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arxiv: 2605.17299 · v1 · pith:VRDMESBDnew · submitted 2026-05-17 · 💰 econ.GN · cond-mat.stat-mech· q-fin.EC· q-fin.RM

Geometric Brownian motion with intermittent entries and exits

Suvam Pal , Viktor Stojkoski , Arnab Pal , Trifce Sandev This is my paper

Pith reviewed 2026-05-19 23:07 UTC · model grok-4.3

classification 💰 econ.GN cond-mat.stat-mechq-fin.ECq-fin.RM
keywords geometric Brownian motionentry and exit ratesstationary distributionmean first-passage timeoptimal exit rateeconomic observablesstochastic processesPoisson processes
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The pith

Geometric Brownian motion with unequal entry and exit rates still reaches a stationary distribution, and an optimal exit rate minimizes the mean time to hit a threshold.

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

The paper studies geometric Brownian motion augmented with independent Poisson processes for new units entering at one rate and existing units exiting at a different rate. It establishes that the distribution of the variable converges to a time-independent stationary form even when the two rates are asymmetric. The analysis further identifies three regimes in the behavior of the moments and derives an explicit expression for the mean first-passage time to a threshold, revealing a value of the exit rate that minimizes this time. These results apply directly to economic settings such as firm populations, labor flows, or income dynamics where growth, volatility, and turnover act together.

Core claim

In the generalized geometric Brownian motion with state-independent Poisson entry and exit at unequal rates, the system relaxes to a stationary distribution despite the asymmetry. The moments exhibit three distinct dynamical regimes determined by the relative sizes of volatility, drift, entry rate, and exit rate. The survival probability and mean first-passage time to a threshold are obtained in closed form, and an optimal exit rate exists that minimizes the mean first-passage time.

What carries the argument

Generalized geometric Brownian motion driven by independent Poisson entry and exit processes at unequal constant rates, used to obtain the stationary distribution and the mean first-passage time via the backward Kolmogorov equation.

If this is right

  • The distribution converges to a stationary form irrespective of entry-exit asymmetry.
  • The moments of the distribution fall into three distinct dynamical regimes governed by volatility, drift, entry rate, and exit rate.
  • Explicit formulas exist for the survival probability and mean first-passage time under the competing entry-exit dynamics.
  • A specific exit rate minimizes the mean first-passage time to any chosen threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Market policies that control exit rates could be tuned to shorten the average time for economic variables to reach desired growth levels.
  • The same entry-exit construction might be applied to non-economic systems such as population models with immigration and emigration.
  • Empirical tests could compare the predicted optimal exit rate against observed turnover statistics in firm or income data.

Load-bearing premise

Entry and exit events are modeled as independent Poisson processes whose rates remain fixed and do not depend on the current value of the geometric Brownian motion variable.

What would settle it

A long-time numerical simulation of the process in which the probability distribution fails to approach any time-independent shape when the entry rate is set different from the exit rate.

Figures

Figures reproduced from arXiv: 2605.17299 by Arnab Pal, Suvam Pal, Trifce Sandev, Viktor Stojkoski.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of income dynamics governed [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Stationary state [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Temporal evolution of the mean and MSD in the presence of entry and exit. Panel (a) indicates the initial exponential [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: III. FIRST-PASSAGE PROPERTIES AND OPTIMAL BEHAVIOR A natural question in open multiplicative systems con￾cerns how the observable x(t); be it a firm size, an indi￾vidual income, a loan balance, or a population count; reaches a prescribed threshold for the first time, and how the entry and exit mechanisms shape this threshold￾crossing statistics. We frame this as a first-passage prob￾lem where the evolving … view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Growth of the boundary as a function of time separat [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Panel (a): Variation of the MFPT (solid lines) with respect to the exit rate [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Scanning the behavior of the speed-up ratio [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Panel (a) illustrates the FPT density of the underlying GBM process (see Eq. (B10)), with different targets [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

We study a generalized geometric Brownian motion framework that incorporates both entries of new units and exit mechanisms for the current population, extending earlier stochastic resetting models where these rates are treated as identical. The model captures realistic features observed in many economic observables, which can be explained as market-driven firm entries/exits, worker inflow/outflow, and income growth/loss. This model is not conservative and, despite the asymmetry in the entry and exit rates, we find that the system eventually relaxes to a stationary distribution. Moreover, our analysis reveals three distinct dynamical regimes in the moments of the distribution, arising from the interplay between volatility, drift, entry, and exit rates. We further derive the survival probability and the mean first-passage time associated with the observed variable reaching certain threshold under the competing entry-exit processes. Interestingly, we identify an optimal exit rate that minimizes the mean first-passage time, providing insights into how entry and exit policies can influence the outcome of the system. These results should be useful for understanding the long-run behavior of economic systems in which growth, volatility, entry, and exit jointly shape the evolution of heterogeneous units.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends geometric Brownian motion by adding independent Poisson entry and exit processes with possibly asymmetric rates. It claims that the system relaxes to a stationary distribution despite non-conservation of probability mass, identifies three dynamical regimes in the moments arising from the interplay of volatility, drift, entry, and exit, derives the survival probability and mean first-passage time (MFPT) to a threshold, and identifies an optimal exit rate that minimizes the MFPT. These results are positioned as relevant to economic observables such as firm dynamics and income processes.

Significance. If the stationary distribution and MFPT results hold after proper accounting for the non-conservative dynamics, the work provides analytical tools for understanding long-run behavior in economic systems with growth, volatility, and turnover. The identification of an optimal exit rate and explicit dynamical regimes could inform policy analysis of entry-exit mechanisms, extending prior stochastic resetting models to asymmetric cases.

major comments (2)
  1. [model definition and derivation of the stationary distribution] Model definition and stationary distribution derivation: The Fokker-Planck equation includes a constant source term proportional to the entry rate and a sink term proportional to the exit rate times the density. When these rates differ, the total mass M(t) satisfies dM/dt = λ_entry - λ_exit M(t) and is not conserved. The manuscript must explicitly construct the stationary solution as the limiting normalized shape p_s(x) = lim P(x,t)/M(t) and verify its normalizability and independence from the exponential prefactor for the reported parameter regimes. Without this step the claim that the system 'eventually relaxes to a stationary distribution' despite asymmetry is not yet supported.
  2. [MFPT and optimal exit rate] MFPT and optimal exit rate section: The derivation of the mean first-passage time and the subsequent identification of an optimal exit rate must be shown to remain valid after the normalization procedure required by the non-conservative dynamics. It is unclear whether the optimality condition is derived from the normalized density or from an unnormalized quantity whose time evolution is affected by the exponential growth/decay of M(t).
minor comments (2)
  1. [dynamical regimes] The three dynamical regimes for the moments should be stated with explicit inequalities involving the drift, volatility, entry, and exit parameters so that the boundaries between regimes are reproducible.
  2. [results] Add a brief numerical verification (e.g., simulation histograms versus the analytic stationary shape) to confirm the claimed relaxation and regime boundaries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comments point by point below, providing clarifications and indicating the revisions made to the manuscript.

read point-by-point responses

  1. Referee: Model definition and stationary distribution derivation: The Fokker-Planck equation includes a constant source term proportional to the entry rate and a sink term proportional to the exit rate times the density. When these rates differ, the total mass M(t) satisfies dM/dt = λ_entry - λ_exit M(t) and is not conserved. The manuscript must explicitly construct the stationary solution as the limiting normalized shape p_s(x) = lim P(x,t)/M(t) and verify its normalizability and independence from the exponential prefactor for the reported parameter regimes. Without this step the claim that the system 'eventually relaxes to a stationary distribution' despite asymmetry is not yet supported.

    Authors: We agree with the referee that an explicit treatment of the non-conservative dynamics strengthens the manuscript. In the revised version, we have added a detailed derivation of the total probability mass M(t), which satisfies the ODE dM/dt = λ_entry - λ_exit M(t) and converges to λ_entry/λ_exit for λ_exit > 0. We then explicitly define the normalized density as p(x, t) = P(x, t)/M(t) and demonstrate that p(x, t) relaxes to a unique stationary distribution p_s(x) as t → ∞. This limit is independent of the transient exponential factors associated with the homogeneous solution of the Fokker-Planck equation. We verify that p_s(x) is normalizable in the parameter regimes analyzed in the paper, consistent with the conditions for the existence of a stationary distribution in geometric Brownian motion with drift. This construction supports the claim of relaxation to stationarity despite asymmetric rates. revision: yes

  2. Referee: MFPT and optimal exit rate section: The derivation of the mean first-passage time and the subsequent identification of an optimal exit rate must be shown to remain valid after the normalization procedure required by the non-conservative dynamics. It is unclear whether the optimality condition is derived from the normalized density or from an unnormalized quantity whose time evolution is affected by the exponential growth/decay of M(t).

    Authors: The mean first-passage time (MFPT) is computed from the underlying stochastic differential equation with the added Poisson entry and exit processes, using the appropriate backward Kolmogorov equation that incorporates the killing (exit) and source (entry) terms. Because the MFPT concerns the expected time for the process to reach the threshold starting from a given state, it is inherently a property of individual trajectories and is unaffected by the overall scaling of the ensemble density M(t). The exponential growth or decay of M(t) factors out in the calculation of the time integrals defining the MFPT. Consequently, the optimality condition for the exit rate, which minimizes this MFPT, holds for the normalized dynamics as well. In the revision, we have included a paragraph clarifying this point and showing that the expressions for the survival probability and MFPT are consistent with the normalized stationary distribution when considering long-time behavior. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no load-bearing reductions to inputs or self-citations

full rationale

The paper starts from the SDE for geometric Brownian motion augmented by independent Poisson entry and exit processes with constant rates, writes the corresponding Fokker-Planck equation containing explicit source and sink terms, and solves the stationary version of that equation for the shape of the distribution after the total mass evolution is factored out. The three dynamical regimes for the moments, the survival probability, and the MFPT (including the reported optimal exit rate) are obtained by direct integration or Laplace-transform methods on the same integro-differential equation. No parameter is fitted to data and then re-used as a prediction, no uniqueness theorem is imported from prior self-work, and the non-conservative character is stated explicitly rather than hidden. All central claims therefore rest on standard stochastic-process manipulations whose validity can be checked independently of the numerical values chosen for the rates.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard assumptions of stochastic calculus for GBM plus independent Poisson entry and exit processes; entry and exit rates function as free parameters that must be chosen or fitted to data.

free parameters (2)
  • entry rate
    Rate parameter controlling addition of new units; appears as a tunable constant in the generalized SDE.
  • exit rate
    Independent rate parameter controlling removal of units; central to the asymmetry and the optimal-rate result.
axioms (2)
  • domain assumption The underlying process obeys a geometric Brownian motion SDE between entry and exit events.
    Standard assumption for multiplicative noise models in finance and economics.
  • domain assumption Entry and exit events occur as independent Poisson processes.
    Enables the use of renewal theory to obtain the stationary distribution and MFPT.

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