On splitting strategies for the numerical solution of stochastic delay differential equations with correlated noises

1. Problem Statement

The paper addresses the numerical solution of a specific class of Scalar Semilinear Stochastic Delay Differential Equations (SDDEs) driven by a pair of correlated Brownian motions. The equation models a system with two feedback channels:

1. Linear Instantaneous Feedback: A term involving the current state $X(t)$ perturbed by noise.
2. Nonlinear Delayed Feedback: A term involving the delayed state $X(t-\tau)$ perturbed by noise.

The governing equation (in Itô form) is:
$$dX(t) = [\mu X(t) + f(X(t-\tau))] dt + \sigma X(t) dW_1(t) + g(X(t-\tau)) dW_2(t)$$
where $W_1$ and $W_2$ are standard Brownian motions with correlation coefficient $\rho \in (-1, 1)$.

The Core Challenge:
Standard numerical methods for SDDEs can be computationally expensive. Splitting methods offer an efficient alternative by decomposing the equation into simpler subsystems (one linear without delay, one nonlinear with delay) that can be solved separately and recombined. However, the authors investigate whether this strategy remains valid and convergent when the noise sources ($W_1, W_2$) are correlated ($\rho \neq 0$).

2. Methodology

2.1. Decomposition Strategy

The authors decompose the SDDE into two subsystems:

• Subsystem 1 (Nonlinear Delay): Contains the nonlinear drift $f$ and diffusion $g$ terms with delay. This is solved numerically using the Euler-Maruyama method.
• Subsystem 2 (Linear No Delay): Contains the linear instantaneous terms. This subsystem admits an exact analytical solution (geometric Brownian motion type).

2.2. Splitting Schemes

Two specific composition strategies are analyzed:

1. Lie-Trotter Splitting: Solves Subsystem 1 then Subsystem 2 over a time step $\delta t$.
   $$X_{n+1} = \Psi^{[2]}(\Psi^{[1]}(X_n))$$
2. Strang Splitting: Solves half-step of Subsystem 1, full step of Subsystem 2, then half-step of Subsystem 1.
   $$X_{n+1} = \Psi^{[1]}(\Psi^{[2]}(\Psi^{[1]}(X_n)))$$

2.3. Theoretical Framework

• Variation of Constants: The authors utilize the method of steps and variation of constants to construct the exact solution, providing a reference for error analysis.
• Moment Bounds: They derive rigorous bounds on the moments of the fundamental solution and the numerical scheme to ensure stability.
• Error Analysis: The main theoretical contribution is a derivation of the mean-square strong error bound for the Lie-Trotter scheme. They employ a Delay Gronwall Inequality to handle the time-delay terms in the error propagation.

3. Key Contributions

3.1. Theoretical Error Bound for Correlated Noise

The paper provides a rigorous upper bound on the global mean-square error for the Lie-Trotter splitting method. The bound is given by:
$$\max_{t \in [0,T]} \mathbb{E}[|X(t) - Y(t)|^2] \leq K_1 \delta t + K_2 |\rho|$$
Where:

• $K_1, K_2$ are constants independent of the step size $\delta t$ and correlation $\rho$.
• Crucial Finding: When $\rho = 0$ (uncorrelated noise), the error is $O(\delta t)$, implying a strong convergence order of $1/2$.
• Critical Limitation: When $\rho \neq 0$, the error bound contains a term $K_2 |\rho|$ that is independent of the step size $\delta t$. This implies that as $\delta t \to 0$, the error does not necessarily vanish if the correlation is non-zero. The method loses its guarantee of strong convergence in the presence of correlation.

3.2. Numerical Verification

The authors conduct extensive numerical experiments on two test cases (one with oscillatory trajectories, one non-oscillatory) to verify the theory:

• Case $\rho = 0$: Both Lie-Trotter and Strang splitting exhibit the expected order 1/2 strong convergence.
• Case $\rho \neq 0$: As the magnitude of the correlation $|\rho|$ increases, the observed order of convergence rapidly drops to zero.
• Strang Splitting: While no theoretical proof is provided for Strang splitting, numerical results show it behaves similarly to Lie-Trotter, suffering the same loss of convergence order when noise is correlated.

4. Results

1. Convergence Order:

   • Uncorrelated ($\rho=0$): Strong convergence order $\gamma = 0.5$.
   • Correlated ($\rho \neq 0$): The convergence order degrades effectively to 0 as $|\rho|$ increases. The global error is bounded by a constant dependent on $|\rho|$ rather than the step size.

2. Error Behavior:

   • The error bound is uniform in $\delta t$ for $\rho \neq 0$. This means refining the time step does not improve accuracy if the correlation is significant; the error is dominated by the correlation term.

3. Comparison of Schemes:

   • Lie-Trotter and Strang splitting perform similarly in this context. Neither strategy successfully mitigates the detrimental effect of correlated noise on convergence rates for this specific decomposition.

5. Significance and Implications

• Warning for Practitioners: The paper serves as a critical warning for numerical analysts. While splitting methods are attractive for SDDEs due to their efficiency (solving linear parts exactly), they are not robust against correlated noises when the decomposition separates the noise sources into different subsystems.
• Theoretical Insight: It highlights that the commutativity of the noise operators (which holds when $\rho=0$) is essential for the convergence of splitting methods in this context. When noise is correlated, the non-commutativity introduces an error that cannot be reduced by mesh refinement.
• Future Directions: The authors suggest that future work should focus on:
  • Developing splitting strategies that preserve the independence of noise perturbations across subsystems.
  • Extending these results to finite-dimensional systems.
  • Investigating dynamic consistency (e.g., preserving oscillatory behavior and stationary distributions) under these conditions.

In conclusion, the paper demonstrates that while splitting methods are theoretically sound for SDDEs with independent noises, correlated noise fundamentally breaks the convergence guarantees of standard Lie-Trotter and Strang splitting strategies, rendering them ineffective for high-precision simulations unless the correlation is negligible.