Solow system driven by $\alpha$-stable L\'evy process

Shenglan Yuan; Yiren Wang

REVIEW 4 major objections 5 minor 16 references

Solow system driven by $\alpha$-stable L\'evy process

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that heavy-tailed jumps, not Gaussian noise, are the right shocks for growth models of volatile economies.

desk verdict Routine Lévy extension with a sign error, an independence mistake, and a confounded empirical comparison; the headline claims don't hold as submitted. read the letter →

arxiv 2607.20997 v1 pith:WN37RW2N submitted 2026-07-23 math.DS

Yiren Wang , Shenglan Yuan This is my paper · ORCID

classification math.DS MSC 91-1060H30

keywords α-stableLévyprocessheavy-tailedshocksgrowthmodelinfinitevariancecapitaladjustmentspeedcrisistrackingtime-varyingelasticitymacroeconomics

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

Macro shocks in emerging economies are not small, smooth, and bell-shaped; they arrive as jumps with heavy tails. This paper claims that a standard growth model driven by symmetric α-stable Lévy noise—rather than a Gaussian OU process—captures those jumps without giving up tractability. It derives the stationary distribution of capital deviations, a closed-form conditional characteristic function, and an integral representation that splits the dynamics into two mean-reverting components, one tied to investment gestation lags and one to endogenous saving feedback. On Argentine data, the Lévy specification yields structural parameters closer to external labor-share benchmarks than the Gaussian benchmark, tracks Argentina's 2018 and 2020 collapses far better, and produces a quarterly capital adjustment speed near 0.05 that repeats in Colombia and the United States. The paper presents this as evidence that heavy-tailed shocks are a robust baseline, with the Gaussian model as a limiting case.

What carries the argument

The central object is the symmetric α-stable Lévy process L_α(τ) with α∈(1,2), a shock process whose increments are discontinuous, heavy-tailed, and have infinite variance. The key identity is the integral representation of capital deviation as a scaled difference of two independent stationary OU-type processes with decay rates a and r(γ), explicitly separating investment gestation lags from endogenous feedback. This representation turns the infinite-variance problem into two tractable stable laws and yields the closed-form conditional characteristic function used for the paper's estimation and forecasting.

What would settle it

Reconstruct the two latent OU components X_a(t) and X_r(t) from a long simulated path and find two histories with identical u(t)=X_a(t)-X_r(t) but different component splits. If E[u(T)|X_a(t), X_r(t)] differs between them, the paper's formula E[u(T)|u(t)] is false. A simpler check: simulate the model with estimated parameters and compare the empirical predictive distribution of one-step-ahead capital changes to the closed-form shifted α-stable law; systematic mismatch would refute the central probabilistic claim.

Watch

Extended reading notes

Core claim

The paper claims that the capital-deviation process driven by a symmetric α-stable Lévy process is strictly stationary and can be written as a linear combination of two independent OU-type integrals with decay rates a and r(γ). From that representation it obtains the conditional characteristic function of future capital deviations in closed form, and hence a conditional mean that decays exponentially toward the balanced growth path. Empirically, it claims that this specification, estimated by mean absolute prediction error and mean matching because variance is infinite, recovers structural parameters substantially closer to externally measured values than those from the Gaussian OU model; th

Load-bearing premise

The paper's conditional-law results rest on the assertion that the past-determined part of future capital is a deterministic function of today's capital deviation alone, even though the model's state is really two-dimensional.

Editorial extensions

If this is right

If the estimates are right, replacing Gaussian shocks with heavy-tailed jumps corrects a systematic bias in structural parameters estimated from volatile economies.
The model's point forecasts match Gaussian benchmarks in calm periods but track crisis downturns markedly better, suggesting that heavy-tailed shocks matter mainly during tail events.
The same quarterly capital adjustment speed of about 0.05 emerges for Argentina, Colombia, and the United States, indicating a stable physical pace of capital formation across very different volatility regimes.
Robustness across tail indices shows that structural estimates remain stable and closer to external benchmarks, so the qualitative conclusions do not hinge on the exact choice of α.
Estimation via mean absolute error and mean matching provides a workable template for macro models whose data have infinite variance.

Reading between the lines

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

Beyond the paper: the closed-form conditional characteristic function is a probability-forecast machine; a natural extension the authors note but do not pursue is to output predictive intervals and density forecasts for crisis scenarios instead of point forecasts only.
Beyond the paper: the claimed invariance of η near 0.05 is a physical claim about capital formation lags. It could be tested independently with micro data on construction, equipment installation, and delivery lags—if micro estimates diverge from 0.05, the interpretation of η as a pure physical speed would need revision.
Beyond the paper: the derivation of the conditional law assumes the past can be summarized by current capital deviation alone. A bivariate extension conditioning on both the capital deviation and the auxiliary shock process would reveal whether this simplification changes the estimates; if it does, the Lévy identification gains might be partly an artifact of the reduced-state assumption.

Share X Bluesky LinkedIn Reddit HN

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

Bottom line: this is a straightforward extension of Brannan's Solow model to α-stable Lévy noise, and the theory section has two or three load-bearing errors that a careful referee would catch quickly. The empirical comparison is also confounded, so the headline claim that the Lévy specification outperforms the Gaussian benchmark is not established.

What the paper does well: the motivation is sensible — Argentina's recurrent crises are a natural test case for heavy-tailed shocks — and the estimation strategy sensibly avoids the infinite-variance problem by using MAE and mean matching rather than variance-matching. The cross-country evidence on η≈0.05 is eye-catching, though it rests on the same shaky theoretical foundation.

Soft spots, in order of severity. First, (2.5) has a sign error: direct integration of (2.2) gives (1−β)σ/(r−a)[∫e^{-a}−∫e^{-r}] dL, not the expression in the paper. That is not a typo in isolation; it flips the conditional mean. Second, (2.9) treats the two OU integrals in u as independent, but they are driven by the same Lévy process L. The characteristic function of a linear combination of integrals of the same process is not the product of the individual characteristic functions — the paper itself uses the correct relationship in (2.12), so this is an internal contradiction. Third, (2.14) claims H(t,T) is a deterministic function of u(t), but u(t) does not identify the two past integrals separately, so conditioning on u(t) instead of the full filtration is unjustified. These are not cosmetic issues: the stationary distribution, conditional characteristic function, and conditional expectation all rest on them.

On the empirical side, the comparison in Tables 1–4 changes three things at once — shock distribution, time-varying capital elasticity, and objective function — so any improvement in s1, s2, η, or crisis tracking could be driven by the time-varying α_k or the robust estimator rather than by Lévy tails. The PWT labor-share data are also used twice: to construct α_t^k and as the benchmark for validation. No uncertainty quantification and no code or data are provided. The authors do list limitations in §6.3, which is honest, but they don't address these identification problems.

Who this is for: readers interested in heavy-tailed Solow models might learn from the empirical setup, but the central claims are unsupported as submitted. The paper needs a careful rewrite of the theory and a properly controlled empirical comparison. I would still send it to peer review because the question is real and the empirical application is new, but I'd tell the referee to look closely at equations (2.5), (2.9), and (2.14) and to demand a cleaner comparison before accepting anything.

Referee Report

4 major / 5 minor

Summary. This paper modifies the Brannan (2019) Solow-type growth model by replacing Gaussian OU noise with a symmetric α-stable Lévy process (1<α<2) and allowing time-varying capital elasticity calibrated from PWT labor shares. It derives a stationary solution, a claimed closed-form conditional characteristic function and conditional mean for capital deviations, and applies a MAE+mean estimation strategy to Argentine data, comparing with the Gaussian OU benchmark. It reports that the Lévy specification yields structural parameters closer to PWT benchmarks and better crisis-period tracking, and that η≈0.05 is stable across Argentina, Colombia, and the United States.

Significance. If correct, the paper would make a useful contribution: an explicit heavy-tailed stochastic extension of a tractable growth model with a computable conditional characteristic function, and an estimation approach that avoids second moments. The cross-country invariance claim for η would be economically interesting. The estimation strategy is clearly described and replicable in principle. However, the theoretical derivations contain several load-bearing errors, and the empirical design does not isolate the effect of the Lévy shock process, so the main claims are not currently supported.

major comments (4)

[§2.2.3, Eq. (2.5)] Direct integration of (2.2) gives u(τ)=((1−β)σ/(r(γ)−a))(X_a(τ)−X_r(τ)); the denominator in (2.5) is a−r(γ), which flips the sign. This error propagates to the definition of H(t,T) in (2.10) and to the conditional mean (2.15), where H is claimed to equal E[u(T)|u(t)]. The stationary CF in (2.9) is protected by the absolute value, but the conditional-mean result is sign-sensitive and needs correction.
[§2.3.2, Eq. (2.9)] Equation (2.9) treats X_a and X_r as independent α-stable OU processes and writes the CF as the product of their individual CFs. But both integrals are driven by the same Lévy process L_α. The correct CF is exp{−C_α |(1−β)σ/(a−r)|^α ∫_0^∞ |e^{-a t}−e^{-r t}|^α dt |θ|^α}, which is not equal to the sum 1/(aα)+1/(rα). This invalidates the stated stationary distribution of u(τ).
[§2.3.3–2.4, Eqs. (2.10), (2.14)] H(t,T) is claimed to be a deterministic function of u(t), and (2.14) claims E[e^{iθu(T)}|u(t)] = exp(iθH(t,T)−...). But u(t)=C(X_a(t)−X_r(t)) does not determine X_a(t) and X_r(t) separately; H(t,T) is F_t-measurable but not σ(u(t))-measurable. The univariate process u is not Markov. The displayed expression is at best the conditional CF given the full filtration (or the two-dimensional state), not the conditional CF given u(t). This undermines the stated probabilistic foundation for prediction.
[§4.5, Tables 1–2, §5.2] The headline comparison is confounded: the Lévy row uses time-varying α_t^k calibrated from PWT labor shares and an MAE+mean objective, while the OU row uses constant α_k and an MSE+variance/autocorrelation objective. Differences in s1, s2, η, and crisis tracking cannot be attributed to the Lévy shock process. Moreover, the PWT reference for α_k is essentially 1−labsh, the same data used to build α_t^k, so the Lévy model's α_k 'closeness' is partly imposed by construction. Finally, the one-step-ahead predictions are conditional means with the same functional form regardless of α; the tail index enters only through parameter estimates. A controlled comparison—e.g., OU with the same time-varying α_t^k and same objective, or Lévy with constant α_k—is required before the abstract's claim can be evaluated.

minor comments (5)

[Eq. (2.12)] The exponent contains 'C_α θ', which appears to be a typo for 'C_α |θ|^α'.
[§4.2 vs §5.1] Section 4.2 sets the baseline tail index at α=1.5, but Section 5.1 states 'we adopt α=1.2' for the baseline results. This inconsistency should be resolved.
[Tables 1–2] The tables do not report the tail index used in the Lévy estimates; given that Table 3 considers several α values, each row should be labeled with the relevant α.
[§2.2.3] The Fubini justification is terse: stationarity plus exponential damping does not by itself establish absolute integrability of the double stochastic integral; a more careful argument or citation is needed.
[§4.7] The text refers to 'Subsection 4.5.2', which does not exist; the k_n-level comparison is in Section 4.6.

Circularity Check

3 steps flagged · score 6.0 of 10

Partially circular: the crisis-tracking 'predictions' are the in-sample MAE objective and the PWT benchmark is also the calibration input; cross-country η stability remains an independent empirical claim.

fitted input called prediction [§4.4 (Estimation strategy) and §4.6 (One-step-ahead predictions), Eqs. (4.27)–(4.28), Figures 1–4]
"Prediction error. We use the mean absolute error (MAE) instead of the mean squared error. ... Define the one-step-ahead prediction error as en(θ)=Eθ[kn|Fn−1]−\hat{k}n ... JMAE(θ)= 1/N ∑|en(θ)|. ... Building on these calibrated parameters, we construct the one-step-ahead predictions of kn and compare them with the observed series."

The parameters used for the 'one-step-ahead predictions' displayed as crisis-tracking evidence are obtained by minimizing J_MAE over exactly the same e_n(θ) series. Thus the plotted Lévy 'predictions' are the fitted conditional means evaluated on the estimation sample; the claimed improvement in crisis-period tracking is partly the in-sample objective renamed as a prediction, not an independent out-of-sample forecast.
fitted input called prediction [§4.1 (time-varying α_t^k) and §4.5/Tables 1–2 (PWT references)]
"Specifically, we set αtk=1−labsh ARG y for each year and assign the same value to all four quarters of that year. ... Note that PWT reference values are taken directly from the Penn World Tables and represent long-run averages over the corresponding sample period."

The same PWT 11.0 labor-share data are used twice: once to construct the time-varying capital elasticity α_t^k (which also enters the reconstructed k_n series via Eq. (4.24)) and once to supply the 'external' PWT reference values against which the model is judged. The α_k benchmark therefore coincides with the calibrated input by construction, and parameters influenced by α_t^k inherit that shared source, so the 'closer to external PWT benchmarks' claim is partly self-referential.

1 more flagged steps

other [§5.1 (Robustness to tail index specification), Table 3]
"For our baseline results, we adopt α=1.2, as it delivers the strongest alignment with PWT benchmarks and the most accurate crisis tracking for Argentina. The optimal choice of α, however, is left for future research."

The tail index used in the headline cross-country estimates was selected after inspecting the two outcomes that the paper then reports as findings—PWT alignment and crisis tracking. This is an ex-post specification choice rather than a derivation. The circularity is limited because Table 3 shows η≈0.05 across α=1.2, 1.5, and 1.8, so the η-stability claim does not rest solely on the chosen α.

full rationale

The theoretical sections are not circular: the stationary-solution and conditional-characteristic-function derivations are carried out in the paper (whatever their mathematical validity—e.g., the H(t,T) measurability issue and the sign in Eq. (2.5) are correctness problems, not circular reductions). The self-citations [4], [7], [9] are used for motivation or standard simulation schemes and are not load-bearing imports of an unproved uniqueness or ansatz. The cross-country η≈0.05 finding is an independent empirical claim not constructed from the PWT labor-share input. However, two parts of the empirical validation are partially circular. First, the 'one-step-ahead predictions' that support the crisis-tracking claim are the same in-sample errors minimized by the MAE objective, so the predictive advantage is partly a fit statistic displayed as a forecast. Second, the PWT data function both as the calibration source for α_t^k—and through Eq. (4.24), for the k_n series being fitted—and as the 'external' benchmark, so the closer-to-PWT conclusion is partly self-referential. The ex-post adoption of α=1.2 adds a specification-search element, though the robustness table mitigates it. Overall, the central crisis-tracking comparison and part of the PWT-benchmark comparison reduce to the paper's own fitting choices, while the η stability retains independent content; hence a partial circularity score of 6.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

The model's central outputs depend on a vector of eight estimated parameters plus a hand-selected tail index α. The time-varying elasticity α_t^k is not estimated but is imported from PWT, and the same PWT data are later used as external benchmarks, creating a partly circular validation. The theoretical conditional-characteristic-function section additionally assumes a Markov reduction to u(t) that is not supplied.

free parameters (9)

α (stable tail index) = 1.5 (§4.2 baseline), 1.2 adopted (§5.1)
Not estimated; chosen by hand from a robustness grid and switched after examining PWT alignment and crisis tracking.
s1 (lower saving-rate bound) = 0.091 (ARG 2004–2014 baseline)
Estimated via combined MAE+mean objective; validated against PWT reference.
s2 (upper saving-rate bound) = 0.159 (ARG 2004–2014 baseline)
Estimated; PWT reference reported as 0.185.
η (quarterly capital adjustment speed) = 0.053 (ARG 2004–2014), ≈0.05 in cross-country table
Central structural parameter; estimated, with no standard errors; claimed stable across countries.
µ (mean reversion of z) = 0.776 (ARG 2004–2014 baseline)
Estimated; varies strongly across α rows in Table 3.
ν (mean reversion of p) = 0.365 (ARG 2004–2014 baseline)
Estimated.
γ (sigmoid saving-rate steepness) = 23.806 (ARG 2004–2014 baseline)
Estimated.
a1 (AR(1) coefficient of Lévy shock) = 0.185 (ARG 2004–2014 baseline)
Estimated; related to continuous-time a by a1=e^{-aΔt}.
σ1 (innovation scale) = 0.305 (ARG 2004–2014 baseline)
Estimated; not a standard deviation under infinite variance.

assumptions (7)

domain assumption Symmetric α-stable Lévy process with α∈(1,2) has finite mean and infinite variance; MAE and mean matching are well-defined objective functions.
Used in §4.4 to justify the estimation strategy; standard property of stable laws.
standard math The integrals in (2.3)-(2.5) are well defined and Fubini interchange is justified by stationarity and exponential decay.
Used in §2.2.3 to obtain the two-OU representation; relies on |f|^α-integrability of the kernels, not fully proven in the paper.
ad hoc to paper The past-determined term H(t,T) is a deterministic function of u(t) alone and u(t) is a sufficient statistic for F_t.
Assumed in §2.3.3–2.4 but false as stated: the system state is (u,Q), and H contains two past integrals not separately determined by u(t).
domain assumption The discrete system (4.19)-(4.22) is a faithful discretization of the continuous system (2.2), with parameters scaled by Δt=0.25.
Used in §4.2-4.3 to construct latent variables and predictions; no formal convergence or approximation error analysis is provided.
domain assumption Capital elasticity α_t^k is known and equals 1−labsh from PWT 11.0, with yearly values repeated quarterly.
Used in §4.1/4.3 to reconstruct k_n and in all empirical estimates; measurement error is acknowledged in §6.3 but not modeled.
domain assumption PWT-based external references for s1, s2, α_k are directly comparable to the model's structural parameters.
Used in §4.5 and Tables 1–4 to judge which specification is closer to benchmarks; the construction of the s1/s2 reference values is not specified.
domain assumption a≠r(γ) so that the (a−r)^{-1} denominators in (2.5), (2.11) are finite.
Assumed implicitly in §2; limiting case a=r is not treated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Solow system driven by $\alpha$-stable L\'evy process." pith.science (2026). https://pith.science/paper/WN37RW2N

@misc{pith2026260720997,
  author       = {Pith},
  title        = {Pith review of: Solow system driven by $\alpha$-stable L\'evy process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WN37RW2N}},
  note         = {Machine review of arXiv:2607.20997}
}

abstract

This paper empirically implements a Solow-type growth model driven by $\alpha$-stable L\'evy shocks with time-varying capital elasticity. We extend the framework with an $\alpha$-stable L\'evy process, thereby capturing three stylized facts of severe macroeconomic fluctuations: heavy-tailed distributions, jump discontinuities, and infinite variance. We derive the stationary distribution of the capital deviation process, obtain its conditional characteristic function in closed form, and provide an integral representation that explicitly reveals a dual mean-reversion structure separating investment gestation lags from endogenous feedback. We design an estimation strategy based solely on well-defined objective functions that respects the probabilistic properties of L\'evy-driven data and circumvents the non-existence of variance. We apply the framework to Argentine quarterly data from 2004 to 2023, with time-varying capital elasticity calibrated from Penn World Table labor shares. Our estimates show that the L\'evy specification delivers structural parameters substantially closer to external PWT benchmarks than the Gaussian Ornstein-Uhlenbeck counterpart and substantially improves crisis-period tracking without sacrificing performance in tranquil periods. Cross-country evidence from Colombia and the United States confirms that the quarterly capital adjustment speed $\eta \approx 0.05$ exhibits striking stability across vastly different volatility regimes. Robustness checks across tail index specifications demonstrate that the L\'evy framework consistently outperforms the Ornstein-Uhlenbeck benchmark for a broad range of empirically relevant tail indices. These findings establish the L\'evy specification as a robust generalization of the Gaussian benchmark, offering a more credible tool for forecasting and structural parameter estimation in both emerging and advanced economies.

Figures

Figures reproduced from arXiv: 2607.20997 by the authors.

**Figure 2.** Observed and Predicted Scaled Capital Stock [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Observed and Predicted GDP: 2004–2014 The solid black line is the observed Argentine quarterly GDP (in levels, after trend removal). The dashed blue line is the one-step-ahead GDP prediction from the Lévy-driven model with time-varying αt . The dotted red line is the corresponding prediction from the Gaussian OU benchmark in [2]. Both predictions are obtained by applying the inverse transformation (4.29) to the resp… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Observed and Predicted GDP: 2008–2023 Same format as [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Capital kn Dynamics: Observed vs. Predictions under Different Tail Indices α processes ranging from moderately heavy to extremely heavy tails—strongly suggests that these parameters are structurally identified and orthogonal to the tail specification. Second, the Lévy…

**Figure 6.** Figure 6: Colombia: Observed vs. Predicted Capital Stock [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: United States: Observed vs. Predicted Capital Stock [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references

[1]

Solow, A contribution to the theory of economic growth.The Quarterly Journal of Economics, 1956, 70(1): 65-94

R.M. Solow, A contribution to the theory of economic growth.The Quarterly Journal of Economics, 1956, 70(1): 65-94

1956
[2]

Brannan, On natural slow time rhythms in economic growth.Physica A: Statistical Mechanics and its Appli- cations, 2019, 535: 122304

J. Brannan, On natural slow time rhythms in economic growth.Physica A: Statistical Mechanics and its Appli- cations, 2019, 535: 122304

2019
[3]

Dave, M.M

C. Dave, M.M. Sorge, Fat-tailed DSGE models: A survey and new results.Journal of Economic Surveys, 2025, 39(1): 146-171

2025
[4]

S. Yuan, Z. Zeng, J. Duan, Stochastic bifurcation for two-time-scale dynamical system withα-stable Lévy noise. Journal of Statistical Mechanics: Theory and Experiment, 2021, 2021(3): 033204

2021
[5]

Applebaum,Lévy processes and stochastic calculus

D. Applebaum,Lévy processes and stochastic calculus. Cambridge university press, 2009

2009
[6]

Protter,Stochastic integration and differential equations

P. Protter,Stochastic integration and differential equations. Springer-Verlag, 2012

2012
[7]

S. Yuan, J. Hu, X. Liu, J. Duan, Slow manifolds for dynamical systems with non-Gaussian stable Lévy noise. Analysis and Applications, 2019, 17(03): 477-511. 21

2019
[8]

S. Yuan, Y . Li, Z. Zeng, Stochastic bifurcations and tipping phenomena of insect outbreak systems driven by α-stable Lévy processes.Mathematical Modelling of Natural Phenomena, 2022, 17: 34

2022

Show all 16 references

[9]

S. Yuan, D. Blömker, Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindricalα-stable Lévy processes.SIAM Journal on Applied Dynamical Systems, 2022, 21(3): 1748-1777

2022
[10]

International Monetary Fund, Argentina: Ex-Post Evaluation of Exceptional Access under the 2022 Extended Fund Facility Arrangement, Country Report, IMF, 25/01, 2025

2022
[11]

Bellocchi, G

A. Bellocchi, G. Travaglini, Can variable elasticity of substitution explain changes in labor shares?Journal of Macroeconomics, 2023, 76: 103518

2023
[12]

Feenstra, R

R.C. Feenstra, R. Inklaar, M.P. Timmer, The next generation of the Penn World Table.American Economic Review, 2015, 105(10): 3150-3182

2015
[13]

Virtanen,et al

P. Virtanen,et al. SciPy 1.0: fundamental algorithms for scientific computing in Python.Nature Methods, 2020, 17(3): 261-272

2020
[14]

Chambers, C.L

J.M. Chambers, C.L. Mallows, B.W. Stuck, A method for simulating stable random variables.Journal of the American Statistical Association, 1976, 71(354): 340-344

1976
[15]

Storn, K

R. Storn, K. Price, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces.Journal of Global Optimization, 1997, 11(4): 341-359

1997
[16]

R.H. Byrd, P. Lu, J. Nocedal, C. Zhu, A limited memory algorithm for bound constrained optimization.SIAM Journal on Scientific Computing, 1995, 16(5): 1190-1208. 22

1995

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Solow, A contribution to the theory of economic growth.The Quarterly Journal of Economics, 1956, 70(1): 65-94

R.M. Solow, A contribution to the theory of economic growth.The Quarterly Journal of Economics, 1956, 70(1): 65-94

1956

[2] [2]

Brannan, On natural slow time rhythms in economic growth.Physica A: Statistical Mechanics and its Appli- cations, 2019, 535: 122304

J. Brannan, On natural slow time rhythms in economic growth.Physica A: Statistical Mechanics and its Appli- cations, 2019, 535: 122304

2019

[3] [3]

Dave, M.M

C. Dave, M.M. Sorge, Fat-tailed DSGE models: A survey and new results.Journal of Economic Surveys, 2025, 39(1): 146-171

2025

[4] [4]

S. Yuan, Z. Zeng, J. Duan, Stochastic bifurcation for two-time-scale dynamical system withα-stable Lévy noise. Journal of Statistical Mechanics: Theory and Experiment, 2021, 2021(3): 033204

2021

[5] [5]

Applebaum,Lévy processes and stochastic calculus

D. Applebaum,Lévy processes and stochastic calculus. Cambridge university press, 2009

2009

[6] [6]

Protter,Stochastic integration and differential equations

P. Protter,Stochastic integration and differential equations. Springer-Verlag, 2012

2012

[7] [7]

S. Yuan, J. Hu, X. Liu, J. Duan, Slow manifolds for dynamical systems with non-Gaussian stable Lévy noise. Analysis and Applications, 2019, 17(03): 477-511. 21

2019

[8] [8]

S. Yuan, Y . Li, Z. Zeng, Stochastic bifurcations and tipping phenomena of insect outbreak systems driven by α-stable Lévy processes.Mathematical Modelling of Natural Phenomena, 2022, 17: 34

2022