Intra-day variability of the stock market activity versus stationarity of the financial time series

We describe the impact of the intra-day activity pattern on the autocorrelation function estimator. We obtain an exact formula relating estimators of the autocorrelation functions of non-stationary process to its stationary counterpart. Hence, we proved that the day seasonality of inter-transaction times extends the memory of as well the process itself as its absolute value. That is, both processes relaxation to zero is longer.

Empirical symptoms of catastrophic bifurcation transitions on financial markets: A phenomenological approach

The principal aim of this work is the evidence on empirical way that catastrophic bifurcation breakdowns or transitions, proceeded by flickering phenomenon, are present on notoriously significant and unpredictable financial markets. Overall, in this work we developed various metrics associated with catastrophic bifurcation transitions, in particular, the catastrophic slowing down (analogous to the critical slowing down). All these things were considered on a well-defined example of financial markets of small and middle to large capitalization. The catastrophic bifurcation transition seems to be connected with the question of whether the early-warning signals are present in financial markets. This question continues to fascinate both the research community and the general public. Interestingly, such early-warning signals have recently been identified and explained to be a consequence of a catastrophic bifurcation transition phenomenon observed in multiple physical systems, e.g. in ecosystems, climate dynamics and in medicine (epileptic seizure and asthma attack). In the present work we provide an analogical, positive identification of such phenomenon by examining its several different indicators in the context of a well-defined daily bubble; this bubble was induced by the recent worldwide financial crisis on typical financial markets of small and middle to large capitalization.

Nucleation, condensation and lambda-transition on a real-life stock market

We fill a void in merging empirical and phenomenological characterisation of the dynamical phase transitions in complex systems by identifying three of them on real-life financial markets. We extract and interpret the empirical, numerical, and semi-analytical evidences for the existence of these phase transitions, by considering the Frankfurt Stock Exchange (FSE), as a typical example of a financial market of a medium size. Using the canonical object for the graph theory, i.e. the Minimal Spanning Tree (MST) network, we observe: (i) The initial phase transition from the equilibrium to non-equilibrium MST network in its nucleation phase, occurring at some critical time. Coalescence of edges on the FSEs transient leader is observed within the nucleation and is approximately characterized by the Lifsthiz-Slyozov growth exponent; (ii) The nucleation accelerates and transforms to the condensation process, in the second phase transition, forming a logarithmically diverging lambda-peak of short-range order parameters at the subsequent critical time - an analogon of such a transition in superfluidity; (iii) In the third phase transition, the peak logarithmically decreases over three quarters of the year, resulting in a few loosely connected sub-graphs. This peak is reminiscent of a non-equilibrium superstar-like superhub or a `dragon king effect, abruptly accelerating the evolution of the leader company. All these phase transitions are caused by the few richest vertices, which drift towards the leader and provide the most of the edges increasing the leaders degree. Thus, we capture an amazing phenomenon, likely of a more universal character, where a peripheral vertex becomes the one which is over dominating the complex network during an exceptionally long period of time.

Dynamic structural and topological phase transitions on the Warsaw Stock Exchange: A phenomenological approach

We study the crash dynamics of the Warsaw Stock Exchange (WSE) by using the Minimal Spanning Tree (MST) networks. We find the transition of the complex network during its evolution from a (hierarchical) power law MST network, representing the stable state of WSE before the recent worldwide financial crash, to a superstar-like (or superhub) MST network of the market decorated by a hierarchy of trees (being, perhaps, an unstable, intermediate market state). Subsequently, we observed a transition from this complex tree to the topology of the (hierarchical) power law MST network decorated by several star-like trees or hubs. This structure and topology represent, perhaps, the WSE after the worldwide financial crash, and could be considered to be an aftershock. Our results can serve as an empirical foundation for a future theory of dynamic structural and topological phase transitions on financial markets.

Structural and topological phase transitions on the German Stock Exchange

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing FSE were found. First transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the markets state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash, to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela-Chakraborti-Kaski-Kertesz for S&P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets.

Fingered growth in channel geometry: A Loewner equation approach

A simple model of Laplacian growth is considered, in which the growth takes place only at the tips of long, thin fingers. In a recent paper, Carleson and Makarov used the deterministic Loewner equation to describe the evolution of such a system. We extend their approach to a channel geometry and show that the presence of the side walls has a significant influence on the evolution of the fingers and the dynamics of the screening process, in which longer fingers suppress the growth of the shorter ones.