arXiv subjects

Irene Tubikanec

Publications and source records attributed to Irene Tubikanec.

10 recordsLinked to original sources

arXiv2026

Stochastic compliance/evasion dynamics in tax models: a piecewise deterministic Markov process approach

This paper introduces a novel stochastic framework for modelling tax evasion dynamics by extending the deterministic model of Bertotti and Modanese (2018) through the use of Piecewise Deterministic Markov Processes (PDMPs). A key limitation of the original model is the static treatment of taxpayer compliance and evasion behaviour. We address this limitation by incorporating two stochastic mechanisms:(i) audits, where random enforcement events shift non-compliant individuals toward compliance, and (ii) imitation, where social influence drives compliant individuals toward evasion. We develop each mechanism as a separate PDMP, proving that both preserve the fundamental conservation laws of population and global income. Numerical simulations show that these mechanisms produce opposing long-term outcomes: pure audits lead to full compliance, while pure imitation leads to full evasion. The central contribution is a combined PDMP model in which both dynamics interact. This model no longer converges to an extreme equilibrium state. Instead, it can exhibit persistent fluctuations around the deterministic trajectory and suggests convergence to a stationary distribution, providing a more realistic representation of compliance-evasion dynamics observed in real economies. The proposed framework offers a versatile approach for integrating behavioural stochasticity into socio-economic models.

physics.soc-ph↗

arXiv2026

Splitting methods for stochastic Hodgkin-Huxley type systems and a localized fundamental mean-square convergence theorem

Existing fundamental theorems for mean-square convergence of numerical methods for stochastic differential equations (SDEs) require globally or one-sided Lipschitz continuous coefficients, while strong convergence results under merely local Lipschitz conditions are largely restricted to Euler-Maruyama type methods. To address these limitations, we introduce a novel localized version of the fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients, which naturally arise in a wide range of applications. Specifically, we show that if a numerical scheme is locally consistent in the mean-square sense of order $q>1$, then it is locally mean-square convergent with rate $q-1$. Building on this result, we further prove that global mean-square convergence follows, provided that both the exact solution and its numerical approximation admit bounded $2p$th moments for some $p>1$. These new convergence results are illustrated on a class of locally Lipschitz SDEs of Hodgkin-Huxley type, characterized by a conditionally linear drift structure. For these systems, we construct different Lie-Trotter and Strang splitting methods exploiting their conditional linearity. The proposed convergence framework is then applied to these schemes, requiring innovative proofs of local consistency and boundedness of moments. In addition, we establish key structure-preserving properties of the splitting methods, in particular state-space preservation and geometric ergodicity. Numerical experiments support the theoretical results and demonstrate that the proposed splitting schemes significantly outperform Euler-Maruyama type methods in preserving the qualitative features of the model.

math.PR↗

arXiv2025

Approximate Bayesian computation for stochastic hybrid systems with ergodic behaviour

Piecewise diffusion Markov processes (PDifMPs) form a versatile class of stochastic hybrid systems that combine continuous diffusion processes with discrete event-driven dynamics, enabling flexible modelling of complex real-world hybrid phenomena. The practical utility of PDifMP models, however, depends critically on accurate estimation of their underlying parameters. In this work, we present a novel framework for parameter inference in PDifMPs based on approximate Bayesian computation (ABC). Our contributions are threefold. First, we provide detailed simulation algorithms for PDifMP sample paths. Second, we extend existing ABC summary statistics for diffusion processes to account for the hybrid nature of PDifMPs, showing particular effectiveness for ergodic systems. Third, we demonstrate our approach on several representative example PDifMPs that empirically exhibit ergodic behaviour. Our results show that the proposed ABC method reliably recovers model parameters across all examples, even in challenging scenarios where only partial information on jumps and diffusion is available or when parameters appear in state-dependent jump rate functions. These findings highlight the potential of ABC as a practical tool for inference in various complex stochastic hybrid systems.

stat.ME↗

arXiv2025

Network inference via approximate Bayesian computation. Illustration on a stochastic multi-population neural mass model

In this article, we propose an adapted sequential Monte Carlo approximate Bayesian computation (SMC-ABC) algorithm for network inference in coupled stochastic differential equations (SDEs) used for multivariate time series modeling. Our approach is motivated by neuroscience, specifically the challenge of estimating brain connectivity before and during epileptic seizures. To this end, we make four key contributions. First, we introduce a 6N-dimensional SDE to model the activity of N coupled neuronal populations, extending the (single-population) stochastic Jansen and Rit neural mass model used to describe human electroencephalography (EEG) rhythms, particularly epileptic activity. Second, we construct a reliable and efficient numerical splitting scheme for the model simulation. Third, we apply the proposed adapted SMC-ABC algorithm to the neural mass model and validate it on different types of simulated data. Compared to standard SMC-ABC, our approach significantly reduces computational cost by requiring fewer model simulations to reach the desired posterior region, thanks to the inclusion of binary parameters describing the presence or absence of coupling directions. Finally, we apply our method to real multi-channel EEG data, uncovering potential similarities in patients' brain activities across different epileptic seizures, as well as differences between pre-seizure and seizure periods.

stat.ME↗

arXiv2024

Inference for the stochastic FitzHugh-Nagumo model from real action potential data via approximate Bayesian computation

The stochastic FitzHugh-Nagumo (FHN) model is a two-dimensional nonlinear stochastic differential equation with additive degenerate noise, whose first component, the only one observed, describes the membrane voltage evolution of a single neuron. Due to its low-dimensionality, its analytical and numerical tractability and its neuronal interpretation, it has been used as a case study to test the performance of different statistical methods in estimating the underlying model parameters. Existing methods, however, often require complete observations, non-degeneracy of the noise or a complex architecture (e.g., to estimate the transition density of the process, "recovering" the unobserved second component) and they may not (satisfactorily) estimate all model parameters simultaneously. Moreover, these studies lack real data applications for the stochastic FHN model. The proposed method tackles all challenges (non-globally Lipschitz drift, non-explicit solution, lack of available transition density, degeneracy of the noise and partial observations). It is an intuitive and easy-to-implement sequential Monte Carlo approximate Bayesian computation algorithm, which relies on a recent computationally efficient and structure-preserving numerical splitting scheme for synthetic data generation and on summary statistics exploiting the structural properties of the process. All model parameters are successfully estimated from simulated data and, more remarkably, real action potential data of rats. The presented novel real-data fit may broaden the scope and credibility of this classic and widely used neuronal model.

stat.CO↗

arXiv2024

Granger Causal Inference in Multivariate Hawkes Processes by Minimum Message Length

Multivariate Hawkes processes (MHPs) are versatile probabilistic tools used to model various real-life phenomena: earthquakes, operations on stock markets, neuronal activity, virus propagation and many others. In this paper, we focus on MHPs with exponential decay kernels and estimate connectivity graphs, which represent the Granger causal relations between their components. We approach this inference problem by proposing an optimization criterion and model selection algorithm based on the minimum message length (MML) principle. MML compares Granger causal models using the Occam's razor principle in the following way: even when models have a comparable goodness-of-fit to the observed data, the one generating the most concise explanation of the data is preferred. While most of the state-of-art methods using lasso-type penalization tend to overfitting in scenarios with short time horizons, the proposed MML-based method achieves high F1 scores in these settings. We conduct a numerical study comparing the proposed algorithm to other related classical and state-of-art methods, where we achieve the highest F1 scores in specific sparse graph settings. We illustrate the proposed method also on G7 sovereign bond data and obtain causal connections, which are in agreement with the expert knowledge available in the literature.

cs.LG↗

arXiv2022

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.

math.NA↗

arXiv2021

Qualitative properties of numerical methods for the inhomogeneous geometric Brownian motion

We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The conditional and asymptotic mean and variance of the IGBM are known and the process can be characterised according to Feller's boundary classification. We compare the frequently used Euler-Maruyama and Milstein methods, two Lie-Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong-Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler-Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler-Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie-Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features.

math.NA↗

arXiv2020

Theoretical analysis and simulation methods for Hawkes processes and their diffusion approximation

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced in Ditlevsen (2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. First, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

math.NA↗

arXiv2019

Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

Approximate Bayesian Computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time dependent, real world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise. First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g., the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography (EEG) data. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.

stat.CO↗