From discrete to continuous time evolutionary finance models
type: Item
instance of: scholarly article
title: From discrete to continuous time evolutionary finance models
zbMATH Open document ID: 1214.91150
author: Jan Palczewski
author: Klaus R. Schenk-Hoppé
publication date: 2010-04-22T00:00:00Z
full work available at URL: Discrete2Cont.pdf
review text: The authors study the wealth dynamics of investors with interacting investment strategies, in discrete and continuous time, in an evolutionary market framework where asset prices are driven by the market interaction of investors. The continuous-time limit of a generalization of the discrete-time evolutionary stock market model by \textit{I.\ V.\ Evstigneev, T.\ Hens, K.\ R.\ Schenk-Hoppé} [J.\ Econ.\ Theory 140, No.\ 1, 197--228 (2008; Zbl 1136.91429)] is obtained. The authors indicate that a numerical simulation of this limit model is possible by proving that suitable discrete approximations converge uniformly on finite time intervals. In the model, in the discrete-time case, time evolves as \(t_{n} = n \nu\) \((n \in \mathbb N)\). There are \(K\) assets with stochastic dividend intensity \(\delta(t)=(\delta_{1}(t), \dots, \delta_{K}(t))\) \((t \geq 0)\) where \(\delta_{k}(t) \geq 0\) \((k \leq K)\); the dependence of \(\delta(t)\) on the random variable is suppressed in notation. Each asset is in positive net supply of 1. The total dividend paid by asset \(k\) at time \(t_{n+1}\) to the investors who hold in part the asset over the time period \([t_{n},t_{n+1})\) is \(D_{k}(t_{n+1}) = \int_{t_{n}}^{t_{n+1}} \delta_{k}(s) \text{d} s\). Dividends are paid in terms of a perishable consumption good, with price normalized to 1. There are \(I\) investors with initial wealth \(V^{i}(0) \geq 0\) \((i \leq I)\). Each investor is represented by a possibly stochastic investment strategy \(\lambda^{i}(t_n) = (\lambda^{i}_{1}(t_n), \dots, \lambda^{i}_{K}(t_n))\) \((n \in \mathbb N)\), representing the investor's budget share invested in asset \(k\) at time \(t_n\). It is assumed that \(\lambda^{i}_{k}(t_n) > 0\) and \(\sum_{i=1}^{I} \lambda^{i}_{k}(t_n) = 1\) \((k \leq K, n \geq 0)\). Each investor consumes the constant fraction \(c\nu\) of his wealth in every time period, and the remainder of wealth is invested in assets, i.e.\ the market value of investor \(i\)'s investment …
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 91G80
Mathematics Subject Classification ID: 91B26
Mathematics Subject Classification ID: 91B69
zbMATH DE Number: 5697077
zbMATH Keywords: evolutionary finance
zbMATH Keywords: market interaction
zbMATH Keywords: wealth dynamics
zbMATH Keywords: self-financing strategy
zbMATH Keywords: endogenous price
zbMATH Keywords: continuous-time limit
Wikidata QID: Q57949609
MaRDI profile type: Publication
OpenAlex ID: W3123392623
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DOI: 10.1016/J.JEDC.2009.12.005
published in: Journal of Economic Dynamics and Control
description: scientific article; zbMATH DE number 5697077