Well-Posedness and Exponential Estimates for the Solutions to Neutral Stochastic Functional Differential Equations with Infinite Delay

Hussein K. ASKER

doi:10.21078/JSSI-2020-434-13

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Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (5) : 434-446. DOI: 10.21078/JSSI-2020-434-13

Well-Posedness and Exponential Estimates for the Solutions to Neutral Stochastic Functional Differential Equations with Infinite Delay

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Abstract

In this work, neutral stochastic functional differential equations with infinite delay (NSFDEwID) have been addressed. By using the Euler-Maruyama scheme and a localization argument, the existence and uniqueness of solutions to NSFDEwID at the state space C_r under the local weak monotone condition, the weak coercivity condition and the global condition on the neutral term have been investigated. In addition, the $L^{2}$ and exponential estimates of NSFDEwID have been studied.

Key words

neutral stochastic functional differential equations / infinite delay / state space C_r / EulerMaruyama scheme

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Hussein K. ASKER. Well-Posedness and Exponential Estimates for the Solutions to Neutral Stochastic Functional Differential Equations with Infinite Delay. Journal of Systems Science and Information, 2020, 8(5): 434-446 https://doi.org/10.21078/JSSI-2020-434-13

1 Introduction

Recently, the neutral stochastic functional differential equations (NSFDEs) addressed by many authors. Among many other references, for infinite delay, we would like to mention^[1-4]. Whereas for finite delay^[5-11]. Many articles studied wellposedness of NSFDEs by imposing Lipschitz condition, for example, see [12-14]. However, it seemed to be frequently strong in the case of the real world. In the last few decades, there has been a growing interest in addressing the existence and uniqueness of stochastic functional differential systems under some weaker assumptions. For example, [15, 16] have studied the existence and uniqueness of solutions with infinite delay at phase space

B C ((- \infty; 0]; R^{d})

with norm

‖ φ ‖ = sup_{- \infty < θ \leq 0} | φ (θ) |

under non-Lipschitz condition. Bao and Hou^[17] have studied the existence and uniqueness of mild solutions to stochastic neutral partial functional differential equations under a non-Lipschitz condition and a weakened linear growth condition. Tan, et al.^[18] by the weak convergence approach have reviewed stability in distribution for NSFDEs. Von Renesse and Scheutzow^[19] have used the Euler-Maruyama approximation for SFDEs to show that only weak one-sided local Lipschitz conditions are sufficient for local existence and uniqueness of strong solutions.

Based on the approach presented above, the purpose of this paper is to investigate the existence and uniqueness under the local weak monotone condition, the weak coercivity condition and the global condition on a neutral term which obtained in the state space

C_{r}

The structure of this paper is as follows. Section 2, recapitulates some basic definitions and notations which has been used to develop our results. Section 3 gives several sufficient conditions to prove the existence and uniqueness for Equation (2.2) and the main result. In Section 4, we prove the

L^{2}

estimate and the exponential estimate of NSFDEwID.

2 Preliminary

Throughout this paper, unless otherwise specified, we use the following notation. Let

R^{d}

denote the usual

d

-dimensional Euclidean space and

| \cdot |

the Euclidean norm. If

A

is a vector or a matrix, its transpose is denoted by

A^{T}

; and

| A | = \sqrt{trace (A^{T} A)}

denotes its trace norm. Denote by

x^{T} y

the inner product of

x

and

y

R^{d}

. Let

C ((- \infty, 0]; R^{d})

denote the family of all continuous functions from

(- \infty, 0]

R^{d} .

We choose the state space with the fading memory defined as follows: For given positive number

r

C_{r} = {φ \in C ((- \infty, 0]; R^{d}) : ∥ φ ∥_{r} = sup_{- \infty < θ \leq 0} e^{r θ} ⏐ φ (θ) ⏐ < \infty} .

(2.1)

Then,

(C_{r}, ∥ \cdot ∥_{r})

is a Polish space. Let

(Ω, F, P)

be a complete probability space with a filtration

{F_{t}}_{t \in [0, + \infty)}

satisfying the usual conditions (i.e., it is right continuous and

F_{0}

contains all P-null sets). Let

1_{B}

denote the indicator function of a set

B

M^{2} ([a, b]; R^{d})

is a family of process

{x (t)}_{a \leq t \leq b}

L^{2} ([a, b]; R^{d})

such that

E \int_{a}^{b} | x (t) |^{2} d t < \infty

. Consider a

d

-dimensional NSFDEwID

d {x (t) - D (x_{t})} = b (x_{t}) d t + σ (x_{t}) d w (t), o n t \geq 0,

(2.2)

with the initial data:

x_{0} = ξ = {ξ (θ) : - \infty < θ \leq 0} \in C_{r},

(2.3)

where

x_{t} = x (t + θ) : - \infty < θ \leq 0,

and

b, D : C_{r} \to R^{d}

;

σ : C_{r} \to R^{d \times m}

are Borel measurable,

w (t)

is an

m

-dimensional Brownian motion. It should be pointed out that

x (t) \in R^{d}

is a point and a continuous adapted process

(x (t))_{t \geq 0}

is called a solution to (2.2) with the initial value

x_{0}

, if P-a.s.

\begin{array}{rcl} x (t) = ξ (0) + D (x_{t}) - D (ξ) + \int_{0}^{t} b (x_{s}) d s + \int_{0}^{t} σ (x_{s}) d w (s) a . s ., \end{array}

while

x_{t} \in C_{r}

is a continuous function on the interval

(- \infty, 0]

taking values in

R^{d}

and we call

(x_{t})_{t \geq 0}

a functional solution to (2.2) with the initial value

x_{0} = ξ \in C_{r}

3 Existence and Uniqueness of Solutions

Considering the NSFDEwID (2.2) and in order to investigate the existence and uniqueness of solutions to NSFDEwID, we impose the local weak monotone condition (A1) and the weak coercivity condition (A2) (the condition on the drift coefficient) by developing the tricks adapted in [13, 19-21]. Under weak local monotonicity and weak coercivity, the existence and uniqueness for path-independent stochastic differential equations are due to Krylov^[22], which extended in [[23], Chapter 3]. Under local weak monotone condition and weak coercive condition, [19] studied wellposedness of path-dependent SDEs with finite memory following Kerylov's approach. The Theorem 3.1 extends the result of [19] to NSFDEwID. We assume:

(A 1)

(local weak monotone condition) There exists a constant

L_{R} > 0

such that for any

ϕ, φ \in C_{r}

with

‖ φ ‖_{r} \lor ‖ ϕ ‖_{r} \leq R

\begin{aligned} 2 ⟨ φ (0) - ϕ (0) - (D (φ) - D (ϕ)), b (φ) - b (ϕ) ⟩ + ‖ σ (φ) - σ (ϕ) ‖_{r}^{2} \leq L_{R} ‖ φ - ϕ ‖_{r}^{2}, \end{aligned}

(3.1)

(A 2)

(Weak coercivity condition) There exist

L > 0

such that:

2 ⟨ φ (0) - D (φ), b (φ) ⟩ \lor | σ (φ) |^{2} \leq L (1 + ‖ φ ‖_{r}^{2}),

(3.2)

(A 3)

There is a

k \in (0, 1)

such that for all

ϕ, φ \in C_{r}

| D (ϕ) - D (φ) | \leq k ∥ ϕ - φ ‖_{r} and D (0) = 0.

(3.3)

Lemma 3.1 For all $φ \in C_{r}$ , $k \in (0, 1)$ ,

| φ (0) - D (φ) |^{2} \leq (1 + k)^{2} ‖ φ ‖_{r}^{2} .

(3.4)

Proof For any

ε > 0

, by the elementary inequality

| a + b |^{p} \leq [1 + ε^{\frac{1}{p - 1}}]^{p - 1} (| a |^{p} + \frac{| b |^{p}}{ε}),

we have:

| φ (0) - D (φ) |^{2} \leq (1 + ε) (| φ (0) |^{2} + \frac{1}{ε} | D (φ) |^{2}) .

Hence, by (A3):

| φ (0) - D (φ) |^{2} \leq (1 + ε) (| φ (0) |^{2} + \frac{k^{2}}{ε} ‖ φ ‖_{r}^{2}) .

Noting that, by letting

ε = k

, and

| φ (0) |^{2} \leq sup_{- \infty < θ \leq 0} e^{2 r θ} | φ (θ) |^{2} = ‖ φ ‖_{r}^{2},

the desired assertion follows.

Theorem 3.1 Under $(A 1)$ , $(A 2)$ and $(A 3)$ , the equation (2.2) admits a unique strong solution. Moreover, there exists constants $C_{2}, C_{3} > 0$ such that

E (sup_{0 < s \leq t \land α_{R}^{(n)}} e^{2 r s} ‖ x_{s}^{n} ‖_{r}^{2}) \leq C_{3} e^{C_{2} t},

where, $C_{2} = 2 C (k δ) C_{2} (L k) + C (k ε_{2})$ and $C_{3} = C (k δ) [(1 + 2 (1 + k)^{2}) E ‖ ξ ‖_{r}^{2} + \frac{73 L}{r} e^{2 r T}]$ .

Proof Throughout the whole proof, we assume that

n, m \geq \frac{r}{\ln 2}

are integers. Define the Euler-Maruyama scheme associated with (2.2) in the form

d [x^{n} (t) - D ({\hat{x}}_{t}^{n})] = b ({\hat{x}}_{t}^{n}) d t + σ ({\hat{x}}_{t}^{n}) d w (t), t > 0, x_{0}^{n} = {\hat{x}}_{0}^{n} = x_{0} = ξ,

(3.5)

where, for each fixed

t \geq 0

{\hat{x}}_{t}^{n} \in C_{r}

is defined in the manner below

{\hat{x}}_{t}^{n} (θ) := x^{n} ((t + θ) \land t_{n}), θ \in (- \infty, 0], t_{n} := \frac{⌊ n t ⌋}{n} .

(3.6)

We point out that all constants are independent of

n \geq 1

. The path-dependent NSFDE (3.5) has a unique solution by solving pice-wisely with the time step length

\frac{1}{n}

. For any

R > 3 ‖ ξ ‖_{r}^{2}

, define the stopping times

τ_{R}^{(n)} := inf {t \geq 0 : | x^{n} (t) | \geq \frac{R}{3}}, α_{R}^{(n)} := inf {t \geq 0 : ∥ x_{t}^{n} ∥_{r} \geq \frac{R}{3}} .

(3.7)

by [21], we have

τ_{R}^{(n)} = α_{R}^{(n)}

. Observe that

∥ x_{t}^{n} ∥_{r} = sup_{- \infty < θ \leq 0} (e^{r θ} | x^{n} (t + θ) |) \geq e^{r (t_{n} - t)} | x^{n} (t_{n}) |,

which, in addition to

n \geq \frac{r}{\ln 2}

, we have

\rme^{\frac{r}{n}} \leq 2

yields that

| x^{n} (t_{n}) | \leq e^{r (t - t_{n})} ‖ x_{t}^{n} ‖_{r} = e^{r (t - \frac{⌊ n t ⌋}{n})} ‖ x_{t}^{n} ‖_{r} = e^{r (\frac{n t - ⌊ n t ⌋}{n})} ‖ x_{t}^{n} ‖_{r} \leq e^{\frac{r}{n}} ‖ x_{t}^{n} ‖_{r} \leq 2 ‖ x_{t}^{n} ‖_{r} .

This, together with

\begin{aligned} ‖ {\hat{x}}_{t}^{n} ‖_{r} & = sup_{- \infty < θ \leq 0} (e^{r θ} {\hat{x}}_{t}^{n} (θ)) = sup_{- \infty < θ \leq 0} (e^{r θ} x^{n} ((t + θ) \land t_{n}) \\ = sup_{- \infty < θ \leq 0} (e^{r θ} x^{n} ((t + θ) \land t_{n}) 1_{{t + θ \leq t_{n}}} + sup_{- \infty < θ \leq 0} (e^{r θ} x^{n} ((t + θ) \land t_{n}) 1_{{t + θ \geq t_{n}}} \\ = sup_{- \infty < θ \leq 0} (e^{r θ} x^{n} (t + θ)) + sup_{- \infty < θ \leq 0} (e^{r θ} x^{n} (t_{n})) \\ \leq ‖ x_{t}^{n} ‖_{r} + | x^{n} (t_{n}) |, \end{aligned}

further leads to

‖ {\hat{x}}_{t}^{n} ‖_{r} \leq 3 ‖ x_{t}^{n} ‖_{r} .

(3.8)

Let

C (R) := sup_{‖ ζ ‖_{r} \leq \frac{R}{3}} | b (ζ) | < \infty . So, | b (x_{t}^{n}) | \leq C (R) R \in (3 ‖ ξ ‖_{r}^{2}, \infty), t \in [0, τ_{R}^{(n)}] .

Then, it follows from (3.8) and the notation of

τ_{R}^{(n)}

that

| b ({\hat{x}}_{t}^{n}) | \leq C (R), t \leq τ_{R}^{(n)} = α_{R}^{(n)} .

(3.9)

To show that

(x_{t}^{n})_{t \geq 0}

converges in probability to some stochastic process

(x_{t})_{t \geq 0}

n \to \infty

, for

n, m \geq \frac{r}{\ln 2}

, set

z^{n, m} (t) := x^{n} (t) - x^{m} (t)

D^{n, m} ({\hat{x}}_{t}) := D ({\hat{x}}_{t}^{n}) - D ({\hat{x}}_{t}^{m})

{\hat{z}}^{n, m} (t) := {\hat{x}}^{n} (t) - {\hat{x}}^{m} (t)

p_{t}^{n} := x_{t}^{n} - {\hat{x}}_{t}^{n}

and

Γ^{n, m} (t) := z^{n, m} (t) - D^{n, m} ({\hat{x}}_{t})

, by using the fact that

x_{t}^{n}

and

x_{t}^{m}

share the initial value with the elementary inequality and (A3), we note that for any

ε_{1} = \frac{k}{1 - k}

and

ε_{2} > \frac{k}{1 - k}

, we have

\begin{aligned} e^{2 r t} ‖ z_{t}^{n, m} ‖_{r}^{2} & = sup_{0 < s \leq t} (e^{2 r s} | z^{n, m} (s) |^{2}) \\ = sup_{0 < s \leq t} (e^{2 r s} | z^{n, m} (s) - D^{n, m} ({\hat{x}}_{s}) + D^{n, m} ({\hat{x}}_{s}) |^{2}) \\ \leq \frac{1}{1 - k} sup_{0 < s \leq t} (e^{2 r s} | z^{n, m} (s) - D^{n, m} ({\hat{x}}_{s}) |^{2}) + k e^{2 r t} ‖ {\hat{x}}_{t}^{n} - {\hat{x}}_{t}^{m} ‖_{r}^{2}, \\ \leq \frac{1}{1 - k} sup_{0 < s \leq t} (e^{2 r s} | Γ^{n, m} (s) |^{2}) + k (1 + ε_{2}) e^{2 r t} ‖ p_{t}^{n} - p_{t}^{m} ‖_{r}^{2} \\ + \frac{k (1 + ε_{2})}{ε_{2}} e^{2 r t} ‖ x_{t}^{n} - x_{t}^{m} ‖_{r}^{2}, \end{aligned}

note that

δ := \frac{k (1 + ε_{2})}{ε_{2}} < 1

, therefore

e^{2 r t} ‖ z_{t}^{n, m} ‖_{r}^{2} \leq \frac{1}{(1 - k) (1 - δ)} sup_{0 < s \leq t} (e^{2 r s} | Γ^{n, m} (s) |^{2}) + \frac{k (1 + ε_{2})}{1 - δ} e^{2 r t} ‖ p_{t}^{n} - p_{t}^{m} ‖_{r}^{2} .

(3.10)

Also, with the definition of

τ_{R}^{(n)}

and

α_{R}^{(n)}

, (3.8) implies that

‖ p_{t}^{n} ‖_{r} \leq \frac{4}{3} R, t \leq τ_{R}^{(n)} .

(3.11)

Now, for

t \leq τ_{R}^{(n)} \land τ_{R}^{(m)} = α_{R}^{(n)} \land α_{R}^{(m)}

, by the Itô formula, we derive from

(A 1)

, (3.9), (3.10) (3.11) and

(A 3)

that

\begin{array}{lll} d (e^{2 r t} | Γ^{n, m} (t) |^{2}) \\ = & e^{2 r t} {2 r | Γ^{n, m} (t) |^{2} + 2 ⟨ Γ^{n, m} (t), b ({\hat{x}}_{t}^{n}) - b ({\hat{x}}_{t}^{m}) ⟩ \\ + ‖ σ ({\hat{x}}_{t}^{n}) - σ ({\hat{x}}_{t}^{m}) ‖_{r}^{2}} d t + d M^{n, m} (t) \\ = & e^{2 r t} {2 r | Γ^{n, m} (t) |^{2} + 2 ⟨ z^{n, m} (t) - D^{n, m} ({\hat{x}}_{t}) + {\hat{z}}^{n, m} (t) - {\hat{z}}^{n, m} (t), b ({\hat{x}}_{t}^{n}) - b ({\hat{x}}_{t}^{m}) ⟩ \\ + ‖ σ ({\hat{x}}_{t}^{n}) - σ ({\hat{x}}_{t}^{m}) ‖_{r}^{2}} d t + d M^{n, m} (t) \\ = & e^{2 r t} {2 r | Γ^{n, m} (t) |^{2} + 2 ⟨ z^{n, m} (t) - {\hat{z}}^{n, m} (t), b ({\hat{x}}_{t}^{n}) - b ({\hat{x}}_{t}^{m}) ⟩ + 2 ⟨ {\hat{z}}^{n, m} (t) - D^{n, m} ({\hat{x}}_{t}), \\ b ({\hat{x}}_{t}^{n}) - b ({\hat{x}}_{t}^{m}) ⟩ + ‖ σ ({\hat{x}}_{t}^{n}) - σ ({\hat{x}}_{t}^{m}) ‖_{r}^{2}} d t + d M^{n, m} (t) \\ \leq & e^{2 r t} {2 r | Γ^{n, m} (t) |^{2} + 2 ⟨ p_{t}^{n} (0) - p_{t}^{m} (0), b ({\hat{x}}_{t}^{n}) - b ({\hat{x}}_{t}^{m}) ⟩ + L_{R} ‖ {\hat{x}}_{t}^{n} - {\hat{x}}_{t}^{m} ‖_{r}^{2}} d t \\ + d M^{n, m} (t) \\ \leq & e^{2 r t} {2 r | Γ^{n, m} (t) |^{2} + 2 | p_{t}^{n} (0) - p_{t}^{m} (0) | \cdot (| b ({\hat{x}}_{t}^{n}) | + | b ({\hat{x}}_{t}^{m}) |) \\ + 2 L_{R} ‖ p_{t}^{n} - p_{t}^{m} ‖_{r}^{2} + 2 L_{R} ‖ z_{t}^{n, m} ‖_{r}^{2}} d t + d M^{n, m} (t) \\ \leq & e^{2 r t} {(2 r + \frac{2 L_{R}}{(1 - k) (1 - δ)}) | Γ^{n, m} (t) |^{2} + 4 C (R) | p_{t}^{n} (0) | + 4 C (R) | p_{t}^{m} (0) | \\ + 2 L_{R} (1 + \frac{k (1 + ε_{2})}{1 - δ}) ‖ p_{t}^{n} - p_{t}^{m} ‖_{r}^{2}} d t + d M^{n, m} (t) \\ \leq & K {sup_{0 < s \leq t} e^{2 r s} | Γ^{n, m} (s) |^{2} + e^{2 r t} (‖ p_{t}^{n} ‖_{r} + ‖ p_{t}^{m} ‖_{r})} d t + d M^{n, m} (t), \end{array}

(3.12)

where

K \geq (2 r + \frac{2 L_{R}}{(1 - k) (1 - δ)}) + (4 C (R) + 4 L_{R} (1 + \frac{k (1 + ε_{2})}{1 - δ})) > 0

d M^{n, m} (t) = 2 e^{2 r t} ⟨ Γ^{n, m} (t), (σ ({\hat{x}}_{t}^{n}) - σ ({\hat{x}}_{t}^{m})) d w (t) ⟩

. Thus, for fixed

T > 0

, each

p \in (0, 1)

and

α > \frac{1 + p}{1 - p}

, in terms of stochastic Gronwall inequality [[19], Lemma 5.4], there exist constant

c > 0

depending on

p, α

such that

\begin{array}{rcl} E {(sup_{0 \leq t \leq T \land τ_{R}^{(n)} \land τ_{R}^{(m)}} e^{2 r t} | Γ^{n, m} (s) |^{2})}^{p} \\ \leq & c {(\int_{0}^{T} E (‖ p_{t}^{n} ‖_{r}^{α} 1_{{t \leq τ_{R}^{(n)}}}) d t)}^{\frac{p}{α}} + c {(\int_{0}^{T} E (‖ p_{t}^{m} ‖_{r}^{α} 1_{{t \leq τ_{R}^{(m)}}}) d t)}^{\frac{p}{α}} . \end{array}

(3.13)

Finally, to estimate

p_{t}^{n}

, for

θ \leq 0

, note that (3.5) implies

\begin{aligned} p_{t}^{n} (θ) & = x^{n} (t + θ) - {\hat{x}}^{n} (t + θ) = x^{n} (t + θ) - x^{n} ((t + θ) \land t_{n}) \\ = {\begin{aligned} 0, & for & t + θ \leq t_{n}, \\ D ({\hat{x}}_{t + θ}^{n}) - D ({\hat{x}}_{t_{n}}^{n}) + \int_{t_{n}}^{t + θ} b ({\hat{x}}_{s}^{n}) d s + \int_{t_{n}}^{t + θ} σ ({\hat{x}}_{s}^{n}) d w (s), & for & t + θ > t_{n}, \end{aligned} \end{aligned}

in view of

(A 3)

, we deduce that

\begin{aligned} ‖ p_{t}^{n} ‖_{r} & = sup_{- \infty < θ \leq 0} (e^{r θ} | p_{t}^{n} (θ) |) \\ = sup_{- \infty < θ \leq 0} (e^{r θ} | x^{n} (t + θ) - x^{n} ((t + θ) \land t_{n}) |) 1_{{t + θ \geq t_{n}}} \\ = e^{- r t} sup_{t_{n} < s \leq t} (e^{r s} | x^{n} (s) - x^{n} (s \land t_{n}) |) \\ \leq e^{- r t} sup_{t_{n} \leq s \leq t} (e^{r s} | D ({\hat{x}}_{s}^{n}) - D ({\hat{x}}_{s \land t_{n}}^{n}) | + e^{r s} \int_{t_{n}}^{s} | b ({\hat{x}}_{u}^{n}) | d u + e^{r s} | \int_{t_{n}}^{s} σ ({\hat{x}}_{u}^{n}) d w (u) |) \\ \leq k e^{- r t} sup_{t_{n} \leq s \leq t} (e^{r s} ‖ {\hat{x}}_{s}^{n} - {\hat{x}}_{s \land t_{n}}^{n} ‖_{r}) + \int_{t_{n}}^{t} | b ({\hat{x}}_{s}^{n}) | d s + sup_{t_{n} \leq s \leq t} | \int_{t_{n}}^{s} σ ({\hat{x}}_{u}^{n}) d w (u) | . \end{aligned}

(3.14)

Note that,

\begin{aligned} ‖ {\hat{x}}_{s}^{n} - {\hat{x}}_{s \land t_{n}}^{n} ‖_{r} & = sup_{- \infty \leq θ \leq 0} (e^{r θ} | {\hat{x}}^{n} (s + θ) - {\hat{x}}^{n} ((s + θ) \land t_{n}) |) 1_{{t_{n} \leq s + θ}} \\ \leq e^{- r s} sup_{t_{n} \leq u \leq s} (e^{r u} | {\hat{x}}^{n} (u) - {\hat{x}}^{n} (t_{n}) |) = 0, \end{aligned}

so that,

\begin{aligned} ‖ p_{t}^{n} ‖_{r} \leq \int_{t_{n}}^{t} | b ({\hat{x}}_{s}^{n}) | d s + sup_{t_{n} \leq s \leq t} | \int_{t_{n}}^{s} σ ({\hat{x}}_{u}^{n}) d w (u) | . \end{aligned}

(3.15)

By means of (3.9), for any

t \geq 0

and

t \leq τ_{R}^{(n)} = α_{R}^{(n)}

, we infer that

\begin{array}{rcl} lim_{n \to \infty} E \int_{t_{n}}^{t \land τ_{R}^{(n)}} | b ({\hat{x}}_{s}^{n}) | d s \leq lim_{n \to \infty} \frac{1}{n} C (R) = 0 . \end{array}

(3.16)

On the other hand, the Burkhold-Davis-Gundy inequality, together with the local boundedness of

σ

, there exists a constant

M_{R} > 0

such that, for any

t \geq 0

and

t \leq τ_{R}^{(n)} = α_{R}^{(n)}

\begin{array}{rcl} lim_{n \to \infty} E (sup_{t_{n} \leq s \leq t \land τ_{R}^{(n)}} | \int_{t_{n}}^{s} σ ({\hat{x}}_{s}^{n}) d w (s) |^{2}) \leq lim_{n \to \infty} \frac{M_{R}}{n} = 0, \end{array}

thus, from this with (3.15) and (3.16), we conclude that

sup_{t \in [0, T]} E (‖ p_{t}^{n} ‖_{r}^{α} 1_{{t \leq τ_{R}^{(n)}}}) \to 0 a s n \to \infty .

(3.17)

Subsequently, for

p = \frac{1}{2}

taking (3.13) and (3.10) into consideration implies that

lim_{n, m \to \infty} P {sup_{0 \leq t \leq T \land τ_{R}^{(n)} \land τ_{R}^{(m)}} ‖ x_{t}^{n} - x_{t}^{m} ‖_{r}} = 0.

(3.18)

Next, to ensure that

x^{n}

converges in probability to a solution of (2.2), it remains to prove

lim_{R \to \infty} \underset{n \to \infty}{lim sup} P (τ_{R}^{(n)} \leq T) = 0.

(3.19)

Indeed, (3.18) and (3.19) yield that

lim_{n, m \to \infty} P {sup_{0 \leq t \leq T} ‖ x_{t}^{n} - x_{t}^{m} ‖_{r} \geq ε} = 0, ε > 0,

whence, by keeping in mind that

C_{r}

is a Polish space under the metric

‖ \cdot ‖_{r}

(completeness of

(C_{r}, ‖ \cdot ‖_{r})

), there exists a continuous adapted process stochastic process

(x_{t})_{t \in [0, T]}

C_{r}

such that

sup_{0 \leq t \leq T} ‖ x_{t}^{n} - x_{t} ‖_{r} \to 0 in probability as n \to \infty .

(3.20)

Then, by following the standard argument we can show that

(x_{t})_{t \in [0, T]}

is the unique functional solution to (2.2) under

(A 1)

(A 2)

and

(A 3)

So, in what follows, it remains to show that (3.19) holds true. Set

Γ^{n} (t) = x^{n} (t) - D ({\hat{x}}_{t}^{n})

. Note, by the elementary inequality and

(A 3)

similar to (3.10) with

x_{0}^{n} = ξ

, for

ε > \frac{3 k^{2}}{1 - 3 k^{2}}

, that

\begin{aligned} e^{2 r t} ‖ x_{t}^{n} ‖_{r}^{2} & \leq ‖ ξ ‖_{r}^{2} + (1 + ε) sup_{0 < s \leq t} (e^{2 r s} | Γ^{n} (s) |^{2}) + \frac{3 k^{2} (1 + ε)}{ε} e^{2 r t} ‖ x_{t}^{n} ‖_{r}^{2}, \end{aligned}

set,

γ := \frac{3 k^{2} (1 + ε)}{ε} < 1

, thus

\begin{aligned} e^{2 r t} ‖ x_{t}^{n} ‖_{r}^{2} & \leq \frac{1}{1 - γ} ‖ ξ ‖_{r}^{2} + \frac{(1 + ε)}{1 - γ} sup_{0 < s \leq t} (e^{2 r s} | Γ^{n} (s) |^{2}) . \end{aligned}

(3.21)

By Itô's formula, we deduce from Lemma 3.1, (A3), (3.8),

(A 2)

, and (3.9) that

\begin{array}{rcl} e^{2 r t} | Γ^{n} (t) |^{2} \\ = & | ξ (0) - D (ξ) |^{2} + \int_{0}^{t} e^{2 r s} [2 r | Γ^{n} (s) |^{2} \\ + 2 ⟨ x^{n} (s) - D ({\hat{x}}_{s}^{n}), b ({\hat{x}}_{s}^{n}) ⟩ + ‖ σ ({\hat{x}}_{s}^{n}) ‖_{r}^{2}] d s + d N^{n} (t) \\ \leq & (1 + k)^{2} ‖ ξ ‖_{r}^{2} + \int_{0}^{t} e^{2 r s} [4 r | x^{n} (s) |^{2} + 4 r k^{2} ‖ {\hat{x}}_{s}^{n} ‖_{r}^{2} \\ + 2 ⟨ x^{n} (s) - D ({\hat{x}}_{s}^{n}) + {\hat{x}}^{n} (s) - {\hat{x}}^{n} (s) b ({\hat{x}}_{s}^{n}) ⟩ + ‖ σ ({\hat{x}}_{s}^{n}) ‖_{r}^{2}] d s + d N^{n} (t) \\ \leq & (1 + k)^{2} ‖ ξ ‖_{r}^{2} + \int_{0}^{t} e^{2 r s} [4 r ‖ x_{s}^{n} ‖_{r}^{2} + 12 r k^{2} ‖ x_{s}^{n} ‖_{r}^{2} + 2 ⟨ x^{n} (s) - {\hat{x}}^{n} (s), b ({\hat{x}}_{s}^{n}) ⟩ \\ + 2 ⟨ {\hat{x}}^{n} (s) - D ({\hat{x}}_{s}^{n}), b ({\hat{x}}_{s}^{n}) ⟩ + ‖ σ ({\hat{x}}_{s}^{n}) ‖_{r}^{2}] d s + d N^{n} (t) \\ \leq & (1 + k)^{2} ‖ ξ ‖_{r}^{2} + \int_{0}^{t} e^{2 r s} [4 r (1 + 3 k^{2}) ‖ x_{s}^{n} ‖_{r}^{2} + 2 | p_{s}^{n} (0) | | b ({\hat{x}}_{s}^{n}) | \\ + L (1 + ‖ {\hat{x}}_{s}^{n} ‖_{r}^{2})] d s + d N^{n} (t) \\ = & (1 + k)^{2} ‖ ξ ‖_{r}^{2} + \frac{L}{2 r} e^{2 r t} + C_{1} (L k) \int_{0}^{t} e^{2 r s} ‖ x_{s}^{n} ‖_{r}^{2} d s \\ + 2 C (R) \int_{0}^{t} e^{2 r s} | p_{s}^{n} (0) | d s + d N^{n} (t), \end{array}

(3.22)

where,

C_{1} (L k) = (4 r (1 + 3 k^{2}) + 3 L)

and

d N^{n} (t) = 2 \int_{0}^{t} e^{2 r s} ⟨ Γ^{n} (s), σ ({\hat{x}}_{s}^{n}) d w (s) ⟩

. Thus for any

t \in [0, T],

from (3.22) we have

\begin{array}{rcl} E (sup_{0 < s \leq t \land τ_{R}^{(n)}} e^{2 r s} | Γ^{n} (s) |^{2}) \\ \leq & (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + \frac{L}{2 r} e^{2 r T} \\ + C_{1} (L k) E \int_{0}^{t} e^{2 r s} ‖ x_{s}^{n} ‖_{r}^{2} d s + 2 C (R) \int_{0}^{t} e^{2 r s} E (| p_{s}^{n} (0) | 1_{{s \leq τ_{R}^{(n)}}}) d s \\ + 2 E (sup_{0 \leq s \leq t \land τ_{R}^{(n)}} \int_{0}^{s} e^{2 r u} ⟨ Γ^{n} (u), σ ({\hat{x}}_{u}^{n}) d w (u) ⟩) . \end{array}

(3.23)

Next, by the Burkholder-Davis-Gundy inequality, we infer that

\begin{array}{lll} 2 E (sup_{0 \leq s \leq t \land τ_{R}^{(n)}} \int_{0}^{s} e^{2 r u} [Γ^{n} (u)]^{T} σ ({\hat{x}}_{u}^{n}) d w (u)) \\ \leq & 8 \sqrt{2} E {(\int_{0}^{t \land τ_{R}^{(n)}} e^{2 r u} | [Γ^{n} (u)]^{T} σ ({\hat{x}}_{u}^{n}) |^{2} d u)}^{\frac{1}{2}} \\ \leq & 12 E {(\int_{0}^{t \land τ_{R}^{(n)}} e^{2 r u} | Γ^{n} (u) |^{2} | σ ({\hat{x}}_{u}^{n}) |^{2} d u)}^{\frac{1}{2}} \\ \leq & 12 E [{(sup_{0 < u \leq t \land τ_{R}^{(n)}} e^{2 r u} | Γ^{n} (u) |^{2})}^{\frac{1}{2}} {(\int_{0}^{t \land τ_{R}^{(n)}} e^{2 r u} | σ ({\hat{x}}_{u}^{n}) |^{2} d u)}^{\frac{1}{2}}] \\ \leq & \frac{1}{2} E (sup_{0 \leq u \leq t \land τ_{R}^{(n)}} e^{2 r u} | Γ^{n} (u) |^{2}) + 72 E \int_{0}^{t \land τ_{R}^{(n)}} e^{2 r u} | σ ({\hat{x}}_{u}^{n}) |^{2} d u \\ \leq & \frac{1}{2} E (sup_{0 < u \leq t \land τ_{R}^{(n)}} e^{2 r u} | Γ^{n} (u) |^{2}) + 72 L E \int_{0}^{t \land τ_{R}^{(n)}} e^{2 r u} (1 + ‖ {\hat{x}}_{u}^{n} ‖_{r}^{2}) d u \\ \leq & \frac{1}{2} E (sup_{0 < u \leq t \land τ_{R}^{(n)}} e^{2 r u} | Γ^{n} (u) |^{2}) + 72 L E \int_{0}^{t \land τ_{R}^{(n)}} e^{2 r u} (1 + 3 ‖ x_{u}^{n} ‖_{r}^{2}) d u \\ \leq & \frac{1}{2} E (sup_{0 < u \leq t \land τ_{R}^{(n)}} e^{2 r u} | Γ^{n} (u) |^{2}) + \frac{36 L}{r} e^{2 r T} + 216 L E \int_{0}^{t} e^{2 r u} ‖ x_{u}^{n} ‖_{r}^{2} d u . \end{array}

(3.24)

Plugging (3.24) into (3.23), we get that

\begin{array}{lll} E (sup_{0 < s \leq t \land τ_{R}^{(n)}} e^{2 r s} | Γ^{n} (s) |^{2}) \\ \leq & (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + \frac{73 L}{2 r} e^{2 r T} + C_{2} (L k) E \int_{0}^{t} e^{2 r s} ‖ x_{s}^{n} ‖_{r}^{2} d s \\ + 2 C (R) \int_{0}^{t} e^{2 r s} E (| p_{s}^{n} (0) | 1_{{s \leq τ_{R}^{(n)}}}) d s \frac{1}{2} E (sup_{0 < s \leq t \land τ_{R}^{(n)}} e^{2 r s} | Γ^{n} (s) |^{2}), \end{array}

(3.25)

where,

C_{2} (L k) = C_{1} (L k) + 216 L

. Consequently,

\begin{array}{lll} E (sup_{0 < s \leq t \land τ_{R}^{(n)}} e^{2 r s} | Γ^{n} (s) |^{2}) \leq & 2 (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + \frac{73 L}{r} e^{2 r T} \\ + 2 C_{2} (L k) E \int_{0}^{t} e^{2 r s} ‖ x_{s}^{n} ‖_{r}^{2} d s \\ + 4 C (R) \int_{0}^{t} e^{2 r s} E (| p_{s}^{n} (0) | 1_{{s \leq τ_{R}^{(n)}}}) d s . \end{array}

(3.26)

Considering (3.21), (3.26) gives that, for any

t \in [0, T]

\begin{array}{lll} E (sup_{0 < s \leq t \land τ_{R}^{(n)}} e^{2 r s} | Γ^{n} (s) |^{2}) \\ \leq & [(2 (1 + k)^{2} + \frac{2 C_{2} (L k) t}{1 - γ}) E ‖ ξ ‖_{r}^{2} + \frac{73 L}{r} e^{2 r T} \\ + 4 C (R) \int_{0}^{t} e^{2 r s} E (| p_{s}^{n} (0) | 1_{{s \leq τ_{R}^{(n)}}}) d s] + C_{2} E \int_{0}^{t} Γ^{n} (s) d s, \end{array}

(3.27)

So, the Gronwall inequality yiel

d

s that

Γ^{n, R} (t) \leq C_{1} e^{C_{2} T},

(3.28)

where,

C_{2} = \frac{2 C_{2} (L k) (1 + ε)}{1 - γ}

, and

\begin{aligned} C_{1} = [(2 (1 + k)^{2} + \frac{2 C_{2} (L k) t}{1 - γ}) E ‖ ξ ‖_{r}^{2} + \frac{73 L}{r} e^{2 r T} + 4 C (R) \int_{0}^{t} e^{2 r s} E (| p_{s}^{n} (0) | 1_{{s \leq τ_{R}^{(n)}}}) d s] . \end{aligned}

According to the notion of

τ_{R}^{(n)}

, the event

{τ_{R}^{(n)} \leq T, sup_{0 \leq t \leq T \land τ_{R}^{(n)}} | x^{n} (t) | < \frac{R}{4}}

is empty set, which, together with (3.17), (3.28) and Chebyshev's inequality yiel

d

s that

\begin{array}{lll} lim_{R \to \infty} lim_{n \to \infty} P (τ_{R}^{(n)} \leq T) & = lim_{R \to \infty} lim_{n \to \infty} P (τ_{R}^{(n)} \leq T, sup_{0 \leq t \leq τ_{R}^{(n)} \land T} | x^{n} (t) | \geq \frac{R}{4}) \\ \leq lim_{R \to \infty} lim_{n \to \infty} P (sup_{0 \leq t \leq τ_{R}^{(n)} \land T} | x^{n} (t) | \geq \frac{R}{4}) \\ \leq lim_{R \to \infty} lim_{n \to \infty} \frac{16 Γ^{n, R} (T)}{R^{2}} = 0. \end{array}

(3.29)

So, (3.19) hol

d

s. Finally, by (3.17), (3.18) and (3.28) and employing Fatous lemma for

n \to \infty

, we obtain there exists constants

C_{3}, C_{4} > 0

such that

E (sup_{0 < s \leq t \land α_{R}^{(n)}} e^{2 r s} ‖ x_{s}^{n} ‖_{r}^{2}) \leq C_{3} e^{C_{2} t},

where

C_{3} = C (k δ) [(1 + 2 (1 + k)^{2}) E ‖ ξ ‖_{r}^{2} + \frac{73 L}{r} e^{2 r T}]

and the proof complete.

4 Exponential Estimate

Let

x (t)

where

t \in [0, \infty)

, be a unique solution to NSFDEwID (2.2). In this section, first we derive the

L^{2}

estimate, then we obtain the exponential estimate for the solution.

Theorem 4.1 Under $(A 2)$ and $(A 3)$ , for $t \geq 0$ there exists constants $C_{4}, C_{5} > 0$ such that the solution $x (t)$ of (2.2), satisfies:

E [sup_{0 \leq s \leq t} | x (s) |^{2}] \leq C_{4} e^{C_{5} t},

(4.1)

where $C_{4} = \frac{1}{1 - 2 k^{2}} [(2 k^{2} e^{- 2 r t} + 4 (1 + k)^{2} + \frac{146 L}{r}) E ‖ ξ ‖_{r}^{2} + 292 L T]$ and $C_{5} = 292 L$ .

Proof By using the inequality

(a + b)^{2} \leq 2 a^{2} + 2 b^{2}

and

(A 3)

, we have

\begin{array}{lll} E (sup_{0 \leq s \leq t} | x (s) |^{2}) & \leq 2 E (sup_{0 < s \leq t} | D (x_{s}) |^{2}) + 2 E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) \\ \leq 2 k^{2} E ∥ x_{t} ∥_{r}^{2} + 2 E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) . \end{array}

(4.2)

Applying Itô formula to

| x (t) - D (x_{t}) |^{2}

, we have:

\begin{aligned} \begin{array}{lll} | x (t) - D (x_{t}) |^{2} = & | x (0) - D (ξ) |^{2} + \int_{0}^{t} [2 [x (s) - D (x_{s})]^{T} b (x_{s}) + | σ (x_{s}) |^{2}] d s \\ + 2 \int_{0}^{s} [x (s) - D (x_{s})]^{T} σ (x_{s}) d w (s), \end{array} \end{aligned}

(4.3)

taking expectation on both sides, we get

\begin{array}{lll} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) = & E | x (0) - D (ξ) |^{2} \\ + E (sup_{0 < s \leq t} \int_{0}^{s} [2 [x (u) - D (x_{u})]^{T} b (x_{u}) + | σ (x_{u}) |^{2}] d u) \\ + 2 E (sup_{0 < s \leq t} \int_{0}^{s} [x (u) - D (x_{u})]^{T} σ (x_{u}) d w (u)) . \end{array}

(4.4)

Note that

\begin{array}{lll} 2 E (sup_{0 < s \leq t} \int_{0}^{s} [x (u) - D (x_{u})]^{T} σ (x_{u}) d w (u)) \\ \leq & 8 \sqrt{2} E {(\int_{0}^{t} | [x (s) - D (x_{s})]^{2} | σ (x_{s}) |^{2} d s)}^{\frac{1}{2}} \\ \leq & 12 E {(\int_{0}^{t} | x (s) - D (x_{s}) |^{2} | σ (x_{s}) |^{2} d s)}^{\frac{1}{2}} \\ \leq & 12 E [{(sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2})}^{\frac{1}{2}} {(\int_{0}^{t} | σ (x_{s}) |^{2} d s)}^{\frac{1}{2}}] \\ \leq & \frac{1}{2} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) + 72 E \int_{0}^{t} | σ (x_{s}) |^{2} d s . \end{array}

(4.5)

Substitute (4.5) into (4.4) and using Lemma 3.1, yiel

d

\begin{array}{lll} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) \\ \leq & (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + E (sup_{0 < s \leq t} \int_{0}^{s} [2 [x (u) - D (x_{u})]^{T} b (x_{u}) + | σ (x_{u}) |^{2}] d u) \\ + \frac{1}{2} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) + 72 E \int_{0}^{t} | σ (x_{s}) |^{2} d s . \end{array}

(4.6)

Applying assumption

(A 2)

, one have

\begin{array}{lll} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) \\ \leq & (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + \frac{1}{2} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) + 73 L E \int_{0}^{t} (1 + ‖ x_{s} ‖_{r}^{2}) d s \\ = & (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + \frac{1}{2} E (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) + 73 L T + 73 L E \int_{0}^{t} ‖ x_{s} ‖_{r}^{2} d s, \end{array}

(4.7)

this implies to

\begin{aligned} E & (sup_{0 < s \leq t} | x (s) - D (x_{s}) |^{2}) \leq 2 (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + 146 L T + 146 L E \int_{0}^{t} ‖ x_{s} ‖_{r}^{2} d s . \end{aligned}

(4.8)

Substituting (4.8) in (4.2), we obtain,

\begin{aligned} E (sup_{0 \leq s \leq t} | x (s) |^{2}) & \leq 2 k^{2} E ∥ x_{t} ∥_{r}^{2} + 4 (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + 292 L T + 292 L E \int_{0}^{t} ‖ x_{s} ‖_{r}^{2} d s . \end{aligned}

(4.9)

Now, correspond to the definition of the norm

‖ \cdot ‖_{r}

, we have:

\begin{aligned} E ‖ x_{t} ‖_{r}^{2} & = E {(sup_{- \infty < θ \leq 0} e^{r θ} | x_{t} (θ) |)}^{2} \leq e^{- 2 r t} E ‖ ξ ‖_{r}^{2} + E (sup_{0 < s \leq t} | x (s) |^{2}) . \end{aligned}

(4.10)

By substituting this into (4.9), we get

\begin{array}{lll} E (sup_{0 \leq s \leq t} | x (s) |^{2}) \leq & 2 k^{2} E ‖ ξ ‖_{r}^{2} + 4 (1 + k)^{2} E ‖ ξ ‖_{r}^{2} + 292 L T \\ + 292 L E \int_{0}^{t} [e^{- 2 r s} E ‖ ξ ‖_{r}^{2} + E (| x (s) |^{2})] d s, \end{array}

(4.11)

this yiel

d

s to,

\begin{aligned} E (| x (t) |^{2}) & \leq [(2 k^{2} + 4 (1 + k)^{2} + \frac{292 L}{2 r}) E ‖ ξ ‖_{r}^{2} + 292 L T] + 292 L E \int_{0}^{t} E (| x (s) |^{2}) d s, \end{aligned}

(4.12)

so, by the Gronwall iniquity we get,

\begin{aligned} E (sup_{0 \leq s \leq t} | x (s) |^{2}) & \leq C_{4} e^{C_{5} t}, \end{aligned}

(4.13)

where,

C_{4} = [(2 k^{2} + 4 (1 + k)^{2} + \frac{292 L}{2 r}) E ‖ ξ ‖_{r}^{2} + 292 L T]

and

C_{5} = 292 L

The proof is complete.

Theorem 4.2 Let assumption $(A 2)$ and $(A 3)$ hold. Then

lim_{t \to \infty} sup \frac{1}{t} \log | x (t) | \leq \hat{C},

(4.14)

where $\hat{C} = \frac{292 L}{2}$ .

Proof From the Theorem 4.1, for each

m = 1, 2, \dots

we get

E [sup_{m - 1 \leq t \leq m} x (s)] \leq C_{4} e^{292 L m},

where

C_{4} = [(2 k^{2} + 4 (1 + k)^{2} + \frac{292 L}{2 r}) E ‖ ξ ‖_{r}^{2} + 292 L T]

. By Chebyshev inequality we further obtain that

P {w : sup_{m - 1 \leq t \leq m} | x (s) |^{2} > e^{(292 L + ϵ) m}} \leq C_{4} e^{- ϵ m} .

(4.15)

As the series

\sum_{m = 1}^{\infty} C_{4} e^{- ϵ m}

is convergent, for almost all

w \in Ω

, the Borel-Cantelli lemma yiel

d

s that there exists a random integer

m_{0} = m_{0} (w)

such that

sup_{m - 1 \leq t \leq m} | x (t) |^{2} \leq e^{(292 L + ϵ) m}, w h e n e v e r m \geq m_{0},

That is, for

m - 1 \leq t \leq m

and

m \geq m_{0}

, we derive

| x (t) | \leq e^{\frac{1}{2} (292 L + ϵ) m} .

Hence for almost all

w \in Ω

m - 1 \leq t \leq m

and

m \geq m_{0}

, then

lim_{t \to \infty} sup \frac{1}{t} \log | x (t) | \leq \frac{1}{2} (292 L + ϵ) m,

the required claim (4.14) follows because

ϵ > 0

is arbitrary.

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Acknowledgement

The author would like to thank Dr Jianhai Bao for his encouragement and kindly advice throughout this work, as well as Professor Chenggui Yuan for his supervision and remarks. This research was supported by Kufa University and the Iraqi Ministry of Higher E $d$ ucation and Scientific Research.

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References

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Footnotes

Acknowledgement

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