Ergodic Network Stochastic Differential Equations

Francesco Iafrate; Stefano Iacus

REVIEW 4 major objections 7 minor 29 references

Ergodic Network Stochastic Differential Equations

T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For network SDEs, drift parameters and the directed graph can both be estimated from high-frequency observations of a single trajectory.

desk verdict Useful new model class, but the advertised growing-network graph-recovery claims are not backed by the proofs; the fixed-dimensional parts are solid. read the letter →

arxiv 2412.17779 v3 pith:V6H3ZQQG submitted 2024-12-23 stat.ME math.STstat.TH

Francesco Iafrate , Stefano Iacus This is my paper · ORCID

classification stat.MEmath.STstat.TH MSC 62M0562F1260H1062J07

keywords networkstochasticdifferentialequationsdirectedgraphrecoveryadaptivelassoquasi-likelihoodestimationhigh-frequencyobservationsergodicdiffusionscalingvolatility

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a network stochastic differential equation (N-SDE) model in which each node evolves according to its own dynamics, feedback from neighbouring nodes, and a stochastic-volatility diffusion term, and it claims that this model can be estimated from high-frequency observations of a single trajectory. When the directed graph is known, the quasi-likelihood estimator is shown to satisfy a non-asymptotic error bound: the squared error is controlled by the number of graph parameters divided by total observation time, up to constants describing model regularity and graph scaling. When the graph is unknown, the paper proposes a two-step adaptive Lasso procedure applied to an augmented model with one weight per candidate directed edge, and proves that the estimated graph equals the true graph with probability tending to one. A closed-form neighbourhood-wise estimator is derived for linear drifts, and simulations plus a real S&P100 data example illustrate the claims. If correct, this gives an interpretable, directed-graph framework for causal and network analysis of high-dimensional continuous-time series.

What carries the argument

The workhorse is the quasi-likelihood contrast $\ell_n(\alpha,\beta)$ built from increments of the observed path, together with the scaled parameter rates $\Gamma_n$, which give $\alpha$ a $1/\sqrt{n}$ rate and $\beta$ a $1/\sqrt{n\Delta_n}$ rate. Assumption C(r) imposes a regular contrast: a gradient bound by a square-integrable random variable $\xi_n$ and a Hessian lower bound $\mu$, which together turn the Taylor expansion of the contrast into the probability bound of Theorem 2. For graph recovery, the machinery is the least-squares-approximation objective $F_n(\theta,w) = \frac12\langle H_n, (\theta-\tilde\theta, w-\tilde w)^{\otimes 2}\rangle + \lambda_n\|(\theta,w)\|_{1,\gamma(n,d)}$ with adaptive weights $\gamma_{n,ij} \propto |\tilde w_{ij}|^{-\delta}$. The KKT equations for zeroed edges give the no-false-inclusion part of Theorem 5, while sign consistency of the nonzero weights gives the no-missing-edge part.

What would settle it

Run the paper's two-step adaptive Lasso on data from model (25) with $d = 10$, $n = 50$, and $\Delta_n = 1/50$ (so $n\Delta_n = 1$ and the 90 candidate edge weights are fewer than the sample size), and repeat with $n = 200, 800$ while holding $d$ and $\Delta_n$ fixed; if the empirical frequency $P(\hat G_n = G)$ does not approach one over many replications, the consistency claim of Theorem 5 fails.

Watch

Extended reading notes

Core claim

The central claim is that the pair consisting of the drift/diffusion parameters and the directed topology of an ergodic N-SDE can be identified from discrete high-frequency data, with explicit rates for both. For a known graph, Theorem 2 states that with high probability $|\hat\theta_n - \theta_0|^2 \le \frac{4\xi_n^2}{\mu^2}\,K\varepsilon$, where $K = \pi_d/|G_d|$ controls how many parameters each edge carries and $\varepsilon = |G_d|/(n\Delta_n)$ is the graph size relative to total observation time; the proof Taylor-expands the quasi-likelihood contrast on an event where the scaled estimation error is bounded, using the regularity constants $\xi_n$ and $\mu$. For an unknown graph, the paper augments the drift with edge weights $w_{ij}$, forms an initial quasi-likelihood estimate $(\tilde\theta, \tilde w)$, and then minimises a least-squares-approximation objective with adaptive $\ell^1$ weights; Theorem 5 shows $P(\hat G_n = G) \to 1$ under rate conditions on the adaptive weights and the information matrix. The linear-drift case reduces each node's estimation to a small regression on its neighbourhood, so computation tracks the graph rather than the full $d^2$ problem.

Load-bearing premise

The graph-recovery results depend on a consistent initial quasi-likelihood estimate of the augmented model containing one weight per possible directed edge, which is only plausible when the number of candidate edges is smaller than the sample size, and the paper does not prove the needed asymptotics when the parameter dimension grows with $n$.

Editorial extensions

If this is right

For a known directed graph, the squared parameter error decays like $|E|/(n\Delta_n)$, so holding the observation window fixed and merely sampling more frequently does not improve precision unless the number of observations grows.
The linear-drift estimator (17) estimates each node's parameters from a regression on that node's neighbours, making estimation feasible for large $d$ when neighbourhoods are small.
When the graph is unknown, the adaptive Lasso recovers the exact directed edge set with probability tending to one, so one-way influence can be read off the support of the estimated edge weights.
The non-asymptotic bound implies a quantitative trade-off: adding edges or adding parameters per edge must be compensated by proportionally longer observation time to keep the error below a fixed level.
The framework extends network autoregressive and graph Ornstein-Uhlenbeck models to nonlinear drifts and state-dependent volatility while keeping parameter estimation interpretable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The graph-recovery theorems require a consistent initial quasi-likelihood estimator in the augmented $d(d-1)$-parameter model, which is only plausible when the number of candidate edges is smaller than the sample size; the paper's S&P100 application ($p \approx 9900$, $n = 1596$) sits outside that regime and should be read as heuristic rather than as covered by Theorem 5.
The exact-recovery result suggests a practical diagnostic: replicating the two-step procedure on subsampled data and checking whether the estimated support stabilises would indicate whether the asymptotic regime is being approached.
Because the adjacency matrix enters the drift, a zero estimated weight means no Granger-type predictive effect in continuous time, but a causal reading of the recovered directed graph still requires the usual assumption that no unobserved common drivers exist.
The error bound depends on the finite-sample identifiability constant $\mu$, so the practical takeaway is that sparsity alone is not enough; the strength of the drift's mean reversion also governs estimation quality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The bottom line: this is a useful model class with a real gap between its advertised growing-network guarantees and what the proofs actually support. The N-SDE framework—nonlinear drift interactions, state-dependent stochastic volatility, directed edges, high-frequency quasi-likelihood—is genuinely new relative to NAR and GrOU models. The fixed-dimensional known-graph part (Theorem 2, linear drift estimates, simulation table) is plausible, and the empirical work is honest. I'd send this to a careful referee, but I wouldn't rely on the graph-recovery theorems as stated.

What's new and good: the model nests NAR and GrOU into a continuous-time setup with nonlinear interaction terms and volatility; Example 2.2's radial-basis ergodicity condition is a nice touch; the explicit linear-drift estimator (17) is a clean contribution. The empirical section tests Erdős–Rényi, directed polymer, and stochastic block models, and the error-bound table aligns with Theorem 2's rate |E|/(nΔ_n). That part looks like real work.

Soft spots: the high-dimensional claims—the advertised 'growing networks' and exact directed graph recovery—rest on Theorem 3, which simply recalls fixed-dimensional QLA asymptotics and applies them to a parameter space of dimension πβ + d(d−1). No growth conditions on d vs n are given. Theorem 5's union bound over O(d^2) null edges shows each exclusion probability goes to zero pointwise; with d growing, you'd need a rate of o(1/d^2), which is not proved. And the S&P100 application runs in a p > n regime (9900 parameters, 1596 observations). The adaptive Lasso step requires a consistent unpenalized initial estimator for the augmented model; in p > n that estimator need not be consistent or unique. The paper does not flag this. The fixed-dimensional parts are not the problem; the load-bearing gap is precisely the growing-network / p > n regime the abstract advertises.

Bottom line: the model and the fixed-d case are worth a serious look. The graph-recovery theory is not supported as written. A referee could push for a restriction to fixed d, or for proper concentration inequalities and growth conditions. I'd accept it for review, but with a note that the advertised scope is currently unmet.

Referee Report

4 major / 7 minor

Summary. The paper proposes a network SDE framework in which each node follows an SDE with a momentum term, a network interaction term, and a state-dependent stochastic volatility term. Estimation is based on high-frequency discrete observations and quasi-likelihood. For a known graph, Theorem 2 gives a non-asymptotic ℓ2 error bound under scaling conditions on the number of edges and observation time. For an unknown directed graph, the paper augments the parameter vector with edge weights and proposes an adaptive Lasso estimator, claiming consistency (Theorem 4) and exact graph recovery (Theorem 5). The paper includes simulations for Erdős–Rényi, polymer, and stochastic block model graphs, and applies the method to S&P100 high-frequency data.

Significance. If the claims were established, the framework would be a notable contribution: it extends network autoregressive and graph Ornstein–Uhlenbeck models to nonlinear drift and stochastic volatility, handles directed edges, and provides finite-sample-type bounds that make the role of graph sparsity and sampling frequency explicit. The fixed-dimensional known-graph analysis and the explicit linear-model estimator in Section 3.1 are useful, and the simulation studies are reasonable. However, the advertised central contribution—parameter and topology estimation for growing networks with unknown graph—is not backed by the proofs. Theorem 3 merely recalls fixed-dimensional QLA results and applies them to a parameter space of dimension πβ + d(d−1) without growth conditions; Theorem 5 relies on a union bound over O(d²) null edges without the necessary uniform rates; and the real-data application operates in a p > n regime where the required initial quasi-likelihood estimator is not justified. The paper is also transparent in Remark 3 that Theorem 2 could be replaced by an estimator-scaling assumption that is essentially the conclusion. These gaps are load-bearing for the paper's main claims.

major comments (4)

[Section 4, Theorem 3] Theorem 3 is stated as a recall of [25, Theorem 13] and [12, Theorem 1, Lemma 4], but those are fixed-dimensional quasi-likelihood asymptotics. In model (18) the augmented parameter vector for (θ, w) has dimension πα + πβ + d(d−1), and the paper explicitly allows d to grow with n (Section 1, contribution (ii)). No growth conditions linking d to n are supplied, and no extension of the polynomial-type large deviation inequalities of [25] to this growing parameter space is proved. Since Theorems 4 and 5 both rely on the Γn-consistency of the initial estimator (19), the graph-recovery guarantees for growing networks rest on an unproven high-dimensional QLA.
[Section 8, Proof of Theorem 5, Step 1] The proof that no null edge is selected uses a union bound over the O(d²) pairs (i, j) not in E. Each summand is shown to tend to 0 pointwise via the KKT conditions, but for d growing the sum tends to 0 only if the individual probabilities are o(1/d²) or if additional structure controls the number of null edges. No such rate is established; condition (L3), which only requires ˇγw_{n,d}√(nΔn) → ∞, does not control the union bound. Consequently, the conclusion P(Ĝn = G) → 1 is not proved for growing d.
[Section 6, real-data application; Section 4, estimator (19)] In the S&P100 application, d = 99 and n = 1596, so the augmented parameter vector for (θ, w) has dimension πd + d² ≈ 9900, exceeding the sample size. The initial quasi-likelihood estimator (19), used to construct the adaptive weights (21)–(22), is an unpenalized QL estimator over this p > n parameter space; it need not be unique, consistent, or even well-defined. The paper gives no first-step regularization or alternative construction of the initial estimator, so the adaptive Lasso procedure is not justified in the reported high-dimensional regime.
[Section 3, Remark 3; Section 8, Proof of Theorem 2] Theorem 2 claims finite-sample bounds for growing networks, but its proof invokes the event {|Γn^{-1}(θ̂n−θ0)| ≤ r} with probability at least 1 − CL/r^L 'due to the results in [25]'. Those results do not provide a uniform bound over the dimension d; the constants CL and L, and the radius r required to make the probability high, may depend on πd. Moreover, Remark 3 acknowledges that Theorem 2 could instead be proved under the estimator-scaling assumption sup_n E|Γn^{-1}(θ̂−θ0)|² ≲ πd, which is essentially the conclusion one would need to prove for the QLE in growing dimension. As written, the known-graph finite-sample guarantee is therefore also not established in the growing-network regime advertised in the abstract.

minor comments (7)

[Section 6] The sentence 'The degree distribution is shown in .' is incomplete; a figure or table reference is missing.
[Section 2, Example 2.1] The condition labelled (8) is referenced before it is defined; equation (8) appears later in Example 2.2. Please renumber or reorder.
[Section 2, Example 2.1] The notation τmax(−µId) is undefined; the largest singular value of −µId is |µ|, not −µ, and the displayed inequality should be clarified.
[Table 1] The column heading 'Bound Mean Error' is ambiguous; specify whether 'Mean Error' is the empirical mean of the ℓ2 error or its square.
[Section 5, model (25)] The function c·tanh(x/c) is a scaled hyperbolic tangent, not a sigmoid; use a consistent term such as 'smooth clipping' throughout.
[Section 8] The proof of Theorem 2 spells 'Cauchy-Schwartz'; this should be 'Cauchy-Schwarz'.
[Table 2 caption] The statement 'No regularization is required on the diffusion part in this example' is unclear because the table has no LASSO column for the α parameters; clarify whether α was unpenalized by construction.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity; flagged Remark 3 (C1′) is an unused shortcut equivalent to Theorem 2's bound, and the growing-dimension gaps are correctness issues.

other [Section 3, Remark 3 (after Theorem 2)]
"The proof of the theorem relies on deriving an error bound on the estimator depending on the number of parameters. In general, given an estimator ˆθn the theorem could be proved under the following modified assumption: (C1′) Estimator scaling : sup_n E|Γ−1_n(ˆθ − θ0)|² ≲ πd."

C1′ is essentially the conclusion of Theorem 2: combining sup_n E|Γ_n^{-1}(θ̂−θ0)|² ≲ πd with (G1)–(G2) (πd/|Gd| ≤ K and |Gd|/(n∆n) ≤ ε) yields E|θ̂−θ0|² ≲ Kε, which is the theorem's bound in expectation. A proof that used C1′ would therefore be circular. However, the actual proof of Theorem 2 does not invoke C1′; it derives the bound from the contrast regularity conditions C(i)/(ii) and the concentration event supplied by [25]. Thus this is an admitted alternative that would be circular, but it is not the load-bearing argument of the paper.

full rationale

The actual derivation chain of Theorem 2 is not circular: the bound |Γ_n^{-1}(θ̂−θ0)| ≤ 2ξ_n√πd/μ is obtained from the Taylor expansion and the regularity/identifiability conditions C(i)–C(ii), and then (G1)–(G2) convert it into the stated Kε bound. The graph-recovery results (Theorems 4–5) are also proved from the KKT conditions and from Theorem 3, which is recalled from external fixed-dimensional quasi-likelihood results [25, 12] rather than from the authors' own conclusions. The adaptive Lasso methodology cites the authors' earlier papers [8, 9, 10], but the proofs are written out in the manuscript and do not reduce to those citations. The main caveats are correctness gaps, not circularity: Theorem 3 adapts fixed-dimensional QLA to d(d−1) growing parameters without growth conditions on d versus n, and Theorem 5's proof uses a union bound over O(d²) null edges whose individual probabilities are shown to vanish pointwise without the o(1/d²) rate needed for growing d. The S&P100 application (p ≈ 9900, n = 1596) therefore lies in an unsupported p > n regime. These are missing-proof issues, not equivalences by construction. No fitted constants are presented as predictions, and no load-bearing self-citation chain forces the conclusions. The only flagged circular-adjacent passage is Remark 3's C1′, which is explicitly not used in the proof; hence the score is low.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central theory rests on standard ergodic diffusion assumptions, plus high-level regular contrast and adaptive Lasso rate conditions that are not verified from primitive model assumptions. The most fragile entries are the requirement of a consistent unpenalized initial estimator in a regime where the parameter count can exceed the sample size, and the unproven extension of fixed-dimensional quasi-likelihood asymptotics to growing parameter dimension.

free parameters (3)

Adaptive Lasso tuning parameter λn = 0.1·λmax (ER graph), λ.5se (polymer, SBM)
Chosen by heuristic or validation loss rather than derived from theory; graph recovery results are sensitive to this choice.
Adaptive weight exponents δ1, δ2, δ3 = δ = 1
Set by hand following standard adaptive Lasso practice; not data-driven.
Sigmoid clipping constant c = 100
Used in simulations to approximate a bounded diffusion; chosen for numerical convenience, not central to the theory.

assumptions (6)

domain assumption (A4) Exponential beta-mixing of the process X
Imposed on the model to ensure ergodicity and to invoke polynomial-type large deviation inequalities. Sufficient conditions are quoted from Pardoux and Veretennikov, but not verified for the general nonlinear class.
ad hoc to paper C(r/nΔn) regular contrast condition: bounded gradient and Hessian lower bound near the estimator
Assumed for Theorem 2 and not derived from (A1)-(A6). The non-asymptotic error bound is vacuous if this condition fails.
domain assumption Fixed-dimensional QLA asymptotics extend to parameter dimension πβ + d(d−1)
Theorem 3 is imported from Yoshida [25] and Kessler [12], which are fixed-dimensional results, but is used to justify Theorems 4 and 5 in the growing-network setting.
ad hoc to paper A consistent non-regularized initial quasi-likelihood estimator (θ̃, w̃) exists for the augmented model
Required by the two-step adaptive Lasso procedure in Section 4. Impossible when d(d−1) exceeds n, as in the S&P100 application with p ≈ 9900 and n = 1596.
domain assumption (L1)-(L3) rate conditions on adaptive weights and information matrix
Standard adaptive Lasso conditions; not verified for the N-SDE setting, particularly the divergence of null weights relative to √(nΔn).
domain assumption (A6) / (A6)′ identifiability of drift and diffusion parameters
Standard identifiability condition, although the text states it with α and β swapped; the intended meaning is clear.

invented entities (1)

Auxiliary edge weights wij
purpose: Latent parameters encoding directed edge presence via Aij = 1(wij ≠ 0) in the augmented model (18), enabling Lasso-based graph recovery.
These are model-introduced parameters, not observable entities; they are estimated from data and have no independent falsifiable handle outside the model.

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Pith. "Pith review of Ergodic Network Stochastic Differential Equations." pith.science (2026). https://pith.science/paper/V6H3ZQQG

@misc{pith2026241217779,
  author       = {Pith},
  title        = {Pith review of: Ergodic Network Stochastic Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V6H3ZQQG}},
  note         = {Machine review of arXiv:2412.17779}
}

read the original abstract

We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (\emph{momentum effect}), feedback from neighboring nodes (\emph{network effect}), and a \emph{stochastic volatility} term modeled by Brownian motion. Our primary objective is to estimate the parameters of the N-SDE system from high-frequency discrete-time observations. The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph. We consider two distinct scenarios: \textit{i) known network structure}: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the linear growth of the parameter space with the number of edges. \textit{ii) unknown network structure}: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models. In this work, we assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems. Through extensive simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.

Figures

Figures reproduced from arXiv: 2412.17779 by the authors.

**Figure 1.** Erd˝os–R´enyi random graph 5.1 Empirical analysis of the error bound We now empirically validate the results of Theorem 2. In particular, [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

**Figure 2.** Sample path from model (25), with true parameter values from Ta [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Polymer graph of a handful of large, connected components with multiple hubs, accompanied by many small isolated components. The degree distribution is shown in . It demonstrates a heavy-tailed pattern, as the most connected nodes have a disproportionately larger number of links. The estimated networks display characteristics that are often observed in power-law graphs ([1]). 7 Conclusions. In this paper we introd… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Cluster identification in a stochastic block model [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]

**Figure 5.** Figure 5: Clusters selected (red) and loss (blue) in a SBM as a function of the [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

**Figure 6.** Figure 6: Estimated graph for the components of S&P100 components stocks. [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]

**Figure 7.** Figure 7: Vertex degree distribution for the S&P100 graph. [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages

[25]

Polynomial type large deviation inequalities and quasi- likelihood analysis for stochastic differential equations

Nakahiro Yoshida. “Polynomial type large deviation inequalities and quasi- likelihood analysis for stochastic differential equations”. In: Annals of the Institute of Statistical Mathematics 63.3 (2011), pp. 431–479. 24

work page 2011
[1]

Power-law partial correlation network models

Matteo Barigozzi, Christian Brownlees, and G´ abor Lugosi. “Power-law partial correlation network models”. In: Electronic Journal of Statistics. 2018 Sep 18; 12 (2): 2905-29 (2018)

work page 2018
[2]

Network granger causality with inherent grouping structure

Sumanta Basu, Ali Shojaie, and George Michailidis. “Network granger causality with inherent grouping structure”. In: The Journal of Machine Learning Research 16.1 (2015), pp. 417–453

work page 2015
[3]

Graph Neural Stochastic Differential Equations

Richard Bergna et al. Graph Neural Stochastic Differential Equations . 2023

work page 2023
[4]

Graph Neural Stochastic Differential Equations for Learning Brownian Dynamics

Suresh Bishnoi et al. Graph Neural Stochastic Differential Equations for Learning Brownian Dynamics. 2023

work page 2023
[5]

On Lasso estimator for the drift function in diffusion models

Gabriela Cio lek, Dmytro Marushkevych, and Mark Podolskij. “On Lasso estimator for the drift function in diffusion models”. In: Bernoulli 31.3 (2025), pp. 1811–1833

work page 2025
[6]

High-frequency estimation of the L´ evy-driven graph ornstein-uhlenbeck process

Valentin Courgeau and Almut ED Veraart. “High-frequency estimation of the L´ evy-driven graph ornstein-uhlenbeck process”. In:Electronic Journal of Statistics 16.2 (2022), pp. 4863–4925

work page 2022
[7]

Likelihood theory for the graph Ornstein-Uhlenbeck process

Valentin Courgeau and Almut ED Veraart. “Likelihood theory for the graph Ornstein-Uhlenbeck process”. In: Statistical Inference for Stochastic Processes 25 (2022), pp. 1–34

work page 2022

Show all 29 references

[8]

Adaptive LASSO-type estimation for multivariate diffusion processes

Alessandro De Gregorio and Stefano M Iacus. “Adaptive LASSO-type estimation for multivariate diffusion processes”. In: Econometric Theory 28.4 (2012), pp. 838–860

2012
[9]

Pathwise optimization for bridge-type estimators and its applications

Alessandro De Gregorio and Francesco Iafrate. “Pathwise optimization for bridge-type estimators and its applications”. In:arXiv preprint arXiv:2412.04047 (2024)

2024 arXiv
[10]

Regularized bridge-type estimation with multiple penalties

Alessandro De Gregorio and Francesco Iafrate. “Regularized bridge-type estimation with multiple penalties”. In: Annals of the Institute of Statis- tical Mathematics 73.5 (2021), pp. 921–951. 23

2021
[11]

Adaptive Elastic-Net estimation for sparse diffusion processes

Alessandro De Gregorio et al. “Adaptive Elastic-Net estimation for sparse diffusion processes”. In: arXiv preprint arXiv:2412.16659 (2024)

2024 arXiv
[12]

Estimation of an ergodic diffusion from discrete obser- vations

Mathieu Kessler. “Estimation of an ergodic diffusion from discrete obser- vations”. In: Scandinavian Journal of Statistics 24.2 (1997), pp. 211–229

1997
[13]

Generalized Network Autoregressive Processes and the GNAR Package

Marina Knight et al. “Generalized Network Autoregressive Processes and the GNAR Package”. In:Journal of Statistical Software 96.5 (2020), pp. 1– 36

2020
[14]

De-Biased Graphical Lasso for High-Frequency Data

Yuta Koike. “De-Biased Graphical Lasso for High-Frequency Data”. In: Entropy 22.4 (2020)

2020
[15]

On the Poisson equation and diffusion approximation. I

´Etienne Pardoux and Yu Veretennikov. “On the Poisson equation and diffusion approximation. I”. In: The Annals of Probability 29.3 (2001), pp. 1061–1085

2001
[16]

Learning Net- works of Stochastic Differential Equations

Jos´ e Pereira, Morteza Ibrahimi, and Andrea Montanari. “Learning Net- works of Stochastic Differential Equations”. In: Advances in Neural In- formation Processing Systems . Ed. by J. Lafferty et al. Vol. 23. Curran Associates, Inc., 2010

2010
[17]

Sparse autoregressive model estimation for learning Granger causality in time series

Jitkomut Songsiri. “Sparse autoregressive model estimation for learning Granger causality in time series”. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing . IEEE. 2013, pp. 3198–3202

2013
[18]

Topology Selection in Graph- ical Models of Autoregressive Processes

Jitkomut Songsiri and Lieven Vandenberghe. “Topology Selection in Graph- ical Models of Autoregressive Processes”. In: Journal of Machine Learning Research 11.91 (2010), pp. 2671–2705

2010
[19]

Penalized least squares approxi- mation methods and their applications to stochastic processes

Takumi Suzuki and Nakahiro Yoshida. “Penalized least squares approxi- mation methods and their applications to stochastic processes”. In:Japanese Journal of Statistics and Data Science 3.2 (2020), pp. 513–541

2020
[20]

Con- centration analysis of multivariate elliptic diffusions

Lukas Trottner, Cathrine Aeckerle-Willems, and Claudia Strauch. “Con- centration analysis of multivariate elliptic diffusions”. In: Journal of Ma- chine Learning Research 24.106 (2023), pp. 1–38

2023
[21]

Adaptive estimation of an er- godic diffusion process based on sampled data

Masayuki Uchida and Nakahiro Yoshida. “Adaptive estimation of an er- godic diffusion process based on sampled data”. In: Stochastic Processes and their Applications 122.8 (2012), pp. 2885–2924

2012
[22]

Bounds for the mixing rate in the theory of stochas- tic equations

A Yu Veretennikov. “Bounds for the mixing rate in the theory of stochas- tic equations”. In: Theory of Probability & Its Applications 32.2 (1988), pp. 273–281

1988
[23]

Unified LASSO estimation by least squares approximation

Hansheng Wang and Chenlei Leng. “Unified LASSO estimation by least squares approximation”. In: Journal of the American Statistical Associa- tion 102.479 (2007), pp. 1039–1048

2007
[24]

Estimation for diffusion processes from discrete ob- servation

Nakahiro Yoshida. “Estimation for diffusion processes from discrete ob- servation”. In: Journal of Multivariate Analysis 41.2 (1992), pp. 220–242

1992
[26]

Quasi-likelihood analysis for nonlinear stochastic pro- cesses

Nakahiro Yoshida. “Quasi-likelihood analysis for nonlinear stochastic pro- cesses”. In: Econometrics and Statistics (2022)

2022
[27]

Simplified quasi-likelihood analysis for a locally asymp- totically quadratic random field

Nakahiro Yoshida. “Simplified quasi-likelihood analysis for a locally asymp- totically quadratic random field”. In: arXiv preprint arXiv:2102.12460 (2021)

2021 arXiv
[28]

Network vector autoregression

Xuening Zhu et al. “Network vector autoregression”. In: The Annals of Statistics 45.3 (2017), pp. 1096–1123

2017
[29]

The adaptive lasso and its oracle properties

Hui Zou. “The adaptive lasso and its oracle properties”. In: Journal of the American statistical association 101.476 (2006), pp. 1418–1429. 25 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18X19 X20 X21 (a) Graph representation of the neighborhood in model (25) f...

2006

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [25]

Polynomial type large deviation inequalities and quasi- likelihood analysis for stochastic differential equations

Nakahiro Yoshida. “Polynomial type large deviation inequalities and quasi- likelihood analysis for stochastic differential equations”. In: Annals of the Institute of Statistical Mathematics 63.3 (2011), pp. 431–479. 24

work page 2011

[2] [1]

Power-law partial correlation network models

Matteo Barigozzi, Christian Brownlees, and G´ abor Lugosi. “Power-law partial correlation network models”. In: Electronic Journal of Statistics. 2018 Sep 18; 12 (2): 2905-29 (2018)

work page 2018

[3] [2]

Network granger causality with inherent grouping structure

Sumanta Basu, Ali Shojaie, and George Michailidis. “Network granger causality with inherent grouping structure”. In: The Journal of Machine Learning Research 16.1 (2015), pp. 417–453

work page 2015

[4] [3]

Graph Neural Stochastic Differential Equations

Richard Bergna et al. Graph Neural Stochastic Differential Equations . 2023

work page 2023

[5] [4]

Graph Neural Stochastic Differential Equations for Learning Brownian Dynamics

Suresh Bishnoi et al. Graph Neural Stochastic Differential Equations for Learning Brownian Dynamics. 2023

work page 2023

[6] [5]

On Lasso estimator for the drift function in diffusion models

Gabriela Cio lek, Dmytro Marushkevych, and Mark Podolskij. “On Lasso estimator for the drift function in diffusion models”. In: Bernoulli 31.3 (2025), pp. 1811–1833

work page 2025

[7] [6]

High-frequency estimation of the L´ evy-driven graph ornstein-uhlenbeck process

Valentin Courgeau and Almut ED Veraart. “High-frequency estimation of the L´ evy-driven graph ornstein-uhlenbeck process”. In:Electronic Journal of Statistics 16.2 (2022), pp. 4863–4925

work page 2022

[8] [7]

Likelihood theory for the graph Ornstein-Uhlenbeck process

Valentin Courgeau and Almut ED Veraart. “Likelihood theory for the graph Ornstein-Uhlenbeck process”. In: Statistical Inference for Stochastic Processes 25 (2022), pp. 1–34

work page 2022