Non-Markovian models for short-scale financial motion

The random nature of financial price fluctuations is considered as resulting from the imbalance of buy and sell orders at each time step. In this work, we explore minimal models of the behavior of the financial agents to study the emergence of short-scale behavior of prices. At a given time, the state of the market is characterized by the set of unrealized highest buy (lowest sell) orders of the order book, which comprises the potential next trading.
%
%
In order to explain the non-Gaussian character of market price fluctuations, we consider extensions in the form of Langevin-type equations with an inertia term. The observed financial motion is described as analogous to a damped harmonic particle embedded in an environment which depicts the accumulated orders in the underlying optimal levels of the order book.
%
%
Memory is a crucial ingredient for the collective properties of markets, especially in the short time regime, and this is account by a non-local exponential kernel in the anticipation of prices by the agents when placing their orders. Using a non-Markovian Langevin description, we consider a random damped harmonic particle in presence of noise.
%
%
The noises, which represent external and/or internal perturbations, are modeled as an Ornstein-Uhlembeck process and/or as a dichotomous process which show to be more amenable to analytic approaches. Indeed, we can picture two subpopulation of trades, buyers and sellers, exchanging particles at rate 1/$\tau$ via some idiosyncratic switching process.
%
%
We provide the expression for the low order expansion in $\tau$ of the effective damping and restoring parameters, in the case where a dichotomous noise process is at play. We also analyze the expected values as well as the dispersion of the log price and the returns analytically and numerically.
Using random damped harmonic oscillator models as a reference tool, we conclude by investigating the intra-day Brazilian stock price series.