Second order stochastic differential models for financial markets

TL;DR

This paper proposes second order, non-Markovian stochastic differential equations for financial prices, supported by agent-based and physical modeling, to better capture market phenomena like cycles and sector shifts.

Contribution

It introduces the necessity of second order models for financial markets, challenging traditional first order approaches, and presents simple models explaining complex market behaviors.

Findings

Second order models better capture boom-bust cycles.

Models explain stochastic quasi-periodic market behavior.

Hot money transfer between sectors is modeled effectively.

Abstract

Using agent-based modelling, empirical evidence and physical ideas, such as the energy function and the fact that the phase space must have twice the dimension of the configuration space, we argue that the stochastic differential equations which describe the motion of financial prices with respect to real world probability measures should be of second order (and non-Markovian), instead of first order models \`a la Bachelier--Samuelson. Our theoretical result in stochastic dynamical systems shows that one cannot correctly reduce second order models to first order models by simply forgetting about momenta. We propose some simple second order models, including a stochastic constrained n-oscillator, which can explain many market phenomena, such as boom-bust cycles, stochastic quasi-periodic behavior, and "hot money" going from one market sector to another.