Stochastic behavioral asset pricing models and the stylized facts

High-frequency financial data are characterized by a set of ubiquitous statistical properties that prevail with surprising uniformity. While these 'stylized facts' have been well-known for decades, attempts at their behavioral explanation have remained scarce. However, recently a new branch of simple stochastic models of interacting traders have been proposed that share many of the salient features of empirical data. These models draw some of their inspiration from the broader current of behavioural finance. However, their design is closer in spirit to models of multi-particle interaction in physics than to traditional asset-pricing models. This reflects a basic insight in the natural sciences that similar regularities like those observed in financial markets (denoted as 'scaling laws' in physics) can often be explained via the microscopic interactions of the constituent parts of a complex system. Since these emergent properties should be independent of the microscopic details of the system, this viewpoint advocates negligence of the details of the determination of individuals' market behavior and instead focuses on the study of a few plausible rules of behavior and the emergence of macroscopic statistical regularities in a market with a large ensemble of traders. This chapter will review the philosophy of this new approach, its various implementations, and its contribution to an explanation of the stylized facts in finance. --

Extensions of the Kuramoto model: from spiking neurons to swarming drones

The Kuramoto model (KM) was initially proposed by Yoshiki Kuramoto in 1975 to model the dynamics of large populations of weakly coupled phase oscillators. Since then, the KM has proved to be a paradigmatic model, demonstrating dynamics that are complex enough to model a wide variety of nontrivial phenomena while remaining simple enough for detailed mathematical analyses. However, as a result of the mathematical simplifications in the construction of the model, the utility of the KM is somewhat restricted in its usual form. In this thesis we discuss extensions of the KM that allow it to be utilized in a wide variety of physical and biological problems. First, we discuss an extension of the KM that describes the dynamics of theta neurons, i.e., quadratic-integrate-and-fire neurons. In particular, we study networks of such neurons and derive a mean-field description of the collective neuronal dynamics and the effects of different network topologies on these dynamics. This mean-field description is achieved via an analytic dimensionality reduction of the network dynamics that allows for an efficient characterization of the system attractors and their dependence not only on the degree distribution but also on the degree correlations. Then, motivated by applications of the KM to the alignment of members in a two-dimensional swarm, we construct a Generalized Kuramoto Model (GKM) that extends the KM to arbitrary dimensions. Like the KM, the GKM in even dimensions continues to demonstrate a transition to coherence at a positive critical coupling strength. However, in odd dimensions the transition to coherence occurs discontinuously as the coupling strength is increased through 0. In contrast to the unique stable incoherent equilibrium for the KM, we find that for even dimensions larger than 2 the GKM displays a continuum of different possible pretransition incoherent equilibria, each with distinct stability properties, leading to a novel phenomenon, which we call `Instability-Mediated Resetting.' To aid the analysis of such systems, we construct an exact dimensionality reduction technique with applicability to not only the GKM, but also other similar systems with high-dimensional agents beyond the GKM