We introduce an unsupervised classification framework that leverages a multi-scale wavelet representation of time-series and apply it to stock price jumps. In line with previous work, we recover the fact that time-asymmetry of volatility is the major feature that separates exogenous, news-induced jumps from endogenously generated jumps. Local mean-reversion and trend are found to be two additional key features, allowing us to identify new classes of jumps. Using our wavelet-based representation, we investigate the endogenous or exogenous nature of co-jumps, which occur when multiple stocks experience price jumps within the same minute. Perhaps surprisingly, our analysis suggests that a significant fraction of co-jumps result from an endogenous contagion mechanism.

E xtreme events and cascades of events are widespread occurrences in both natural and social systems (1). Examples include earthquakes, volcanic eruptions, hurricanes, epileptic crises (2, 3), epidemic spread, financial crashes (4-6), economic crises (7, 8), book sales shocks (9, 10), riot propagation (11, 12) or failures in socio-technical systems (13). Understanding the origin of such events is essential for forecasting and possibly stabilizing their dynamics.

A widely studied question is the reflexive, self-exciting nature of those shocks. The concept of financial market reflexivity was introduced by Soros in ( 14), to describe the idea that price dynamics are mostly endogenous and arise from internal feedback mechanisms, as was first surmised by Cutler, Poterba and Summers in 1988 (15) (see also ( 16)). Extreme events, in particular, often arise from feedback mechanisms within the system's structure (1, 17, 18). Quantifying the extent of endogeneity in a complex system and distinguishing events caused by external shocks from those provoked endogenously, and more generally identifying different classes of events, are crucial questions.

Prior research has proposed to differentiate between endogenous and exogenous dynamics by analyzing the profile of activity around the shock (9, 10, 19, 20), in particular in the context of financial markets (21-23). It has been observed that endogenous shocks are preceded by a growth phase mirroring the post event powerlaw relaxation, in contrast to exogenous shocks that are strongly asymmetric. The universality of this result is quite intriguing as they have been observed in various contexts: intra-day book sales on Amazon (9, 10), daily views of YouTube videos (20) and intra-day financial market volatility and price jumps (23, 24). Meanwhile, Wu et al. (25) differentiate exogenous and endogenous bursts of comment posting on social media using the analysis of collective emotion dynamics and time-series distributions of comment arrivals.

Furthermore, in complex systems, events can propagate along two directions: temporally and towards other elements of the system. Financial markets offer an attractive setting for studying multi-dimensional shocks due to the abundance of available data, the frequent occurrence of financial shocks and price jumps and the inter-connectivity of markets. In fact, a recent study by Lillo et al. (26, 27) demonstrates the frequent occurrence of "co-jumps", defined as simultaneous jumps of multiple stocks (as illustrated in Fig. 1) and establishes a correlation between their prevalence and the inter-connectivity of different markets.

In this paper, we address the problem of classifying financial price jumps (and co-jumps), in particular measuring their self-exciting character, by analyzing their time-series using wavelets. We introduce an unsupervised classification based on an embedding Φ(x) of each jump time-series of returns x(t) into a low dimensional-space more appropriate to clustering. Such embedding, composed of wavelet scattering coefficients (see (28) and below), relies on wavelet coefficients of the time-series at the time of the jump t = 0 and wavelet coefficients of volatility. Such coefficients are

Cascades of events and extreme occurrences have garnered significant attention across diverse domains like seismology, neuroscience, economics, finance, and other social sciences. Such events may arise from internal system dynamics (endogenous) or external shocks (exogenous). Devising rigorous methods to distinguish between them is vital for professionals and regulators to create early warning systems and effective responses. Understanding these dynamics could improve the stability and resilience of crisisprone socio-economic systems. We show how wavelets can be used for the unsupervised separation of shocks in financial time-series, based on time-asymmetry around the shock. Additionally, we highlight the significant role contagion mechanisms play in financial markets.