Stochastic Delay Differential Equations

• SDDEs are stochastic differential equations that incorporate time delays, capturing memory effects and the interaction between randomness and historical states.
• They are applied across engineering, biology, and machine learning to model systems where both the current state and past influences determine behavior.
• Advanced numerical methods, such as Euler–Maruyama variants and implicit schemes, ensure stability and strong convergence despite infinite-dimensional phase spaces.

A stochastic delay differential equation (SDDE) is a stochastic differential equation whose evolution depends not only on the current state but also on one or more delayed states, with randomness typically modeled by Brownian motion or general Wiener processes. SDDEs arise across mathematics, probability, engineering, biology, and machine learning when both stochasticity and memory effects are inherent in the system dynamics. They present unique analytical, stability, and numerical challenges due to their infinite-dimensional (history-dependent) state space and the interaction between noise and delay.

1. Formulation and Well-Posedness of SDDEs

Consider the $d$-dimensional SDDE with a single constant delay $\tau>0$: $$dX(t) = f\bigl(X(t), X(t-\tau)\bigr)\,dt + g\bigl(X(t), X(t-\tau)\bigr)\,dW(t), \quad t\ge 0,$$ with initial data $X(t)=\xi(t)$ on $t\in[-\tau,0]$, $f:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}^d$ (drift), $$g:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}^{d\times m}$$ (diffusion), and $W(t)$ an $m$-dimensional Brownian motion. The theory extends to multiple or state-dependent delays, variable coefficients, and even to mixed or non-Brownian drivers (Shevchenko, 2013).

Existence and uniqueness follow under a combination of local (or global) Lipschitz conditions in the state and delay arguments, and Lyapunov/Khasminskii-type moment bounds. For polynomial or super-linear coefficients, generalized Khasminskii–type and one-sided monotonicity conditions control potential moment explosion or finite-time blow-up (Song et al., 2020, Guo et al., 2017, Liu et al., 2022).

The phase space is $C([-\tau,0];\mathbb{R}^d)$. The solution $X(t)$ is non-Markovian in finite dimensions, but the associated segment process $X_t(\theta) = X(t+\theta)$, $\theta\in[-\tau,0]$, is a Markov process in the path space. In the presence of jumps or mixed rough noise, existence and uniqueness require further structural assumptions (e.g., Hölder continuity, Fréchet differentiability in the functional argument (Shevchenko, 2013)).

2. Stability, Contractivity, and Long-Term Behavior

The stability theory for SDDEs encompasses mean-square stability, contractivity, and exponential decay—often using invariant functionals or direct Itô-type arguments.

• Mean-square stability: Under local one-sided Lipschitz in the instantaneous variable, Lipschitz in the delayed variable, and appropriate control of the diffusion, solutions are stable:

$$\mathbb{E}|x(t)-y(t)|^2 \leq C\,\sup_{s\in[-\tau,0]}\mathbb{E}|\xi(s)-\eta(s)|^2$$

for two solutions with initial data $\xi,\eta$ (Gan et al., 2013). The constants are explicit in terms of bounds on $f$ and $g$.

• Asymptotic contractivity: Given $\lim_{t\to\infty}(t-\tau(t))=+\infty$ and combined negative drift-diffusion coefficients, any solution pair converges to the same path $\mathbb{E}|x(t)-y(t)|^2\to0$ as $t\to\infty$ (Gan et al., 2013).
• Exponential mean-square stability: For equations with delayed and current-state dissipation, exponential decay holds, e.g., $E|X(t)|^2 \leq C e^{-\alpha t}$ (Song et al., 2020). For $G$-Brownian motion–driven SDDEs, explicit delay-dependent conditions link allowable delay to stochastic Lyapunov inequalities for almost sure and mean stability (Fei et al., 2020). Near-optimal stability bounds are achievable for variable delays and neutral terms.
• Blow-up (Finite-time Explosion): For scalar SDDEs, the existence (with positive probability) of finite-time blow-up is equivalent to that for the undelayed SDE, under monotonicity bounding of the drift and regularity of the diffusion (Busse, 17 Dec 2024).

3. Numerical Methods and Strong Convergence Theory

The delay and stochasticity in SDDEs prevent direct application of classic SDE numerical techniques. Key developments include:

• Euler–Maruyama (EM) and Variants: Direct EM exists for globally Lipschitz SDDEs, but superlinear coefficients require modifications:
  • Truncated Euler–Maruyama (TEM): Coefficient truncation controls growth outside compact sets. Strong (mean-square) convergence order $1/2$ is established under a Khasminskii-type dissipativity and weakly local Lipschitz bounds (Song et al., 2020, Guo et al., 2017). Moment bounds are uniform, and exponential mean-square stability is preserved for sufficiently small step (Song et al., 2020).
  • Partially Truncated EM: For variable delay and switching diffusions, the super-linear part is truncated while preserving stability for the linear part (Cong et al., 2018).
  • Adaptive EM Schemes: Step size adaptation based on local drift magnitude maintains stability and $1/2$ convergence order for non-globally Lipschitz, time-dependent delay coefficients (Liu et al., 16 Apr 2024).
  • Neutral SDDEs: EM schemes converge with $1/2$ order for polynomially growing neutral/drift/diffusion, and nearly $1/2$ for compound Poisson noise (Ji et al., 2015).
• Implicit Methods:
  • Backward Euler–Maruyama (BEM): Provides unconditional mean-square stability in presence of super-linear growth or stiff drift (one-sided monotonicity), with $1/2$ order global strong convergence (Liu et al., 2022, Wang et al., 19 May 2025). BEM can handle infinite-horizon (ergodic) limits and convergence to invariant measures (Wang et al., 19 May 2025, Wang et al., 20 May 2025).
  • Split-Step Backward Euler (SSBE): Allows for variable delay via interpolation at unaligned time points. Exponential mean-square stability is achieved without step size constraint under one-sided Lipschitz drift (Wang et al., 2011).
• Magnus-Type Integrators and Milstein Schemes: For linear and semilinear SDDEs (especially with multiple delays), Magnus–Euler–Maruyama (MEM) and Magnus–Milstein (MM) schemes allow for order $1/2$ and $1$ strong convergence, leveraging matrix exponentials on the homogeneous part and properly computed iterated stochastic integrals. Augmented meshes guarantee order $1$ for multiple indivisible delays (Griggs et al., 21 Aug 2025, Griggs et al., 20 Jun 2025).
• Segment Process and Invariant Measures: Markov segment chains (discrete or continuous in the segment $\theta\mapsto X(t+\theta)$) allow for ergodic analysis, strong convergence results (uniform-in-time), and numerical approximation of invariant laws (Wang et al., 19 May 2025, Wang et al., 20 May 2025).

The following table overviews convergence properties for several principal algorithms:

Method	Strong Order	Handles Superlinear	Exponential Stability	Variable/State Delay	Invariant Measure	Reference
EM	$1/2$	No	No	Only constant	No	(Guo et al., 2017)
TEM	$1/2$	Yes	Yes	Yes	Yes	(Song et al., 2020)
BEM	$1/2$	Yes	Yes	Yes	Yes	(Liu et al., 2022)
SSBE	$1/2$	No	Yes (uncond.)	Yes	No	(Wang et al., 2011)
MEM/MM	$1/2,\,1$	No	Yes	Yes	No	(Griggs et al., 20 Jun 2025)
Adaptive EM	$1/2^*$	Yes	Yes	Yes	No	(Liu et al., 16 Apr 2024)

4. Advanced Topics: Singular, Mixed, and Structured SDDEs

Recent work addresses SDDEs beyond Itô/Brownian settings:

• Singular SDDEs and Rough Path Analysis: When diffusion depends only on delayed states, classical Itô-flow techniques fail due to lack of continuity in the path-norm. Such equations generate random dynamical systems (RDS) in a measurable field of Banach spaces and admit a full Multiplicative Ergodic Theorem (MET) and Lyapunov spectrum (Varzaneh et al., 2019).
• Mixed Wiener–Fractional SDDEs: SDDEs driven by both Brownian motion and Hölder-continuous rough noise possess robust well-posedness and moment bounds provided coefficients and fractional signals satisfy suitable regularity and growth (Shevchenko, 2013). Key results include continuous dependence on rough drivers and limit theorems for approximating solutions.
• SDDEs with Jumps or Nonlinear Expectation: Marcus SDDEs capture stochastic systems with jumps and memory. The traditional Fokker–Planck equation is unavailable due