Stability of Numerical Methods for Stochastic Delay Differential Equations: Theoretical Framework, Analysis, and Implications for Biological and Engineering Systems

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Stability of Numerical Methods for Stochastic Delay Differential Equations: Theoretical Framework, Analysis, and Implications for Biological and Engineering Systems

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Stability of Numerical Methods for Stochastic Delay Differential Equations

Stochastic delay differential equations frequently occur in biological and engineering systems; however, their numerical analysis has received less attention than that of immediate-value stochastic differential equations. This study establishes stability conditions for several numerical schemes applied to stochastic delay differential equations, identifying circumstances in which delays stabilize or destabilize numerical solutions.

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Abstract

Stochastic delay differential equations (SDDEs) constitute a fundamental class of mathematical models that capture the joint effects of randomness and time-lagged feedback in complex dynamical systems. Despite their prevalence in mathematical biology, control engineering, finance, and neuroscience, the numerical analysis of SDDEs—particularly the question of how classical and modern schemes behave in terms of stability—has received considerably less systematic attention than the corresponding theory for ordinary stochastic differential equations. This article develops a rigorous theoretical framework for the stability analysis of several numerical methods applied to SDDEs, with emphasis on mean-square stability and almost-sure exponential stability. We examine four principal schemes: the Euler–Maruyama method, the Milstein method, the stochastic theta method, and a split-step backward Euler variant. For each scheme, stability conditions are derived in terms of the step size, the drift and diffusion coefficients, and the delay parameter. A central finding is that the presence of a delay can either stabilize or destabilize a numerical solution depending on the sign and magnitude of the delay coefficient relative to the diffusion intensity—a phenomenon with direct implications for model calibration in biological systems. We validate the theoretical predictions through parameter-space analysis and asymptotic argument, and we discuss the implications of these stability regions for practitioners choosing among competing numerical strategies. This work contributes new stability criteria and extends existing results from the theory of stochastic analysis to the delayed setting.

Keywords: stochastic delay equations, numerical stability, stochastic analysis, numerical methods, mathematical biology, mean-square stability, Euler–Maruyama, Milstein method, stochastic theta method, Lyapunov functional

1. Introduction

Differential equations that incorporate both stochastic forcing and time-lagged terms are among the most realistic yet mathematically challenging models available to applied scientists. In ecology, for example, predator–prey interactions do not respond instantaneously to changes in population density; in neural circuits, axonal propagation introduces synaptic delays of tens to hundreds of milliseconds; in engineering feedback loops, measurement and actuation lags are unavoidable. When these inherently delayed dynamics are further perturbed by environmental noise—fluctuations in temperature, resource availability, sensor error, or turbulence—the modeling framework of choice is the stochastic delay differential equation.

The theoretical foundations of SDDEs as infinite-dimensional stochastic processes were laid by Mohammed (1984), who extended the theory of stochastic functional differential equations to a rigorous functional-analytic setting. Subsequent work by Mao (2007) provided accessible yet comprehensive treatments of existence, uniqueness, and moment bounds for solutions under global and local Lipschitz conditions. Meanwhile, the deterministic theory of delay differential equations—pioneered by Hale and Lunel (1993) and Kolmanovskii and Myshkis (1992)—had already demonstrated that delays can induce oscillatory behavior, bifurcations, and instability even in systems whose delay-free counterparts are perfectly stable. Combining stochastic forcing with delay thus creates a rich but subtle landscape of possible long-term behaviors.

Given that analytical solutions to SDDEs are rarely available in closed form, numerical simulation is indispensable for both theoretical investigation and practical prediction. The natural starting point is the Euler–Maruyama (EM) method, a stochastic analog of the classical Euler scheme, adapted to the delayed setting by Buckwar (2000) and Baker and Buckwar (2000). These authors established strong convergence of order one-half and weak convergence of order one for the EM scheme under global Lipschitz and linear growth conditions. Hu et al. (2004) extended the analysis to allow more general coefficient functions, while subsequent contributions by Higham (2000) and Saito and Mitsui (1996) clarified the distinct roles of the drift and diffusion terms in determining numerical stability for standard SDEs—results that motivate analogous investigations in the delayed case.

Stability of numerical methods is not merely an academic concern. A scheme that is formally convergent may still produce trajectories that diverge in mean square or almost surely, giving the practitioner a completely misleading picture of the system's long-term behavior. For SDDEs, this problem is compounded because the stability region depends on the delay in a nontrivial way. Zhang and Gan (2010) demonstrated that the stochastic theta method can preserve mean-square stability for standard SDEs under conditions inaccessible to explicit schemes, and Wang and Chen (2012) extended parts of this analysis to the semi-implicit Euler method for a class of SDDEs. Nevertheless, a unified treatment that systematically compares stability regions across multiple schemes—Euler–Maruyama, Milstein, stochastic theta, and split-step backward Euler—and that explicitly tracks how the delay modulates these regions, has been lacking.

The present article fills that gap. We proceed as follows. Section 2 establishes the theoretical background, introducing the class of SDDEs under study, the relevant stability concepts, and the Lyapunov functional methodology. Section 3 presents the four numerical schemes and derives the discrete recurrences they generate. Section 4 is the analytical core: we derive necessary and sufficient stability conditions for each scheme applied to a linear test equation, then extend the analysis to a class of nonlinear equations satisfying a one-sided Lipschitz condition. Section 5 synthesizes the results through a detailed parameter-space analysis and compares stability regions. Section 6 discusses implications for mathematical biology and engineering, including the stabilizing and destabilizing roles of delay, and Section 7 concludes with a summary and directions for future research.

2. Theoretical Background

2.1 Stochastic Delay Differential Equations

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geq 0}, \mathbb{P})$ be a complete filtered probability space satisfying the usual conditions, and let $W(t)$ be a scalar standard Brownian motion adapted to $\{\mathcal{F}_t\}$ . We consider the scalar SDDE

$dx(t) = f\bigl(x(t),\, x(t - \tau)\bigr)\,dt + g\bigl(x(t),\, x(t-\tau)\bigr)\,dW(t), \quad t \in [0, T], \tag{1}$

with deterministic initial history $x(t) = \phi(t)$ for $t \in [-\tau, 0]$ , where $\tau > 0$ is a fixed constant delay and $\phi : [-\tau, 0] \to \mathbb{R}$ is a given continuous function. The functions $f : \mathbb{R}^2 \to \mathbb{R}$ and $g : \mathbb{R}^2 \to \mathbb{R}$ are the drift and diffusion coefficients, respectively.

Existence and uniqueness of a strong solution to Equation (1) are guaranteed under the following standard assumptions (Mao, 2007).

Assumption 1 (Global Lipschitz and Linear Growth). There exist constants $L > 0$ and $K > 0$ such that for all $x_1, x_2, y_1, y_2 \in \mathbb{R}$ ,

$|f(x_1, y_1) - f(x_2, y_2)|^2 \vee |g(x_1, y_1) - g(x_2, y_2)|^2 \leq L\bigl(|x_1 - x_2|^2 + |y_1 - y_2|^2\bigr), \tag{2}$ $|f(x_1, y_1)|^2 \vee |g(x_1, y_1)|^2 \leq K\bigl(1 + |x_1|^2 + |y_1|^2\bigr). \tag{3}$

Under Assumption 1, Equation (1) has a unique $\mathcal{F}_t$ -adapted strong solution with continuous sample paths and finite moments of all orders on $[0, T]$ .

2.2 The Linear Test Equation

For the purpose of stability analysis, it is standard practice—following Higham (2000) and Buckwar (2000)—to study a linear test equation that isolates the key parameters governing stability. We adopt

$dx(t) = \bigl(\alpha\, x(t) + \beta\, x(t - \tau)\bigr)\,dt + \bigl(\mu\, x(t) + \sigma\, x(t-\tau)\bigr)\,dW(t), \quad t \geq 0, \tag{4}$

where $\alpha, \beta, \mu, \sigma \in \mathbb{R}$ are real constants. This four-parameter family captures the essential features of both the deterministic (delay-driven) and stochastic (diffusion-driven) instability mechanisms. When $\beta = \sigma = 0$ , Equation (4) reduces to the standard linear SDE test equation studied by Saito and Mitsui (1996). When $\mu = \sigma = 0$ , it reduces to the deterministic linear delay differential equation with characteristic root analysis.

2.3 Stability Concepts

We distinguish three notions of stability that are central to the analysis that follows.

Definition 1 (Mean-Square Stability). The zero solution of Equation (1) is said to be mean-square stable (MS-stable) if

$\lim_{t \to \infty} \mathbb{E}\bigl[|x(t)|^2\bigr] = 0 \tag{5}$

for every initial condition $\phi$ . If additionally there exist constants $C > 0$ and $\lambda > 0$ such that $\mathbb{E}[|x(t)|^2] \leq C e^{-\lambda t}$ , the solution is said to be exponentially mean-square stable .

Definition 2 (Almost-Sure Exponential Stability). The zero solution is almost-surely exponentially stable if

$\limsup_{t \to \infty} \frac{1}{t} \ln |x(t)| < 0 \quad \mathbb{P}\text{-almost surely.} \tag{6}$

Definition 3 (Numerical MS-Stability). A discrete-time approximation $\{X_n\}$ to $x(t)$ is numerically MS-stable if $\mathbb{E}[|X_n|^2] \to 0$ as $n \to \infty$ when the exact solution is MS-stable, for step sizes $h$ in some non-empty region $\mathcal{R} \subset (0, \infty)$ called the stability region .

A critical distinction is between A-stability (the stability region contains all step sizes) and conditional stability (stability requires $h < h^*$ for some critical step size). As we show below, explicit methods such as EM and Milstein are at most conditionally stable for the SDDE test equation, while the stochastic theta method achieves A-stability for $\theta \geq 1/2$ under appropriate parameter constraints.

2.4 Lyapunov Functionals for SDDEs

The principal analytical tool for establishing stability of both exact and numerical solutions is the Lyapunov functional method. Unlike ordinary differential equations, where a scalar Lyapunov function $V(x)$ suffices, delay systems require a functional $V(x_t)$ that depends on the entire history segment $x_t = \{x(t + s) : s \in [-\tau, 0]\}$ (Kolmanovskii & Myshkis, 1992). For the stochastic setting, the Itô formula for functionals (Mao, 2007, p. 152) states that if $V : \mathcal{C}([-\tau, 0];\mathbb{R}) \to \mathbb{R}^+$ is sufficiently smooth, then

$dV(x_t) = \mathcal{L}V(x_t)\,dt + V'(x_t) g(x_t(0), x_t(-\tau))\,dW(t), \tag{7}$

where $\mathcal{L}$ is the generator of the process. For a quadratic functional $V(x_t) = x(t)^2 + c \int_{t-\tau}^{t} x(s)^2\,ds$ with $c > 0$ , this generates the condition

$\mathbb{E}\bigl[\mathcal{L}V(x_t)\bigr] \leq -\lambda\, \mathbb{E}\bigl[V(x_t)\bigr] \tag{8}$

for some $\lambda > 0$ , which implies exponential MS-stability. The value of the parameter $c$ is chosen to eliminate the delayed terms from the generator, a procedure we carry out explicitly in Section 4.

3. Numerical Schemes for Stochastic Delay Differential Equations

Let the step size be $h = T/N$ for some positive integer $N$ , and suppose for simplicity that $m = \tau/h$ is also a positive integer, so the delay spans exactly $m$ grid steps. We denote the approximation to $x(t_n) = x(nh)$ by $X_n$ , and the increments of the Brownian motion by $\Delta W_n = W(t_{n+1}) - W(t_n) \sim \mathcal{N}(0, h)$ .

3.1 The Euler–Maruyama Method

The most elementary scheme, introduced to the SDDE context by Buckwar (2000), applies the Euler discretization to the drift and the Maruyama discretization to the diffusion:

$X_{n+1} = X_n + h\,f(X_n, X_{n-m}) + g(X_n, X_{n-m})\,\Delta W_n. \tag{9}$

Applied to the linear test Equation (4), this becomes

$X_{n+1} = \bigl(1 + \alpha h\bigr)X_n + \beta h\, X_{n-m} + \bigl(\mu X_n + \sigma X_{n-m}\bigr)\Delta W_n. \tag{10}$

The EM method has strong order of convergence $1/2$ and weak order $1$ under Assumption 1, as established by Baker and Buckwar (2000).

3.2 The Milstein Method

The Milstein scheme improves upon EM by adding a correction term that captures the leading-order contribution of the diffusion coefficient's spatial variation. For Equation (1), this takes the form (Hu et al., 2004)

$X_{n+1} = X_n + h\,f(X_n, X_{n-m}) + g(X_n, X_{n-m})\,\Delta W_n + \frac{1}{2}\,\mathcal{D}g(X_n, X_{n-m})\bigl[(\Delta W_n)^2 - h\bigr], \tag{11}$

where $\mathcal{D}g$ denotes the directional derivative of $g$ with respect to the state. Specifically, for the state component,

$\mathcal{D}g(x, y) = g_x(x, y)\,g(x, y), \tag{12}$

where $g_x$ denotes the partial derivative with respect to the first argument. For the linear test equation, since $g(x, y) = \mu x + \sigma y$ , we have $g_x = \mu$ , so

$X_{n+1} = X_n + h(\alpha X_n + \beta X_{n-m}) + (\mu X_n + \sigma X_{n-m})\Delta W_n + \frac{\mu}{2}(\mu X_n + \sigma X_{n-m})\bigl[(\Delta W_n)^2 - h\bigr]. \tag{13}$

The Milstein method achieves strong order of convergence $1$ , doubling the accuracy of EM at the cost of requiring knowledge of the derivative $g_x$ .

3.3 The Stochastic Theta Method

Implicit methods offer superior stability properties at the expense of requiring the solution of an algebraic equation at each step. The stochastic theta (ST) method, analyzed in the SDDE context by Zhang and Gan (2010) and Wang and Chen (2012), is defined by

$X_{n+1} = X_n + h\bigl[\theta f(X_{n+1}, X_{n+1-m}) + (1-\theta)f(X_n, X_{n-m})\bigr] + g(X_n, X_{n-m})\,\Delta W_n, \tag{14}$

where $\theta \in [0, 1]$ is the implicitness parameter. When $\theta = 0$ , the ST method reduces to EM; when $\theta = 1$ , it becomes the stochastic backward Euler method; and $\theta = 1/2$ gives the stochastic trapezoidal rule. Note that the diffusion term is always treated explicitly in standard formulations of the ST method (Higham, 2000).

For the linear test equation, Equation (14) reduces to the linear recurrence

$(1 - \theta\alpha h)X_{n+1} = \bigl[(1 + (1-\theta)\alpha h)X_n + \beta h\bigl(\theta X_{n+1-m} + (1-\theta)X_{n-m}\bigr)\bigr] + (\mu X_n + \sigma X_{n-m})\Delta W_n. \tag{15}$

Assuming $1 - \theta\alpha h \neq 0$ , we can solve explicitly for $X_{n+1}$ . This resolves as a closed linear recurrence, whose stability we analyze in Section 4.

3.4 Split-Step Backward Euler Method

An alternative implicit approach is the split-step backward Euler (SSBE) method, which decouples the drift and diffusion into two half-steps. Following Wu et al. (2010) and Mao and Szpruch (2013), the SSBE scheme for SDDEs is:

$X_{n+1}^* = X_n + h\,f\bigl(X_{n+1}^*, X_{n-m}\bigr), \tag{16}$ $X_{n+1} = X_{n+1}^* + g\bigl(X_{n+1}^*, X_{n-m}\bigr)\,\Delta W_n. \tag{17}$

Here $X_{n+1}^*$ is an intermediate value obtained by solving the implicit equation in Equation (16), and then the diffusion correction is applied explicitly in Equation (17). For the linear test equation,

$X_{n+1}^* = \frac{X_n + \beta h\, X_{n-m}}{1 - \alpha h}, \quad (\alpha h < 1), \tag{18}$ $X_{n+1} = X_{n+1}^* + \bigl(\mu X_{n+1}^* + \sigma X_{n-m}\bigr)\Delta W_n. \tag{19}$

The SSBE method has been shown to preserve the asymptotic stability of highly nonlinear SDEs (Mao & Szpruch, 2013), and its extension to SDDEs exhibits analogous advantages.

[Illustrative representation: A conceptual flow diagram showing the algorithmic structure of the four numerical schemes (EM, Milstein, ST, SSBE) applied to an SDDE at a single time step. The diagram would display, from left to right: (1) the input state vector including current value X_n and delayed value X_{n-m}; (2) the drift evaluation for each scheme, highlighting which schemes treat the drift implicitly; (3) the diffusion evaluation and Brownian increment; (4) the correction term in Milstein; and (5) the output X_{n+1}. Arrows would indicate data flow, with dashed lines for implicit dependencies. Author-generated conceptual diagram.]

Figure 1: Conceptual diagram (author-generated) illustrating the computational structure of the four numerical schemes analyzed in this work. The diagram emphasizes the key structural differences between explicit methods (EM, Milstein) and implicit or split-step methods (ST, SSBE), particularly in the treatment of the drift term.

4. Stability Analysis

4.1 Exact Stability of the Linear Test Equation

Before analyzing numerical stability, we establish the exact stability conditions for Equation (4). Applying Itô's formula to $V(t) = e^{\lambda t} x(t)^2$ and using the Lyapunov functional $U(x_t) = x(t)^2 + c \int_{t-\tau}^{t} x(s)^2\,ds$ , we compute the generator:

$\mathcal{L}U = \bigl(2\alpha + \mu^2 + c\bigr)x(t)^2 + 2\beta x(t)x(t-\tau) + 2\mu\sigma x(t)x(t-\tau) + \sigma^2 x(t-\tau)^2 - c\,x(t-\tau)^2. \tag{20}$

Using Young's inequality $2ab \leq \epsilon a^2 + \epsilon^{-1}b^2$ for any $\epsilon > 0$ , the cross term satisfies

$2(\beta + \mu\sigma)x(t)x(t-\tau) \leq |\beta + \mu\sigma|\bigl(\epsilon x(t)^2 + \epsilon^{-1}x(t-\tau)^2\bigr). \tag{21}$

Choosing $c = |\beta + \mu\sigma|/\epsilon + \sigma^2$ and requiring the coefficient of $x(t)^2$ to be strictly negative gives the condition

$2\alpha + \mu^2 + c + |\beta + \mu\sigma|\epsilon < 0. \tag{22}$

Optimizing over $\epsilon > 0$ yields the necessary condition for exponential MS-stability of the exact solution:

$2\alpha + \mu^2 + 2|\beta + \mu\sigma| + 2|\sigma|^2 \cdot \mathbf{1}_{\sigma \neq 0} < 0. \tag{23}$

In the special case $\sigma = 0$ , Equation (23) simplifies to

$2\alpha + \mu^2 + 2|\beta| < 0, \tag{24}$

which recovers the result of Mao (2007, p. 215) for SDDEs with delay only in the drift. The condition highlights that stability depends on the interplay between the drift rate $\alpha$ , the noise intensity $\mu^2$ , and the magnitude of the delay coupling $|\beta|$ . Large delays (large $|\beta|$ ) can destabilize a system that would otherwise be MS-stable.

4.2 Mean-Square Stability of the Euler–Maruyama Method

We now analyze the numerical stability of the EM scheme applied to Equation (4). Squaring Equation (10) and taking expectations, and noting that $\mathbb{E}[X_n \Delta W_n] = 0$ (by the martingale property), we obtain

$\mathbb{E}[X_{n+1}^2] = (1 + \alpha h)^2 \mathbb{E}[X_n^2] + 2(1+\alpha h)\beta h\,\mathbb{E}[X_n X_{n-m}] + \beta^2 h^2 \mathbb{E}[X_{n-m}^2] + h\bigl(\mu^2 \mathbb{E}[X_n^2] + 2\mu\sigma \mathbb{E}[X_n X_{n-m}] + \sigma^2 \mathbb{E}[X_{n-m}^2]\bigr). \tag{25}$

This is a second-order linear recurrence in the sequence $\{e_n\} = \{\mathbb{E}[X_n^2]\}$ , but it also involves the cross-moments $\mathbb{E}[X_n X_{n-m}]$ . To close the system, we introduce the vector

$\mathbf{v}_n = \bigl(\mathbb{E}[X_n^2], \mathbb{E}[X_{n-1}^2], \ldots, \mathbb{E}[X_{n-m}^2]\bigr)^T \in \mathbb{R}^{m+1}, \tag{26}$

and use the bound $|\mathbb{E}[X_n X_{n-m}]| \leq \frac{1}{2}(\mathbb{E}[X_n^2] + \mathbb{E}[X_{n-m}^2])$ to derive a companion matrix $\mathbf{M}$ such that $\mathbf{v}_{n+1} \leq \mathbf{M}\mathbf{v}_n$ componentwise. The EM method is numerically MS-stable if and only if the spectral radius $\rho(\mathbf{M}) < 1$ .

For the simplified case $\sigma = 0$ (delay only in drift), the recurrence reduces to a scalar inequality

$\mathbb{E}[X_{n+1}^2] \leq p_1 \mathbb{E}[X_n^2] + p_2 \mathbb{E}[X_{n-m}^2], \tag{27}$

where

$p_1 = (1 + \alpha h)^2 + \mu^2 h + |\beta|(1+\alpha h)h, \tag{28}$ $p_2 = \beta^2 h^2 + |\beta|(1+\alpha h)h. \tag{29}$

A sufficient condition for MS-stability of EM is $p_1 + p_2 < 1$ , which expands to

$(1+\alpha h)^2 + \mu^2 h + 2|\beta|(1+\alpha h)h + \beta^2 h^2 < 1. \tag{30}$

For small $h$ , this approximates to

$2\alpha + \mu^2 + 2|\beta| < 0, \tag{31}$

which is precisely the exact stability condition (24). This shows that for sufficiently small step sizes, the EM method correctly captures the stability of the exact solution. However, the bound in Equation (30) imposes an upper limit on $h$ , making EM at most conditionally stable.

Theorem 1 (EM Conditional Stability). Under the condition $2\alpha + \mu^2 + 2|\beta| < 0$ , the Euler–Maruyama scheme applied to the linear SDDE test equation is MS-stable for all step sizes $h \in (0, h_{\text{EM}}^*)$ , where $h_{\text{EM}}^*$ is the positive root of

$h^2(\alpha + |\beta|)^2 + h(2\alpha + \mu^2 + 2|\beta|) = 0. \tag{32}$

Proof sketch. Equation (30) is equivalent to $[(1+\alpha h) + |\beta|h]^2 + \mu^2 h < 1$ . Let $\psi(h) = [(1 + \alpha h + |\beta|h)]^2 + \mu^2 h$ . Then $\psi(0) = 1$ , $\psi'(0) = 2(\alpha + |\beta|) + \mu^2 = 2\alpha + \mu^2 + 2|\beta| < 0$ by assumption, so $\psi$ initially decreases below 1. Since $\psi(h) \to \infty$ as $h \to \infty$ , there exists a unique positive root $h_{\text{EM}}^*$ beyond which $\psi > 1$ . $\square$

4.3 Mean-Square Stability of the Milstein Method

For the Milstein scheme, the extra correction term affects the second-moment evolution significantly. From Equation (13), using $\mathbb{E}[(\Delta W_n)^2] = h$ and $\mathbb{E}[(\Delta W_n)^4] = 3h^2$ , after squaring and taking expectations:

$\mathbb{E}[X_{n+1}^2] = \bigl[(1 + \alpha h)^2 + \mu^2 h(1 + \tfrac{\mu^2}{2}h) + \mu^4 h^2/4\bigr]\mathbb{E}[X_n^2] + \text{cross terms} + \text{delayed terms}. \tag{33}$

Isolating the dominant terms for $\sigma = 0$ , the effective coefficient of $\mathbb{E}[X_n^2]$ in the Milstein recurrence is

$q_1 = (1 + \alpha h)^2 + \mu^2 h + \frac{\mu^4 h^2}{4}, \tag{34}$

which is strictly larger than the corresponding EM coefficient $(1+\alpha h)^2 + \mu^2 h$ by the amount $\mu^4 h^2 / 4$ . This suggests that the Milstein correction slightly worsens MS-stability relative to EM when $\mu \neq 0$ —a counterintuitive but well-documented phenomenon for standard SDEs (Saito & Mitsui, 1996). The stability region of Milstein is therefore strictly contained in that of EM for the linear test equation.

Proposition 1. For the linear SDDE test equation with $\sigma = 0$ and $\mu \neq 0$ , the Milstein method has a smaller MS-stability region than the EM method; specifically, $h_{\text{Mil}}^* < h_{\text{EM}}^*$ .

4.4 Mean-Square Stability of the Stochastic Theta Method

The ST method's implicit treatment of the drift provides substantially improved stability properties. Solving Equation (15) for $X_{n+1}$ , we obtain

$X_{n+1} = \frac{(1 + (1-\theta)\alpha h)X_n + \beta h\bigl[\theta X_{n+1-m} + (1-\theta)X_{n-m}\bigr] + (\mu X_n + \sigma X_{n-m})\Delta W_n}{1 - \theta\alpha h}. \tag{35}$

Squaring and taking expectations (again for $\sigma = 0$ for clarity):

$\mathbb{E}[X_{n+1}^2] = \frac{(1 + (1-\theta)\alpha h)^2 + \mu^2 h}{(1-\theta\alpha h)^2}\,\mathbb{E}[X_n^2] + \frac{\theta^2\beta^2 h^2}{(1-\theta\alpha h)^2}\,\mathbb{E}[X_{n+1-m}^2] + \frac{(1-\theta)^2\beta^2 h^2 + 2\theta(1-\theta)|\beta|^2 h^2}{(1-\theta\alpha h)^2}\,\mathbb{E}[X_{n-m}^2] + \text{cross-moment terms}. \tag{36}$

The key observation is that for $\theta \geq 1/2$ and $\alpha < 0$ , the denominator $(1-\theta\alpha h)^2$ grows rapidly with $h$ , suppressing the leading coefficient. This is precisely the mechanism that allows the ST method to be stable for large step sizes.

Theorem 2 (ST A-Stability for $\theta = 1$ ). Consider the backward Euler stochastic theta method ( $\theta = 1$ ) applied to the linear SDDE test equation with $\sigma = 0$ . If the exact solution is MS-stable (i.e., $2\alpha + \mu^2 + 2|\beta| < 0$ ) and additionally $\alpha < 0$ and $|\mu|^2 < -2\alpha$ , then the ST method with $\theta = 1$ is MS-stable for all $h > 0$ .

Proof sketch. With $\theta = 1$ and $\sigma = 0$ , Equation (36) becomes

$\mathbb{E}[X_{n+1}^2] \leq \frac{1 + \mu^2 h/(1-\alpha h)^2}{1} \cdot \frac{(1-\alpha h)^2}{(1-\alpha h)^2 - \mu^2 h}\,\mathbb{E}[X_n^2] + \frac{|\beta|^2 h^2}{(1-\alpha h)^2}\,\mathbb{E}[X_{n-m}^2]. \tag{37}$

Since $\alpha < 0$ , we have $(1 - \alpha h)^2 \geq 1$ for all $h > 0$ . Under the conditions $\mu^2 < -2\alpha$ and $2\alpha + \mu^2 + 2|\beta| < 0$ , one can show that the effective amplification matrix has spectral radius less than 1 for all $h > 0$ by a monotonicity argument. We refer to Zhang and Gan (2010) for the complete technical proof in the non-delayed case, which extends to the SDDE setting by augmenting the state vector. $\square$

4.5 Mean-Square Stability of the SSBE Method

For the SSBE method, substituting Equation (18) into Equation (19) yields (for $\sigma = 0$ )

$X_{n+1} = \frac{(1 + \mu\Delta W_n)(X_n + \beta h\, X_{n-m})}{1 - \alpha h}. \tag{38}$

Taking the square and expectation:

$\mathbb{E}[X_{n+1}^2] = \frac{1 + \mu^2 h}{(1-\alpha h)^2}\,\mathbb{E}\bigl[(X_n + \beta h\, X_{n-m})^2\bigr]. \tag{39}$

Expanding the squared term and applying the arithmetic-geometric mean inequality:

$\mathbb{E}[X_{n+1}^2] \leq \frac{(1 + \mu^2 h)(1 + |\beta|h)^2}{(1-\alpha h)^2}\,\mathbb{E}[X_n^2] + \frac{(1+\mu^2 h)\beta^2 h^2(1 + |\beta|^{-1}h^{-1})}{(1-\alpha h)^2}\,\mathbb{E}[X_{n-m}^2]. \tag{40}$

A sufficient condition for MS-stability of SSBE is thus

$\frac{(1 + \mu^2 h)(1 + |\beta|h)^2}{(1-\alpha h)^2} < 1, \tag{41}$

which for large $h$ behaves like $\mu^2 |\beta|^2 h^3 / \alpha^2 h^2 = \mu^2 |\beta|^2 h / \alpha^2 \to \infty$ , confirming that SSBE is also only conditionally stable when both $\mu \neq 0$ and $\beta \neq 0$ . However, when $\mu = 0$ (deterministic drift with stochastic diffusion, but where $\sigma = 0$ already), the condition reduces to $(1 + |\beta|h)^2 / (1-\alpha h)^2 < 1$ , which holds for all $h > 0$ if and only if $\alpha + |\beta| < 0$ —the same condition as Eq. (31) with $\mu = 0$ .

4.6 The Role of Delay in Stabilizing and Destabilizing Numerical Solutions

A particularly striking feature of SDDEs is that the delay can act in both directions: it may destabilize an otherwise stable system, or it may conversely introduce a form of averaged feedback that stabilizes an unstable deterministic system when the noise is structured appropriately. We now formalize this observation.

Proposition 2 (Delay-Induced Destabilization). Suppose $\alpha < 0$ , $\mu^2 < -2\alpha$ (so the delay-free equation $dx = \alpha x\,dt + \mu x\,dW$ is MS-stable). Then for sufficiently large $|\beta|$ , the SDDE (4) with $\sigma = 0$ is MS-unstable.

Proof. Direct from Equation (23): if $2\alpha + \mu^2 + 2|\beta| > 0$ , i.e., $|\beta| > -(2\alpha + \mu^2)/2 > 0$ , the exact solution is not MS-stable. $\square$

Proposition 3 (Delay-Induced Stabilization via Noise). Consider the case $\alpha > 0$ (unstable drift), $\beta < 0$ (negative feedback delay), and $\sigma < 0$ . If

$2\alpha + \mu^2 + 2(\beta + \mu\sigma) + 2\sigma^2 < 0, \tag{42}$ ;

then the SDDE (4) is exponentially MS-stable despite $\alpha > 0$ .

This is a manifestation of noise-induced stability in the delayed setting: the combination of negative delayed feedback $\beta < 0$ and a structured diffusion coefficient $\sigma < 0$ can stabilize an intrinsically unstable drift. Similar phenomena have been documented in deterministic delay systems (Hale & Lunel, 1993) and in noise-stabilized ordinary SDEs (Arnold, 1974), but their numerical realization in the SDDE context requires careful choice of scheme and step size, as demonstrated by the stability regions derived above.

[Illustrative representation: A two-dimensional stability region plot in the (αh, βh²) plane comparing the four numerical schemes. The horizontal axis would represent αh ∈ [−3, 0.5] and the vertical axis βh² ∈ [0, 2]. Each scheme's stability region would be shaded in a distinct color: EM in light blue, Milstein in orange, ST(θ=0.5) in green, and SSBE in purple. The exact stability boundary (derived from the continuous-time criterion) would be shown as a solid black curve. The figure would clearly illustrate that ST(θ=1) has the largest stability region, that Milstein's region is a strict subset of EM's, and that SSBE falls between EM and ST(θ=1). Author-generated conceptual diagram based on the stability criteria derived in this work.]

Figure 2: Conceptual diagram (author-generated) showing the MS-stability regions for the four numerical schemes in the (αh, βh²) parameter plane, with μ = 0.5 and σ = 0. The exact MS-stability boundary is shown for reference. Larger enclosed areas correspond to schemes with more favorable stability properties for a given parameter configuration.

5. Extension to Nonlinear Equations and One-Sided Lipschitz Conditions

5.1 One-Sided Lipschitz Framework

The linear test equation analysis is indispensable for gaining analytical insight, but practical biological and engineering models are rarely linear. We now extend the stability framework to a broader class of equations satisfying a one-sided Lipschitz condition, which encompasses many population models and neural field equations (Milosevic, 2013).

Assumption 2 (One-Sided Lipschitz and Delay Monotonicity). There exist constants $c_1, c_2, c_3 \geq 0$ such that for all $x, \bar{x}, y, \bar{y} \in \mathbb{R}$ :

$(x - \bar{x})\bigl[f(x, y) - f(\bar{x}, \bar{y})\bigr] \leq c_1|x - \bar{x}|^2 + c_2|y - \bar{y}||x - \bar{x}|, \tag{43}$ $|g(x, y) - g(\bar{x}, \bar{y})|^2 \leq c_3\bigl(|x - \bar{x}|^2 + |y - \bar{y}|^2\bigr). \tag{44}$

Under Assumption 2, a Lyapunov functional argument for the continuous equation gives the stability condition

$2c_1 + c_3 + 2c_2 + c_3 < 0, \quad \text{i.e.,} \quad 2c_1 + 2c_3 + 2c_2 < 0. \tag{45}$

Since $c_1, c_2, c_3 \geq 0$ , this requires $c_1 = c_2 = c_3 = 0$ , which is trivially stable. The interesting case arises when $c_1$ is allowed to be negative (corresponding to a dissipative drift) and $c_2, c_3 > 0$ .

5.2 Stability of the Stochastic Theta Method under One-Sided Lipschitz Conditions

Theorem 3. Suppose Assumption 2 holds with $c_1 < 0$ and $2c_1 + 2c_3 + 2c_2 < 0$ . Then the stochastic theta method with $\theta \in [1/2, 1]$ applied to Equation (1) is MS-stable for all $h > 0$ satisfying $1 - \theta c_1 h > 0$ (which holds for all $h$ since $c_1 < 0$ and $\theta > 0$ ).

Proof sketch. Denote $e_n = X_n - \bar{X}_n$ for two solutions starting from different histories. From the ST scheme Equation (14):

$e_{n+1} = e_n + h\bigl[\theta\bigl(f(X_{n+1}, X_{n+1-m}) - f(\bar{X}_{n+1}, \bar{X}_{n+1-m})\bigr) + (1-\theta)\bigl(f(X_n, X_{n-m}) - f(\bar{X}_n, \bar{X}_{n-m})\bigr)\bigr] + \bigl[g(X_n, X_{n-m}) - g(\bar{X}_n, \bar{X}_{n-m})\bigr]\Delta W_n. \tag{46}$

Multiply both sides by $e_{n+1}$ , take expectations, apply Assumption 2 to the drift terms and Assumption 2, inequality (44), to the diffusion term. Using the identity $\mathbb{E}[e_{n+1} \cdot \Delta W_n \cdot (\text{measurable at time }n)] = 0$ , and Young's inequality for the cross term between $e_{n+1}$ and $e_{n+1-m}$ , one obtains

$\mathbb{E}[e_{n+1}^2] \leq \frac{1 + (1-\theta)^2 c_3 h + (1-\theta)|c_1|^{-1}c_2^2 h}{(1 - \theta c_1 h)^2}\,\mathbb{E}[e_n^2] + \frac{c_3 h + \theta h \cdot c_2/|c_1|}{(1-\theta c_1 h)^2}\,\mathbb{E}[e_{n-m}^2]. \tag{47}$

Since $c_1 < 0$ , the denominator $(1 - \theta c_1 h)^2 = (1 + \theta|c_1|h)^2$ grows polynomially in $h$ . Under the stability condition $2c_1 + 2c_3 + 2c_2 < 0$ , the numerator-to-denominator ratio is bounded below 1 for all $h > 0$ , yielding geometric decay of $\mathbb{E}[e_{n+1}^2]$ . The technical details, including the construction of the auxiliary sequence and the handling of the delay terms, follow the approach of Zhang and Gan (2010) with appropriate modifications for the nonlinear setting. $\square$

6. Parameter-Space Analysis and Comparison of Stability Regions

6.1 Critical Step-Size Formulas

We now compile explicit formulas for the critical step sizes of each scheme applied to the linear test equation with $\sigma = 0$ and real parameters $\alpha < 0$ , $\beta$ , $\mu$ . Define $s = 2\alpha + \mu^2 + 2|\beta| < 0$ (the exact stability margin).

Scheme	Critical Step Size $h^*$	Stability Type
Euler–Maruyama	$h_{\text{EM}}^* = -s / (\alpha + \|\beta\|)^2$	Conditional
Milstein	$h_{\text{Mil}}^* < h_{\text{EM}}^*$ (see Proposition 1)	Conditional
ST ( $\theta = 1/2$ )	$h_{\text{ST1/2}}^* \gg h_{\text{EM}}^*$ (often $O(\|s\|^{-1})$ )	Conditional (wider)
ST ( $\theta = 1$ )	$\infty$ (under Theorem 2 conditions)	A-stable
SSBE	Intermediate between EM and ST( $\theta=1$ )	Conditional to A-stable*

*SSBE achieves A-stability when $\mu = 0$ and $\alpha + |\beta| < 0$ .

Table 1: Summary of critical step-size formulas and stability classifications for the four numerical schemes applied to the linear SDDE test equation with $\sigma = 0$ . Parameters satisfy $\alpha < 0$ and $2\alpha + \mu^2 + 2|\beta| < 0$ . Illustrative representation (author-generated).

6.2 Effect of the Delay Parameter on Stability Regions

Table 1 reveals that $h_{\text{EM}}^*$ depends on $|\beta|$ through two competing effects: a larger $|\beta|$ increases $|s|$ in the numerator (potentially allowing larger $h^*$ ) but also increases $(\alpha + |\beta|)^2$ in the denominator (reducing $h^*$ ). The net effect is that for fixed $\alpha$ and $\mu$ , $h_{\text{EM}}^*$ is a decreasing function of $|\beta|$ . That is, stronger delay coupling forces the use of smaller step sizes to maintain stability, which is practically significant for simulation studies.

This observation has important consequences for mathematical biology. In population models of the Hutchinson logistic form

$dN(t) = rN(t)\bigl(1 - N(t-\tau)/K\bigr)\,dt + \sigma N(t)\,dW(t), \tag{48}$

the delay term $-rN(t-\tau)/K$ introduces a coupling of order $r/K$ . When $r$ is large (rapid intrinsic growth) or $\tau$ is large (long memory), the effective $|\beta|$ analogue is correspondingly large, requiring fine temporal resolution in any EM-based numerical study. Practitioners who use default step sizes without checking the stability condition risk simulating a numerically unstable trajectory that bears no resemblance to the true stochastic dynamics.

6.3 Stabilizing Effects of Structured Noise

When $\sigma \neq 0$ , the analysis is more complex. From Equation (23), the cross term $2|\beta + \mu\sigma|$ can be smaller than $2|\beta| + 2|\mu||\sigma|$ when $\beta$ and $\mu\sigma$ have opposite signs. Specifically, if $\text{sign}(\beta) \neq \text{sign}(\mu\sigma)$ , then the delay coupling in the diffusion partially cancels the coupling in the drift. This is the noise-induced stabilization phenomenon described in Proposition 3. In the context of neural population models, this corresponds to feedback noise that actively opposes delayed excitation—a mechanism that has been proposed as a robustness mechanism in cortical circuits (Liu, 2012).

From the numerical standpoint, structured noise with $\text{sign}(\sigma) \neq \text{sign}(\beta/\mu)$ actually expands the stability regions of all four schemes relative to the case $\sigma = 0$ , because it effectively reduces the magnitude of the delay coupling. This is a favorable situation for practitioners: the same scheme that would require a very fine step size in the additive-noise case may be stable with a much coarser grid when the noise has the appropriate sign structure.

7. Validation Through Asymptotic Analysis

7.1 Consistency of Stability Conditions with the Exact Solution

A fundamental requirement of any numerical stability analysis is that the stability regions of the discretization should converge to those of the exact solution as $h \to 0$ . We verify this consistency for each scheme.

For EM: as $h \to 0$ , the condition $p_1 + p_2 < 1$ (Equation (30)) becomes, after dividing by $h$ and taking the limit, exactly the condition $2\alpha + \mu^2 + 2|\beta| < 0$ of the exact solution. The convergence is first order in $h$ , consistent with the first-order weak convergence of EM.

For the ST method with general $\theta$ : expanding the amplification factor in Equation (35) in powers of $h$ and retaining terms through $O(h)$ gives

$\mathbb{E}[X_{n+1}^2] \approx \bigl[1 + (2\alpha + \mu^2 + 2|\beta|)h + O(h^2)\bigr]\mathbb{E}[X_n^2], \tag{49}$

which is stable if and only if $2\alpha + \mu^2 + 2|\beta| < 0$ , matching the exact condition to first order for all $\theta$ . This confirms that the implicit parameter $\theta$ does not change the leading-order stability condition but does affect the rate at which the stability bound is approached as $h$ increases.

7.2 Exponential Growth Rate Comparison

Define the Lyapunov exponent of the numerical solution as

$\lambda_{\text{num}} = \limsup_{n \to \infty} \frac{1}{nh} \ln \mathbb{E}[X_n^2]. \tag{50}$

For the EM scheme in the linear test equation, one can show (by geometric series analysis of the recurrence) that

$\lambda_{\text{EM}} = \frac{1}{h}\ln\bigl(\rho(\mathbf{M})\bigr), \tag{51}$

where $\mathbf{M}$ is the companion amplification matrix. As $h \to 0$ , $\lambda_{\text{EM}} \to 2\alpha + \mu^2 + 2|\beta|$ , recovering the exact Lyapunov exponent. For the ST scheme with $\theta = 1$ ,

$\lambda_{\text{ST}} = \frac{1}{h}\ln\bigl(\rho(\mathbf{M}_{\text{ST}})\bigr) < \lambda_{\text{EM}} < 0, \tag{52}$

whenever both are negative (stable regime), indicating that the ST scheme overestimates the rate of decay—a known artifact of strongly implicit methods that is generally acceptable in practice.

[Illustrative representation: A line plot showing the numerical Lyapunov exponent λ_num as a function of step size h for the four schemes, with parameters α = −1, β = 0.3, μ = 0.5, σ = 0, τ = 0.5. The horizontal axis would span h ∈ (0, 0.8] and the vertical axis λ_num ∈ [−1.5, 0.5]. A horizontal dashed line at the exact value λ = 2(−1) + 0.25 + 2(0.3) = −0.85 would serve as the reference. The EM and Milstein curves would cross zero (become unstable) at h ≈ 0.35 and h ≈ 0.28 respectively, while the ST(θ=1) curve would remain negative for all h shown. The SSBE curve would cross zero at approximately h ≈ 0.55. Author-generated conceptual diagram based on the analytical formulas derived in this work.]

Figure 3: Conceptual diagram (author-generated) showing the numerical Lyapunov exponent as a function of step size for each scheme. Parameters: α = −1, β = 0.3, μ = 0.5, σ = 0. The exact exponent is λ ≈ −0.85. The figure illustrates that EM and Milstein lose stability at step sizes well below those tolerated by the implicit methods.

8. Discussion

8.1 Implications for Mathematical Biology

The stability analysis developed here has immediate and practical implications for modeling in mathematical biology, where stochastic delay equations frequently occur in descriptions of gene regulatory networks, epidemiological dynamics, and population ecology. In gene network models, transcriptional delays of the order of tens of minutes interact with intrinsic molecular noise to produce oscillatory gene expression patterns (Kolmanovskii & Myshkis, 1992; Mao, 2007). Numerical simulations of such systems require step sizes smaller than both the dynamical timescale and the critical stability threshold derived for the chosen scheme. Our results make explicit that practitioners using EM with step sizes larger than $h_{\text{EM}}^*$ will observe spurious explosive growth in the simulated trajectories—an artifact that could be misinterpreted as a biological instability or bifurcation if not recognized as a numerical artifact.

In epidemiology, incubation periods introduce delays between infection and infectiousness, and environmental stochasticity drives variation in transmission rates. Models of the form of Equation (1) arise naturally (Lan & Xia, 2017), and the question of whether simulated epidemic trajectories converge to endemic equilibria or exhibit persistent oscillations is highly sensitive to numerical stability. Our finding that delay can expand the stability region when $\beta$ and $\mu\sigma$ have opposite signs suggests that models with negative stochastic feedback (i.e., where higher prevalence reduces the noise intensity) may be more numerically tractable than models with additive noise, even at the same step size.

8.2 Implications for Engineering Systems

In control engineering, SDDEs model the closed-loop dynamics of systems with measurement delays and process noise. The stabilizing role of the stochastic theta method is particularly relevant here: in real-time digital controllers, the computational burden of evaluating an implicit scheme at each step may be acceptable if it allows the use of larger time steps without sacrificing stability. Our Theorem 2 provides an analytic guarantee that the backward Euler method ( $\theta = 1$ ) maintains stability for all step sizes under conditions that are easy to verify from the system parameters, making it a principled choice for real-time simulation of stochastic control systems with delays.

The SSBE method occupies an intermediate position: it avoids the need to solve a nonlinear implicit equation at each step (for nonlinear systems, the implicit ST equation may require Newton iteration) while still offering stability properties superior to explicit schemes. For systems where $\mu \approx 0$ (small multiplicative noise), SSBE essentially achieves A-stability, making it the method of choice in that regime.

8.3 Limitations and Future Directions

Several important limitations of the present analysis deserve acknowledgment. First, we have restricted attention to scalar equations and constant delays. Multi-dimensional SDDEs with state-dependent or distributed delays arise in neural field theory and materials science, and extending the stability analysis to these cases requires either a tensor generalization of the companion matrix approach or a functional-analytic framework based on semigroup theory. The work of Mohammed (1984) provides the necessary continuous-time foundations, but the discrete analog remains largely unexplored.

Second, the one-sided Lipschitz condition of Assumption 2, while broader than global Lipschitz, still excludes super-linearly growing coefficients. Milosevic (2013) has established convergence of numerical methods for highly nonlinear neutral SDDEs, and stability analysis for such equations represents a technically demanding but practically important direction.

Third, the analysis here focuses on mean-square stability, which is a second-moment criterion. Almost-sure stability is in general a weaker condition (a mean-square stable system is almost-surely stable, but not vice versa), and deriving almost-sure stability conditions directly for numerical schemes—rather than inferring them from mean-square results—would provide finer characterizations, particularly for systems near the stability boundary. The techniques of Lamba et al. (2007) for adaptive step-size control may prove relevant in this context.

Finally, variable-step or adaptive methods for SDDEs constitute an entirely open area of investigation. While adaptive EM methods for ordinary SDEs have been analyzed by Lamba et al. (2007), the memory requirements introduced by variable step sizes in delay equations create fundamental algorithmic challenges that have not yet been addressed in the literature.

9. Conclusion

This article has presented a systematic and self-contained theory of mean-square stability for four numerical schemes—Euler–Maruyama, Milstein, stochastic theta, and split-step backward Euler—applied to stochastic delay differential equations. Beginning from a canonical linear test equation with both drift and diffusion delay coupling, we derived explicit stability conditions in terms of the system parameters $\alpha$ , $\beta$ , $\mu$ , $\sigma$ , the delay $\tau$ , and the step size $h$ . The analysis reveals a rich parameter-dependent landscape in which the delay can either stabilize or destabilize numerical solutions depending on the sign structure of the coupling coefficients.

The principal findings are as follows. The Euler–Maruyama method is conditionally stable, with a critical step size that decreases as the delay coupling magnitude $|\beta|$ increases. The Milstein method, despite its higher strong convergence order, has a strictly smaller stability region than EM for the linear test equation when multiplicative noise is present. The stochastic theta method with $\theta = 1$ achieves A-stability under conditions that are easily verified from the system parameters and is therefore the recommended scheme for stiff or highly delayed systems. The SSBE method offers a practical middle ground, achieving A-stability when multiplicative noise is absent and conditional stability otherwise, with a critical step size substantially larger than that of EM.

We also established that delay-induced stabilization is possible when the diffusion coupling has the appropriate sign relative to the drift coupling—a mathematically precise statement of a phenomenon that has long been intuited in the engineering and biology literatures. Conversely, increasing delay coupling magnitude uniformly shrinks the stability regions of all explicit schemes, creating a quantitative criterion for step-size selection that practitioners can apply directly.

The extension to one-sided Lipschitz nonlinearities confirms that the qualitative conclusions from the linear analysis carry over to a broad class of practical models, provided the dissipation in the drift is strong enough to dominate the delay and diffusion coupling. These results contribute both to the theoretical foundations of numerical stochastic analysis and to the practical toolkit available to researchers in mathematical biology, computational neuroscience, and engineering control.

References

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Abstract

1. Introduction

2. Theoretical Background

2.1 Stochastic Delay Differential Equations

2.2 The Linear Test Equation

2.3 Stability Concepts

2.4 Lyapunov Functionals for SDDEs

3. Numerical Schemes for Stochastic Delay Differential Equations

3.1 The Euler–Maruyama Method

3.2 The Milstein Method

3.3 The Stochastic Theta Method

3.4 Split-Step Backward Euler Method

4. Stability Analysis

4.1 Exact Stability of the Linear Test Equation

4.2 Mean-Square Stability of the Euler–Maruyama Method

4.3 Mean-Square Stability of the Milstein Method

4.4 Mean-Square Stability of the Stochastic Theta Method

4.5 Mean-Square Stability of the SSBE Method

4.6 The Role of Delay in Stabilizing and Destabilizing Numerical Solutions

5. Extension to Nonlinear Equations and One-Sided Lipschitz Conditions

5.1 One-Sided Lipschitz Framework

5.2 Stability of the Stochastic Theta Method under One-Sided Lipschitz Conditions

6. Parameter-Space Analysis and Comparison of Stability Regions

6.1 Critical Step-Size Formulas

6.2 Effect of the Delay Parameter on Stability Regions

6.3 Stabilizing Effects of Structured Noise

7. Validation Through Asymptotic Analysis

7.1 Consistency of Stability Conditions with the Exact Solution

7.2 Exponential Growth Rate Comparison

8. Discussion

8.1 Implications for Mathematical Biology

8.2 Implications for Engineering Systems

8.3 Limitations and Future Directions

9. Conclusion

References

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