Segment convergence for super-linear stochastic functional differential equations by the truncated Euler-Maruyama method

Abstract: Most existing literature focuses on pointwise convergence (i.e., convergence at a fixed time point) of numerical solutions for Stochastic functional differential equations (SFDEs). In contrast, this paper investigates the strong segment convergence (i.e., the strong order of convergence of the numerical segment process). For SFDEs with super-linear drift and diffusion coefficients, we employ the explicit truncated Euler-Maruyama (EM) scheme. First, we establish the uniform moment boundedness of the truncated EM solution over a finite time interval. Second, we derive the $L^2$-error estimate between the continuous numerical segment and the step numerical segment. Finally, we prove the strong convergence order of the numerical segment generated by the truncated EM. The results can be used to analyze invariant measures and ergodicity of numerical segment, and have important applications in practical problems such as path-dependent financial options. We also provide a numerical example to support the theoretical results.

• The paper establishes strong segment convergence rates for SFDEs with super-linear drift and diffusion using a truncated Euler-Maruyama scheme.
• It derives uniform moment bounds and L2 error estimates, enabling rigorous error propagation across the solution segment.
• The method guarantees numerical stability and efficiency, making it valuable for path-dependent option pricing and ergodicity analysis.

Segment Convergence Analysis for Super-Linear SFDEs via the Truncated Euler-Maruyama Method

Introduction and Problem Setting

The paper addresses the strong segment convergence of numerical solutions to stochastic functional differential equations (SFDEs) with super-linear drift and diffusion. While much of the existing literature emphasizes pointwise convergence results—a fixed-time strong error analysis for numerical approximations—segment convergence analyses the numerical approximation of the entire solution segment over an interval $[T-\tau,T]$. This is particularly relevant in applications such as path-dependent option pricing, ergodicity analysis, and invariant measure approximation.

The equation under consideration is

$$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$

with $X_t$ the solution segment on $[t-\tau, t]$, and $f, g$ allowed to exhibit super-linear growth.

Numerical Scheme: Truncated Euler-Maruyama

The core numerical approach is the truncated Euler-Maruyama (EM) method, where the standard explicit EM update is combined with a truncation mapping that restricts coefficients to a bounded domain. This is crucial for handling super-linearities, preventing numerical blowup and allowing moment bounds which are unrecoverable with standard EM schemes for such problems.

The discrete update is:

• Set $Y(k)$ to be the projection of $\hat Y(k)$ onto a ball with radius determined by the step size and the initial condition.
• Iterate:

$$\hat Y(k+1) = Y(k) + f(Y_k)\,\Delta + g(Y_k)\,\Delta B_k,\quad Y_{k}(\cdot) = \textrm{interpolated segment from last }m \textrm{ values}$$

where the truncation is step-size dependent and designed to ensure the coefficients are locally Lipschitz within the domain of the numerical trajectory.

Main Results: Segment Strong Convergence

The main theoretical contribution is a strong segment convergence rate for the truncated EM scheme. Specifically, for the segment process at terminal time $T$,

$$E\left[\|X_T - \bar Y_T\|^2\right] \leq C(T)\, \Delta^{\hat\gamma}$$

with explicit, parameter-dependent convergence order $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$0 given in closed form (see Theorem 4.3 in the paper).

This result is obtained in steps:

• Uniform moment bounds: Uniform-in-step-size moment estimates for the numerical scheme using truncated coefficients.
• $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$1 estimates for the error between continuous and stepwise interpolated segments.
• Strong segment convergence: Combining local error controls and using the Gronwall lemma to propagate bounds globally.
• Comparison with prior work: Previous results (e.g., [Li, Mao, Song, 2024, Stochastic Processes Appl.]) yielded strong order $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$2 for pointwise error for super-linear SFDEs, but did not establish segment process error. This work generalizes the analysis from the one-point case to the pathwise/segmental case.

For specific parameter choices, the method recovers rates $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$3 or $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$4, depending on the nonlinearity degree.

Implications and Practical Relevance

Ergodic Properties and Invariant Measures

Segment convergence is essential for analysis of ergodic properties and accurate approximation of the invariant measure of SFDEs. The result rigorously justifies Monte Carlo simulations for SFDEs in applications to long-time statistical behavior, where the process's entire path segment (not just terminal value) determines its statistical law. This is directly relevant to ergodicity and numerical invariant measures, as shown in related work such as [Shi et al., J. Diff. Eq., 2024].

Path-Dependent Option Valuation

The segment process approximation is critical for path-dependent options (e.g., Asian, barrier, lookback) [Higham & Mao, J. Comput. Finance, 2005]. The established strong pathwise error guarantees the reliability of Monte Carlo methods based on truncated EM for super-linear SFDE market models.

Numerical Stability and Efficiency

By offering an explicit scheme, the truncated EM possesses superior computational efficiency versus fully implicit (e.g., backward EM) schemes, which are often used in super-linear regimes but require nonlinear solves at each step. The truncated EM achieves strong mean-square stability and boundedness without such computational overhead.

Numerical Experiment

The paper provides simulation results on a prototypical stochastic volatility SFDE with super-linear coefficients. Empirical convergence rates exceed the conservative theoretical bound (observed: $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$5, theory: $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$6 or $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$7 in certain regimes), reflecting the typical sub-optimality of analytical rates due to worst-case estimates in proofs.

Links with Broader Numerical SDE Theory

This work integrates several modern trends in SDE numerics:

• Use of taming/truncation for super-linear SDEs (see, e.g., [Mao, J. Comput. Appl. Math., 2015] for ODEs/SDEs).
• Segment-based analysis for SFDEs, extending theory from pointwise to pathwise convergence, as critical for delay equations and path-functional functionals.
• Explicit order estimates with explicit rate dependencies on local Lipschitz and growth constants, capturing the influence of nonlinearity and delay.

Future Directions

The gap between observed and theoretical rates suggests potential for improved analysis. The derivation of sharper, possibly step size-adaptive or pathwise-adaptive, error bounds is an open problem. Extensions to higher-order explicit methods, models with jumps or regime-switching, and more general noise (e.g., $$\begin{cases} dX(t) = f(X_t)\,dt + g(X_t)\,dB(t), \quad t > 0,\ X(t) = \xi(t), \quad t \in [-\tau, 0], \end{cases}$$8-Brownian) are also natural directions, as are weak approximation error analysis for path-dependent observables.

Conclusion

The paper rigorously establishes strong segment convergence—via explicit mean-square error bounds—of the truncated Euler-Maruyama method applied to SFDEs with super-linear coefficients. This bridges a major gap and lays a robust foundation for practical simulation and statistical estimation in scientific and financial applications of delay SDEs, with theoretical guarantees for a class of explicit schemes that are both efficient and reliable in the super-linear, non-Lipschitz regime (2604.21704).