Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii-Type Conditions

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic di ﬀ erential equations with piecewise continuous arguments (cid:3) SEPCAs (cid:4) . The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic di ﬀ erential equations (cid:3) SDEs (cid:4) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.


Introduction
Stochastic modeling has come to play an important role in many branches of science and industry.Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance.Most stochastic differential equations are nonlinear and cannot be solved explicitly, but it is very important to research the existence and uniqueness of solution of stochastic differential equations.Many authors have studied the problem of SDEs.The classical existence-and-uniqueness theorem requires the coefficients f x t and g x t to satisfy the local Lipschitz condition and the linear growth condition see 1 .However, there are many SDEs that do not satisfy the linear growth condition, so more general conditions have been introduced to replace theirs.Khasminskii 2 has studied Khasminskii's test for SDEs which are the most powerful conditions.Similarly, the classical existence-and-uniqueness theorem for stochastic differential delay equations SDDEs requires the coefficients f x t , x t−τ and g x t , x t− τ to satisfy the local Lipschitz condition and the linear growth condition see 3-6 .Mao 7 has proved Khasminskii-type theorem, and this is a natural generalization of the classical Khasminskii test.
In recent years, differential equations with piecewise continuous arguments EPCAs had attracted much attention, and many useful conclusions were obtained.These systems have applications in certain biomedical models, control systems with feedback delay in the work of L. Cooke and J. Wiener 8 .The general theory and basic results for EPCAs have by now been thoroughly investigated in the book of Wiener 9 .A typical EPCA contains arguments that are constant on certain intervals.The solutions are determined by a finite set of initial data, rather than by an initial function, as in the case of general functional differential equation.A solution is defined as a continuous, sectionally smooth function that satisfies the equation within these intervals.Continuity of a solution at a point joining any two consecutive intervals leads to recursion relations for the solution at such points.Hence, EPCAs represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations.
However, up to now, there are few people who have considered the influence of noise to EPCAs.Actually, the environment, and accidental events may greatly influence the systems.Thus, analyzing SEPCAs is an interesting topic both in theory and applications.In this paper, we give the Khasminskii-type theorems for SEPCAs, which shows that SEPCAs have nonexplosion global solutions under local Lipschitz condition without the linear growth condition.
On the other hand, there is in general no explicit solution to an SEPCA, whence numerical solutions are required in practice.Numerical solutions to SDEs have been discussed under the local Lipschitz condition and the linear growth condition by many authors see 5 .Mao 10 gives the convergence in probability of numerical solutions to SDDEs under Khasminskii-Type conditions.Dai and Liu 11 give the mean-square stability of the numerical solutions of linear stochastic differential equations with piecewise continuous arguments.However, SEPCAs do not have the convergence results.The other main aim of this paper is to establish convergence of numerical solution for SEPCAs under the differential conditions.
The paper is organized as follows.In Section 2, we introduce necessary notations and Euler method.In Section 3, we obtain the existence and uniqueness of solution to stochastic differential equations with piecewise continuous arguments under Khasminskiitype conditions.Then the convergence in probability of numerical solutions to stochastic differential equations with piecewise continuous arguments under the same conditions is established.Finally, an example is provided to illustrate our theory.

Preliminary Notation and Euler Method
In this paper, unless otherwise specified, let |x| be the Euclidean norm in x ∈ R n .If A is a vector or matrix, its transpose is defined by A T .If A is a matrix, its trace norm is defined by |A| trace A T A .For simplicity, we also have to denote by a ∧ b min{a, b}, a ∨ b max{a, b}.
Let Ω, F, P be a complete probability space with a filtration {F t } t≥0 , satisfying the usual conditions.L 1 0, ∞ , R n and L 2 0, ∞ , R n denote the family of all real-valued F tadapted process f t t≥0 , such that for every T > 0, with initial data x 0 c 0 , where f : R n × R n → R n , g : R n × R n → R n×d , c 0 is a vector, and • denotes the greatest-integer function.By the definition of stochastic differential, this equation is equivalent to the following stochastic integral equation: Moreover, we also require the coefficients f x t , x t and g x t , x t to be sufficiently smooth.
To be precise, let us state the following conditions.
H1 The local Lipschitz condition For every integer i ≥ 1, there exists a positive constant L i such that H2 Linear growth condition There exists a positive constant K such that 4

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H3 There are a function V ∈ C 2 R n ; R and a positive constant α such that lim inf Let us first give the definition of the solution.
Definition 2.1 see 11 .An R n -valued stochastic process {x t } is called a solution of 2.3 on 0, ∞ if it has the following properties:

2.9
Let h 1/m be a given stepsize with integer m ≥ 1, and let the gridpoints t n be defined by t n nh n 0, 1, 2, . . . .For simplicity, we assume that T Nh.We consider the Euler-Maruyama method to 2.3 ,

Convergence in Probability of the Euler-Maruyama Method
In this section, we concentrate on 2.3 under the local Lipschitz condition H1 without the linear growth condition H2 to establish the generalized existence and uniqueness theorem for stochastic differential equations with piecewise continuous arguments.We then give the convergence in probability of the EM method to 2.3 under the local Lipschitz condition H1 and some additional conditions H3 .
In what follows, we will prove η ∞ ∞ almost surely and assertion 3.1 .By the Itô formula and condition 2.8 , we derive that dV x t LV x t , x t dt V x x t g x t , x t dB t ≤ α 1 V x t V x t dt V x x t g x t , x t dB t ,

3.4
We take the expectations in both sides of 3.4 , where It is easy to compute

3.7
Now the Gronwall inequality yields that So we have

3.9
Defining denoting I A as the indicator function of a set A, we compute

3.13
Now let us prove η ∞ > 2, for t 2 ∈ 1, 2 , and we can integrate both sides of 3.3 from 1 to η i ∧ t 2 and take the expectations

3.15
Now the Gronwall inequality yields that

3.16
Hence, we have 3.17 By 3.10 and 3.17 , we compute

3.19
From 3.16 and 3.19 , we yield

3.23
Therefore, we must have η ∞ ∞ almost surely as well as the required assertion 3.1 .The proof is completed.
Theorem 3.2.Under the conditions (H1) and (H3), if ε ∈ 0, 1 and T > 0, then there exists a sufficiently large integer i, dependent on ε and T such that

3.24
Proof.By Theorem 3.1, we have

3.25
Choose j large enough for j > T. From 3.25 , we get

3.26
It follows from 3.10 and 3.26 that 3.27 while by H3 , γ i → ∞ as i → ∞.Thus, there is a sufficiently large integer i such that

3.28
Therefore, we get that The proof is completed.
The following lemma shows that both y t and z t are close to each other.

3.33
Substituting 3.32 and 3.33 into 3.31 gives Let n t km l for t ∈ t km l , t km l 1 , then we have that

3.35
while by the Doob martingale inequality, we have

3.37
Thus, we obtain

3.38
where The proof is completed.
Lemma 3.4.Under the condition (H1), for any T > 0, there exists a positive constant C 2 i dependent on i and independent of h such that where Proof.It follows from 2.4 and 2.12 that

3.40
By the H ölder inequality, we obtain

3.41
This implies that for any 0

3.42
By Doob martingale inequality, it is not difficult to show that

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Note from H1 and Lemma 3.3 that

3.44
Similarly, we obtain that

3.46
By the Gronwall inequality, we must get where Journal of Applied Mathematics 13 Lemma 3.5.Under the conditions (H1) and (H3) if ε ∈ 0, 1 and T > 0, then there exists a sufficiently large integer i (dependent on ε and T ) and sufficiently small h such that

3.75
The following theorems describe the convergence in probability of the EM method to 2.3 under the local Lipschitz condition H1 and some additional conditions H3 .Theorem 3.6.Under the conditions (H1) and (H3), for arbitrarily small σ ∈ 0, 1 , for any T > 0.
Proof.For arbitrarily small σ, ε ∈ 0, 1 .We set 3.77 By Theorem 3.2 and Lemma 3.5, there exists a pair of i and h such that

3.79
By Lemma 3.4, we get

3.81
For all sufficiently small h, we obtain

3.82
From 3.79 and 3.82 , we see that for all sufficiently small h, P Ω ≤ ε, 3.83 which proves the theorem.
Of course, z t is computable but y t is not, so the following theorem is much more useful in practice.

3.85
In the same way as Theorem 3.6, we can see that

3.86
But by Lemma 3.3, we get 3.87 therefore,

3.88
For all sufficiently small h, we obtain

3.89
From 3.86 and 3.89 , we see that for all sufficiently small h, P Ω ≤ ε, 3.90 which proves the assertion 3.84 .

T
0 |f t | 2 dt < ∞ almost surely, respectively.For any a, b ∈ R with a < b, denote C a, b ; R n as the family of continuous functions φ from a, b to R n with the norm φ sup a≤θ≤b |φ θ |.
Proof.Applying the standard truncation technique to 2.3 , we obtain the unique maximal local solution x t exists on 0, η e under the local Lipschitz condition in a similar way as the proof of 10, Theorem 3.15, page 91 , where η e is the explosion time.For each integer i ≥ |c 0 |, define the stopping time ∈ n, n 1 , we can integrate both sides of 3.51 from n to t ∧ θ i and take the expectations