arXiv subjects

Max-Olivier Hongler

Publications and source records attributed to Max-Olivier Hongler.

16 recordsLinked to original sources

arXiv2025

Skew-Normal Diffusions

We construct a class of stochastic differential equations driven by White Gaussian noise sources whose solutions can be drawn from skewed Gaussian probability laws, here referred as skew-Normal diffusion (SKN) processes. The non-Gaussian character results from implementing a nonlinear and time-inhomogneous drift constructed via ad-hoc changes of probability measure (i.e. Doob's $h$-transform). The SKN processes can be alternatively constructed as dynamic censoring models. While explicitly non-Gaussian, the SKN processes share several properties of Gaussian processes, in particular the invariance under linear transformations. This result allows us to discuss analytically the characteristics of this class of stochastic dynamics. As an illustration, we show how linear noisy monitoring of SKN processes yields a solvable finite dimensional and non-linear stochastic filtering which naturally extends the Kalman-Bucy Gaussian case.

math.PR↗

arXiv2024

Nonlinear Economic State Equilibria via van der Waals Modeling

The renowned van der Waals (VDW) state equation quantifies the equilibrium relationship between pressure $P$, volume $V$ and temperature $k_{B}T$ of a real gas. We assign new variable interpretations adapted to the economic context: $P \rightarrow Y$, representing price; $V \rightarrow X$, representing demand; and $k_{B}T \rightarrow κ$, representing income, to describe an economic state equilibrium. With this reinterpretation, the price elasticity of demand (PED) and the income elasticity of demand (YED) are non-constant factors and may exhibit a singularity of the cusp-catastrophe type. Within this economic framework, the counterpart of VDW liquid-gas phase transition illustrates a substitution mechanism where one product or service is replaced by an alternative substitute. The conceptual relevance of this reinterpretation is discussed qualitatively and quantitatively via several illustrations ranging from transport (carpooling), medical context (generic versus original medication) and empirical data drawn from the electricity market in Germany.

nlin.AO↗

arXiv2023

An exact bandit model for the risk-volatility tradeoff

We revisit the two-armed bandit (TAB) problem where both arms are driven by diffusive stochastic processes with a common instantaneous reward. We focus on situations where the Radon-Nikodym derivative between the transition probability densities of the first arm with respect to the second is explicitly known. We calculate how the corresponding Gittins' indices behave under such a change of probability measure. This general framework is used to solve the optimal allocation of a TAB problem where the first arm is driven by a pure Brownian motion and the second is driven by a centered super-diffusive non-Gaussian process with variance quadratically growing in time. The probability spread due to the super-diffusion introduces an extra risk into the allocation problem. This drastically affects the optimal decision rule. Our modeling illustrates the interplay between the notions of risk and volatility.

math.PR↗

arXiv2023

Stochastic Pairwise Preference Convergence in Bayesian Agents

Beliefs inform the behavior of forward-thinking agents in complex environments. Recently, sequential Bayesian inference has emerged as a mechanism to study belief formation among agents adapting to dynamical conditions. However, we lack critical theory to explain how preferences evolve in cases of simple agent interactions. In this paper, we derive a Gaussian, pairwise agent interaction model to study how preferences converge when driven by observation of each other's behaviors. We show that the dynamics of convergence resemble an Ornstein-Uhlenbeck process, a common model in nonequilibrium stochastic dynamics. Using standard analytical and computational techniques, we find that the hyperprior magnitudes, representing the learning time, determine the convergence value and the asymptotic entropy of the preferences across pairs of agents. We also show that the dynamical variance in preferences is characterized by a relaxation time $t^\star$, and compute its asymptotic upper bound. This formulation enhances the existing toolkit for modeling stochastic, interactive agents by formalizing leading theories in learning theory, and builds towards more comprehensive models of open problems in principal-agent and market theory.

nlin.AO↗

arXiv2022

Can one hear the shape of a target zone?

We develop an exchange rate target zone model with finite exit time and non-Gaussian tails. We show how the tails are a consequence of time-varying investor risk aversion, which generates mean-preserving spreads in the fundamental distribution. We solve explicitly for stationary and non-stationary exchange rate paths, and show how both depend continuously on the distance to the exit time and the target zone bands. This enables us to show how central bank intervention is endogenous to both the distance of the fundamental to the band and the underlying risk. We discuss how the feasibility of the target zone is shaped by the set horizon and the degree of underlying risk, and we determine a minimum time at which the required parity can be reached. We prove that increases in risk after a certain threshold can yield endogenous regime shifts where the ``honeymoon effects'' vanish and the target zone cannot be feasibly maintained. None of these results can be obtained by means of the standard Gaussian or affine models. Numerical simulations allow us to recover all the exchange rate densities established in the target zone literature. The generality of our framework has important policy implications for modern target zone arrangements.

econ.GN↗

arXiv2019

Imitation, proximity, and growth -- A collective swarm dynamics approach

This paper is based on the premise that economic growth is driven by an interplay between innovation and imitation in an economy composed of interacting firms operating in a stochastic environment. A novel approach to modeling imitation is presented, based on range-dependent processes that describe how firms consider proximity when imitating peers who are found in a given neighborhood in terms of productivity. Using a particularly tractable approach, we are able to analyze how drastically different economic growth scenarios emerge from different imitation strategies. These emerging scenarios range from diffusive growth where the variance of productivity grows indefinitely, to balanced growth described by a traveling wave with fixed variance. The latter scenario is sustained only when imitation strength among firms exceeds a critical bifurcation threshold.

nlin.PS↗

arXiv2019

Opinion formation dynamics -- Swift collective disillusionment triggered by unmet expectations

We propose a microscopic model to describe how individual opinions shared between interacting agents initiate excessive collective expectations about a new idea or an innovation, followed by a swift collapse towards a dramatic collective disillusionment. The basic assumption which underlies the dynamics is that the information gathering process is not instantaneous but requires maturation. Agents steadily refine and update their personal opinion via a recurrent consultation of a public pool which stores information tokens (ITs). The expectation for the innovative idea is monitored in real-time by counting the number of stored ITs. The flow dynamics of ITs is assimilated to a single node queuing system (QS) with feedback loop. It incorporates the information pool (the waiting room), an IT inflow, and a service outflow that stylizes the information gathering process. Contrary to basic queuing theory, here the ITs roaming the QS are endowed with time-dependent internal variables. This additional dynamic information is used to construct the information maturation process. Such a maturation of the information introduces response delays into the dynamics, which ultimately generates the collective disillusionment trough. We illustrate the introduced generic modeling framework by considering in details the hype cycle dynamics, a key managerial topic when dealing with diffusion of innovation. In a second part of the paper, we introduce a stylized framework to detect, as soon as possible, the onset of the collective disillusionment phase, while minimizing the frequency of false alarms.

nlin.AO↗

arXiv2018

Exactly solvable Gaussian and non-Gaussian mean-field games and collective swarms dynamics

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are modulated by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between drastically collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a swarm soliton propagating wave is observable. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show the the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. The MFG's equilibria resulting from the optimisation of individual utility functions are solitons that are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules ultimately producing similar collective effects can be explicitly constructed. For both Gaussian and non-Gaussian environments our exact results unveil new classes of exactly solvable mean-field games dynamics.

cond-mat.stat-mech↗

arXiv2018

Increasing Risk: Dynamic Mean-Preserving Spreads

We extend the celebrated Rothschild and Stiglitz (1970) definition of Mean-Preserving Spreads to a dynamic framework. We adapt the original integral conditions to transition probability densities, and give sufficient conditions for their satisfaction. We then prove that a specific nonlinear scalar diffusion process, super-diffusive ballistic noise, is the unique process that satisfies the integral conditions among a broad class of processes. This process can be generated by a random superposition of linear Markov processes with constant drifts. This exceptionally simple representation enables us to systematically revisit, by means of the properties of Dynamic Mean-Preserving Spreads, four workhorse economic models originally based on White Gaussian Noise.

math.PR↗

arXiv2017

Unravelling brittle fracture statistics out of self-healing patterns forming during femtosecond laser exposure

Femtosecond laser written patterns at the surface of brittle materials may show a regenerative random transition from self-organized to disordered structures. Here, we show that this random intermittent behaviour carries relevant fracture statistics information, such as the so-called Weibull parameters. Furthermore, we draw a phenomenological analogy with idle and busy periods arising in queueing systems that we used to establish that these successive laser generated cycles are statistically independent. Based on this analogy and together with microscopic observations, we propose an experimental method bypassing the need for many specimens to build-up statistically relevant ensembles of fracture tests. This method is potentially generic as it may apply to a broad number of brittle materials.

cond-mat.mtrl-sci↗

arXiv2016

On Jump-Diffusive Driving Noise Sources: Some Explicit Results and Applications

We study some linear and nonlinear shot noise models where the jumps are drawn from a compound Poisson process with jump sizes following an Erlang-$m$ distribution. We show that the associated Master equation can be written as a spatial $m^{\rm th}$ order partial differential equation without integral term. This differential form is valid for state-dependent Poisson rates and we use it to characterize, via a mean-field approach, the collective dynamics of a large population of pure jump processes interacting via their Poisson rates. We explicitly show that for an appropriate class of interactions, the speed of a tight collective traveling wave behavior can be triggered by the jump size parameter $m$. As a second application we consider an exceptional class of stochastic differential equations with nonlinear drift, Poisson shot noise and an additional White Gaussian Noise term, for which explicit solutions to the associated Master equation are derived.

math-ph↗

arXiv2015

Exact soliton-like probability measures for interacting jump processes

The cooperative dynamics of a 1-D collection of Markov jump, interacting stochastic processes is studied via a mean-field approach. In the time-asymptotic regime, the resulting nonlinear master equation is analytically solved. The nonlinearity compensates jumps induced diffusive behavior giving rise to a soliton-like stationary probability density. The soliton velocity and its sharpness both intimately depend on the interaction strength. Below a critical threshold of the strength of interactions, the cooperative behavior cannot be sustained leading to the destruction of the soliton-like solution. The bifurcation point for this behavioral phase transition is explicitly calculated.

math.PR↗

arXiv2012

Co-evolving agents subject to local versus nonlocal barycentric interactions

The mean-field dynamics of a collection of stochastic agents with local versus nonlocal interactions is studied via analytically soluble models. The nonlocal interactions result from a barycentric modulation of the observation range of the agents. Our modeling framework is based on a discrete two-velocity Boltzmann dynamics which can be analytically discussed. Depending on the span and the modulation of the interaction range, we analytically observe a transition from a purely diffusive regime without definite pattern to a flocking evolution represented by a solitary wave traveling with constant velocity.

nlin.AO↗

arXiv2011

Spatio-Temporal Patterns for a Generalized Innovation Diffusion Model

We construct a model of innovation diffusion that incorporates a spatial component into a classical imitation-innovation dynamics first introduced by F. Bass. Relevant for situations where the imitation process explicitly depends on the spatial proximity between agents, the resulting nonlinear field dynamics is exactly solvable. As expected for nonlinear collective dynamics, the imitation mechanism generates spatio-temporal patterns, possessing here the remarkable feature that they can be explicitly and analytically discussed. The simplicity of the model, its intimate connection with the original Bass' modeling framework and the exact transient solutions offer a rather unique theoretical stylized framework to describe how innovation jointly develops in space and time.

nlin.SI↗

arXiv1998

Semi-Markov Random Walks and Universality in Ising-like Chains

We exhibit a one to one correspondence between some universal probabilistic properties of the ordering coordinate of one-dimensional Ising-like models and a class of continuous time random walks. This correspondence provides an new qualitative picture of the properties of the ordering coordinate of the Ising model.

cond-mat.stat-mech↗

arXiv1994

A Solvable Nonlinear Reaction-Diffusion Model

We construct a coupled set of nonlinear reaction-diffusion equations which are exactly solvable. The model generalizes both the Burger equation and a Boltzman reaction equation recently introduced by Th. W. Ruijgrok and T. T. Wu.

chao-dyn↗