We study a random version of the population-market model proposed by Arlot, Marmi and Papini in Arlot et al. (2019). The latter model is based on the Yoccoz–Birkeland integral equation and describes a time evolution of livestock commodities prices which exhibits endogenous deterministic stochastic behaviour. We introduce a stochastic component inspired from the Black-Scholes market model into the price equation and we prove the existence of a random attractor and of a random invariant measure. We compute numerically the fractal dimension and the entropy of the random attractor and we show its convergence to the deterministic one as the volatility in the market equation tends to zero. We also investigate in detail the dependence of the attractor on the choice of the time-discretization parameter. We implement several statistical distances to quantify the similarity between the attractors of the discretized systems and the original one. In particular, following a work by Cuturi Cuturi (2013), we use the Sinkhorn distance. This is a discrete and penalized version of the Optimal Transport Distance between two measures, given a transport cost matrix.

### The Yoccoz-Birkeland livestock population model coupled with random price dynamics

#### Abstract

We study a random version of the population-market model proposed by Arlot, Marmi and Papini in Arlot et al. (2019). The latter model is based on the Yoccoz–Birkeland integral equation and describes a time evolution of livestock commodities prices which exhibits endogenous deterministic stochastic behaviour. We introduce a stochastic component inspired from the Black-Scholes market model into the price equation and we prove the existence of a random attractor and of a random invariant measure. We compute numerically the fractal dimension and the entropy of the random attractor and we show its convergence to the deterministic one as the volatility in the market equation tends to zero. We also investigate in detail the dependence of the attractor on the choice of the time-discretization parameter. We implement several statistical distances to quantify the similarity between the attractors of the discretized systems and the original one. In particular, following a work by Cuturi Cuturi (2013), we use the Sinkhorn distance. This is a discrete and penalized version of the Optimal Transport Distance between two measures, given a transport cost matrix.
##### Scheda breve Scheda completa Scheda completa (DC)
2022
Settore SECS-S/06 - Metodi mat. dell'economia e Scienze Attuariali e Finanziarie
population dynamics; delays dynamical systems; strange attractor; chaotic livestock commodities cycles; Sinkhorn distance; statistical distances
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Descrizione: Submitted version available on ArXiv: https://arxiv.org/abs/2205.09796
Tipologia: Accepted version (post-print)
Licenza: Creative Commons
Dimensione 3.87 MB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11384/125424`
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