From discrete to continuous time evolutionary finance models

Abstract

This paper aims to open a new avenue for research in continuous-time financial market models with endogenous prices and heterogenous investors. To this end we introduce a discrete-time evolutionary stock market model that accommodates time periods of arbitrary length. The dynamics is timeconsistent and allows the comparison of paths with different frequency of trade. The main result in this paper is the derivation of the limit model as the length of the time period tends to zero. The resulting model in continuous time generalizes the workhorse model of mathematical finance by introducing asset prices that are driven by the market interaction of investors following self-financing trading strategies. Our approach also offers a numerical scheme for the simulation of the continuous-time model that satisfies constraints such as market clearing at every time step. An illustration is provided.

The Interval Market Model in Mathematical Finance

“…where μ and σ are known, deterministic parameters, or time functions, called "drift" and "volatility," respectively, and B(•) is a standard Brownian motion (or Wiener process). Following these prestigious forerunners, most of the literature in mathematical finance relies on Samuelson's model, although notable exceptions have existed ever since, for example, [56,57,77,87,109,117,122,124,133].…”

Evolutionary Behavioral Finance

“…The time interval under consideration has to be divided into subintervals during which the "slow" variables must be kept frozen, while the "fast" ones rapidly reach a unique state of equilibrium. In this connection, discrete-time settings in our field are most natural for modeling purposes, and attempts to realize similar ideas in continuous-time frameworks face serious conceptual and technical difficulties [51,52].…”

Market selection of constant proportions investment strategies in continuous time

“…An alternative derivation of the above model is provided in Palczewski and Schenk-Hoppé [23]. They obtain the dynamics (4) as the limit of the discrete-time evolutionary finance model (Evstigneev et al [10,11,12]) as the length of the time-step tends to zero.…”

“…The continuoustime setting overcomes the problem of a priori setting a frequency of trade. It also defines a benchmark for the specification of discrete-time models in which the wealth dynamics is consistent over different trading frequencies (see Palczewski and Schenk-Hoppé [23]). …”