Some results of solutions to neutral stochastic functional operator-differential equations

Ling Hu; Zheng Wu; Lianglong Wang

doi:10.1515/nleng-2025-0111

Article Open Access

Some results of solutions to neutral stochastic functional operator-differential equations

Ling Hu , Zheng Wu and Lianglong Wang

Published/Copyright: April 17, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

In this article, we shall consider the solution to neutral stochastic functional operator differential equations with finite delay r > 0 : d D ( t , x t ) = f ( t , x t ) d t + g ( t , x t ) d B ( t ) , t 0 ≤ t ≤ T . By using Banach fixed point theorem, we obtain the existence of the solutions. We also obtain some properties of the solution.

Keywords: operator functional differential equations; atomic

1 Introduction

To the best of our knowledge, there exist extensive literature on the existence of solutions and qualitative properties for deterministic partial neutral functional differential equations. But in natural world, stochastic phenomenon is everywhere. In models of biological, chemical, physical, and economical systems, there are many stochastic facts. So it is very important and necessary to study stochastic functional differential equations (SFDEs).

Recently, existence, uniqueness, stability, invariant measures, and other quantitative and qualitative properties of solutions to stochastic functional differential equations (FDEs) have been extensively investigated by many authors. Some of these topics have been solved by the semigroup approach [1–4] and others have been solved by the variational one [5,6]. Although SFDEs with finite delays also seem very important as stochastic models of biological, chemical, physical, and economical systems, the corresponding properties of these systems have not been studied in great detail [7,8]. As a matter of fact, there are a lot of studies on the related topics for deterministic functional differential equations with finite delays [9]. There are also some similar topics for stochastic ordinary functional differential equations with finite delays that have already been investigated by various authors [10,11].

Mao [12] discussed the neutral stochastic functional differential equations (NSFDE) with finite delay as follows:

(1.1) d [ x ( t ) − D ( x t ) ] = f ( t , x t ) d t + g ( t , x t ) d B ( t ) , t 0 ≤ t ≤ T , x t 0 = ξ ∈ L ℱ t 0 2 ( Ω , C ( [ − r , 0 ] ; R n ) ) , r ≥ 0 .

By using the well-known Picard iterative method, the existence, uniqueness, and asymptotic behavior of solutions were obtained.

In this study, we shall discuss a class of NSFDEs with finite delays,

(1.2) d D ( t , x t ) = f ( t , x t ) d t + g ( t , x t ) d B ( t ) , t 0 ≤ t ≤ T , x t 0 = φ ∈ L ℱ t 0 2 ( Ω , C ( [ − r , 0 ] ; R n ) ) , r ≥ 0 ,

where

D : [ t 0 , T ] × C ( [ − r , 0 ] ; R n ) → R n , f : [ t 0 , T ] × C ( [ − r , 0 ] ; R n ) → R n , g : [ t 0 , T ] × C ( [ − r , 0 ] ; R n ) → R n × m .

As usual, we are still working on the given complete probability space ( Ω , ℱ , P ) with the filtration { ℱ t } t ≥ 0 satisfying the usual conditions, and B ( t ) is the given m-dimensional Brown motion defined on the space. This class of NSFDEs are more general. Let x ( t ) − D ( x t ) = U ( t , x t ) the equation in the study by Mao [12] is given as Eq. (1.2).

The contents of this study are organized as follows. First, we shall give some preliminary results which are fundamental for the subsequent developments in Section 2. Then, in Section 3 we shall investigate the existence of local solutions to the above equations. We also discuss the continuation and continuous dependence of solutions in Sections 4 and 5. Finally, in Section 6, we give the stability of solutions.

2 Materials and methods

Definition 2.1

An R n -valued stochastic process x ( t ) on t 0 − r ≤ t ≤ T is called a solution to Eq. (1.2) if it has the following properties:

it is continuous and { x t } t 0 ≤ t ≤ T is ℱ t -adapted;
{ f ( t , x t ) } ∈ ℒ 1 ( [ t 0 , T ] , R n ) and { g ( t , x t ) } ∈ ℒ 2 ( [ t 0 , T ] , R n × m ) ;
x t 0 = φ and (2.1) holds for every t 0 ≤ t ≤ T .
(2.1) D ( t 0 + t , x t 0 + t ) = D ( t 0 , φ ) + ∫ t 0 t 0 + t f ( u , x u ) d u + ∫ t 0 t 0 + t g ( u , x u ) d B ( u ) .

A solution x ( t ) is said to be unique if any other solution x ¯ ( t ) is indistinguishable from it, i.e.,

P { x ( t ) = x ¯ ( t ) } = 1 , t 0 − r ≤ t ≤ T .

Definition 2.2

Suppose B is a bounded subset of Banach space Z , α ( B ) = inf { d : B has finite covers whose diameters are less than d } is called the Kuratowski measure of noncompactness of B if

α ( B ) = 0 if and only if B is compact;
α ( A ∪ B ) = max ( α ( A ) , α ( B ) ) ;
α ( cov ¯ A ) = α ( A ) ;
α ( A + B ) ≤ α ( A ) + α ( B ) .

Definition 2.3

Suppose Z is a Banach space and α is the Kuratowski measure of noncompactness of the subset of Z . T : Z → Z is continuous. If ∃ K ∈ (0, 1), s.t. for any bounded set B ⊂ Z , α ( T B ) ≤ K α ( B ) , then T is an α -contraction.

Lemma 2.1

Suppose Γ is a close subset of Banach space Z , T : Γ → Γ is an α -contraction, then T has at least a fixed point.

Lemma 2.2

(2.1) is equal to

(2.2) D ( t 0 + t , φ ^ t + Z t ) = D ( t 0 , φ ) + ∫ 0 t f ( t 0 + s , φ ^ s + Z s ) d s + ∫ 0 t g ( t 0 + s , φ ^ s + Z s ) d B ( s ) .

Proof

Let u = t 0 + s , then (2.1) can be written as

D ( t 0 + t , x t 0 + t ) = D ( t 0 , φ ) + ∫ 0 t f ( t 0 + s , x t 0 + s ) d u + ∫ 0 t g ( t 0 + s , x t 0 + s ) d B ( s ) ,

extending the definition of φ ( t ) , we set

φ ^ ( t ) = φ ( t ) , t ∈ [ − r , 0 ] φ ( 0 ) , t ≥ 0

and

x ( t 0 + t ) = φ ^ ( t ) + Z ( t ) ,

so x t 0 + t = φ ^ t + Z t , we obtain (2.2).□

Obviously, the operator D is crucial for the equation. If we can solve x ( t ) from D ( t , x t ) , the problem will be solved easily. So, if the equation has solutions, the operator D must have some characteristics.

Definition 2.4

Suppose Ω ⊆ R × C is an open set, D ( t , φ ) : Ω → R is a nonlinear continuous operator. It is Fréchet derivable with respect to φ and continuous with respect to t . Fréchet derivation D φ ( t , φ ) ∈ ℒ ( C , R n ) . By the Riesz theorem, there exist a bounded matrix of variation μ ( t , φ , θ ) , ( t , φ ) ∈ Ω , θ ∈ [ − r , 0 ] , which make

D φ ( t , φ ) ψ = ∫ − r 0 [ d θ μ ( t , θ , φ ) ] ψ ( θ ) , ψ ∈ C .

For any ( σ , φ 0 ) ∈ Ω , θ 0 ∈ [ − r , 0 ] , if

det [ μ ( σ , φ 0 , θ 0 + ) − μ ( σ , φ 0 , θ 0 − ) ] ≠ 0 ,

D is called atomic at θ 0 on ( σ , φ 0 ) . If it is satisfied at θ 0 on any ( t , φ ) ∈ H ⊆ Ω , D is called atomic at θ 0 on H .

Lemma 2.3

Suppose D is atomic at 0, then

(2.3) D φ ( t , φ ) ψ = A ( t , φ ) ψ ( 0 ) + ∫ − r 0 − [ d θ μ ( t , θ , φ ) ] ψ ( θ ) , ψ ∈ C ,

where A ( t , φ ) = μ ( t , 0 , φ ) − μ ( t , 0 − , φ ) .

Proof

Because D φ ( t , φ ) is a continuous linear operator, by the Riesz theorem, there exist a bounded matrix of variation μ ( t , θ , φ ) , ( t , φ ) ∈ Ω , θ ∈ [ − r , 0 ] , which make

D φ ( t , φ ) ψ = ∫ − r 0 [ d θ μ ( t , θ , φ ) ] ψ ( θ ) , ψ ∈ C .

It is known that D is atomic at 0, then

det A ( t , φ ) = det ( μ ( t , 0 , φ ) − μ ( t , 0 − , φ ) ) ≠ 0 ,

so, we obtain (2.3).□

Lemma 2.4

Operators S, U are defined as follows, then Z in Lemma 2.1 can be written as

(2.4) Z = S Z + U Z ,

where

S : ( S Z ) ( t ) = 0 , t ∈ [ − r , 0 ] A ( t 0 + t , φ t ^ ) ( S Z ) ( t ) = − ∫ − r 0 − [ d θ μ ( t 0 + t , θ , φ t ^ ) ] Z ( t + θ ) + [ D ( t 0 , φ ) − D ( t 0 + t , φ t ^ ) ] − [ D ( t 0 + t , φ t ^ + Z t ) − D ( t 0 + t , φ t ^ ) − D φ ( t 0 + t , φ t ^ ) Z t ] , t ∈ [ 0 , d ] ,

U : ( U Z ) ( t ) = 0 , t ∈ [ − r , 0 ] , A ( t 0 + t , φ t ^ ) ( U Z ) ( t ) = ∫ 0 t f ( t 0 + s , φ s ^ + Z s ) d s + ∫ 0 t g ( t 0 + s , φ s ^ + Z s ) d B ( s ) , t ∈ [ 0 , d ] .

Proof

By use of Lemmas 2.1 and 2.3, we can obtain the conclusion from direct computation.□

3 Existence of solutions

Theorem 3.1

Suppose D , f , g are continuous. D is atomic at 0 and has second-order Fréchet derivation. Then, (2.1) has a solution through ( t 0 , φ ) .

We first give four lemmas for the proof of Theorem 3.1.

Lemma 3.1

Suppose α , β are constants.

A ( α , β ) = { ξ : ξ ∈ C ( [ − r , α ] , R n ) , E ∣ ξ 0 ∣ 2 = 0 , E sup ‖ ξ t ‖ 2 ≤ β , t ∈ [ 0 , α ] } ,

then S + U : A ( α , β ) → A ( α , β ) .

Proof

Select α 0 > 0 to make A ( t 0 + t , φ t ^ ) ≠ 0 , the norm of matrix is recorded as ∣ ⋅ ∣ , assume

M = sup { ∣ A − 1 ( t 0 + t , φ t ^ ) ∣ : t ∈ [ 0 , α 0 ] } .

Because D ( t , φ ) is continuous, ∃ α 1 ≤ α 0 s.t.

(3.1) E sup 0 ≤ t ≤ α 1 ∣ D ( t 0 , φ ) − D ( t 0 + t , φ t ^ ) ∣ 2 < β 25 M .

From the definition of Fréchet derivation, ∃ α 2 ≤ α 0 , when t ∈ [ 0 , α 2 ] Z ∈ A ( α 2 , β ) , we have

(3.2) E sup 0 ≤ t ≤ α 2 ∣ D ( t 0 + t , φ t ^ + Z t ) − D ( t 0 + t , φ t ^ ) − D φ ( t 0 + t , φ t ^ ) Z t ∣ 2 < β 25 M .

Because D φ is smooth in measure, that is, ∃ r ( t , s , φ ) ∈ R , r is continuous about ( t , φ ) ∈ [ t 0 , T ] × C ( [ − r , 0 ] , R n ) , s ∈ R , and r ( t , 0 , φ ) = 0 , s.t. for α ∈ [ 0 , r ] , ( t , φ ) ∈ [ t 0 , T ] × C ( [ − r , 0 ] , R n ) ,

E sup ∫ − α 0 − [ d θ μ ( t , θ , φ ) ] ψ ( θ ) 2 ≤ E sup ( r ( t , α , φ ) ∣ ψ ∣ ) 2 ,

then ∃ α 3 ≤ α 0 , when t ∈ [ 0 , α 3 ] Z ∈ A ( α 3 , β ) , we have

(3.3) E sup 0 ≤ t ≤ α 3 ∫ − α 0 − [ d θ μ ( t 0 + t , θ , φ t ^ ) ] Z ( t + θ ) 2 ≤ E sup ( r ( t 0 + t , α 3 , φ t ^ ) ‖ Z t ‖ ) 2 < β 25 M ,

because of the continuity of f , as long as α 4 , β are small enough. When t ∈ [ 0 , α 4 ] Z ∈ A ( α 4 , β ) , we have

(3.4) E sup 0 ≤ t ≤ α 4 ∫ 0 t f ( t 0 + s , φ t ^ ) + Z s d s 2 < β 25 M ,

because of the continuity of g , as long as α 5 , β are small enough. When t ∈ [ 0 , α 5 ] Z ∈ A ( α 5 , β ) , we have

(3.5) E sup 0 ≤ t ≤ α 5 ∫ 0 t g ( t 0 + s , φ t ^ ) + Z s d B ( s ) 2 ≤ E ∫ 0 α 4 ∣ g ( t 0 + s , φ t ^ ) ∣ 2 d s < β 25 M ,

select α = min ( α 1 , α 2 , α 3 , α 4 , α 5 ) , from (1)–(5). When t ∈ [ 0 , α ] Z ∈ A ( α , β ) , we have

□ E sup 0 ≤ t ≤ α ∣ ( S Z ) ( t ) + ( U Z ) ( t ) ∣ 2 < 5 M β 25 M + β 25 M + β 25 M + β 25 M + β 25 M = β .

Lemma 3.2

S is a contraction map on A ( α , β ) .

Proof

Note D ( t 0 + t , φ t ^ + Z t ) − D ( t 0 + t , φ t ^ ) − D φ ( t 0 + t , φ t ^ ) Z t = h ( t 0 + t , φ t ^ , Z t ) , because D ( t , φ ) has second-order derivation about φ , so ∃ ε ( t , β , φ ) , which is continuous and satisfies ε ( t , 0 , φ ) = 0 . When E ‖ ψ ‖ 2 ≤ β , E ‖ ξ ‖ 2 ≤ β we have

E sup ∣ h ( t , φ , ψ ) − h ( t , φ , ξ ) ∣ 2 ≤ E sup ( ε ( t , β , φ ) ‖ ψ − ξ ‖ ) 2 .

So, when E ‖ Z t ‖ 2 ≤ β , E ‖ y t ‖ 2 ≤ β , we have

E sup ∣ h ( t 0 + t , φ t ^ , Z t ) − h ( t 0 + t , φ t ^ , y t ) ∣ 2 ≤ E sup ( ε ( t 0 + t , β , φ t ^ ) ‖ Z t − y t ‖ ) 2 .

From above, we can obtain that

E sup 0 ≤ t ≤ α ∣ ( S z ) ( t ) − ( S y ) ( t ) ∣ 2 ≤ 2 M 2 { E sup 0 ≤ t ≤ α ∫ − α 0 − [ d θ μ ( t 0 + t , θ , φ t ^ ) ] ∣ Z ( t + θ ) − y ( t + θ ) ∣ 2 + E sup 0 ≤ t ≤ α ( ε 2 ( t 0 + t , β , φ t ^ ) ‖ Z t − y t ‖ 2 ) } ≤ 2 M 2 E sup 0 ≤ t ≤ α [ r 2 ( t 0 + t , α , φ t ^ ) + ε 2 ( t 0 + t , β , φ t ^ ) ] ‖ Z t − y t ‖ 2

when α , β are small enough,

2 M 2 E sup 0 ≤ t ≤ α [ r 2 ( t 0 + t , α , φ t ^ ) + ε 2 ( t 0 + t , β , φ t ^ ) ] ≤ k < 1 ,

so S is contract on A ( α , β ) .□

Lemma 3.3

U B ¯ ( B ⊆ A ( α , β ) ) is a compact set.

Proof

It is obvious that U : A ( α , β ) → A ( α , β ) is continuous. Next we will prove for bounded set B ⊆ A ( α , β ) , U B is compact.

First, B ⊆ A ( α , β ) , U B ⊆ A ( α , β ) , and A ( α , β ) is close, so U B ¯ ⊆ A ( α , β ) , then the functions in U B ¯ are uniform bound.

Second, assume Z ∈ U B ¯ ( Z ∈ U B or Z ∈ ( U B ) ′ ) ,

if Z ∈ U B , suppose

∣ f ( t 0 + t , φ t ^ + Z t ) ∣ ≤ M * , ∣ g ( t 0 + t , φ t ^ + Z t ) ∣ ≤ M * , t ∈ [ 0 , α ] ,

when t − τ is sufficiently small,

E sup 0 ≤ t ≤ α ∣ ( U z ) ( t ) − ( U z ) ( τ ) ∣ 2 ≤ 2 M 2 ( E sup 0 ≤ t ≤ α ∣ ∫ τ t f ( t 0 + s , φ s ^ + Z s ) d s ∣ 2 + E sup 0 ≤ t ≤ α ∣ ∫ τ t g ( t 0 + s , φ s ^ + Z s ) d B ( s ) ∣ 2 ) ≤ 2 M 2 ( t − τ ) 2 M * 2 + 2 M 2 ( t − τ ) 2 4 M * 2 < ε 9 .

if Z ∈ U B ′ , we can find Z n ∈ U B , Z n → Z , n → ∞ , then ∀ ε > 0 , when n is sufficiently small,

E sup 0 ≤ t ≤ α ∣ ( U z ) ( t ) − ( U z ) ( τ ) ∣ 2 ≤ 3 E sup 0 ≤ t ≤ α ∣ ( U z ) ( t ) − ( U z n ) ( t ) ∣ 2 + 3 E sup 0 ≤ t ≤ α × ∣ ( U z n ) ( t ) − ( U z n ) ( τ ) ∣ 2 + 3 E sup 0 ≤ t ≤ α ∣ ( U z n ) ( τ ) − ( U z ) ( τ ) ∣ 2 < ε 3 + ε 3 + ε 3 = ε .

So the functions in U B ¯ are equicontinuous. So U B ¯ is compact.□

Lemma 3.4

S + U is α -contraction.

Proof

Because S + U : A ( α , β ) → A ( α , β ) , ∀ B ⊆ A ( α , β ) , ( S + U ) B ⊆ A ( α , β ) , so ( S + U ) B is bounded. It is known that U is completely continuous and U B ¯ is compact. So α ( U B ) = 0 .

From Lemma 3.2, S is a contraction, so ∃ k ∈ ( 0 , 1 ) , α ( S B ) ≤ k α ( B ) , then

α [ ( S + U ) B ] ≤ α ( S B ) + α ( U B ) = α ( S B ) ≤ k α ( B ) ,

so S + U is contract.

Proof of Theorem 3.1

From Lemmas 2.2 and 2.4, we can know that if x ( t ) determined by x ( t 0 + t ) = φ ^ ( t ) + Z t satisfies (1.2), then Z satisfies (2.2), and then Z ∈ C ( [ − r , a ] , R n ) satisfies (2.4).

So, our aim is to prove that Z satisfies Darbo fixed point theorem. From Lemmas 3.1–3.4 and Darbo theorem S + U has a fixed point on A ( α , β ) , which is just the solution of (2.1).□

4 Continuation of solutions

We first give the definition of the continuation.

Definition 4.1

If x ( t ) is a solution of (2.1) on an interval [ σ , a ) , a > σ . We say x ^ is a continuation of x if there is b > a such that x ^ is defined on [ σ − r , b ) , coincides with x on [ σ − r , a ) , and x satisfies (2.1) on [ σ , b ) . A solution is noncontinuable if no such continuation exists, that is, the interval [ σ , a ) is the maximal interval of existence of the solution.

Definition 4.2

Let Λ be an open subset of a metric space. We say L : Λ → ℒ ( C , R n ) has smoothness on the measure if for any β ∈ R n , there is a scalar function γ ( λ , s ) continuous for λ ∈ Λ , s ∈ R , γ ( λ , 0 ) = 0 such that if

L ( λ ) ϕ = ∫ − r 0 d η [ ( λ , θ ) ] ϕ ( θ ) , λ ∈ Λ , 0 < s ,

then

lim h → 0 + ∫ β + h β + s + ∫ β − s β − h d [ η ( λ , θ ) ] ϕ ( θ ) ≤ γ ( λ , s ) ∣ ϕ ∣ .

Theorem 4.1

Suppose W ⊂ Ω is a close bounded set. There exits a territory W ( δ ) ⊂ Ω , if

f , g : W → bounded set of R n and R n × m ,
D is uniform atomic at 0 on W ,
D ( t , φ ) , D φ ( t , φ ) are uniform continuous and D φ has smoothness on the measure,
x ( t ) is a noncontinuable solution of (1.2) on [ σ − r , b ) ,

then there exist t ′ ∈ [ σ , b ) s.t. ( t ′ , x ′ ) ∉ W .

Proof

Suppose r > 0 , b > σ are constants, otherwise the theorem is established. Next we give two conditions:

(i) If there exist t k → b − , x t k → ψ ∈ C , when t → b − , x ( t ) → ψ ( 0 ) .

Define x ( b ) = ψ ( 0 ) , then ( b , x b ) is on the border of Ω , otherwise x ( t ) can be extended to the right. Note that x t is continuous about t and W ( δ ) ⊂ Ω , so dis ( ( b , x b ) , W ) > 0 , then ∃ t w < b , s . t . when t w < t < b , ( t , x t ) ∈ W .

(ii) If t k → b − does not exist, there are two subsequences:

When V = { ( t , x t ) : t ∈ [ σ , b ) } is unbounded, the theorem is established because W is bounded.

When V = { ( t , x t ) : t ∈ [ σ , b ) } is bounded, the following is reduced to absurdity.

If the theorem is not true, for all t ∈ [ σ , b ) , ( t , x t ) ∈ W , V ⊆ W . We want to prove that ∃ α > 0 s . t . x ( t ) is uniform continuous, so V is relatively compact, then ∃ t k → b , this is a contradiction.

Because V ⊆ W , so the conditions about W in the theorem are also established about V : from (2), ∃ N > 0 and continuous function γ ( s ) , η ( β ) , and γ ( 0 ) = η ( 0 ) = 0 , ∀ ( t , φ ) ∈ V ,

E ‖ ψ ‖ 2 ≤ β , 0 ≤ β ≤ m , 0 ≤ s ≤ r ,

the following are established:

D ( t , φ + ψ ) = D ( t , φ ) + D φ ( t , φ ) ψ + h ( t , φ , ψ ) , E sup ∣ A − 1 ( t , 0 , φ ) ∣ 2 ≤ N , E sup ∣ D φ ( t , φ ) ∣ 2 ≤ N , E sup ∣ h ( t , φ , ψ ) ∣ 2 ≤ η ( β ) E ‖ ψ ‖ 2 , E sup ∫ − s 0 − [ d θ μ ( t , θ , φ ) ] ψ ( θ ) 2 ≤ γ ( s ) E ‖ ψ ‖ 2 ,

E sup ∣ D ( t , φ + ψ ) − D ( t , φ ) ∣ 2 = E sup ∣ D φ ( t , φ ) ψ + h ( t , φ , ψ ) ∣ 2 = E sup ∣ A ( t , 0 , φ ) ψ ( 0 ) + ∫ − r 0 − [ d θ μ ( t , θ , φ ) ] ψ ( θ ) + h ( t , φ , ψ ) ∣ 2 = E sup ∣ A ( t , 0 , φ ) ψ ( 0 ) + ∫ − r − s [ d θ μ ( t , θ , φ ) ] ψ ( θ ) + ∫ − s 0 − [ d θ μ ( t , θ , φ ) ] ψ ( θ ) + h ( t , φ , ψ ) ∣ 2 ≥ E ∣ ψ ( 0 ) ∣ 2 N − N E ‖ ψ ‖ 2 − γ ( s ) E ‖ ψ ‖ 2 − η ( β ) E ‖ ψ ‖ 2 .

If x ( t ) is not uniform continuous on [ σ − r , b ) , there exist a constant ε 0 and series { t k } , { δ k } s . t . t k , t k − δ k ∈ [ σ , b ) and ∀ k ,

E sup ∣ x ( t k ) − x ( t k − δ k ) ∣ 2 ≥ ε 0 ,

select s > 0 , x ( t ) is uniform continuous on [ σ − r , b − s ] , ∀ ε > 0 , ∃ δ > 0 , when ∣ t − t ′ ∣ < δ , t , t ′ ∈ [ σ − r , b − s ] ,

E sup ∣ x ( t ) − x ( t ′ ) ∣ 2 ≤ ε ,

from condition (3), D ( t , φ ) is uniform continuous on V .

Select δ small enough s . t . when ∣ t − t ′ ∣ < δ , ( t , φ ) ∈ V , ( t ′ , φ ) ∈ V ,

E sup ∣ D ( t , φ ) − D ( t ′ , φ ) ∣ 2 ≤ ε .

Select β ∈ ( 0 , m ) , ε < min ( β , ε 0 ) , set positive integral k 0 large enough, s.t. when k ≥ k 0 , δ k < δ , for k ≥ k 0 , make { S k } as follows:

S k = inf { t ∈ [ σ , b ) : E sup ∣ x ( t ) − x ( t − δ k ) ∣ 2 ≥ min ( β , ε 0 ) }

because E sup ∣ x ( t ) − x ( t − δ k ) ∣ 2 ≥ ε 0 ≥ min ( β , ε 0 ) , { S k } exist, so

E sup ∣ D ( s k , x s k ) − D ( s k − δ k , x s k − δ k ) ∣ 2 ≥ E sup ∣ D ( s k , x s k ) − D ( s k , x s k − δ k ) ∣ 2 − E sup ∣ D ( s k , x s k − δ k ) − D ( s k − δ k , x s k − δ k ) ∣ 2 ≥ E sup ∣ D ( s k , x s k ) − D ( s k , x s k − δ k ) ∣ 2 − ε ≥ min ( β , ε 0 ) ⁄ N − N ε − γ ( s ) β − η ( β ) min ( β , ε 0 ) = ε 1 ,

choosing β , s , ε properly can make ε 1 > 0 , so D ( t , x t ) is not uniform continuous on [ σ , b ) .

But on the other hand, from condition (1) ∃ M > 0 , s.t. ∀ t ∈ [ σ , b ) ,

E sup ∣ f ( t , x t ) ∣ 2 < M ⁄ 2 , E sup ∣ g ( t , x t ) ∣ 2 < M ⁄ 2 ,

so when t , t + τ ∈ [ σ , b ) ,

E sup ∣ D ( t + τ , x t + τ ) − D ( t , x t ) ∣ 2 = E sup ∣ ∫ t t + τ f ( s , x s ) d s + ∫ t t + τ g ( s , x s ) d B ( s ) ∣ 2 ≤ M ∣ τ ∣ .

So D ( t , x t ) is uniform continuous on [ σ , b ) . It is a contradiction from which ∃ α > 0 s.t. x ( t ) is uniform continuous on [ b − α , b ) .□

5 Continuous dependence of solutions

In this section, we discuss the disturbed stochastic differential equation.

Lemma 5.1

Suppose Γ is a bounded close convex set of Banach space Z . Λ is a subset of a Banach space Y . T : Γ × Λ → Γ is a mapping, which satisfies the following:

T ( ⋅ , λ ) is continuous for all λ ∈ Λ , ∃ λ 0 ∈ Λ s.t. for every x ∈ Γ , T ( x , λ ) is continuous at ( x , λ 0 ) .
For ∀ Γ ′ ⊆ Γ , the Kuratowski measure α ( Γ ′ ) > 0 , there exists an open territory of λ 0 , B = B ( Γ ′ ) , for any relatively compact set Λ ′ ⊆ Λ ∩ B , α ( T ( Γ ′ , Λ ′ ) ) < α ( Γ ′ ) .
When λ = λ 0 , equation x = T ( x , λ ) has a unique solution x ( λ 0 ) in Γ , then any solution x ( λ ) is continuous at λ = λ 0 .

Lemma 5.2

Γ , Λ are the same as that in Lemma 5.1, T : Γ × Λ → Γ satisfies (1) and (3), T = S + U , and for every Λ ′ ⊆ Λ ,

S ( ⋅ , λ ) is a contraction on Γ and uniform contract for λ ∈ Λ ′ .
U ( Γ , Λ ′ ) is uniform compact.

Theorem 5.1

Assume Ω ⊆ R × C is an open set. Λ is a subset of a Banach space. D , f , : Ω × Λ → R n , g : Ω × Λ → R n × m satisfy

∀ ( t , φ ) ∈ Ω , D ( t , φ , λ ) is atomic at 0 on Ω , which is uniform about λ ∈ Λ .
∀ λ ∈ Λ , D ( t , φ , λ ) , f ( t , φ , λ ) , g ( t , φ , λ ) is continuous in ( t , φ ) ∈ Ω , ∀ ( t , φ ) ∈ Ω , D , f , g is continuous at ( t , φ , λ 0 ) .
Equation
d D ( t , x t , λ 0 ) = f ( t , x t , λ 0 ) d t + g ( t , x t , λ 0 ) d B ( t )
has an unique solution through ( σ , φ ) on [ σ − r , b ] .
Then, there exist a territory of ( σ , φ , λ 0 ) , N ( σ , φ , λ 0 ) s . t . ∀ ( σ ′ , φ ′ , λ ′ ) ∈ N ( σ , φ , λ 0 ) , equation
d D ( t , x t , λ ′ ) = f ( t , x t , λ ′ ) d t + g ( t , x t , λ ′ ) d B ( t )
has a solution through ( σ , φ ) on [ σ − r , b ] and when t ∈ [ σ , b ] , ( σ ′ , φ ′ , λ ′ ) ∈ N ( σ , φ , λ 0 ) , x t ( σ ′ , φ ′ , λ ′ ) are continuous at ( t , σ ′ , φ ′ , λ 0 ) .

Proof

To modify the proof of Theorem 3.1, define operators S , U : A ( α , β ) × Λ → A ( α , β ) . Let T = S + U , it is easy to prove that S ( z , λ ′ ) , U ( z , λ ′ ) satisfy conditions (4) and (5) in Lemma 5.2.

From assumption (3), we obtain (1) and (3) in Lemma 5.1. The solution z ( λ ′ ) of equation z = T ( z , λ ′ ) is continuous at λ ′ = λ 0 , so the solution x ( σ ′ , φ ′ , λ ′ ) ( t ) is continuous at λ 0 , x t ( σ ′ , φ ′ , λ ′ ) is continuous at ( t , σ ′ , φ ′ , λ ′ ) .□

6 Moment estimate of solutions

In this section, we try to establish the exponential estimate for the solution of (1.2). Assume x ( t ) is the unique global solution of (1.2). It is much more difficult to establish L p -estimate for NSFDE than SFDE. So, we need some lemmas.

Lemma 6.1

Let p ≥ 2 and ε , a , b > 0 . Then,

a p − 1 b ≤ ( p − 1 ) ε a p p + b p p ε p − 1

and

a p − 2 b 2 ≤ ( p − 2 ) ε a p p + 2 b p p ε ( p − 2 ) ⁄ 2 .

Theorem 6.1

Let p ≥ 2 and E ‖ φ ‖ < ∞ . If

there is a constant K > 1 , ( t , x ) ∈ [ t 0 , T ] × C ( [ − r , 0 ] ; R n ) ,
∣ f ( t , x ) ∣ 2 ∨ ∣ g ( t , x ) ∣ 2 ≤ K ( 1 + ‖ x ‖ 2 ) ;
there is a constant k ∈ ( 0,1 ) such that
‖ x t ‖ ≤ ∣ D ( t , x t ) ∣ ≤ ( 1 + k ) ‖ x t ‖ .
Then,
E ( sup t 0 − r ≤ s ≤ t ∣ x ( s ) ∣ p ) ≤ ( 1 + C ¯ E ‖ φ ‖ p ) e C ( t − t 0 ) ,
where C = 2 C 1 + 64 K p 2 ( 1 + k ) p − 2 , C ¯ = 1 + 2 ( 1 + k ) p .

Proof

By Itô’s formula, we can show that

(6.1) ∣ D ( t , x t ) ∣ p ≤ ∣ D ( t 0 , φ ) ∣ p + ∫ t 0 t [ p ∣ D ( s , x s ) ∣ p − 1 ∣ f ( s , x s ) ∣ + p ( p − 1 ) 2 ∣ D ( s , x s ) ∣ p − 2 ∣ g ( s , x s ) ∣ 2 d s + p ∫ t 0 t ∣ D ( s , x s ) ∣ p − 2 ( D ( s , x s ) ) T g ( s , x s ) d B ( s ) .

Applying Lemma 6.1 and conditions (i) and (ii), we easily see that for any ε > 0 ,

∣ D ( s , x s ) ∣ p − 1 ∣ f ( s , x s ) ∣ ≤ ( p − 1 ) ε ( 1 + k ) p p ‖ x s ‖ p + K p 2 p ε p − 1 ( 1 + ‖ x s ‖ 2 ) p 2 ≤ ( p − 1 ) ε ( 1 + k ) p p + ( 2 K ) p 2 p ε p − 1 ( 1 + ‖ x s ‖ p ) .

Letting ε = 2 K ⁄ ( 1 + k ) yield that

∣ D ( s , x s ) ∣ p − 1 ∣ f ( s , x s ) ∣ ≤ 2 K ( 1 + k ) p − 1 ( 1 + ‖ x s ‖ p ) .

Similarly, we can show that

∣ D ( s , x s ) ∣ p − 2 ∣ g ( s , x s ) ∣ ≤ 2 K ( 1 + k ) p − 2 ( 1 + ‖ x s ‖ p ) .

We therefore obtain from above that

(6.2) E ( sup t 0 ≤ s ≤ t ∣ D ( s , x s ) ∣ p ) ≤ ( 1 + k ) p E ‖ φ ‖ p + C 1 ∫ t 0 t ( 1 + E ‖ x s ‖ p ) d s + p E sup t 0 ≤ s ≤ t ∫ t 0 s ∣ D ( r , x r ) ∣ p − 2 ( D ( r , x r ) ) T g ( r , x r ) d B ( r )

where C 1 = p 2 K ( 1 + k ) p − 1 + p ( p − 1 ) K ( 1 + k ) p − 2 .

On the other hand, by the Burkholder-Davis-Gundy inequality and the assumptions, we derive that

(6.3) p E ( sup t 0 ≤ s ≤ t ∫ t 0 s ∣ D ( r , x r ) ∣ p − 2 ( D ( r , x r ) ) T g ( r , x r ) d B ( r ) ) ≤ 4 p 2 E ∫ t 0 t ∣ D ( s , x s ) ∣ 2 p − 2 ∣ g ( s , x s ) ∣ 2 d s 1 2 ≤ 4 p 2 E ( sup t 0 ≤ s ≤ t ∣ D ( s , x s ) ∣ p ) ∫ t 0 t ∣ D ( s , x s ) ∣ p − 2 ∣ g ( s , x s ) ∣ 2 d s 1 2 ≤ 1 2 E ( sup t 0 ≤ s ≤ t ∣ D ( s , x s ) ∣ p ) + 16 p 2 E ∫ t 0 t ∣ D ( s , x s ) ∣ p − 2 ∣ g ( s , x s ) ∣ 2 d s ≤ 1 2 E ( sup t 0 ≤ s ≤ t ∣ D ( s , x s ) ∣ p ) + 32 K p 2 ( 1 + k ) p − 2 ∫ t 0 t ( 1 + E ‖ x s ‖ p ) d s

Substituting (6.3) in (6.2) yields that

(6.4) E ( sup t 0 ≤ s ≤ t ∣ D ( s , x s ) ∣ p ) ≤ 2 ( 1 + k ) p E ‖ φ ‖ p + C ∫ t 0 t ( 1 + E ‖ x s ‖ p ) d s ,

where C = 2 C 1 + 64 K p 2 ( 1 + k ) p − 2 .

Applying (ii) we can directly obtain

E ( sup t 0 ≤ s ≤ t ∣ x s ∣ p ) ≤ 2 ( 1 + k ) p E ‖ φ ‖ p + C ∫ t 0 t ( 1 + E ‖ x s ‖ p ) d s .

Consequently,

1 + E ( sup t 0 − r ≤ s ≤ t ∣ x s ∣ p ) ≤ 1 + E ‖ φ ‖ p + E ( sup t 0 ≤ s ≤ t ∣ x s ∣ p ) ≤ 1 + C ¯ E ‖ φ ‖ p + C ∫ t 0 t [ 1 + E ( sup t 0 − r ≤ τ ≤ s ∣ x τ ∣ p ) ] d s ,

where C ¯ = 1 + 2 ( 1 + k ) p .

Finally, by the Gronwall inequality, we obtain that

1 + E ( sup t 0 − r ≤ s ≤ t ∣ x s ∣ p ) ≤ [ 1 + C ¯ E ‖ φ ‖ p ] exp [ C ( t − t 0 ) ] .

The proof is complete.□

7 Discussion

Some other properties of neutral stochastic functional operator-differential equations will be discussed in the future.

8 Conclusion

We have the existence, continuation, continuous dependence, and moment estimate of solutions of neutral stochastic functional operator-differential equations. In financial modeling, stochastic differential equations are widely used in financial risk management, derivatives pricing, and portfolio optimization. For example, in an interest rate model, stochastic differential equations are used to describe the process of interest rate movements. By solving these stochastic differential equations, we can predict the future direction of interest rates, which in turn can provide decision support for financial institutions and investors.

# These authors contributed equally to this work.

Funding information: This study was supported by the key Projects of Natural Science Research in Colleges and Universities in Anhui Province (No. 2023AH051363), the Natural Science Foundation of Huangshan University (2021xkjq005).
Author contributions: Ling Hu was mainly engaged in the writing the draft, Zheng Wu was mainly responsible for inspecting the article, and Lianglong Wang (Hu Ling’s supervisor) revised the article.
Conflict of interest: The authors declared no potential conflicts of interest with respect to the research, author-ship, and publication of this article.
Data availability statement: The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

References

[1] Chojnowska-Michalik A. Stochastic differential equations in Hilbert spaces and their applications. Ph.D. thesis, Institute of Mathematics, Polish Academy of Sciences. 1976. Search in Google Scholar

[2] Da Prato G, Iannelli M, Tubaro L. Semilinear stochastic differential equations in Hilbert spaces. Boll Unione Mat Ital. 1979;16A:168–85. Search in Google Scholar

[3] Dawson DA. Stochastic evolution equations and related measured processes. J Multivariate Anal. 1975;5:1–52. 10.1016/0047-259X(75)90054-8Search in Google Scholar

[4] Kotelenez P. On the semigroup approach to stochastic evolution equations. in Stochastic space time models and limit theorems. Dordrecht: Reidel; 1985. 10.1007/978-94-009-5390-1_6Search in Google Scholar

[5] Krylov NV, Rozovskii BL. Stochastic evolution equations. J Soviet Math. 1981;16:1233–77. 10.1007/BF01084893Search in Google Scholar

[6] Pardoux E, Stochastic partial differential equations and filtering of diffusion processes. Stochastics. 1979;3:127–67. 10.1080/17442507908833142Search in Google Scholar

[7] Caraballo T. Asymptotic exponential stability of stochastic partial differential equations with delay. Stochastics. 1990;33:27–47. 10.1080/17442509008833662Search in Google Scholar

[8] Taniguchi T. Almost sure exponential stability for stochastic partial functional differential equations. Stochastic Anal. Appl. 1998;16(5):965–75. 10.1080/07362999808809573Search in Google Scholar

[9] Wu J. Theory and applications of partial functional differential equations, applied mathematics science. Vol. 119. New York: Springer-Verlag; 1996. 10.1007/978-1-4612-4050-1Search in Google Scholar

[10] Ky Tran Q, Dang Nguyen H. Exponential stability of impulsive stochastic differential equations with Markovian switching. Syst Control Lett. 2022;162:105–78. 10.1016/j.sysconle.2022.105178Search in Google Scholar

[11] Barrimi Oussama EI. Stability results for stochastic differential equations driven by an additive fractional Brownian sheet. Random Operat Stochastic Equ. 2023;3:257–72. 10.1515/rose-2023-2013Search in Google Scholar

[12] Mao X. Stochastic differential equations and applications. Chichester, UK: Horwood Publishing; 2007. 10.1533/9780857099402Search in Google Scholar

Received: 2024-12-05

Revised: 2025-01-13

Accepted: 2025-02-27

Published Online: 2025-04-17

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Keywords for this article

operator functional differential equations; atomic

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