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Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion

Alexander Abreu, Lisandro Fermin, Ernesto Mordecki, Soledad Torres

arXiv:2609.37019v1math.PRcs.NAmath.NA

TL;DR

For fBm-driven SFDEs with distributed memory, the interaction between memory quadrature, pathwise fractional calculus, and implicit drift evaluation makes a backward scheme’s analysis delicate. The paper establishes consistency and proves uniform global pathwise convergence for a backward-Euler method. Its convergence order can be taken arbitrarily close to 2H −1, matching the explicit Euler scheme’s rate.

  • Problem

    Combining distributed memory with fBm is delicate because pathwise fractional calculus, memory quadrature over the delay window, and implicit drift evaluation must be handled together.

    Source: §1 · p. 2

  • Method

    The paper analyzes a backward-Euler scheme that shares the explicit scheme’s left-point diffusion and memory discretizations but evaluates the drift at the right endpoint.

    Source: §1 · p. 3

  • Results

    The scheme has uniform global pathwise convergence of order H −ρ−β for β ∈(1−H, 1/2) and 0 < ρ < H−β, arbitrarily close to 2H −1.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The backward-Euler scheme attains the same convergence rate as the explicit Euler scheme, arbitrarily close to 2H −1.

    Source: §1 · p. 3

    The implicit solve requires hL2 < 1, an additional step-size restriction absent from the explicit scheme.

    Source: §1 · p. 3

Abstract

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We study the backward-Euler scheme for a class of stochastic functional differential equations (SFDEs) with memory driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first establish the local consistency of the method, with a local truncation error of order H -$\rho$- $\beta$ + 1, and then, as the main result, prove the uniform global pathwise convergence of the numerical approximation, on a set of probability one, with order H -$\rho$ -$\beta$, for $\beta$ $\in$ (1-H, 1/2) and 0 < $\rho$ < H -$\beta$. A uniform high-probability formulation with deterministic constants follows as a corollary. In particular, the convergence order can be taken arbitrarily close to 2H -1, matching the order obtained for the explicit Euler scheme. The convergence proof relies on an exact integral representation of the scheme at mesh points, an a priori bound for the numerical solution, fractional calculus estimates, and a Gronwall inequality for weakly singular kernels.

1 Introduction

The paper extends pathwise numerical analysis for fBm-driven SFDEs to a backward-Euler scheme with distributed memory, where implicit drift evaluation adds proof obligations. It establishes consistency and global convergence while retaining the explicit scheme’s convergence rate.

1 Introduction

  • Combining distributed memory with fBm is delicate because the noise requires pathwise fractional calculus, memory quadrature spans the delay window, and the drift is evaluated implicitly.The scheme extends the pathwise analysis of the explicit Euler method to this combination.Source: §1 · pp. 2–3
  • The analysis establishes well-posedness of each implicit step and couples estimates for the state, its fractional increments, and memory-functional discretization error.The convergence proof uses an exact mesh-point integral representation and uniform pathwise estimates.Source: §1 · p. 3
  • Unlike the explicit scheme, each backward-Euler step requires an implicit solve, and its analysis must control the right-endpoint drift projection.The schemes share the left-point diffusion discretization and distributed-memory quadrature.Source: §1 · pp. 2–3
  • The implicit well-posedness condition hL2 < 1 adds a step-size restriction absent from the explicit scheme.The a priori argument also requires smallness conditions to absorb mesh-dependent remainders.Source: §1 · p. 3
  • The backward-Euler scheme is consistent and converges uniformly pathwise at rate H −ρ−β, for β ∈(1−H, 1/2) and 0 < ρ < H−β.Its order can be chosen arbitrarily close to 2H −1.Source: §1 · p. 3
  • The paper includes a numerical study illustrating the practical behaviour of both schemes on a concrete example.Stronger stability results under monotonicity or one-sided Lipschitz conditions are not required for the convergence theorem.Source: §1 · p. 3

2 The Stochastic Functional Differential Equation

The section formulates an fBm-driven SFDE with distributed memory and states regularity assumptions used for its analysis. Under the stated hypotheses, the equation has a unique solution with pathwise Hölder regularity.

2 The Stochastic Functional Differential Equation

  • The equation is driven by fractional Brownian motion with H > 1/2, and its stochastic integral is interpreted as a generalized Lebesgue-Stieltjes integral.The drift and diffusion depend on both the state and the distributed-memory functional.Source: §2 · p. 4
  • The distributed-memory functional is built from a continuous kernel K acting on past states over the delay interval.The coefficient hypotheses include regularity conditions on the kernel and the state-dependent coefficients.Source: §2 · p. 4
  • The functional coefficient assumptions imply the regularity of σ(t,x,y) needed in the numerical estimates, rather than imposing it as an additional scheme assumption.The kernel’s differential regularity supplies the four-point estimate used to control memory-term increments.Source: §2 · pp. 4–6
  • Under the stated hypotheses, the SFDE has a unique solution whose extended path is Hölder continuous of order H −ρ for every 0 < ρ < H, almost surely.The corresponding bound has an almost surely finite random constant.Source: §2 · p. 6

3 Backward-Euler Scheme for the SFDE

The scheme uses a left-endpoint diffusion evaluation and memory quadrature, with the drift evaluated implicitly at the next mesh point. Auxiliary interpolations and memory estimates support analysis between mesh points and control the scheme’s errors.

3 Backward-Euler Scheme for the SFDE

  • The numerical estimates fix β ∈(1−H, 1/2) and 0 < ρ < H−β, and use left- and right-endpoint mesh projections to represent diffusion and drift evaluations.The left projection selects an interval’s left endpoint and the right projection its right endpoint.Source: §3 · p. 7
  • The backward-Euler update evaluates the drift at the next mesh point, while the diffusion uses the left endpoint and the memory term depends on already computed values.The implicitness is confined to the drift variable.Source: §3 · p. 7
  • The auxiliary continuous extension contains a future nodal value inside each interval and is therefore generally not adapted; convergence is formulated pathwise in the uniform norm.The mesh-point values, not the interior extension, constitute the numerical scheme.Source: §3 · p. 8

3.1 Estimates for the Numerical Scheme

  • The estimates relate the memory-functional approximation to the exact memory term and bound differences between the scheme’s memory approximation and the exact-solution-based approximation.The latter difference is controlled by the difference between the exact solution and the backward-Euler scheme.Source: §3.1 · pp. 8–9

4 Consistency and Convergence

The section analyzes the backward-Euler method’s local consistency and uniform pathwise convergence, developing pathwise regularity bounds, error estimates, and the representations and estimates used to establish convergence at rates below 2H − 1.

4.1 Local Error of the Backward-Euler Method

  • O(h^(H-rho-beta+1)) is the uniform local truncation-error bound, so the backward-Euler method is consistent of order H-rho-beta.The bound’s constant is independent of the mesh size and step index.Source: §4.1 · p. 10, §4.2 · p. 11

4.2 Convergence of the Backward-Euler Scheme

  • The convergence proof combines an exact integral representation at mesh points, a priori bounds, fractional estimates for the fBm integrals, and a weakly singular-kernel Gronwall inequality.The estimates control pointwise error, increments, and the memory term.Source: §4.2 · pp. 11–12
  • When hL2 < 1, each implicit step has a unique solution, established by expressing the step as a contraction and applying the Banach fixed-point theorem.The discrete initial condition then determines the full sequence by induction.Source: §4.2 · p. 12
  • The mesh-point representation shows that backward Euler shifts only the drift evaluation to the right endpoint; the diffusion remains evaluated at the left endpoint.This separates estimates inherited from explicit Euler from those requiring adaptation for the implicit drift.Source: §4.2 · p. 13
  • The auxiliary interpolation is pathwise (H - rho)-Hölder continuous on [−r, T], with constants independent of h for sufficiently small mesh sizes.The stated regularity bounds cover both [0, T] and the prescribed history interval [−r, 0].Source: §4.2 · pp. 19–20

4.2.1 Error Functionals

  • The error functional Eh tracks differences between the exact solution and approximation, unlike phi_h, and convergence is proved by jointly controlling Zh, theta_h, and kappa_h.The target is a closed integral inequality for Eh with a free term of order h^(H - rho - beta).Source: §4.2.1 · p. 21

4.2.2 Global Error Estimates

  • The global error argument combines estimates for the delay functional, diffusion, drift, and memory, then applies weakly singular Gronwall to a closed inequality for Eh.The pathwise estimates hold for sufficiently small h, with thresholds and constants depending on the sample path.Source: §4.2.2 · p. 25
  • On a probability-one set, the interpolation converges uniformly with order H - rho - beta for beta in (1 - H, 1/2) and 0 < rho < H - beta, attaining every rate below 2H - 1.The endpoint rate 2H - 1 is not established; it is approached as beta decreases to 1 - H and rho decreases to zero.Source: §4.2.2 · pp. 25–26

5 Numerical Study

The numerical study illustrates the backward-Euler scheme on a scalar distributed-delay SFDE and compares it with explicit Euler using a fine-mesh numerical reference. Over the tested meshes, backward Euler has the larger empirical convergence slope, though the reference and finite mesh range limit quantitative interpretation.

5.1 Model Considered

  • The example uses affine drift and nonconstant affine diffusion with a linear distributed-memory kernel, and all coefficients satisfy the stated hypotheses.The parameters are H = 0.7, r = T = 1, a = 1, c = 0.3, κ = 0.5, σ0 = 0.25, σ1 = 0.15, and x0 = 1.Source: §5.1 · p. 26

5.2 Simulation Design

  • The comparison uses the same subsampled fractional Brownian path for both schemes and averages maximum mesh-point errors over 24 realizations.The reference is backward Euler on a fine mesh with Nfine = 2048 because no closed-form exact solution is available; using a same-family reference may partially favor backward Euler.Source: §5.2 · p. 27

5.3 Numerical Results

  • The empirical log-log slopes are 1.18 for backward Euler and 0.84 for explicit Euler.For ρ = 0.02 and β = 0.35, the guaranteed exponent is 0.33 and the limiting value is 2H −1 = 0.4; the theorem does not claim the guaranteed exponent is exact for every coefficient choice.Source: §5.3 · p. 27
  • Across the tested range, both errors decay regularly and backward Euler has a smaller discrepancy from the numerical reference than explicit Euler.The observed slopes may reflect pre-asymptotic effects and reference error, so they are not estimates of an exact asymptotic rate.Source: §5 · p. 26, §5.3 · p. 27

A Technical Details for the Error Transfer

The appendix makes explicit which singular-integration estimates depend only on the left-point projection and thus remain unchanged under backward drift evaluation.

A Technical Details for the Error Transfer

A.1 Fractional integral estimate

The fractional-integral estimate has a finite constant on trajectories with the stated Hölder regularity when the exponent parameters satisfy the given bounds.

A.1 Fractional integral estimate

  • The estimate's constant is finite when the path has finite H−ρ Hölder seminorm and 1−H < β < H−ρ.Source: §A.1 · p. 28

A.2 Seven-term decomposition of the delay increment

The appendix bounds delay increments by decomposing the memory interval into regions and applying Hölder regularity and singular-integral estimates.

A.2 Seven-term decomposition of the delay increment

  • The increment argument uses second increments of K and state-increment products; its algebraic identity requires no estimate of the backward drift at that stage.Source: §A.2 · pp. 28–29
  • Initial-history and origin-crossing regions are controlled using Hölder continuity and the bound u−Ch(u) ≤ h.Source: §A.2 · p. 30
  • The most singular factors on A1 and short v-intervals integrate to order h^(H−ρ−β), using H−ρ−β > 0.Source: §A.2 · p. 30
  • On A5, splitting the memory interval into three pieces yields a κh(u) term and bounds the remaining terms by C(ω)h^(H−ρ).Source: §A.2 · p. 30
  • On A6, Hölder regularity bounds Υh(u, v, s) by C(ω)(u−v)^(H−ρ), and the v-interval has length at most h.Source: §A.2 · p. 30

A.3 Diffusion transfer through the left-point projection

The diffusion error is decomposed into state, time-projection, and within-cell interpolation components, then bounded to transfer estimates through the left-point projection. The decomposition uses only the projection map C_h and applies identically to the explicit and backward schemes.

A.3 Diffusion transfer through the left-point projection

  • The decomposition separates state, time-projection, and within-cell interpolation errors, uses only C_h, and is identical for the explicit and backward schemes.This structure supports the diffusion-transfer estimates used in the backward analysis.Source: §A.3 · p. 31
  • The functional Hypothesis 2 permits applying Lemma 2.2 to the four exact-state and interpolated-state pairs at s and v.The resulting terms, combined with pathwise a priori bounds, supply the contributions in (45).Source: §A.3 · p. 31
  • The fractional increment integral splits at C_h(s): the first region uses time-Lipschitz bounds, while the second uses its length-at-most-h bound and Hölder estimates.The latter estimates apply to the interpolation error eX_h and eY.Source: §A.3 · p. 31
  • Applying the increment estimate on [v, u] and integrating against (u − v)^−β−1 yields two Volterra kernels by Fubini’s theorem.The shifted-interval calculation also establishes the corresponding estimate over −C_h(v) ≤ s ≤ 0.Source: §A.3 · p. 31
  • The shifted-interval bounds contribute C(ω)h^(H−ρ−β) without introducing any occurrence of C_+h.The absence of C_+h is identified as the transfer property used in the backward analysis.Source: §A.3 · p. 31
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