From discrete to continuous time evolutionary finance models
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 in asset \(k\) held between time \(t_n\) and \(t_{n+1}\) is given by \((1-c\nu)\lambda^{i}_{k}(t_n)V^{i}(t_n)\). The discrete-time dynamics is as follows: the wealth of investor \(i\) evolves as \[ V^{i}(t_{n+1}) = \sum_{k=1}^{K} \theta^{i}_{k}(t_{n})[S_{k}(t_{n+1})+D_{k}(t_{n+1})], \] where \(\theta^{i}_{k}(t_{n}) = (1-c\nu)\lambda^{i}_{k}(t_n)V^{i}(t_n)/S_{k}(t_{n})\) is the portion of asset \(k\) owned by investor \(i\) at time \(t_n\); \(S_{k}(t_{n}) = (1-c\nu) \sum_{i=1}^{I}\lambda^{i}_{k}(t_n)V^{i}(t_n)\) is the price of asset \(k\), which is determined by the condition \(\sum_{i=1}^{I}\theta^{i}_{k}(t_n)=1\) \((k \leq K, n \geq 0)\), i.e.\ that the market for each asset clears at each time \(t_{n}\) \((n \geq 0)\). So for the wealth dynamics of the investors, one gets the implicit formula \[ V^{i}(t_{n+1}) = \sum_{k=1}^{K} \frac{\lambda^{i}_{k}(t_n)V^{i}(t_n)}{\sum_{i=1}^{I}\lambda^{i}_{k}(t_n)V^{i}(t_n)}\Big[(1-c\nu)\sum_{i=1}^{I}\lambda^{i}_{k}(t_{n+1})V^{i}(t_{n+1}) + D_{k}(t_{n+1})\Big]~(i \leq I). \] Thus the market dynamics is driven by the interaction of investment strategies, which are taken as fundamental characteristics of the investors. In the continuous-time case, the authors obtain the analogous dynamics \[ \text{d} V^{i}(t) = \sum_{k=1}^{K} \frac{\lambda^{i}_{k}(t)V^{i}(t)}{\sum_{i=1}^{I}\lambda^{i}_{k}(t)V^{i}(t)}\Big[\text{d}\Big(\sum_{i=1}^{I}\lambda^{i}_{k}(t)V^{i}(t)\Big) + \delta_{k}(t)\text{d}t\Big]-cV^{i}(t)\text{d}t ~(i \leq I), \] which can be written as \[ \text{d} V^{i}(t) = \sum_{k=1}^{K} \theta^{i}_{k}(t)[\text{d} S_{k}(t) + \delta_{k}(t)\text{d}t]-cV^{i}(t)\text{d}t ~(i \leq I). \] This latter form is to be compared to the standard wealth dynamics of a trader employing a self-financing strategy with consumption in a market with \(K\) assets. In both of the discrete-time and continuous-time cases, the authors represent this evolutionary stock market model as a stochastic dynamical system; this complements the implicit formulation of the dynamics presented above with an explicit one, which is a step toward the further analysis and numerical simulation of the model. Under mild assumptions, it is shown that every non-trivial initial condition gives rise to a global solution of the market model; moreover, the solutions of the discrete-time model converge to that of the continuous-time model as \(\nu\) tends to zero. A numerical illustration of this result is also presented.
scientific article; zbMATH DE number 5697077