Conditional Local Existence and Uniqueness for Impulsive Stochastic Differential Equations with State-Dependent Delay

Abstract

We consider impulsive stochastic delay differential equations with state-dependent impulse times, where the k-th impulse occurs at a time $\tau$ satisfying $\tau ={\tau }_k\left(u\left({\tau }^{-}\right)\right)$. In generalized Lipschitz and growth conditions for coefficients, establishing existence and uniqueness is conditional on the successive approximations sharing a common finite impulse-time structure. The proof of our main result is based on the successive approximation scheme along with Itô’s isometry and Bihari’s inequality. We prove that the approximating sequence is bounded and convergent in a suitable Banach space, which gives a unique local solution.

1. Introduction

Impulsive delay differential equations provide a natural framework for describing systems whose evolution depends on past states and is interrupted by sudden changes. When stochastic perturbations are included, the analysis must simultaneously account for memory effects, random forcing, and discontinuities produced by the impulsive mechanism. Accordingly, the existence, uniqueness, continuation, stability, and controllability of solutions have been investigated for several classes of deterministic and stochastic impulsive systems.

For clarity, it is useful to distinguish between state-dependent delays and state-dependent impulse times. In an equation with a state-dependent delay, the delayed argument itself depends on the current state or on the history of the solution. By contrast, in a system with state-dependent impulses, the occurrence time of an impulse is determined implicitly by the trajectory. In the present problem, the coefficients depend on the history segment $u_{\tau }$ over the fixed interval $[-T,0]$, while the k-th impulse is triggered by the condition

$$\tau ={\tau }_k\left(u\left({\tau }^{-}\right)\right).$$

Thus, the explicit state dependence in the impulsive mechanism concerns the impulse occurrence times.

In the deterministic setting, Dubeau and Karrakchou [1] studied delay differential equations with infinitely many state-dependent impulses. They established existence and uniqueness by transforming the impulsive problem into an equivalent delay equation without impulses and then applying a fixed-point argument with a suitable norm. Hernández, Sakthivel, and Tanaka Aki [2,3] obtained existence results for mild solutions of impulsive evolution equations with state-dependent delay. Rezapour et al. [4] investigated the existence of mild solutions for second-order non-autonomous integro-differential evolution equations with infinite state-dependent delay and extended their analysis to the neutral case by using resolvent operators and fixed-point theorems.

The stochastic setting has also been examined from several perspectives. Alwan, Liu, and Xie [5,6] studied stochastic impulsive systems with time delay whose impulse times depend on the state. They established local and global existence, forward continuation, and uniqueness results for adapted solutions. Ren and Xiong [7] considered stochastic impulsive switched systems with deterministic state-dependent impulses and switches and obtained results related to well-posedness and stability. For fractional and infinite-dimensional models, Kalamani et al. [8,9] derived existence conditions for impulsive fractional neutral stochastic integro-differential equations with nonlocal conditions and state-dependent delay. Diop et al. [10] investigated the existence of mild solutions and optimal control problems for impulsive stochastic integro-differential equations with state-dependent delay. More recently, You, Shu, and Shu [11] established existence and approximate controllability results for second-order neutral stochastic evolution systems with random impulses and state-dependent delay, whereas Raghavendran et al. [12] studied controllability for fractional impulsive neutral Volterra–Fredholm integro-differential equations with state-dependent delay.

The specific contribution of this paper is to isolate, precisely, the obstruction that arises when a stochastic functional differential equation has both a state-dependent delay and state-dependent impulse times simultaneously: because the impulse times ${\tau }_k\left(u\left({\tau }^{-}\right)\right)$ depend on the very solution being constructed, each stage of the Picard iteration can in principle trigger impulses at different times than the previous stage and than the limiting process. We isolate this obstruction as an explicit working hypothesis, Assumption 1 below, rather than leaving it implicit, and we prove a conditional local existence and uniqueness theorem under Assumption 1 together with generalized Lipschitz/growth hypotheses on the coefficients. Our proof follows the successive-approximation framework for stochastic differential equations described in LaSalle and Mao [13,14], adapted to the impulsive, state-dependent setting. Since the coefficients satisfy only a generalized, non-Lipschitz modulus-of-continuity condition governed by a concave function K, we repeatedly pass between $K(\mathbb{E}\parallel {\dot{\parallel }}^2)$ and $\mathbb{E}K(\parallel {\dot{\parallel }}^2)$ via the concavity inequality of Jensen [15], and uniqueness is then obtained not from a direct Gronwall-type estimate but from the nonlinear comparison inequality of Bihari [16]. A fully worked one-dimensional example (Section 4) verifies all the hypotheses explicitly, including a numerical check of the smallness conditions required by the main theorem.

We emphasize from the outset and return to this point in the Conclusions—that Assumption 1 is a working condition, not a consequence of the model that we derive from Hypotheses 1–4. It is imposed in order to adapt the classical successive-approximation argument, developed for fixed impulse times, to the state-dependent setting. A complete derivation of a common impulse family directly from the stochastic dynamics is not established in the present paper; Remark 4 below discusses two possible directions for removing this restriction in future work. Readers should therefore regard Theorem 1 as establishing existence and uniqueness conditional on Assumption 1, and not as a fully unconditional state-dependent-impulse existence theory.

2. Preliminaries

Throughout this work, we write ${\mathbb{R}}_{+}:=[0,\infty )$. We fix an initial time ${\tau }_0\in {\mathbb{R}}_{+}$, a delay length $T>0$, and a local horizon $\alpha >0$, and we set

$$J:=[{\tau }_0,{\tau }_0+\alpha ].$$

Let D be a nonempty open subset of $R^n$. We denote the Euclidean norm in ${\mathbb{R}}^n$ by $|·|$, and the same notation is used for the induced operator norm in ${\mathbb{R}}^{n\times m}$ whenever no confusion can arise.

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a complete probability space equipped with a filtration ${\left\{{\mathcal{F}}_{\tau }\right\}}_{\tau \ge {\tau }_0}$ satisfying the usual conditions, and suppose that

$$W\left(\tau \right)={\left(W_1\left(\tau \right),\dots ,W_m\left(\tau \right)\right)}^{\top },\tau \ge {\tau }_0,$$

is an m-dimensional standard Brownian motion adapted to ${\left\{{\mathcal{F}}_{\tau }\right\}}_{\tau \ge {\tau }_0}$.

We study the following stochastic impulsive differential equation incorporating a state-dependent delay.

$$\left\{\begin{array}{cc}{\displaystyle du\left(\tau \right)=f\left(\tau ,u_{\tau }\right)d\tau +\eta \left(\tau ,u_{\tau }\right)dW\left(\tau \right),}\hfill & \tau \in J,\tau \ne {\tau }_k\left(u\left({\tau }^{-}\right)\right),\hfill \\ u\left(\tau \right)-u\left({\tau }^{-}\right)=I\left(\tau ,u_{{\tau }^{-}}\right),\hfill & \tau ={\tau }_k\left(u\left({\tau }^{-}\right)\right),k\ge 1,\hfill \\ u_{{\tau }_0}=\phi ,\hfill & \end{array}\right.$$

(1)

where the coefficients are the following measurable mappings:

$$\begin{array}{cc}\hfill f& :J\times PC\left([-T,0];D\right)⟶{\mathbb{R}}^n,\hfill \\ \hfill \eta & :J\times PC\left([-T,0];D\right)⟶{\mathbb{R}}^{n\times m},\hfill \\ \hfill I& :J\times PC\left([-T,0];D\right)⟶{\mathbb{R}}^n,\hfill \end{array}$$

and for each $k\ge 1$, ${\tau }_k:D\to {\mathbb{R}}_{+}$ is the k-th state-dependent impulse time function.

The expression $\tau ={\tau }_k\left(u\left({\tau }^{-}\right)\right)$ represents the left-hand limit of u at the point $\tau$. In other words, it describes the value that $u\left(s\right)$ approaches as s gets closer to $\tau$ from the left. The difference $u\left(\tau \right)-u\left({\tau }^{-}\right)$ then measures the size of the jump of the solution at the impulse time $\tau ={\tau }_k$. For any $\tau \in J$, we also define the segment function $u_{\tau }\in PC([-T,0];D)$ by shifting the function around $\tau$. Specifically,

$$u_{\tau }\left(\theta \right):=u(\tau +\theta ),\theta \in [-T,0],$$

and captures the entire past history of the solution over the window $[\tau -T,\tau ]$.

The initial datum $\phi \in PC\left(\right[-T,0];D)$ is an ${\mathcal{F}}_{{\tau }_0}$-measurable random variable satisfying

$${\mathbb{E}\parallel \phi \parallel }_T^2<\infty .$$

where $PC\left(\right[-T,0],D)$ is defined as the set of functions $\xi :[-T,0]\to D$ that are right-continuous with left limits, and the norm is defined as ${\parallel \xi \parallel }_T:=su{p}_{\theta \in [-T,0]}\left|\xi \left(\theta \right)\right|.$

We work in the following Banach space of càdlàg adapted processes:

$$S_{\alpha }:=\left\{u:[{\tau }_0-T,{\tau }_0+\alpha ]⟶L^2\left(\Omega ;PC([-T,0];{\mathbb{R}}^n)\right)|\begin{array}{c}u\mathrm{is}{\mathcal{F}}_{\tau }-\mathrm{adapted},\hfill \\ u\mathrm{is}\mathrm{c}à\mathrm{dl}à\mathrm{g},\mathrm{and}\hfill \\ {\parallel u\parallel }_{S_{\alpha }}<\infty \hfill \end{array}\right\},$$

(2)

equipped with the norm

$${\parallel u\parallel }_{S_{\alpha }}:=\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}{\left(\mathbb{E}\parallel u_{\tau }{\parallel }_T^2\right)}^{1/2}.$$

(3)

The pair $(S_{\alpha },\parallel ·{\parallel }_{S_{\alpha }})$ is a Banach space.

A process $u\in S_{\alpha }$ is said to be a local solution of system (S) on the interval $[{\tau }_0-T,{\tau }_0+\alpha ]$ if it satisfies the following integral equation, $\mathbb{P}$—almost surely, for every $\tau \in [{\tau }_0-T,{\tau }_0+\alpha ]$

$$u\left(\tau \right)=\left\{\begin{array}{cc}\phi (\tau -{\tau }_0),\hfill & \tau \in [{\tau }_0-T,{\tau }_0],\hfill \\ {\displaystyle \phi \left(0\right)+{\int }_{{\tau }_0}^{\tau }f\left(\nu ,u_{\nu }\right)d\nu +{\int }_{{\tau }_0}^{\tau }\eta \left(\nu ,u_{\nu }\right)dW\left(\nu \right)}\hfill & \\ {\displaystyle +\sum _{\begin{array}{c}k\ge 1:\\ {\tau }_k\left(u\left({\tau }_k^{-}\right)\right)\in ({\tau }_0,\tau ]\end{array}}I\left({\tau }_k\left(u\left({\tau }_k^{-}\right)\right),u_{{\tau }_k^{-}}\right),}\hfill & \tau \in ({\tau }_0,{\tau }_0+\alpha ].\hfill \end{array}\right.$$

(4)

Remark 1.

The equivalence between system (1) and the integral Equation (4) follows by integrating the stochastic differential between consecutive impulse times and summing the resulting contributions together with the accumulated jumps. The summation index condition ${\tau }_k\left(u\left({\tau }_k^{-}\right)\right)\in ({\tau }_0,\tau ]$ automatically counts only those impulses whose state-dependent times have been triggered by time τ, consistently with the state-dependent nature of the impulsive mechanism.

Remark 2.

The integral formulation (4) is the natural working form for the successive approximation scheme developed in the proof of the main theorem. Both integrals run from the fixed lower limit ${\tau }_0$, which allows Itô’s isometry, the Cauchy–Schwarz inequality, and Bihari’s inequality to be applied directly over the full interval $[{\tau }_0,{\tau }_0+\alpha ]$ without splitting at individual impulse times.

2.1. Hypothesis

Hypothesis 1.

Let $K:{\mathbb{R}}_{+}⟶{\mathbb{R}}_{+}$ be a continuous, nondecreasing, and concave function satisfying

$$K\left(0\right)=0,K\left(v\right)>0\mathit{for}\mathit{every}v>0,$$

and

$${\int }_{0^{+}}{\displaystyle \frac{dv}{K\left(v\right)}}=\infty .$$

Moreover, for all $\tau \in [{\tau }_0,{\tau }_0+\alpha ]$ and all $\xi ,\theta \in PC$,

$${\parallel f(\tau ,\xi )-f(\tau ,\theta )\parallel }^2\le K\left({\parallel \xi -\theta \parallel }_T^2\right).$$

In addition, there exists a constant ${\kappa }_0>0$ such that

$${\parallel f(\tau ,0)\parallel }^2\le {\kappa }_0,\forall \tau \in [{\tau }_0,{\tau }_0+\alpha ].$$

Hypothesis 2.

The mapping $\eta :J\times PC([-T,0];D)⟶{\mathbb{R}}^{n\times m}$ is jointly measurable. Moreover, for every $\tau \in J$ and every $\xi ,\theta \in PC\left(\right[-T,0];D)$,

$${\parallel \eta (\tau ,\xi )-\eta (\tau ,\theta )\parallel }^2\le K\left({\parallel \xi -\theta \parallel }_T^2\right).$$

In addition,

$${\parallel \eta (\tau ,0)\parallel }^2\le {\kappa }_0,\tau \in J.$$

Hypothesis 3.

Assume further that there exist non-negative constants $h_k$ such that

$$\parallel I({\tau }_k,\xi )-I({\tau }_k,\theta ){\parallel }^2\le h_k{\parallel \xi -\theta \parallel }_T^2$$

for all $\xi ,\theta \in \mathcal{PC}$ and for every impulse time ${\tau }_k\in ({\tau }_0,{\tau }_0+\alpha ]$.

Also assume that

$$\parallel I({\tau }_k,0){\parallel }^2\le {\kappa }_0$$

for every ${\tau }_k\in ({\tau }_0,{\tau }_0+\alpha ]$.

Hypothesis 4.

Assume that, for each $k=1,2,\dots$, the function

$${\tau }_k:D\to {\mathbb{R}}_{+}$$

belongs to $C^1(D,{\mathbb{R}}_{+})$ and satisfies

$$0<{\tau }_1\left(u\right)<{\tau }_2\left(u\right)<\dots ,\underset{k\to \infty }{lim}{\tau }_k\left(u\right)=+\infty$$

uniformly in $u\in D$.

Moreover, assume that for every $\xi \in PC\left(\right[-T,0];D)$ with $\xi \left(0^{-}\right)=\xi \left(0\right)$ and for every $k\ge 1$,

$$\xi \left(0\right)+I\left({\tau }_k\left(\xi \left(0\right)\right),\xi \right)\in D$$

and

$${\tau }_k\left(\xi \left(0\right)+I\left({\tau }_k\left(\xi \left(0\right)\right),\xi \right)\right)\le {\tau }_k\left(\xi \left(0\right)\right).$$

Assume that for each $k\ge 1$ and for every $(\tau ,\xi )\in [{\tau }_0,{\tau }_0+\alpha ]\times PC([-T,0];D)$,

$$\nabla {\tau }_k\left(\xi \left(0\right)\right)·f(\tau ,\xi )<1.$$

2.2. Additional Conditional Assumption (A)

Assumption 1.

On the local interval $[{\tau }_0-T,{\tau }_0+\alpha ]$, there exist at most N ordered impulse times in $({\tau }_0,{\tau }_0+\alpha ]$.

$${\left\{{\tau }_k\right\}}_{k=1}^N\subset ({\tau }_0,{\tau }_0+\alpha ]$$

such that the successive approximations ${\left\{u^n\right\}}_{n\ge 0}$ and the limit process u are all represented with respect to this same family. More precisely, for every $\tau \in [{\tau }_0,{\tau }_0+\alpha ]$,

$$N\left(\tau \right):=\#\{k:{\tau }_k\in ({\tau }_0,\tau ]\}\le N,$$

and the impulsive terms in the approximation scheme are written with the same ordered family ${\left\{{\tau }_k\right\}}_{k=1}^N$.

Remark 3.

Assumption 1 is imposed in order to adapt the successive approximation argument from the fixed-impulse stochastic framework to the present state-dependent setting. A complete derivation of a common impulse family from the stochastic dynamics is not established in the present proof.

Remark 4.

Discussing why Assumption 1 is hard in general. The crux is that ${\tau }_k\left(u\left({\tau }^{-}\right)\right)$ depends on the very solution being constructed, so at each stage of Picard iteration the impulse times ${\tau }_k\left(u^{n-1}\left({\tau }^{-}\right)\right)$ can differ from stage to stage and from the limit. This is worth saying explicitly rather than leaving it implicit.

The restriction is inherent to the successive-approximation technique as currently used: because the impulse times ${\tau }_k\left(u^{n-1}\left({\tau }^{-}\right)\right)$ depend on the previous iterate, each stage of the Picard iteration can in principle trigger impulses at different times, and the argument as given needs these to coincide (or be assumed to coincide via Assumption 1 in order to compare $u^n$ and $u^{n-1}$ on a common partition. We now note two possible directions for relaxing Assumption 1 in future work: (i) a fixed-point argument formulated directly on the space of finite impulse-time sequences, tracking the impulse times themselves as part of the unknown, rather than assuming them to be fixed in advance; and (ii) a continuity/stability argument for the impulse-time map ${\tau }_k\left( \right)$ that would let one show the iterates’ impulse times converge together with the iterates themselves, so that Assumption 1 emerges as a consequence rather than a hypothesis. We state explicitly that neither direction is pursued in the present paper and that Assumption 1 is used here as a working condition that isolates the successive-approximation argument from this additional difficulty.

3. Results

Theorem 1

(Conditional local existence and uniqueness). Assume Hypotheses 1–4 and the additional conditional assumption Assumption 1. Suppose further that

$$\rho <1\mathit{and}3N{\displaystyle \sum _{k=1}^N}{h}_k<1,$$

where

$$\rho :=8b(\alpha +1)\alpha +8N{\displaystyle \sum _{k=1}^N}{h}_k.$$

Then the sequence of successive approximations ${\left\{u^n\right\}}_{n\ge 0}$ is bounded in mean square on $[{\tau }_0-T,{\tau }_0+\alpha ]$ and converges in $S_{\alpha }$ to a local solution u of system (1) on $[{\tau }_0-T,{\tau }_0+\alpha ]$.

Moreover, if $u_1$ and $u_2$ are two local solutions corresponding to the same initial history φ and associated with the same ordered finite impulse family ${\left\{{\tau }_k\right\}}_{k=1}^N$, then

$$u_1\left(\tau \right)=u_2\left(\tau \right)a.s.\mathit{for}\mathit{all}\tau \in [{\tau }_0-T,{\tau }_0+\alpha ].$$

Lemma 1.

Under Hypotheses 1–3 and Assumption 1, let $v,w:[{\tau }_0-T,{\tau }_0+\alpha ]\to L^2(\Omega ;{\mathbb{R}}^n)$ be two adapted càdlàg processes such that $v_{\tau },w_{\tau }\in PC([-T,0];D)$ for every τ. Then, for every $\tau \in [{\tau }_0,{\tau }_0+\alpha ]$,

$$\begin{array}{cc}\hfill \mathbb{E}\parallel & {\int }_{{\tau }_0}^{\tau }\left(f(\nu ,v_{\nu })-f(\nu ,w_{\nu })\right)d\nu +{\int }_{{\tau }_0}^{\tau }\left(\eta (\nu ,v_{\nu })-\eta (\nu ,w_{\nu })\right)dW\left(\nu \right)\hfill \\ \hfill & {+\sum _{{\tau }_0<{\tau }_k\le \tau }\left(I({\tau }_k,v_{{\tau }_k^{-}})-I({\tau }_k,w_{{\tau }_k^{-}})\right)\parallel }^2\hfill \\ \hfill & \le 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(\mathbb{E}\parallel v_{\nu }-w_{\nu }{\parallel }_T^2\right)d\nu +3N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)\underset{{\tau }_0\le u\le \tau }{sup}\mathbb{E}{\parallel v_u-w_u\parallel }_T^2.\hfill \end{array}$$

Proof. 

The proof (given once, then invoked at each application site) combines the elementary inequality ${\parallel a+b+c\parallel }^2\le {3(\parallel a\parallel }^2+{\parallel b\parallel }^2+{\parallel c\parallel }^2)$ with the Cauchy–Schwarz inequality applied to the deterministic term, Itô’s isometry applied to the stochastic term, and the inequality $\parallel {\sum }_k{y}_k{\parallel }^2\le N{\sum }_k{\parallel y_k\parallel }^2$ applied to the impulsive term. Hypotheses 1–3 are then used to bound each difference term by $K(·)$ or $h_k(·)$, and the concavity of K together with Jensen’s inequality is used to pass from $K(\mathbb{E}\parallel ·{\parallel }^2)$ to $\mathbb{E}K(\parallel ·{\parallel }^2)$. □

Proof of Theorem 1. 

For simplicity of notation, throughout the proof, ${\left\{{\tau }_k\right\}}_{k=1}^N$ denotes the common ordered finite impulse family given by assumption Assumption 1. By Assumption 1, this family is well defined and finite on every interval $({\tau }_0,t]\subset ({\tau }_0,{\tau }_0+\alpha ]$. Hence we write

$$N\left(\tau \right):=\#\{k:{\tau }_k\in ({\tau }_0,\tau ]\}.$$

Following the classical method of successive approximations LaSalle [13], we define

$$u^0\left(\tau \right)=\left\{\begin{array}{cc}\phi (\tau -{\tau }_0),\hfill & \tau \in [{\tau }_0-T,{\tau }_0],\hfill \\ \phi \left(0\right),\hfill & \tau \in ({\tau }_0,{\tau }_0+\alpha ],\hfill \end{array}\right.$$

and for every $n\ge 1,$

$$u^n\left(\tau \right)=\left\{\begin{array}{cc}\phi (\tau -{\tau }_0),\hfill & \tau \in [{\tau }_0-T,{\tau }_0],\hfill \\ \phi \left(0\right)+{\int }_{{\tau }_0}^{\tau }f(\nu ,u_{\nu }^{n-1})d\nu +{\int }_{{\tau }_0}^{\tau }\eta (\nu ,u_{\nu }^{n-1})dW\left(\nu \right)+\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}I({\tau }_k,u_{{\tau }_k^{-}}^{n-1}),\hfill & \tau \in ({\tau }_0,{\tau }_0+\alpha ].\hfill \end{array}\right.$$

Step 1:

$$\phi \left(0\right)+{\int }_{{\tau }_0}^{\tau }f(\nu ,u_{\nu }^{n-1})d\nu +{\int }_{{\tau }_0}^{\tau }\eta (\nu ,u_{\nu }^{n-1})dW\left(\nu \right)+\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}I({\tau }_k,u_{{\tau }_k^{-}}^{n-1}),$$

apply Lemma 1 which immediately gives

$$\begin{array}{cc}\hfill \parallel u^n{\left(\tau \right)\parallel }^2\le & {4\parallel \phi \left(0\right)\parallel }^2\hfill \\ \hfill & +8(\tau -{\tau }_0){\int }_{{\tau }_0}^{\tau }\left(\parallel f(\nu ,u_{\nu }^{n-1})-f(\nu ,0){\parallel }^2+{\parallel f(\nu ,0)\parallel }^2\right)d\nu \hfill \\ \hfill & +8{∥{\int }_{{\tau }_0}^{\tau }\left(\eta (\nu ,u_{\nu }^{n-1})-\eta (\nu ,0)\right)dW\left(\nu \right)∥}^2\hfill \\ \hfill & +8{∥{\int }_{{\tau }_0}^{\tau }\eta (\nu ,0)dW\left(\nu \right)∥}^2\hfill \\ \hfill & +8N\left(\tau \right)\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}\left(\parallel I({\tau }_k,u_{{\tau }_k^{-}}^{n-1})-I({\tau }_k,0){\parallel }^2+{\parallel I({\tau }_k,0)\parallel }^2\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2\hfill \\ \hfill & +8(\tau -{\tau }_0){\int }_{{\tau }_0}^{\tau }E\left(\parallel f(\nu ,u_{\nu }^{n-1})-f(\nu ,0){\parallel }^2+{\parallel f(\nu ,0)\parallel }^2\right)d\nu \hfill \\ \hfill & +8E{∥{\int }_{{\tau }_0}^{\tau }\left(\eta (\nu ,u_{\nu }^{n-1})-\eta (\nu ,0)\right)dW\left(\nu \right)∥}^2\hfill \\ \hfill & +8E{∥{\int }_{{\tau }_0}^{\tau }\eta (\nu ,0)dW\left(\nu \right)∥}^2\hfill \\ \hfill & +8N\left(\tau \right)\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}E\left(\parallel I({\tau }_k,u_{{\tau }_k^{-}}^{n-1})-I({\tau }_k,0){\parallel }^2+{\parallel I({\tau }_k,0)\parallel }^2\right).\hfill \end{array}$$

Then immediately apply Itô isometry

$$E{∥{\int }_{{\tau }_0}^{\tau }\left(\eta (\nu ,u_{\nu }^{n-1})-\eta (\nu ,0)\right)dW\left(\nu \right)∥}^2={\int }_{{\tau }_0}^{\tau }E{\parallel \eta (\nu ,u_{\nu }^{n-1})-\eta (\nu ,0)\parallel }^{2}d\nu ,$$

$$E{∥{\int }_{{\tau }_0}^{\tau }\eta (\nu ,0)dW\left(\nu \right)∥}^2={\int }_{{\tau }_0}^{\tau }E{\parallel \eta (\nu ,0)\parallel }^{2}d\nu .$$

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2\hfill \\ \hfill & +8(\tau -{\tau }_0){\int }_{{\tau }_0}^{\tau }E\left(\parallel f(\nu ,u_{\nu }^{n-1})-f(\nu ,0){\parallel }^2+{\parallel f(\nu ,0)\parallel }^2\right)d\nu \hfill \\ \hfill & +8{\int }_{{\tau }_0}^{\tau }E{\parallel \eta (\nu ,u_{\nu }^{n-1})-\eta (\nu ,0)\parallel }^{2}d\nu \hfill \\ \hfill & +8{\int }_{{\tau }_0}^{\tau }E{\parallel \eta (\nu ,0)\parallel }^{2}d\nu \hfill \\ \hfill & +8N\left(\tau \right)\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}E\left(\parallel I({\tau }_k,u_{{\tau }_k^{-}}^{n-1})-I({\tau }_k,0){\parallel }^2+{\parallel I({\tau }_k,0)\parallel }^2\right).\hfill \end{array}$$

Now we use hypotheses,

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2\hfill \\ \hfill & +8(\tau -{\tau }_0){\int }_{{\tau }_0}^{\tau }\left(EK(\parallel u_{\nu }^{n-1}{\parallel }_T^2)+{\kappa }_0\right)d\nu \hfill \\ \hfill & +8{\int }_{{\tau }_0}^{\tau }\left(EK(\parallel u_{\nu }^{n-1}{\parallel }_T^2)+{\kappa }_0\right)d\nu \hfill \\ \hfill & +8N\left(\tau \right)\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}\left(h_{k}E{\parallel u_{{\tau }_k^{-}}^{n-1}\parallel }_T^2+{\kappa }_0\right).\hfill \end{array}$$

Since $h_k\ge 0$

$$N\left(\tau \right)\le N:=\left|\left\{k:{\tau }_k\in ({\tau }_0,{\tau }_0+\alpha ]\right\}\right|,\sum _{\{k:{\tau }_k\in ({\tau }_0,\tau ]\}}{h}_k\le {\displaystyle \sum _{k=1}^N}{h}_k,$$

Since $0\le \tau -{\tau }_0\le \alpha$ and $N\left(\tau \right)\le N$, it follows that

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2\hfill \\ \hfill & +8(\alpha +1){\int }_{{\tau }_0}^{\tau }\left(EK(\parallel u_{\nu }^{n-1}{\parallel }_T^2)+{\kappa }_0\right)d\nu \hfill \\ \hfill & +8N{\displaystyle \sum _{k=1}^N}\left(h_{k}E{\parallel u_{{\tau }_k^{-}}^{n-1}\parallel }_T^2+{\kappa }_0\right).\hfill \end{array}$$

Since $K(·)$ is concave and satisfies $K\left(0\right)=0$, it follows that there exist positive constants $a>0$ and $b>0$ such that

$$K\left(u\right)\le a+bu,\forall u\ge 0.$$

Moreover, by Jensen’s inequality and the concavity of K, we have

$$EK\left(\parallel u_{\nu }^{n-1}{\parallel }_T^2\right)\le K\left(E\parallel u_{\nu }^{n-1}{\parallel }_T^2\right)\le a+bE{\parallel u_{\nu }^{n-1}\parallel }_T^2.$$

If we plug this into the earlier inequality gives

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2\hfill \\ \hfill & +8(\alpha +1){\int }_{{\tau }_0}^{\tau }\left(a+bE\parallel u_{\nu }^{n-1}{\parallel }_T^2+{\kappa }_0\right)d\nu \hfill \\ \hfill & +8N{\displaystyle \sum _{k=1}^N}\left(h_{k}E{\parallel u_{{\tau }_k^{-}}^{n-1}\parallel }_T^2+{\kappa }_0\right).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2+8(\alpha +1)(a+{\kappa }_0)(\tau -{\tau }_0)\hfill \\ \hfill & +8b(\alpha +1){\int }_{{\tau }_0}^{\tau }E\parallel u_{\nu }^{n-1}{\parallel }_T^{2}d\nu +8N{\displaystyle \sum _{k=1}^N}{h}_{k}E{\parallel u_{{\tau }_k^{-}}^{n-1}\parallel }_T^2+8N^2{\kappa }_0.\hfill \end{array}$$

Step 2:

By definition,

$$u^0\left(\tau \right)=\left\{\begin{array}{cc}\phi (\tau -{\tau }_0),\hfill & \tau \in [{\tau }_0-T,{\tau }_0],\hfill \\ \phi \left(0\right),\hfill & \tau \in ({\tau }_0,{\tau }_0+\alpha ].\hfill \end{array}\right.$$

Thus, for any $\tau \in ({\tau }_0,{\tau }_0+\alpha ]$, so we get that

$$\parallel u^0{\left(\tau \right)\parallel }^2={\parallel \phi \left(0\right)\parallel }^2.$$

Taking expectation, we obtain

$$E\parallel u^0{\left(\tau \right)\parallel }^2=E{\parallel \phi \left(0\right)\parallel }^2<\infty .$$

Step 3: Introduce the recursive bound.

Define

$$\parallel u_{\tau }^n{\parallel }_T:=\underset{\tau -T\le s\le \tau }{sup}\parallel u^n\left(s\right)\parallel ,U_n:=\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}E{\parallel u_{\tau }^n\parallel }_T^2.$$

Consequently, for each $\nu \in [{\tau }_0,{\tau }_0+\alpha ]$, one obtains

$$E\parallel u_{\nu }^{n-1}{\parallel }_T^2\le U_{n-1},E{\parallel u_{{\tau }_k^{-}}^{n-1}\parallel }_T^2\le U_{n-1},$$

and

$${\int }_{{\tau }_0}^{\tau }E{\parallel u_{\nu }^{n-1}\parallel }_T^{2}d\nu \le \alpha U_{n-1}.$$

Therefore,

$$\begin{array}{cc}\hfill E\parallel u^n{\left(\tau \right)\parallel }^2\le & {4E\parallel \phi \left(0\right)\parallel }^2+8\alpha (\alpha +1)(a+{\kappa }_0)\hfill \\ \hfill & +8b(\alpha +1)\alpha U_{n-1}+8N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)U_{n-1}+8N^2{\kappa }_0.\hfill \end{array}$$

Set

$$Q_3:=4E{\parallel \phi \left(0\right)\parallel }^2+8\alpha (\alpha +1)(a+{\kappa }_0)+8N^2{\kappa }_0,$$

and

$$\rho :=8b(\alpha +1)\alpha +8N{\displaystyle \sum _{k=1}^N}{h}_k.$$

Then

$$E\parallel u^n{\left(\tau \right)\parallel }^2\le Q_3+\rho U_{n-1}.$$

Moreover,

$$\parallel u^n{\parallel }_{\tau }^2\le {\parallel \phi \parallel }_T^2+\underset{{\tau }_0\le u\le \tau }{sup}{\parallel u^n\left(u\right)\parallel }^2.$$

Taking expectation and supremum over $\tau \in [{\tau }_0,{\tau }_0+\alpha ]$, we obtain

$$U_n\le E{\parallel \phi \parallel }_T^2+Q_3+\rho U_{n-1}.$$

If we denote

$$Q_4:=E{\parallel \phi \parallel }_T^2+Q_3,$$

then the recurrence relation becomes

$$U_n\le Q_4+\rho U_{n-1},n\ge 1.$$

Step 4: Iterate the recursive inequality.

From the previous step, we have

$$U_n\le Q_4+\rho U_{n-1},n\ge 1.$$

Now we iterate this estimate.

For $n=1$,

$$U_1\le Q_4+\rho U_0.$$

For $n=2$,

$$U_2\le Q_4+\rho U_1\le Q_4+\rho (Q_4+\rho U_0)=Q_4(1+\rho )+{\rho }^2{U}_0.$$

For $n=3$,

$$U_3\le Q_4+\rho U_2\le Q_4+\rho \left(Q_4(1+\rho )+{\rho }^2{U}_0\right)=Q_4(1+\rho +{\rho }^2)+{\rho }^3{U}_0.$$

Proceeding inductively, we obtain

$$U_n\le Q_4{\displaystyle \sum _{j=0}^{n-1}}{\rho }^j+{\rho }^n{U}_0,n\ge 1.$$

If, in addition, we assume that

$$\rho <1,$$

then

$${\displaystyle \sum _{j=0}^{n-1}}{\rho }^j\le {\displaystyle \sum _{j=0}^{\infty }}{\rho }^j={\displaystyle \frac{1}{1-\rho }}.$$

Hence,

$$U_n\le {\displaystyle \frac{Q_4}{1-\rho }}+{\rho }^n{U}_0\le {\displaystyle \frac{Q_4}{1-\rho }}+U_0.$$

Therefore, there exists

$$C_{\alpha }:={\displaystyle \frac{Q_4}{1-\rho }}+U_0$$

satisfying

$$U_n\le C_{\alpha },\forall n\ge 0.$$

Recalling the definition of $U_n$, it follows that

$$\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}E{\parallel u^n\parallel }_{\tau }^2\le C_{\alpha },\forall n\ge 0.$$

Equivalently,

$$E\parallel u^n{\parallel }_{\tau }^2\le C_{\alpha }<\infty ,\forall n\ge 0,\forall \tau \in [{\tau }_0,{\tau }_0+\alpha ].$$

Thus, the sequence of successive approximations $\left\{u^n\right\}$ is bounded in mean square on $[{\tau }_0,{\tau }_0+\alpha ]$.

• Step 5: We aim to show that $\left\{u^n\right\}$ is a Cauchy sequence.

Apply Lemma 1 with $v=u^q,w=u^p:$

$$\begin{array}{cc}\hfill \mathbb{E}\parallel u^{q+1}\left(\tau \right)-u^{p+1}{\left(\tau \right)\parallel }^2& \le 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(\mathbb{E}\parallel u_{\nu }^q-u_{\nu }^p{\parallel }_T^2\right)d\nu \hfill \\ \hfill & +3N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)\underset{{\tau }_0\le u\le \tau }{sup}\mathbb{E}{\parallel u_u^q-u_u^p\parallel }_T^2,\hfill \end{array}$$

hence

$$D_{q+1,p+1}\left(\tau \right)\le 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(D_{q,p}\left(\nu \right)\right)d\nu +3N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)D_{q,p}\left(\tau \right).$$

Now define

$$D\left(\tau \right):=\underset{q,p\to \infty }{lim\; sup}{D}_{q,p}\left(\tau \right).$$

Passing to the limit superior on both sides and using reverse Fatou’s lemma, we obtain

$$D\left(\tau \right)\le 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(D\left(\nu \right)\right)d\nu +3N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)D\left(\tau \right).$$

Assume that

$$3N{\displaystyle \sum _{k=1}^N}{h}_k<1.$$

Then

$$\left(1-3N{\displaystyle \sum _{k=1}^N}{h}_k\right)D\left(\tau \right)\le 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(D\left(\nu \right)\right))d\nu .$$

Hence,

$$D\left(\tau \right)\le {\displaystyle \frac{3(\alpha +1)}{1-3N{\sum }_{k=1}^N{h}_k}}{\int }_{{\tau }_0}^{\tau }K\left(D\left(\nu \right)\right)d\nu .$$

Set

$$\widehat{C}:={\displaystyle \frac{3(\alpha +1)}{1-3N{\sum }_{k=1}^N{h}_k}}.$$

Then

$$D\left(\tau \right)\le \widehat{C}{\int }_{{\tau }_0}^{\tau }K\left(D\left(\nu \right)\right)d\nu .$$

Since $D\left({\tau }_0\right)=0$ and

$${\int }_{0^{+}}{\displaystyle \frac{du}{K\left(u\right)}}=\infty ,$$

An application of Bihari’s inequality yields [16]

$$D\left(\tau \right)=0,\tau \in [{\tau }_0,{\tau }_0+\alpha ].$$

Therefore,

$$\underset{q,p\to \infty }{lim}{D}_{q,p}\left(\tau \right)=0,\tau \in [{\tau }_0,{\tau }_0+\alpha ],$$

Therefore, $\left\{u^n\right\}$ is a Cauchy sequence.

that is,

$$\underset{q,p\to \infty }{lim}\underset{{\tau }_0\le u\le \tau }{sup}E{\parallel u_u^q-u_u^p\parallel }_T^2=0.$$

Thus, $\left\{u^n\right\}$ is a Cauchy sequence in the space

$$S_{\alpha }:=\left\{u:[{\tau }_0,{\tau }_0+\alpha ]\to L^2\left(\Omega ;PC([-T,0];{\mathbb{R}}^n)\right);u\mathrm{is}{\mathcal{F}}_{\tau }-\mathrm{adapted}\mathrm{and}\mathrm{c}à\mathrm{dl}à\mathrm{g},\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}E{\parallel u_{\tau }\parallel }_T^2<\infty \right\},$$

equipped with the norm

$${\parallel u\parallel }_{S_{\alpha }}:=\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}{\left(E\parallel u_{\tau }{\parallel }_T^2\right)}^{1/2}.$$

Then $(S_{\alpha },\parallel ·{\parallel }_{S_{\alpha }})$ is a Banach space.

By completeness of $S_{\alpha }$, the Cauchy sequence $\left\{u^n\right\}$ admits a limit $u\in S_{\alpha }$ such that

$$\underset{n\to \infty }{lim}{\parallel u^n-u\parallel }_{S_{\alpha }}=0.$$

Equivalently,

$$\underset{n\to \infty }{lim}\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}E{\parallel u_{\tau }^n-u_{\tau }\parallel }_T^2=0.$$

On the other hand, from the previous estimate, we already know that

$$\underset{{\tau }_0\le \tau \le {\tau }_0+\alpha }{sup}E{\parallel u_{\tau }^n\parallel }_T^2\le C_{\alpha },\forall n\ge 0.$$

Hence, for each fixed $\tau \in [{\tau }_0,{\tau }_0+\alpha ]$,

$$E\parallel u_{\tau }^n{\parallel }_T^2\le C_{\alpha },\forall n\ge 0.$$

Letting $n\to \infty$ and invoking Fatou’s lemma yields

$$E\parallel u_{\tau }{\parallel }_T^2\le \underset{n\to \infty }{lim\; inf}E{\parallel u_{\tau }^n\parallel }_T^2\le C_{\alpha },\tau \in [{\tau }_0,{\tau }_0+\alpha ].$$

Therefore,

$$E\parallel u_{\tau }{\parallel }_T^2\le C_{\alpha }<\infty .$$

Thus, the limit process u also belongs to $S_{\alpha }$ and inherits the same uniform mean-square bound as the approximating sequence.

Therefore, the limit process u is a local solution of system (1) on $[{\tau }_0-T,{\tau }_0+\alpha ]$.

□

3.1. Passage to the Limit in the Approximation Formula

We now show that the limit process u satisfies the integral equation associated with system (1) by using Dominated Convergence Theorem for Itô Integrals apply Lemma 1 with $v=u^{n-1},w=u$ (the limit process):

$$\begin{array}{cc}\hfill & \mathbb{E}\parallel u^n\left(\tau \right)-\phi \left(0\right)+{\int }_{{\tau }_0}^{\tau }f(\nu ,u_{\nu })d\nu +{\int }_{{\tau }_0}^{\tau }\eta (\nu ,u_{\nu })dW\left(\nu \right)+\sum _{{\tau }_0<{\tau }_k\le \tau }I\left({\tau }_k,u_{{\tau }_k^{-}}\right)\left(\tau \right){\parallel }^2\le \hfill \\ \hfill & 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(\mathbb{E}\parallel u_{\nu }^{n-1}-u_{\nu }{\parallel }^2\right)d\nu +3N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)\underset{{\tau }_0\le \nu \le \tau }{sup}\mathbb{E}{\parallel u_{\nu }^{n-1}-u_{\nu }\parallel }_T^2⟶0,\hfill \end{array}$$

since ${sup}_{\tau }\mathbb{E}{\parallel u_{\tau }^{n-1}-u_{\tau }\parallel }_T^2\to 0$ by convergence in $S_{\alpha }$ and K is continuous with $K\left(0\right)=0$.

Hence, u satisfies the integral equation associated with system (1). Therefore, u is a local solution of system (1) on $[{\tau }_0-T,{\tau }_0+\alpha ]$.

3.2. Uniqueness of the Local Solution

Let $u_1,u_2\in {\mathcal{S}}_{\alpha }$ be two local solutions of system (1) with the same initial history $\phi$ on $[{\tau }_0-T,{\tau }_0]$. Then apply Lemma 1 with $v=u_1,w=u_2$:

$$D\left(\tau \right)\le 3(\alpha +1){\int }_{{\tau }_0}^{\tau }K\left(D\left(\nu \right)\right)d\nu +3N\left({\displaystyle \sum _{k=1}^N}{h}_k\right)D\left(\tau \right),$$

exactly the inequality from Step 5, hence $D\left(\tau \right)=0$ by the same Bihari argument already established.

And therefore

$$u_1\left(\tau \right)=u_2\left(\tau \right)\mathrm{a}.\mathrm{s}.\mathrm{for}\mathrm{all}\tau \in [{\tau }_0-T,{\tau }_0+\alpha ].$$

Thus, the local solution of system (1) is unique.

4. Example

We present a concrete one-dimensional stochastic example with a nontrivial diffusion term, adapted to the conditional local framework developed above.

Let ${\tau }_0=0$, let $T>0$ be fixed, and let

$$D:=(-1,\infty ).$$

Consider the local interval

$$[0,\alpha ],\alpha :={\displaystyle \frac{1}{2}},$$

and let the initial history be the constant function

$$\phi \left(\theta \right)\equiv 1,\theta \in [-T,0].$$

We study the stochastic impulsive delay system

$$\left\{\begin{array}{cc}du\left(\tau \right)={\displaystyle \frac{1}{4}}\left(1+u\left(\tau \right)\right)d\tau +{\displaystyle \frac{1}{4}}\left(1+u\left(\tau \right)\right)dW\left(\tau \right),\hfill & \tau \ne {\tau }_k\left(u\left({\tau }^{-}\right)\right),\hfill \\ \Delta u\left(\tau \right)={\displaystyle \frac{1}{2}},\hfill & \tau ={\tau }_k\left(u\left({\tau }^{-}\right)\right),\hfill \\ u_0=\phi ,\hfill & \end{array}\right.$$

where the state-dependent impulse times are given by

$${\tau }_k\left(x\right):=k+{\displaystyle \frac{1}{4}}{e}^{-x},x\in D,k=1,2,\dots .$$

We verify that this system satisfies all the assumptions of the conditional local existence and uniqueness.

• Step 1: Verification of Hypothesis 1.

Define

$$f(\tau ,\xi ):={\displaystyle \frac{1}{4}}\left(1+\xi \left(0\right)\right),(\tau ,\xi )\in [0,\alpha ]\times PC([-T,0];D).$$

Let

$$K\left(u\right):={\displaystyle \frac{1}{16}}u,u\ge 0.$$

Then K is continuous, nondecreasing, concave, and satisfies

$$K\left(0\right)=0,K\left(u\right)>0\mathrm{for}u>0.$$

Moreover,

$${\int }_{0^{+}}{\displaystyle \frac{du}{K\left(u\right)}}=16{\int }_{0^{+}}{\displaystyle \frac{du}{u}}=\infty .$$

Now, for any $\xi ,\zeta \in PC\left(\right[-T,0];D)$, we have

$${|f(\tau ,\xi )-f(\tau ,\zeta )|}^2={\left|{\displaystyle \frac{1}{4}}\left(\xi \left(0\right)-\zeta \left(0\right)\right)\right|}^2={\displaystyle \frac{1}{16}}{|\xi \left(0\right)-\zeta \left(0\right)|}^2\le {\displaystyle \frac{1}{16}}{\parallel \xi -\zeta \parallel }_T^2={K(\parallel \xi -\zeta \parallel }_T^2).$$

Also,

$${\left|f(\tau ,0)\right|}^2={\left({\displaystyle \frac{1}{4}}\right)}^2={\displaystyle \frac{1}{16}}.$$

Hence Hypothesis 1 holds.

• Step 2: Verification of Hypothesis 2.

Define

$$\eta (\tau ,\xi ):={\displaystyle \frac{1}{4}}\left(1+\xi \left(0\right)\right),(\tau ,\xi )\in [0,\alpha ]\times PC([-T,0];D).$$

Thus the diffusion term is nontrivial. For any $\xi ,\zeta \in PC\left(\right[-T,0];D)$,

$${|\eta (\tau ,\xi )-\eta (\tau ,\zeta )|}^2={\left|{\displaystyle \frac{1}{4}}\left(\xi \left(0\right)-\zeta \left(0\right)\right)\right|}^2={\displaystyle \frac{1}{16}}{|\xi \left(0\right)-\zeta \left(0\right)|}^2\le {\displaystyle \frac{1}{16}}{\parallel \xi -\zeta \parallel }_T^2={K(\parallel \xi -\zeta \parallel }_T^2).$$

Moreover,

$${\left|\eta (\tau ,0)\right|}^2={\left({\displaystyle \frac{1}{4}}\right)}^2={\displaystyle \frac{1}{16}}.$$

Therefore Hypothesis 2 is satisfied.

• Step 3: Verification of Hypothesis 3.

Define the impulse operator by

$$I(\tau ,\xi )\equiv {\displaystyle \frac{1}{2}}.$$

Then, for all $\xi ,\zeta \in PC\left(\right[-T,0];D)$ and every impulse time ${\tau }_k$,

$$|I({\tau }_k,\xi )-I({\tau }_k,\zeta ){|}^2=0.$$

Hence we may choose

$$h_k=0,k\ge 1.$$

Also,

$$|I({\tau }_k,0){|}^2={\left({\displaystyle \frac{1}{2}}\right)}^2={\displaystyle \frac{1}{4}}.$$

Thus Hypothesis 3 is fulfilled.

• Step 4: Verification of Hypothesis 4.

For each $k\ge 1$, define

$${\tau }_k\left(u\right):=k+{\displaystyle \frac{1}{4}}{e}^{-u},u\in D=(-1,\infty ).$$

Then ${\tau }_k\in C^1(D,{\mathbb{R}}_{+})$, and for every $u\in D$,

$$0<{\tau }_1\left(u\right)<{\tau }_2\left(u\right)<\dots ,$$

since

$${\tau }_{k+1}\left(u\right)-{\tau }_k\left(u\right)=1.$$

Moreover, since

$${\tau }_k\left(u\right)\ge k,$$

we have

$$\underset{k\to \infty }{lim}{\tau }_k\left(u\right)=+\infty$$

uniformly in $u\in D$.

Next, let $\xi \in PC\left(\right[-T,0];D)$ with $\xi \left(0^{-}\right)=\xi \left(0\right)$. Then

$$\xi \left(0\right)+I({\tau }_k\left(\xi \left(0\right)\right),\xi )=\xi \left(0\right)+{\displaystyle \frac{1}{2}}>-1,$$

so the post-impulse state remains in D. Furthermore,

$${\tau }_k\left(\xi \left(0\right)+I({\tau }_k\left(\xi \left(0\right)\right),\xi )\right)=k+{\displaystyle \frac{1}{4}}{e}^{-\left(\xi \right(0)+1/2)}\le k+{\displaystyle \frac{1}{4}}{e}^{-\xi \left(0\right)}={\tau }_k\left(\xi \left(0\right)\right).$$

Hence the post-impulse condition in Hypothesis 4 is satisfied.

Finally,

$${\tau }_k^{\prime }\left(u\right)=-{\displaystyle \frac{1}{4}}{e}^{-u},$$

and therefore

$$\nabla {\tau }_k\left(\xi \left(0\right)\right)·f(\tau ,\xi )=-{\displaystyle \frac{1}{4}}{e}^{-\xi \left(0\right)}·{\displaystyle \frac{1}{4}}(1+\xi \left(0\right))=-{\displaystyle \frac{1}{16}}{e}^{-\xi \left(0\right)}(1+\xi \left(0\right))<1,$$

for every $(\tau ,\xi )\in [0,\alpha ]\times PC\left(\right[-T,0];D)$, because $\xi \left(0\right)>-1$. Thus the transversality condition is also verified.

• Step 5: Verification of the additional conditional Assumption 1.

We claim that no impulse occurs on the local interval $[0,\alpha ]=[0,{\displaystyle \frac{1}{2}}]$. Indeed, for every $u\in D$,

$${\tau }_1\left(u\right)=1+{\displaystyle \frac{1}{4}}{e}^{-u}>1>{\displaystyle \frac{1}{2}}=\alpha .$$

Hence there is no impulse time in $(0,\alpha ]$.

Therefore, no effective impulse occurs on $(0,\alpha ]$, although the upper bound for the number of impulses is taken as $N=1$.

Consequently, Assumption 1 is automatically satisfied on the chosen local interval.

• Step 6: Verification of the smallness conditions.

Since no effective impulse occurs on $(0,\alpha ]$ and $h_1=0$, we immediately obtain

$$3N{\displaystyle \sum _{k=1}^N}{h}_k=0<1,4N{\displaystyle \sum _{k=1}^N}{h}_k=0<1.$$

Moreover, in Theorem 1,

$$\rho =8b(\alpha +1)\alpha +8N{\displaystyle \sum _{k=1}^N}{h}_k.$$

Here

$$b={\displaystyle \frac{1}{16}},\alpha ={\displaystyle \frac{1}{2}},N=1,$$

so

$$\rho =8·{\displaystyle \frac{1}{16}}·\left({\displaystyle \frac{1}{2}}+1\right)·{\displaystyle \frac{1}{2}}=8·{\displaystyle \frac{1}{16}}·{\displaystyle \frac{3}{2}}·{\displaystyle \frac{1}{2}}={\displaystyle \frac{3}{8}}<1.$$

Thus all the smallness assumptions required in the conditional local existence, uniqueness, and stability results are satisfied.

• Step 7: Explicit form of the local solution.

Since no impulse occurs on $[0,\alpha ]$, the system reduces there to

$$du\left(\tau \right)={\displaystyle \frac{1}{4}}\left(1+u\left(\tau \right)\right)d\tau +{\displaystyle \frac{1}{4}}\left(1+u\left(\tau \right)\right)dW\left(\tau \right),u\left(0\right)=1.$$

Set

$$y\left(\tau \right):=1+u\left(\tau \right).$$

Then

$$dy\left(\tau \right)={\displaystyle \frac{1}{4}}y\left(\tau \right)d\tau +{\displaystyle \frac{1}{4}}y\left(\tau \right)dW\left(\tau \right),y\left(0\right)=2.$$

This is a geometric Brownian motion, and its unique solution is

$$y\left(\tau \right)=2exp\left[\left({\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{16}}\right)\tau +{\displaystyle \frac{1}{4}}W\left(\tau \right)\right]=2exp\left[{\displaystyle \frac{7}{32}}\tau +{\displaystyle \frac{1}{4}}W\left(\tau \right)\right].$$

Therefore

$$u\left(\tau \right)=-1+2exp\left[{\displaystyle \frac{7}{32}}\tau +{\displaystyle \frac{1}{4}}W\left(\tau \right)\right],0\le \tau \le {\displaystyle \frac{1}{2}}.$$

In particular,

$$u\left(\tau \right)>-1\mathrm{a}.\mathrm{s}.\mathrm{for}\mathrm{all}\tau \in [0,\frac{1}{2}],$$

so the solution remains in D.

5. Conclusions

The above system is a concrete stochastic state-dependent impulsive delay model with nonzero diffusion term. It satisfies all the Hypothesis 1–4, as well as the additional conditional Assumption 1 on the local interval $[0,{\displaystyle \frac{1}{2}}]$. Hence, via Theorem 1, it admits a unique local solution on $[-T,{\displaystyle \frac{1}{2}}]$.

We reiterate, as discussed in the Introduction, that the existence and uniqueness result proved here (Theorem 1) is conditional on Assumption 1: the requirement that the successive approximations and the limiting process share a common finite ordered family of impulse times. This assumption is imposed as a working hypothesis to isolate the successive-approximation argument from the additional difficulty created by the state-dependence of the impulse times—it is not derived from the coefficients f, $\eta$, I or from Hypotheses 1–4, and in the worked example above it holds only because no impulse occurs at all on the chosen local interval. Consequently, the present result should be read as a conditional local existence and uniqueness theorem for state-dependent impulsive stochastic delay equations, rather than as an unconditional one.

Removing Assumption 1 is, in our view, the main open problem raised by this work. As discussed in Remark 4, two natural directions are: (i) reformulating the fixed-point argument on the space of finite impulse-time sequences, so that the impulse times are tracked as part of the unknown of the iteration rather than fixed in advance; and (ii) establishing a continuity/stability property of the impulse-time map ${\tau }_k(·)$ under Hypothesis 4 that would guarantee that the impulse times of successive Picard iterates converge together with the iterates themselves, so that Assumption 1 would emerge as a consequence of Hypotheses 1–4 rather than as an independent hypothesis. We leave both directions for future work.