Emergence of statistical financial factors by a diffusion process.

Factor models provide a representation of the joint behavior of large cross sections of financial assets through a reduced set of underlying drivers. In this work, we propose a network-based dynamical framework in which statistical factors emerge endogenously from interactions among assets, rather than being imposed exogenously or extracted purely statistically. We model asset returns as a system of coupled nonlinear maps influenced by a network of interactions encoded in a coupling matrix. This matrix is constructed via an orthogonal transformation of a Laplacian operator with prescribed nullity, allowing us to control the system's effective dimensionality. Under appropriate coupling conditions, the dynamics reduce from a high-dimensional phase space to a lower-dimensional invariant manifold, where synchronization modes of co-movement arise. We show that the number of emergent statistical factors is directly related to the nullity of the coupling matrix, consistent with a stability analysis performed here. Simulations demonstrate that the model reproduces key stylized features of financial returns while generating factors with balanced loadings across assets. Finally, we show that this model reproduces features that are observed in an empirical portfolio. These findings suggest that statistical factors can be interpreted as emergent synchronization modes of an interacting nonlinear system, providing a complementary perspective to the existing economic and statistical factor models.