A NOTE ON THE APPROXIMATE SOLUTIONS TO STOCHASTIC DIFFERENTIAL DELAY EQUATION

Abstract

The main aim of this paper is to discuss the difference between the Euler-Maruyama's approximate solutions and the accurate solution to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and weakened linear growth condition. Furthermore, we give the pth moment continuous of the approximate solution for the delay equation.

1. Introduction

In the study of stochastic system, a more realistic model would include some of the past states of the system. Stochastic functional differential equation gives a mathematical formulation for such system. In addition, in the study of the stochastic differential delay equations, If there is not any explicit solution then how we can obtain the approximate solution is a very important matter. One of the special but important class of stochastic functional differential equations is the stochastic differential delay equations. In 2016, Kim [5] considered the following stochastic differential delay equation

on t ∈ [t0, T] and defined the Euler-Maruyama approximation to the delay equation (1) as follows: For each integer n ≥ 1/τ, define xn(t) on [−τ, T] by

and

for t0 + k/n < t ≤ [t0 + (k + 1)/n] ∧ T, k = 0, 1, 2, · · · .

In [5], by employing non-uniform Lipschitz condition and weakened linear growth condition, Kim established the following results for the second moment to stochastic differential delay equation. The following theorem shows that the Euler-Maruyama sequence (2) converges to the unique solution of the equation (1) and gives an estimate for difference between the approximate solution xn(t) and the accurate solution x(t).

Theorem 1.1 ([5]). Assume that there exists a constant K and a concave function κ such that

(i) (non-uniform Lipschitz condition) For all t ∈ [t0, T], and all

where κ(·) is a concave nondecreasing function from ℝ+ to ℝ+ such that κ(0) = 0, κ(u) > 0 for u > 0 and ∫0+ du/κ(u) = ∞.

(ii) (weakened linear growth condition) there is a K > 0 such that for all (x, y, t) ∈ Rd × Rd × [t0, T],

Also, assume that δ(·) is Lipschitz continuous, that is there is a positive constant α such that

if t0 ≤ s < t ≤ T. Then, for every n > 1 + α, the difference between the Euler-Maruyama approximate solution xn(t) defined by (2) and the accurate solution x(t) of equation (1) can be estimate as

where γ = (T − t0)(T − t0 + 4),

and C3 is defined in [5].

For results related to the stochastic differential equation, see [1]-[12], and references therein for details. By using the non-uniform Lipschitz condition and weakened growth condition, Kim [5] studied the difference between the approximate and the accurate solution to stochastic differential delay equation (SDDEs). Motivated by the results, we established some exponential estimate for the pth moment and estimated on difference between the approximate solutions and the unique solution to stochastic differential delay equation that can be obtained from the conditions. When we try to carry over this procedure to the this delay equation, we used the Euler-Maruyama sequence approximation procedure.

 

2. Preliminary

Assume that B(t) is an m-dimensional Brownian motion defined on complete probability space (Ω, F, P) with a filtration {Ft}t≥t0 satisfying the usual conditions (i.e. it is right continuous and Ft0 contains all P-null sets), where B(t) = (B1(t), B2(t), ..., Bm(t))T. And let | · | denote Euclidean norm in Rn. If A is a vector or a matrix, its transpose is denoted by AT ; if A is a matrix, its trace norm is represented by

Also, let C([−τ, 0];Rd) denote the family of continuous Rd-valued functions φ defined on [−τ, 0] with norm ∥φ∥ = sup−τ≤θ≤0 |φ|.

In the result [9], they considered the following non-Lipschitz condition and non-linear growth condition:

(iii) (Non-Lipschitz condition) For any φ, ψ ∈ BC((−∞, 0];Rd) and t ∈ [t0, T], it follows that

where κ(·) is a concave nondecreasing function from ℝ+ to ℝ+ such that κ(0) = 0, κ(u) > 0 for u > 0 and ∫0+ du/κ(u) = ∞.

(iv) (Non-linear growth condition) f(0, t), g(0, t) ∈ L2 and for all t ∈ [t0, T], it follows that

where K > 0 is a constant. Moreover, the authors established the following results for d-dimensional stochastic functional differential equation.

Theorem 2.1 ([9]). Assume that the non-Lipschitz condition and non-linear growth condition hold. Then, there exists a unique solution to the equation

with initial data.

For more results related to some stochastic differential delay equation, see [2], [3], [6] - [12], and references therein for details.

On the other hand, we consider a special class of stochastic functional differential delay equation

on t ∈ [t0, T], where F : Rd×Rd×[t0, T] → Rd and G : Rd×Rd×[t0, T] → Rd×m are Borel measurable. If we define

for (φ, t) ∈ C([−τ, 0];Rd) × [t0, T], then equation (4) can be written as the equation (3). So we can apply the existence-and-uniqueness theorem established in the previous theorem to the delay equation (4).

Let us now prepare a few lemmas in order to show the main result.

Lemma 2.2 (Moment inequality, [7]). If p ≥ 2, g ∈ M2([0, T];Rd×m) such that then

In particular, ds when p = 2.

Lemma 2.3 (Moment inequality, [7]). Under the same assumptions as Lemma 2.2, we have

 

3. Approximate solutions

Let us begin with the discussion of the following stochastic differential delay equation

on t ∈ [t0, T], where F : Rd×Rd×[t0, T] → Rd and G : Rd×Rd×[t0, T] → Rd×m are Borel measurable. Moreover, the initial value is followed:

BC([−τ, 0];Rd) − value random variable such that ξ ∈ M2([−τ, 0];Rd). Moreover, we impose the non-uniform Lipschitz condition and weakened linear growth condition:

(v) (Non-uniform Lipschitz condition) For all t ∈ [t0, T], and all

where κ(·) is a concave nondecreasing function from ℝ+ to ℝ+ such that κ(0) = 0, κ(u) > 0 for u > 0 and ∫0+ du/κ(u) = ∞.

(vi) (Weakened linear growth condition) There is a K > 0 such that for all (x, y, t) ∈ Rd × Rd × [t0, T],

Let us now turn to the Euler-Maruyama approximation procedure. Consider the stochastic differential delay equation (5) with initial data (6). It is in this spirit we define the Euler-Maruyama approximation procedure as follows: For each integer n ≥ 1/τ, define xn(t) on [t0 − τ, T] by

and

for t0 + k/n < t ≤ [t0 + (k + 1)/n] ∧ T, k = 0, 1, 2, · · · . Moreover, if we define

for t0 + k/n < t ≤ [t0 + (k + 1)/n] ∧ T, k = 0, 1, 2, · · · , it then follows from (9) that

From now on, xn(t) means the Euler-Maruyama approximation (9). The following lemma shows that the Euler-Maruyama approximation sequence is bounded in Lp.

Lemma 3.1. Let (7) and (8) hold and p ≥ 2. Then, for all n ≥ 1/τ, we have

for all t ≥ t0, where C1 = 6p−1(2(p−2)/2αp/2 + Kp/2) and C2 = (T − t0)p + [(p3/2(p − 1))p/2](T − t0)p/2.

Proof. Fix n ≥ 1 arbitrarily. It is easy to see from the equation (10) that

for t0 ≤ t ≤ T. By Hölder’s inequality and Lemma 2.3, it is easy to see from (12) that for t0 ≤ t ≤ T,

By the condition (7) and (8), we obtain

Given that κ(·) is concave and κ(0) = 0, we can find a positive constant α such that κ(u) ≤ α(1+u) for all u ≥ 0 and recalling the definition of we then see that

where C1 = 6p−1(2(p−2)/2αp/2 + Kp/2) and C2 = (T − t0)p + [(p3/2(p − 1))p/2](T − t0)p/2. Consequently

An application of the Gronwall inequality implies that

and the desired inequality follows immediately. Thus the proof is complete. □

As an application of Lemma 3.1 we show the continuity of the p-th moment of the Euler-Maruyama’s approximate solution.

Theorem 3.2. Let (7) and (8) hold and p ≥ 2. Then, for any t0 ≤ s < t ≤ T with t − s < 1, we have

where Ck is defined in Lemma 3.1 and C3 = 1 + (p(p − 1)/2)p/2(t − s)−p/2.

Proof. It is easy to see from the equation (10) that

By Hölder’s inequality and Lemma 2.2, it is easy to note that for t0 ≤ t ≤ T,

By the condition (7) and (8), we obtain

where C3 = 1 + (p(p − 1)/2)p/2(t − s)-p/2.

Given that κ(·) is concave and κ(0) = 0, we can find a positive constant α such that κ(u) ≤ α(1 + u) for all u ≥ 0. Therefore

Hence, by Lemma 3.1,

and the desired inequality follows immediately. Thus the proof is complete. □

Moreover, under non-uniform Lipschitz condition (7) and weakened linear growth condition (8), we are still able to show that the solution of the delay equation (5) is bounded in Lp, that is, the pth moment of the solution satisfies

In view of Theorem 3.2, we could know that the continuity of the pth moment of the solution of equation (5) satisfies

This means that the pth moment of the solution is continuous. But the details are left to the reader.

The following theorem shows that the Euler-Maruyama approximate solution of the equation (9) gives an estimate for the difference between the approximate solution xn(t) and the accurate solution x(t).

Theorem 3.3. Let (7) and (8) hold and p ≥ 2. Assume that the initial data ξ = {ξ(θ) : −τ ≤ θ ≤ 0} is uniformly Lipschitz Lp-continuous, that is, there is a positive constant β such that

if −τ ≤ θ1 < θ2 ≤ 0. Then, the difference between the Euler-Maruyama approximate solution xn(t) and the accurate solution x(t) of equation (5) can be estimate as

where C2 = (T − t0)p + [(p3/2(p − 1))p/2](T − t0)p/2,

Proof. By Hölder’s inequality, we can derive that

By Lemma 2.3, the condition (7) and (8), we then see that

Given that κ(·) is concave and κ(0) = 0, we can find a positive constant α such that κ(u) ≤ α(1 + u) for all u ≥ 0. Therefore

Define and it then follows from (17) that

where

An application of the Gronwall inequality implies that

We now estimate J1 and J2. By the condition (15), we can estimate

Also, by the condition (15) and (16), we can estimate

It is easy to show that

if −τ ≤ s < t ≤ τ, t − s ≤ 1.

Substituting (19) and (20) into (18) yields that

Thus the proof is complete. □

In the case when both functions F and G are independent of t, the Euler-Maruyama approximate solutions can be defined by a simpler form, that is (5) can be replaced by

and

for t0 + k/n < t ≤ [t0 + (k + 1)/n] ∧ T, k = 0, 1, 2, · · · .

Let us second discuss the Euler-Maruyama approximation procedure. Consider the following stochastic differential delay equation

on t ∈ [t0, T] with initial data, where δ : [t0, T] → [0, τ ], F : Rd × Rd × [t0, T] → Rd and G : Rd × Rd × [t0, T] → Rd×m are Borel measurable. In the case when the time delay function δ(t) is Lipschitz continuous, the Euler-Maruyama approximate sequence of the equation (22) can be definde as follows: For each integer n ≥ 1, define yn(t) on [t0 − τ, T] by

and

for t0 + k/n < t ≤ [t0 + (k + 1)/n] ∧ T, k = 0, 1, 2, · · · .

Moreover, under non-uniform Lipschitz condition (7) and weakened linear growth condition (8), we are still able to show that the Euler-Maruyama approximation sequence (23) is bounded in L2.

From now on, yn(t) means the Euler-Maruyama approximation (23). The following lemma shows that the Euler-Maruyama approximation sequence is bounded in Lp.

Lemma 3.4. Let (7) and (8) hold and p ≥ 2. Then, for all n ≥ 1/τ, we have

for all t ≥ t0, where Ck is defined in Lemma 3.1.

Proof. The proof is similar to the proof of Lemma 3.1, but the details are left to the reader. □

As an application of Lemma 3.4 we show the continuity of the p-th moment of the Euler-Maruyama’s approximate sequence.

Theorem 3.5. Let (7) and (8) hold and p ≥ 2. Then, for any t0 ≤ s < t ≤ T with t − s < 1, we have

where Ck is defined in Lemma 3.1 and C3 = 1 + (p(p − 1)/2)p/2(t − s)−p/2.

Proof. The proof is similar to the proof of Theorem 3.2, but the details are left to the reader. □

Moreover, under non-uniform Lipschitz condition (7) and weakened linear growth condition (8), we are still able to show that the solution of the delay equation (22) is bounded in Lp, that is, the pth moment of the solution satisfies

In view of Theorem 3.5, we could know that the continuity of the pth moment of the solution of equation (22) satisfies

This means that the pth moment of the solution is continuous.

The following theorem estimates the difference between Euler-Maruyama approximate sequence and the accurate solution of equation (22).

Theorem 3.6. In addition to the assumptions of Theorem 3.3. Then the difference between the Euler-Maruyama approximate solution yn(t) defined by (23) and the accurate solution y(t) of equation (22) can be estimate as

where C2 = (T − t0)p + [(p3/2(p − 1))p/2](T − t0)p/2,

Proof. This theorem can be proved in the same way as in the proof of Theorem 3.3 with a little bit careful consideration on the estimation of the integral. If we define by Hölder’s inequality, we can derive that

By Lemma 2.3, the condition (7) and (8), we then see that

Given that κ(·) is concave and κ(0) = 0, we can find a positive constant α such that κ(u) ≤ α(1 + u) for all u ≥ 0. Therefore

Define and It then follows from (27) that

where

An application of the Gronwall inequality implies that

We now estimate M1 and M2. By the condition (26), we can estimate

Also, by the condition (15) and (26), we can estimate

Substituting (29) and (30) into (28) yields that

Thus the proof is complete. □