Stationary Distributions of the Mode-switching Chiarella Model

Jean-Philippe Bouchaud; Jutta G. Kurth

REVIEW 2 major objections 4 minor 24 references

Stationary Distributions of the Mode-switching Chiarella Model

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper shows that, in the slow-trend limit of the extended Chiarella model, the stationary mispricing distribution is a Gaussian–cosh whose uni-to-bimodal transition occurs at κ = 2β²/σ² — not at the previously claimed Hopf condition κ =

desk verdict Corrects two published claims in the Chiarella literature with a solid asymptotic analysis; the slow-trend bifurcation condition holds up, but there are fixable slips and sparse numerics. read the letter →

arxiv 2511.13277 v2 pith:H7WA7KON submitted 2025-11-17 q-fin.TR physics.data-an

Jutta G. Kurth , Jean-Philippe Bouchaud This is my paper · ORCID

classification q-fin.TRphysics.data-an MSC 60H1091G8037H20

keywords ChiarellamodelmispricingdistributionstationaryP-bifurcationFokker-PlancktrendfollowingmeanreversionFurutsu-Novikovtheorem

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 derives stationary distributions for the extended Chiarella model, a stochastic dynamical system that pits mean-reverting value investors against saturating trend followers. In the slow-trend, large-saturation limit it obtains a closed-form Gaussian–cosh mispricing distribution and shows that this distribution turns from unimodal to bimodal when mean reversion is slow, with a critical point that depends on the total noise and quadratically on the trend feedback — not on the deterministic Hopf bifurcation condition that earlier work had proposed. In the fast-trend limit it shows that weak coupling produces a unimodal Gaussian mispricing distribution, and it disproves the claim that a bimodal trend distribution necessarily implies a bimodal mispricing distribution. The paper's own caveat is that the fast-trend critical threshold is indicative and possibly a lower bound, and that the exact solution in the strongly coupled regime remains out of reach.

What carries the argument

The workhorse is a quasi-static (adiabatic) factorization p(x) = ∫ p(x|y) p(y) dy, where x is the mispricing and y the slowly-varying trend variable; p(x|y) is taken as the stationary solution of the frozen-y Fokker-Planck equation, and p(y) as the Gaussian OU marginal. In the slow-trend, large-γ limit the conditional distribution is approximated by replacing cosh^n(γ(αx+y)) with cosh(nγ(αx+y)) (a saddle-point replacement), which makes all integrals explicit and yields the Gaussian–cosh distribution. In the fast-trend limit the Furutsu-Novikov theorem is used to compute the variance that enters the effective Gaussian, and the effective mean-reversion speed κ_eff = κZ(Θ) with Z(Θ) = 1 − 2Θ² +

What would settle it

Run the Euler-Maruyama simulation of the full two-dimensional system (Eq. 2) across a grid of κ values around 2β²/σ² for, say, α = 10^−5, β = 0.05, γ = 5×10^4, σ_N = 0.2, σ_V = 0.1, and measure the number of modes of the stationary p(δ) histogram. If the transition point moves systematically with α or γ, or if it deviates from 2β²/σ² by more than the finite-sample resolution, the claimed boundary is not established.

Watch

Extended reading notes

Core claim

The central claim is a set of stationary distributions for the mode-switching Chiarella model that correct two earlier results. In the slow-trend limit κ≫α with large saturation γ, the stationary mispricing distribution is p(δ) ∝ cosh(2βδ/σ²)·exp(−κδ²/σ²), i.e. a Gaussian–cosh distribution whose modality is governed by the sign of κ − 2β²/σ²: unimodal when mean reversion is fast, bimodal when it is slow. This differs from the Hopf condition κ = α(βγ−1) of the deterministic system and thereby refutes the earlier claim that the P-bifurcation coincides with the dynamical bifurcation. In the fast-trend limit α≫κ, the paper uses the Furutsu-Novikov theorem to show that weak coupling produces a un

Load-bearing premise

The sharp boundary κ = 2β²/σ² rests on two stacked approximations — treating the slow variable as frozen (adiabatic factorization) and replacing cosh^n with cosh(n·) at large γ — and the paper validates them numerically at only a single parameter point without error bars, so if the approximation error varies across the boundary the critical value could shift.

Editorial extensions

If this is right

If correct, the P-bifurcation boundary for slow trends is κ = 2β²/σ²: the critical mean-reversion strength is independent of the trend time scale α and saturation γ, and grows only quadratically in the trend feedback β.
Strong noise from either price or value shocks (σ) can wipe out bimodality, so empirically observed bimodal mispricing requires both weak noise and slow mean reversion.
In the fast-trend, weak-coupling regime, the mispricing distribution is Gaussian to leading order; bimodality can only arise when the coupling parameter Θ exceeds a threshold that the paper estimates, making the telegraphic trend switches rare enough to let the mispricing settle in one of two states.
The counterexample to the 'bimodal trend implies bimodal mispricing' claim means empirical studies that infer mispricing multimodality from trend multimodality need a separate check on the feedback strength.

Reading between the lines

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

A testable extension is to calibrate the model's parameters on real price and value data and check whether the empirical mispricing modality follows the κ = 2β²/σ² boundary rather than the Hopf line; the paper's boundary predicts that the modality transition can shift substantially when noise is heterogeneous.
The large-γ saddle-point replacement could be relaxed to finite γ, suggesting the boundary may acquire a weak γ-dependence at moderate saturation; the paper does not explore this, but the derivation in Appendix C shows the next-order terms are suppressed by powers of 1/γ.
The decoupling of trend and mispricing modalities in the fast-trend regime implies a similar decoupling could appear in other two-timescale stochastic systems with saturating feedback, e.g. opinion dynamics or predator-prey models with telegraphic switching.
The Θ_c estimate being a lower bound means the true transition could lie noticeably higher; a numerical scan across β for fixed α, σ_N would either confirm Θ_c ≈ 0.798 or push it upward, which the paper itself flags as an open point.

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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

This paper does real work. It derives stationary distributions for the extended Chiarella model in three regimes and, if I read it right, corrects two published claims: the P-bifurcation in the slow-trend regime is governed by κ = 2β²/σ², not the Hopf condition, and a bimodal trend distribution does not force a bimodal mispricing distribution in the fast-trend case. The Gaussian-cosh result and the Furutsu-Novikov treatment are genuinely new in this context, and the numerics, while sparse, support the qualitative conclusions.

I gave the central approximation some scrutiny because the stress-test note worries about the replacement coshⁿ ≈ cosh(n·) failing near y = 0 where the curvature decides the bifurcation. That worry does not land. In the slow-trend limit you have α → 0, γ → ∞ with αγ = O(1), which makes the typical scaled argument γy of order γ√α → ∞. The near-zero region in y carries negligible probability mass, and the exact n = 2 normalization confirms the leading-order result. The derivation is an honest adiabatic-plus-saddle-point computation, not a hidden fitting step. The qualitative refutation of the Hopf-coincidence claim is robust.

The soft spots are real but manageable. First, Eq. (26) and Appendix D disagree on the O(Θ) variance term by what looks like a factor π; that is probably a typo, but it must be fixed before publication. Second, the abstract says the critical point is “established,” which is defensible for the slow-trend boundary but sits awkwardly next to the text’s own caveat that Θc ≈ 0.798 is “only indicative and possibly a lower bound.” The abstract should be more precise about which regime is established. Third, the numerical verification is thin: one γ, one n, no error bars. A parameter sweep across the predicted boundary would considerably strengthen confidence. No code or data is shipped, which is a pity.

None of this breaks the central claims. The paper is honest about where it cannot solve the problem, and that honesty should count in its favor. The audience is people who use the Chiarella model or cite the earlier claim that P-bifurcation coincides with Hopf bifurcation; for them this is a needed correction. I would bring it to a reading group and would cite the slow-trend result. It deserves a serious referee, not a desk reject.

My recommendation: send it to peer review with a request for the typo fix, a sharper abstract, and one additional numerical convergence check around the boundary.

Referee Report

2 major / 4 minor

Summary. The paper studies stationary distributions of the generalized Chiarella model (Eqs. 1–2) in three parameter regimes. In the linear/small-γ regime (Sec. III) the joint stationary density is a bivariate Gaussian obtained from the Lyapunov equation. In the slow-trend regime κ≫α with large γ (Sec. IV) the paper derives a Gaussian-cosh mispricing distribution and claims a P-bifurcation threshold κ=2β²/σ², independent of α and γ, which it contrasts with the deterministic Hopf condition κ=α(βγ−1). In the fast-trend regime α≫κ (Sec. V), the paper uses a Furutsu–Novikov argument for weak coupling to obtain a Gaussian distribution, and for stronger coupling argues—with the help of an approximate quasi-static calculation and Monte Carlo simulation—that a bimodal trend distribution does not imply a bimodal mispricing distribution; the estimated threshold Θ_c≈0.798 is explicitly labeled as indicative and possibly a lower bound. Appendices contain the detailed derivations and the exact normalization for integer n in the slow-trend case.

Significance. If the central claims hold, the paper is a useful contribution: it provides explicit stationary distributions in several regimes and challenges the common identification of the P-bifurcation condition with the deterministic Hopf condition. The linear-regime Lyapunov solution is verified algebraically against the FPE, the fast-trend variance is derived rather than fitted, and the authors are commendably explicit about the breakdown of the quasi-static assumption in Appendix E and about the absence of an exact solution in the strong-coupling fast-trend case. Monte Carlo histograms are provided for each regime. However, the headline slow-trend threshold rests on an uncontrolled asymptotic replacement that is least accurate at the very point where the bifurcation condition is decided. The quantitative claim therefore needs additional support before the paper can be accepted as establishing the critical point.

major comments (2)

[Sec. IV.2 / Appendix C, Eqs. (C6)–(C7) and (22)–(23)] Eq. (22) is obtained by replacing [cosh(γ(αx+y))]^n with cosh(nγ(αx+y)) in Eqs. (C6)–(C7). This approximation is not uniform: the bifurcation condition Eq. (23) is decided by p''(0), i.e. by the integrand at x=0 and, for α small, y≈0, where the argument of the cosh is not large. For n=2, cosh²(u)=(cosh(2u)+1)/2, so the replacement drops a term of the same order and changes the small-argument curvature. Because the exact A(y) for integer n is known (Eq. 19), the one-dimensional integral (20) can be evaluated and compared with Eq. (22); the paper gives only a single n=2 check (Fig. 2) with no error bars. Until the exact integral is shown to reproduce the curvature at x=0, the 'established' threshold κ=2β²/σ² and its independence of α,γ are not supported.
[Fig. 2 and Sec. IV.3] The numerical confirmation of the slow-trend P-bifurcation consists of three histograms (κ=0.2, 0.06, 0.01) with no error bars, no quantitative goodness-of-fit measure, and no scan across the claimed boundary κ=2β²/σ². Because that boundary is the central quantitative claim, please add a direct numerical estimate of p''(0) or a threshold scan, and compare with the exact integral for integer n.

minor comments (4)

[Eq. (14)] The second line appears inconsistent with the exact change of variables in Appendix B: from Eq. (B5), dy = -α y dt - α² x dt + α σ_V dW_V, not dy = -α y dt + α² dx + α σ_V dW_V. Please correct the typo or explain the intended ordering.
[Eq. (26) / App. D] The typesetting of the O(Θ) term is ambiguous. It should read 2ασ_N²/(√π κ(α+κ_eff)) Θ to match Eq. (D52); if the √π is misplaced, the expression differs by a factor π.
[Sec. V.2 / Fig. 4] The text says p(δ) is still unimodal at Θ=1.01 even though Θ_c≈0.798; this is consistent with the 'lower bound' caveat, but the figure caption or text should state explicitly that the analytic Θ_c is not a confirmation of the observed onset.
[Eq. (E34)] The symbol T is used for the noise temperature T=σ_N²α/2, while T is also used for simulation time in figures; please distinguish these to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stationary distributions are derived from the stated Fokker-Planck equations with explicit approximations and checked against independent Monte Carlo; self-citations are contextual.

full rationale

The derivation chain for each regime starts from the model SDEs and the corresponding Fokker-Planck equation, not from the result being claimed. In the slow-trend regime, p(x|y) is the stationary Maxwell-Boltzmann solution of the frozen-y FPE (Eqs. 16-17), p(y) is the alpha->0 OU marginal, and Eq. (22) follows by integrating over y after the stated large-gamma replacement in Appendix C. That replacement is an approximation and a potential correctness risk, not an identity between input and output; it does not presuppose the bifurcation boundary. The critical condition Eq. (23) is then obtained by evaluating p''(0) of the derived density, not by assuming it. In the fast-trend regime, the Gaussian result Eq. (25) is derived via the Furutsu-Novikov theorem, and the decoupling claim in Sec. V.2 is supported by independent numerical simulation. The authors explicitly flag that Theta_c is 'only indicative and, as argued in Appendix E, possibly a lower bound' (Sec. V.2), so that overclaim is acknowledged rather than concealed. No parameter is fitted to the quantity being predicted, and no load-bearing conclusion is justified solely by self-citation: refs. [3] and [4] provide model background and the Hopf comparison condition, which is independently standard and also appears in the external literature [5]. The acknowledged limitations of the large-gamma approximation and of Theta_c are correctness risks, not circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

All results are derived from the model SDEs via Fokker-Planck/Lyapunov/Furutsu-Novikov machinery; the only 'inputs' beyond the model parameters are asymptotic approximations the paper spells out. No data fitting occurs; the parameters in the figures are simulation inputs, not fits. The ledger records the approximation regimes the derivations stand on.

assumptions (8)

standard math Ito interpretation and Fokker-Planck equation for the SDE system (1)–(2)
Sec. III, Eq. (6): the stationary FPE is the device for all results.
domain assumption Deterministic stability κ > α(βγ−1) is required for a stationary distribution to exist; in the linear regime this is always satisfied as γ→0
Sec. III: 'Σ is positive semi-definite when κ > α(βγ−1), which is the bifurcation condition in the deterministic system... always true in the considered limit.'
ad hoc to paper Quasi-static / adiabatic elimination: for α≪κ, x equilibrates at fixed y, so p(x|y) is the frozen-y stationary solution of Eq. (16), and p(y) is the α→0 OU Gaussian
Sec. IV.1: 'a quasi-static approximation can be assumed, whereby the standard Maxwell-Boltzmann equilibrium is reached before y has had time to vary much' (Eq. 17). The −α²x coupling in ẏ (App. B, Eq. B5) is dropped.
ad hoc to paper Large-γ saddle-point reduction: cosh^n(γ(αx+y)) ≈ cosh(nγ(αx+y)), n = 2β/(αγσ²), keeping only the leading exponential order
Sec. IV.2 and App. C2, Eqs. (C6)–(C7): 'the leading exponential order of the cosh overwhelms all others'.
ad hoc to paper Telegraphic-noise approximation: for α≫κ and γσ²_N ↛ 0, tanh of a fast OU process acts as a two-state ±1 noise, and the x-dynamics is linearized to first order in κx and O(β³)
Sec. V.1, App. D, Eqs. (D1)–(D9): 'the encapsulated hyperbolic tangent may be approximated by auto-correlated telegraphic noise'.
ad hoc to paper Quasi-static assumption for x in the α≫κ strong-coupling regime, used to derive p(M) and the Θ_c estimate, and acknowledged to break down
Sec. V.2, App. E: 'the quasi-stationary assumption is only approximate... possibly a lower bound'; App. E.b 'Quasi-static Assumption Break-down' shows the separation of time scales fails as Θ grows.
domain assumption The two Brownian motions dW_N and dW_V are independent
Eq. (1) states 'standard Brownian Motions'; the diffusion matrix D in Eq. (5) treats them as uncorrelated.
standard math Furutsu-Novikov theorem for Gaussian white-noise functionals
App. D, Eq. (D32), applied to compute ⟨tanh(X_s)ξ_{s'}⟩; cited to [12].

invented entities (1)

Effective telegraphic noise ξ^tele ∈ {±1}
purpose: Approximate the action of the fast, saturating trend term βtanh(γM) on the mispricing in the α≫κ limit, enabling closed-form variance (Eq. 26)
Mathematical approximation (App. D), not a physical mechanism; its autocorrelation (2/π)arcsin(e^{−α|t−s|}) is derived, not measured.

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Cite this review

Pith. "Pith review of Stationary Distributions of the Mode-switching Chiarella Model." pith.science (2026). https://pith.science/paper/H7WA7KON

@misc{pith2026251113277,
  author       = {Pith},
  title        = {Pith review of: Stationary Distributions of the Mode-switching Chiarella Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H7WA7KON}},
  note         = {Machine review of arXiv:2511.13277}
}

read the original abstract

We derive the stationary distribution in various regimes of the extended Chiarella model of financial markets. This model is a stochastic nonlinear dynamical system that encompasses dynamical competition between a (saturating) trending and a mean-reverting component. We find the so-called mispricing distribution and the trend distribution to be unimodal Gaussians in the small noise, small feedback limit. Slow trends yield Gaussian-cosh mispricing distributions that allow for a P-bifurcation: unimodality occurs when mean-reversion is fast, bimodality when it is slow. The critical point of this bifurcation is established and refutes previous ad-hoc reports and differs from the bifurcation condition of the dynamical system itself. For fast, weakly coupled trends, deploying the Furutsu-Novikov theorem reveals that the result is again unimodal Gaussian. For the same case with higher coupling we disprove another claim from the literature: bimodal trend distributions do not generally imply bimodal mispricing distributions. The latter becomes bimodal only for stronger trend feedback. The exact solution in this last regime remains unfortunately beyond our proficiency.

Figures

Figures reproduced from arXiv: 2511.13277 by the authors.

**Figure 1.** Grey: Numerical histograms of the stationary distribution in the case γ small. Simulation parameters are (κ, β, α, σN , σV ) = (0.1, 0.2, 0.2, 0.2, 0.1) and γ as detailed in the plot. T = γ × 109 , dt = γ/2 and g = 0. Coloured: Corresponding analytical stationary distributions p(δ) according to Eq.(13). The distributions with γ > 10−4 are shifted by multiples of 1 on the abscissa and of 0.5 on the ordinate. dynamics… view at source ↗

**Figure 2.** Same as Fig. 1 but for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Same as Fig. 1 but in the case [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Same as Fig. 1 but for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 1 linked inside Pith

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Attraction/convergence of price towards funda- mental value
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It is common belief that in the presence of noise the distribution ofmispricings(i.e

Oscillation of price around the fundamental value. It is common belief that in the presence of noise the distribution ofmispricings(i.e. the difference between price and value) is unimodal in case 1 and bimodal in case 2, when the price stochastically quasi-oscillates around the fundamental value [3, 5, 6]. This means that the phenomenological P-bifurcati...
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Co-existence of trend and value in financial markets: Estimating an extended Chiarella model

Majewski AA, Ciliberti S, Bouchaud JP. Co-existence of trend and value in financial markets: Estimating an extended Chiarella model. Journal of Economic Dy- namics and Control. 2020;112:103791

2020
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polarize

would have falsely predictedp(δ)to be unimodal via the Hopf-bifurcation condition of the dynamical system, while Eq.(23) correctly predicts the bimodality, which has further been confirmed on edge cases. V. F AST TRENDS: THEα≫κLIMIT Considering the Langevin equation corresponding to Eqs. (14), we see thatytracksxclosely whenα≫κ. In this case, and assuming...
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The dynamical mechanisms that lead to either uni- or bimodality are clearly established

and is correct in some limiting cases, the present paper disproves the result in general and provides the correct stationary mispricing distributions in many pos- arXiv:2511.13277v1 [q-fin.TR] 17 Nov 2025 2 sible scenarios. The dynamical mechanisms that lead to either uni- or bimodality are clearly established. Several extensions to the Chiarella model an...

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The dynamics of speculative behaviour

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Black was right: Price is within a factor 2 of Value

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Revisit- ing the Excess Volatility Puzzle Through the Lens of the Chiarella Model

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IV, Eq.(19)

Normalisation F unctionA(y) In this section we derive the normalisation functionA(y)given in Sec. IV, Eq.(19). As detailed in the main text, Eq.(18), the normalisation function reads A(y) = Z ∞ −∞ e− 1 σ2 κx2 cosh(γ(αx+y)) n dx(C1) and can generally only be calculated for inte...
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For that,A(y)must first be determined

Large-γ-Limit Derivations: Normalisation and Stationary Distribution In the limitγ→ ∞, whileαγ=const.∈R, the stationary distribution can be derived. For that,A(y)must first be determined. When the exponents inA(y)are large, the leading order term substantially overwhelms all o...
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Thus, whether the distribution is uni- or bimodal is independent ofαandγin this limit

The distribution lost itsγ-dependence (as expected) but this leads to the stationary distribution being independent of the trend time scaleα, too. Thus, whether the distribution is uni- or bimodal is independent ofαandγin this limit
[21]

In the bimodal case, interestingly, the position of the maxima will not only depend onβandκbut also on the noise strengthσ 2

There is always an extremum atx= 0, which is either unique and a maximum (unimodality), or a minimum accom- panied by two maxima symmetrically placed around it (bimodality) forpis even. In the bimodal case, interestingly, the position of the maxima will not only depend onβandκ...
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First, letY t be the underlying OU-process

Autocovariance T elegraphic Noise In this section the autocovariance of the telegraphic noise will be derived. First, letY t be the underlying OU-process. Henceforth, it is assumed thatYis stationary, such that ⟨YtYs⟩ ∼e−α|t−s|.(D10) 14 Further, for the OU-process is a Gaussia...
[23]

The variance of the process is thus given by ⟨x2⟩=⟨A 2⟩+⟨B 2⟩+⟨C 2⟩+ 2⟨AB⟩=⟨A 2⟩+ σ2 N +σ 2 V 2κeff + 2⟨AB⟩(D19) because all terms are centered

V ariance of the Process The integrated version of the Langevin equation (using an integrating factoreκeff t) reads x(t) = Z t 0 e−κeff (t−s) h βeff ξtele s +σ N ξN s −σ V ξV s i ds=:A(t) +B(t) +C(t),(D18) where the termsA,B,Care defined by the three integrals. The variance of...
[24]

Leading orders of⟨A 2⟩ We have found (in the limitα≫κ) the closed-form solution ⟨A2⟩= β2 √πκ2 √π− Γ( α+κeff 2α ) Γ( 2α+κeff 2α ) ! .(D53) Let us determine the leading order inΓwhenα≫κ. Definingϵ= κ 2α, the argument of the Gamma-fuction in the numerator is1 2 +ϵand of the denom...

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Attraction/convergence of price towards funda- mental value

[2] [2]

It is common belief that in the presence of noise the distribution ofmispricings(i.e

Oscillation of price around the fundamental value. It is common belief that in the presence of noise the distribution ofmispricings(i.e. the difference between price and value) is unimodal in case 1 and bimodal in case 2, when the price stochastically quasi-oscillates around the fundamental value [3, 5, 6]. This means that the phenomenological P-bifurcati...

[3] [3]

Co-existence of trend and value in financial markets: Estimating an extended Chiarella model

Majewski AA, Ciliberti S, Bouchaud JP. Co-existence of trend and value in financial markets: Estimating an extended Chiarella model. Journal of Economic Dy- namics and Control. 2020;112:103791

2020

[4] [4]

polarize

would have falsely predictedp(δ)to be unimodal via the Hopf-bifurcation condition of the dynamical system, while Eq.(23) correctly predicts the bimodality, which has further been confirmed on edge cases. V. F AST TRENDS: THEα≫κLIMIT Considering the Langevin equation corresponding to Eqs. (14), we see thatytracksxclosely whenα≫κ. In this case, and assuming...

[5] [5]

The dynamical mechanisms that lead to either uni- or bimodality are clearly established

and is correct in some limiting cases, the present paper disproves the result in general and provides the correct stationary mispricing distributions in many pos- arXiv:2511.13277v1 [q-fin.TR] 17 Nov 2025 2 sible scenarios. The dynamical mechanisms that lead to either uni- or bimodality are clearly established. Several extensions to the Chiarella model an...

arXiv 2025

[6] [6]

The dynamics of speculative behaviour

Chiarella C. The dynamics of speculative behaviour. Annals of Operations Research. 1992;37(1):101-23

1992

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