(0)∈𝑅,𝑑𝑞=𝑎𝑝𝑑𝑡+𝑏𝑝∘𝑑𝐵,𝑞=𝑞₀∈𝑅, (16) where𝑎and𝑏are constants and𝐵(𝑡)is a standard one-dimensional Brownian motion. The small circle “∘” before𝑑𝐵(𝑡)denotes stochastic differential equations of Stratonovich type. Using (14), the exact solution to (16) is illustrated as follows.

Theorem 4. The exact solution of (16) is given by 𝑝(𝑡)=𝑝₀cos(𝑎𝑡+𝑏𝐵(𝑡))−𝑞₀((𝑡,𝑞(𝑡)sin𝑎𝑡+𝑏𝐵))=𝑝₀sin(𝑎𝑡+𝑏𝐵(𝑡))+𝑞₀cos(𝑎𝑡+𝑏𝐵(𝑡)). (17) Further, the exact finite-difference scheme for (16) can be constructed by applying (17). Assumeℎis the time increment andΔ𝐵_𝑛=𝐵(𝑡_𝑛+1)−𝐵(𝑡_𝑛)is independent𝑁(0,ℎ)-distributed Gaussian random variables.𝑃_𝑛and𝑄_𝑛(𝑛=0,1,2,…) are the exact discrete values to𝑝(𝑡)and𝑞(𝑡)when𝑡=𝑡_𝑛=𝑛ℎ. The exact finite-difference scheme of (16) is given by 𝑃_𝑛+1=𝑝𝑡_𝑛+1=𝑝₀cos𝑎𝑡_𝑛+1+𝑏𝐵𝑡_𝑛+1−𝑞₀sin𝑎𝑡_𝑛+1+𝑏𝐵𝑡_𝑛+1=𝑝₀cos𝑎𝑡_𝑛+𝑏𝐵𝑡_𝑛+𝑎ℎ+𝑏Δ𝐵_𝑛−𝑞₀sin𝑎𝑡_𝑛+𝑏𝐵𝑡_𝑛+𝑎ℎ+𝑏Δ𝐵_𝑛=𝑝₀cos𝑎_𝑛cos𝑏_𝑛−𝑝₀sin𝑎_𝑛sin𝑏_𝑛−𝑞₀sin𝑎_𝑛cos𝑏_𝑛−𝑞₀cos𝑎_𝑛sin𝑏_𝑛=𝑝₀cos𝑎_𝑛−𝑞₀sin𝑎_𝑛cos𝑏_𝑛−𝑝₀sin𝑎_𝑛+𝑞₀cos𝑎_𝑛sin𝑏_𝑛=𝑃_𝑛cos𝑏_𝑛−𝑄_𝑛sin𝑏_𝑛,𝑄_𝑛+1=𝑞𝑡_𝑛+1=𝑝₀sin𝑎𝑡_𝑛+1+𝑏𝐵𝑡_𝑛+1+𝑞₀cos𝑎𝑡_𝑛+1+𝑏𝐵𝑡_𝑛+1=𝑝₀sin𝑎𝑡_𝑛+𝑏𝐵𝑡_𝑛+𝑎ℎ+𝑏Δ𝐵_𝑛+𝑞₀cos𝑎𝑡_𝑛+𝑏𝐵𝑡_𝑛+𝑎ℎ+𝑏Δ𝐵_𝑛=𝑝₀sin𝑎_𝑛cos𝑏_𝑛+𝑝₀cos𝑎_𝑛sin𝑏_𝑛+𝑞₀cos𝑎_𝑛cos𝑏_𝑛−𝑞₀sin𝑎_𝑛sin𝑏_𝑛=𝑝₀sin𝑎_𝑛+𝑞₀cos𝑎_𝑛cos𝑏_𝑛+𝑝₀cos𝑎_𝑛−𝑞₀sin𝑎_𝑛sin𝑏_𝑛=𝑄_𝑛cos𝑏_𝑛+𝑃_𝑛sin𝑏_𝑛, (18) where𝑎_𝑛=𝑎𝑡_𝑛+𝑏𝐵(𝑡_𝑛)and𝑏_𝑛=𝑎ℎ+𝑏Δ𝐵_𝑛.
Due to 𝑑𝛾𝑝(𝑡),𝑞(𝑡)=𝑑𝑝(𝑡)𝑑𝑡₂+𝑞(𝑡)²(𝑡)𝑝̇(𝑡)(𝑡)𝑞̇(𝑡)(𝑡)(𝑡)(𝑡)∘𝐵̇(𝑡)(𝑡(𝑡)(𝑡)∘𝐵̇(𝑡𝑑𝑡=2𝑝+2𝑞=2𝑝−𝑎𝑞−𝑏𝑞+2𝑞)𝑎𝑝+𝑏𝑝)=0, (19) then𝛾(𝑝,𝑞)=𝑝²+𝑞²is conservative along the phase flow of (16). That is,𝛾(𝑝,𝑞)is a first integral of (16), indicating that a phase trajectory of (16) is a circle with the center at(0,0)and with the radius𝐻(𝑝₀,𝑞₀). It can be proved that scheme (18) can preserve the first integral𝛾(𝑝,𝑞)exactly.

Theorem 5. Scheme (18) for solving (16) has the property𝛾(𝑃_𝑛+1,𝑄_𝑛+1)=𝛾(𝑃_𝑛,𝑄_𝑛)for any𝑛=0,1,2,….

Proof. Substitution of (18) into𝛾(𝑝,𝑞)yields 𝛾𝑃_𝑛+1,𝑄_𝑛+1=𝑃²_𝑛+1+𝑄²_𝑛+1=𝑃_𝑛cos𝑏_𝑛−𝑄_𝑛sin𝑏_𝑛²+𝑄_𝑛cos𝑏_𝑛+𝑃_𝑛sin𝑏_𝑛²=𝑃²_𝑛cos²𝑏_𝑛+𝑄²_𝑛sin²𝑏_𝑛+𝑄²_𝑛cos²𝑏_𝑛+𝑃²_𝑛sin²𝑏_𝑛=𝑃²_𝑛+𝑄²_𝑛. (20) This completes the proof.

It is obvious that (16) is a stochastic Hamiltonian system with𝐻(𝑝,𝑞)=𝑎(𝑝²+𝑞²)/2and𝐻₁(𝑝,𝑞)=𝑏(𝑝²+𝑞²)/2, and thus the phase flow of (16) preserves the symplectic structure𝑑𝑝(𝑡)∧𝑑𝑞(𝑡)=𝑑𝑝₀∧𝑑𝑞₀for all𝑡≥0. A good analytical and numerical study of stochastic Hamiltonian systems can be found in [34, 35]. The following theorem elicits whether scheme (18) is symplectic.

Theorem 6. Scheme (18) for solving (16) preserves the symplectic structure; that is,𝑑𝑃_𝑛+1∧𝑑𝑄_𝑛+1=𝑑𝑃_𝑛∧𝑑𝑄_𝑛.

Proof. Differentiating (18), it is known that 𝑑𝑃_𝑛+1=cos𝑏_𝑛𝑑𝑃_𝑛−sin𝑏_𝑛𝑑𝑄_𝑛,𝑑𝑄_𝑛+1=cos𝑏_𝑛𝑑𝑄_𝑛+sin𝑏_𝑛𝑑𝑃_𝑛. (21) From the exterior product, it can be derived as 𝑑𝑃_𝑛+1∧𝑑𝑄_𝑛+1=cos²𝑏_𝑛𝑑𝑃_𝑛∧𝑑𝑄_𝑛−sin²𝑏_𝑛𝑑𝑄_𝑛∧𝑑𝑃_𝑛=𝑑𝑃_𝑛∧𝑑𝑄_𝑛. (22) This completes the proof.

4. Numerical Experiments

A linear stochastic differential equation of Itô type can be written as (𝑡)(𝑡)(𝑡)(𝑡)[],𝑥(0)𝑑𝑥=2𝑥𝑑𝑡+𝑥𝑑𝐵,𝑡∈0,1=1. (23) The exact solution to (23) is given by 𝑥(𝑡)=exp(1.5𝑡+𝐵(𝑡)). (24) By (8), the exact finite-difference scheme of (23) is 𝑥_𝑛+1=exp1.5ℎ+Δ𝐵_𝑛𝑥_𝑛. (25) Figure 1 exhibits the exact solution of a sample phase trajectory of (23) simulated by (25).

Figure 1 

The exact solution of (23) simulated by (25) with fixed step sizeℎ=2−8.

Next, the exact finite-difference scheme (18) can be applied to solve the Kubo oscillator (16). The coefficients of (16) are chosen as𝑎=0.2,𝑏=0.01,𝑝₀=1,𝑞₀=0,ℎ=0.02, and𝑡∈[0,400]. Figure 2 exhibits the numerical solutions of a sample phase trajectory of (16) simulated by (18). Figure 3 shows that the numerical solutions created by (18) could preserve the first integral𝛾(𝑝,𝑞)of (16).

Figure 2 

Applying (18) to simulate (16) with fixed step sizeℎ=0.02.

Figure 3 

Preservation of first integral𝛾(𝑝,𝑞)by numerical solutions produced by (18).

5. Conclusions

In this paper, the authors extend the exact finite-difference schemes to linear stochastic differential equations with constant coefficients. The exact finite-difference schemes have been calculated for general𝑑-dimensional Itô and Stratonovich type stochastic differential equations. By using the exact finite-difference schemes to solve the Kubo oscillator, the authors have proven that the findings illustrated in this paper can preserve the symplectic structure and first integral. Numerical examples demonstrate the validity of the exact finite-difference schemes in this paper.

Acknowledgment

This work is supported by the National Science Council of the Republic of China, under Project No. NSC 71031003.