The Numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion

1. Introduction

• Numerical simulations typically display discretized sample trajectories, motivating the question of whether their Onsager-Machlup functional is more relevant than the one for the exact solution.The introduction asks whether numerical schemes preserve the Onsager-Machlup structure of the original fractional Brownian motion-driven SDEs.Source: §1 · p. 2
• The setup studies limits of small-ball probability ratios for Euler approximations and families of numerical reference paths on a uniform partition.The partition is on [0, 1] with nodes t_i = i/n.Source: §1 · p. 2
• The numerical Onsager-Machlup action functional is stated for singular and regular cases for numerical reference paths satisfying K_H dot(phi_n) = phi_n - x.The discrete functional notably has no divergence correction term, unlike the continuous functional.Source: §1 · p. 3
• The derivation reorganizes coupled double sums from the discretized Volterra kernel and decomposes fractional operators, proving nuclearity of corresponding integral operators.The estimates address singular or regular kernel behavior and coupling across partition intervals.Source: §1 · p. 3
• The paper derives numerical fractional Euler-Lagrange equations describing most probable paths of the discretized solution process and presents experiments supporting the theoretical results.Source: §1 · p. 3

2.1. Fractional calculus.

• Fractional calculus preliminaries define right-sided Riemann-Liouville integrals and recall composition formulas, derivative representations, and an integral norm estimate.The integral extends the usual iterated integral when its order is a positive integer.Source: §2.1 · pp. 3–4

2.2. The Wiener integral with respect to fractional Brownian motion.

• The fractional Brownian motion is represented in law using standard Brownian motion and the kernel operator K_H, whose mapping properties and fractional-integral representation are recalled.The text also specifies conditions on p for the operator's continuity and image space.Source: §2.2 · pp. 4–5

3.1. The structure of numerical reference path.

• The reference paths are constrained through the K_H operator, with their fractional derivative in Lp and p > 1/H.The setting assumes a drift b in C2 and defines the discretized SDE; the reference paths also satisfy a stated fractional-space condition.Source: §3.1 · p. 5

3.2. Girsanov transform for discretized stochastic differential equation.

• A change of measure justified by Novikov's condition makes the transformed Brownian motion standard and the transformed fractional Brownian motion retain its fractional Brownian law.Under the transformed measure, the centered Euler approximation becomes fractional Brownian motion, reformulating the small-ball probability problem.Source: §3.2 · p. 6

4. The numerical Onsager-Machlup action functional for discretized SDEs

• The numerical Onsager–Machlup action functional is derived for both singular H < 1/2 and regular H > 1/2 regimes.The section presents the corresponding result statements separately for the two regimes.Source: §4 · p. 7
• In the singular regime, the reference paths are assumed to satisfy ϕ^n − x ∈ H_p for p > 1, and the theorem’s estimates are summarized as establishing the action-functional result.Source: §4 · pp. 7, 11
• For 1/2 < H < 1, the action functional exists for the supremum norm and for Hölder norms with 0 < β < H − 1/4, assuming ϕ^n − x ∈ H_2 and b ∈ C_b^2(R).Source: §4 · pp. 7, 11
• The regular-regime proof handles the Hölder norm directly and states that the proof for the supremum norm is the same.Source: §4 · p. 7

22. Recall

• In the diagonal-term calculation, P^n_3(s,s) is zero, and the resulting diagonal expression is written as an integral of the terms P^n_1, P^n_2, and P^n_3.Source: §22 · p. 18
• The residual term is introduced for estimating quadratic variation, with b′′ assumed Lipschitz in the ensuing bound.Source: §22 · p. 18
• The preceding inequalities and Lemma A.8 yield the section’s concluding expression.Source: §22 · p. 19

• Numerical experiments are used to verify the theoretical results.Source: §5 · p. 20
• The most probable path of the numerical solution process satisfies the variational condition obtained from the numerical Onsager–Machlup action functional.The section derives its Euler–Lagrange equations using the calculus of variations.Source: §5 · p. 20
• The derivation also identifies a jump in the path derivative at a node in the other case.Source: §5 · p. 21
• The Euler–Lagrange analysis distinguishes singular and regular cases for the fractional equation.The section derives an explicit discrete form for the equation in each case.Source: §5 · pp. 21, 23
• In the singular case, the jump condition applies to the residual A_n; a finite jump of the path derivative at nodes is not asserted.Source: §5 · p. 23

• The example uses a periodic potential with local minima at x = 2m + 1 and maxima at x = 2m, representing metastable states.The potential barrier at x = 0 separates the two metastable states.Source: §6 · p. 23
• For the scalar Euler scheme, the section presents the corresponding Onsager–Machlup action functional and fractional Euler–Lagrange equation.Source: §6 · p. 23
• As n increases from 10 to 100 at H = 0.35, the numerical paths become smoother and more stable.Figure 1 also compares most probable paths for n = 2, 10, 100 at H = 0.35 and for H = 0.35, 0.4, 0.45 at n = 20.Source: §6 · p. 24, Figure 1 · p. 24

• The estimate |u + v + w|^p ≤ 3^(p−1)(|u|^p + |v|^p + |w|^p) bounds a three-term sum by separate terms.Source: §A.1 · p. 25

• The Euler-scheme convergence proof uses the scheme's definition to estimate the process over s ∈ [0, 1).Source: §A.1 · p. 25
• The estimates apply Hölder’s inequality, Tonelli’s theorem, and the Lipschitz continuity of the drift coefficient.Source: §A.1 · p. 25
• Gronwall’s inequality gives a bound with constant C_{p,L} independent of n, completing the convergence proof.Source: §A.1 · p. 25

A.2. Approximate limits in Wiener space.

• Finite-rank orthogonal projections Q on H1 define QW independently of the chosen orthonormal basis, and approximating projection sequences increase strongly to the identity.Source: §A.2 · pp. 25–26
• Measurable seminorms are defined through convergence in probability of N(Q_nW) to a finite random variable with positive small-ball probabilities; a measurable norm is such a seminorm that is a norm.Source: §A.2 · p. 26
• The appendix introduces measurable-norm and nuclear-kernel results as basic ingredients for computing the numerical Onsager-Machlup action functional.Source: §A.2 · p. 26
• NH and NH,β are measurable norms whose associated fractional Brownian motion quantities are its supremum and β-Hölder norms, respectively.Source: §A.2 · p. 27

A.3. Technical lemmas.

• Lemma A.9 states that the process η defined by (3.13) is adapted.Source: §A.3 · p. 27
• The proof decomposes terms into bounded linear functionals and rank-one nuclear operators, then combines them into finite sums of nuclear operators.Source: §A.3 · pp. 28–29
• Under the stated ϕn − x ∈ Hp condition for p > 1, a symmetrized kernel operator is shown to be nuclear.Source: §A.3 · p. 29
• A finite-dimensional decomposition of fn yields a finite-rank operator K(fn), so its symmetrization is also finite-rank and nuclear.Source: §A.3 · p. 30

Conflicts of interests

• The authors declare no conflicts of interest.Source: Conflicts of interests · p. 31