# A fractional reaction-diffusion description of supply and demand

## TL;DR

This paper introduces a fractional reaction-diffusion model for financial market liquidity, capturing heterogeneous agent behaviors and reproducing empirical impact decay with analytical and numerical validation.

## Contribution

It develops a novel fractional reaction-diffusion framework for latent liquidity, linking agent heterogeneity to impact decay and market efficiency.

## Key findings

- Impact is a concave function of volume, consistent with the square-root law.
- Impact kernel decay varies with diffusion type, matching empirical decay exponent.
- Numerical simulations support analytical predictions.

## Abstract

We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics of latent liquidity in financial markets, where agents are very heterogeneous in terms of their characteristic frequencies. Several features of our model are amenable to an exact analytical treatment. We find in particular that the impact is a concave function of the transacted volume (aka the "square-root impact law"), as in the normal diffusion limit. However, the impact kernel decays as $t^{-\beta}$ with $\beta=1/2$ in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent $\beta$ takes any value in $[0,1/2]$, and can be tuned to match the empirical value $\beta \approx 1/4$. Numerical simulations confirm our theoretical results. Several extensions of the model are suggested.