Modelling Asset Price Dynamics with Investor Inertia: Diffusion with Advection and Fourth-Order Extension

Standard models of asset price dynamics, such as geometric Brownian motion (see, for example, Osborne, 1959, Samuelson, 2016), do not formally incorporate investor inertia. This paper presents a two-stage framework for modelling this behaviour. First, we establish a microfoundation for the classic diffusion-with-advection model by representing the asset's log price as a three-state random walk (up, down or neutral). While this derivation offers a clear behavioural origin for drift and volatility, it is ultimately limited by its Gaussian nature and fails to capture the heavy tails (leptokurtosis) observed in financial markets. To address this issue, we introduce and apply a fourth-order extension inspired by diffusion-with-retention models (Bevilacqua, 2011), where a more complex representation of inertia generates non-Gaussian dynamics. Through an empirical application using Brazilian PETR4.SA data, we demonstrate that this extended model significantly outperforms the original in fitting the real distribution of returns. Our findings suggest that investor inertia is a dual concept capable of explaining both standard market trends and extreme events.