Regime switches induced by supply-demand equilibrium: a model for power-price dynamics

Regime-switching models can be used to describe stochastic movements of electricity prices in deregulated markets. This paper shows that regime-switching dynamics arise quite naturally in an equilibrium context in which the functional form of the supply curve is described by a two-state Markov process. This mechanism is responsible for random switches between regimes and it allows one to describe the main features of the price-formation process. With the interplay between demand and supply, the proposed methodology can be used to capture shortages in electricity generation, forced outages, and peaks in electricity demand. As an example of application, a two-regime model specification is proposed, and it will be shown that the empirical analysis, performed by estimating using the model on the California power market, offers an interesting agreement with observed data.     

Modelling spikes in electricity markets using excitable dynamics

We present a general methodology to model spikes in deregulated electricity markets using excitable dynamics in a multi-regime switching approach. In particular, we propose a two-regime switching model and a three-regime switching model in which the spikes phenomenon is described by a FitzHugh-Nagumo excitable dynamics. Both models seems to be interesting candidates for describing the main characteristics of electricity price dynamics as the occurrence of stable periods in which prices fluctuate around some long-run mean, and turbulent periods in which prices experience jumps and spikes of very large magnitude. In agreement with market data, both models can produce probability distributions of price returns with positive skewness and very high values of kurtosis.     

Modeling electricity spot and futures price dependence: A multifrequency approach

Electricity prices are known to exhibit multifractal properties. We accommodate this finding by investigating multifractal models for electricity prices. In this paper we propose a flexible Copula-MSM (Markov Switching Multifractal) approach for modeling spot and weekly futures price dynamics. By using a conditional copula function, the framework allows us to separately model the dependence structure, while enabling use of multifractal stochastic volatility models to characterize fluctuations in marginal returns. An empirical experiment is carried out using data from Nord Pool. A study of volatility forecasting performance for electricity spot prices reveals that multifractal techniques are a competitive alternative to GARCH models. We also demonstrate how the Copula-MSM model can be employed for finding optimal portfolios, which minimizes the Conditional Value-at-Risk.     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

Regime-switching characterization of electricity prices dynamics

Stochastic models of electricity prices have been used extensively during the last few years to describe prices fluctuations in deregulated power markets. Regime-switching models seem good candidates to capture the main features of electricity prices dynamics as the mean-reversion property as well the presence of jumps and spikes. Since they offer the possibility to introduce various mean-reversion rates, volatility and jumps, depending on the state of the system, such models allow to describe the properties of the stable motion and of the spike dynamics in a very flexible way. In this paper, two-regime and three-regime models are discussed, and a comparison performed on market data, is proposed.     

Domino mechanisms in photoinduced phase transitions

Photoinduced structural phase transitions via excited electronic states are discussed theoretically using a one-dimensional model composed of localized electrons and lattices under the adiabatic or diabatic approximation. We show that the global structural change by photoexcitation only at a site is possible, and we clarify conditions for the occurrence of such phenomena. Spatiotemporal dynamics of nonequilibrium first-order phase transitions is also investigated in detail in terms of photoinduced nucleations and domino processes of the domain boundaries (domain walls), which are in striking contrast to the mean-field dynamics. In the adiabatic regime, after the spontaneous emission of a photon, an initial local structural change (i) remains locally, (ii) induces cooperatively a global structural change, or (iii) disappears and returns to the initial phase. Dynamical features of the case (ii) are characterized by the deterministic (semichaotic) domino process; domain walls between the two phases move determinis-tically at a constant velocity (with changing speed) without further spontaneous emissions in the case of strong (weak) dissipation. In the diabatic regime, similar three types of structural change exist. The domain-wall dynamics is described as the stochastic domino process, which is accompanied by the successive radiative transitions. A new theoretical treatment is also proposed to study crossover between the adiabatic and diabatic regimes.     

A structural model of market dynamics,and why it matters

In this paper, we explore an approach to understanding price fluctuations within a market via considerations of functional dependencies between asset prices. Interestingly, this approach suggests a class of models of a type used earlier to describe the dynamics of real and artificial neural networks. Statistical physics approaches turn out to be suitable for an analysis of their collective properties. In this paper, we first motivate the basic phenomenology and modelling arguments before moving on to discussing some major issues with inference and empirical verification. In particular, we focus on the natural creation of market states through the inclusion of interactions and how these then interfere with inference. This is primarily addressed in a synthetic setting. Finally we investigate real data to test the ability of our approach to capture some key features of the behaviour of financial markets.     

A theoretical model for the collective motion of proteins by means of principal component analysis

A coarse grained model in the frame work of principal component analysis is presented. We used a bath of harmonic oscillators approach, based on classical mechanics, to derive the generalized Langevin equations of motion for the collective coordinates. The dynamics of the protein collective coordinates derived from molecular dynamics simulations have been studied for the Bovine Pancreatic Trypsin Inhibitor. We analyzed the stability of the method by studying structural fluctuations of the C _a atoms obtained from a 20 ns molecular dynamics simulation. Subsequently, the dynamics of the collective coordinates of protein were characterized by calculating the dynamical friction coefficient and diffusion coefficients along with time-dependent correlation functions of collective coordinates. A dual diffusion behavior was observed with a fast relaxation time of short diffusion regime 0.2–0.4 ps and slow relaxation time of long diffusion about 1–2 ps. In addition, we observed a power law decay of dynamical friction coefficient with exponent for the first five collective coordinates varying from −0.746 to −0.938 for the real part and from −0.528 to −0.665 for its magnitude. It was found that only the first ten collective coordinates are responsible for configuration transitions occurring on time scale longer than 50 ps.     

Occupation times of the Ornstein–Uhlenbeck process: Functional PCA and evidence from electricity prices

We discuss the functional principal component analysis (FPCA) of the occupation times of the Ornstein–Uhlenbeck process. For the eigenvalue problem of the covariance operator of the occupation times we derive the corresponding integral equation in the large time limit and we solve numerically for the principal components. The formulation applies the path-integral approach of Feynman and Kac. The principal components are compared with those from empirical electricity price processes on energy markets. The results indicate that FPCA of the occupation times is a suitable tool in stochastic energy modeling to generate moderately-sized scenario trees.     

Noise-induced dynamical transition in systems with symmetric absorbing states

We investigate the effect of noise strength on the macroscopic ordering dynamics of systems with symmetric absorbing states. Using an explicit stochastic microscopic model, we present evidence for a phase transition in the coarsening dynamics, from an Ising-like to a voter-like behavior, as the noise strength is increased past a nontrivial critical value. By mapping to a thermal diffusion process, we argue that the transition arises due to locally-absorbing states being entered more readily in the high-noise regime, which in turn prevents surface tension from driving the ordering process.     

Transition Effect Matrices and Quantum Markov Chains

A transition effect matrix (TEM) is a quantum generalization of a classical stochastic matrix. By employing a TEM we obtain a quantum generalization of a classical Markov chain. We first discuss state and operator dynamics for a quantum Markov chain. We then consider various types of TEMs and vector states. In particular, we study invariant, equilibrium and singular vector states and investigate projective, bistochastic, invertible and unitary TEMs.     

Clustering of volatility as a multiscale phenomenon

The dynamics of prices in financial markets has been studied intensively both experimentally (data analysis) and theoretically (models). Nevertheless, a complete stochastic characterization of volatility is still lacking. What is well known is that absolute returns have memory on a long time range, this phenomenon is known as clustering of volatility. In this paper we show that volatility correlations are power-laws with a non-unique scaling exponent. This kind of multiscale phenomenology has some analogies with fully developed turbulence and disordered systems and it is now pointed out for financial series. Starting from historical returns series, we have also derived the volatility distribution, and the results are in agreement with a log-normal shape. In our study, we consider the New York Stock Exchange (NYSE), daily composite index closes (January 1966 to June 1998) and the US Dollar/Deutsche Mark (USD-DM) noon buying rates certified by the Federal Reserve Bank of New York (October 1989 to September 1998). Received 1 February 2000     

Direct observation of dynamical bifurcation between two driven oscillation states of a Josephson junction

We performed a novel phase-sensitive microwave reflection experiment which directly probes the dynamics of the Josephson plasma resonance in both the linear and the nonlinear regime. When the junction was driven below the plasma frequency into the nonlinear regime, we observed for the first time the transition between two different dynamical states predicted for nonlinear systems. In our experiment, this transition appears as an abrupt change in the reflected signal phase at a critical excitation power. This controlled dynamical switching can form the basis of a sensitive amplifier, in particular, for the readout of superconducting qubits.     

Dynamics of oscillators with periodic dichotomous noise

The dynamics of bistable oscillators driven by periodic dichotomous noise is described. The stochastic differential equation governing the flow implies smooth trajectories between noise switching events. The dynamics of the two-branched map induced by this flow is a Markov process. Harmonic and quartic models of the bistable potential are studied in the overdamped limit. In the linear (harmonic) case the dynamics can be reduced to a stochastic one-dimensional map with two branches. The moments decay exponentially in this case, although the invariant measure may be multifractal. For strong damping, relaxation induces a cascade leading to a Cantor set and anomalous decay of the density in this case is modeled by a Markov chain. For the physically more realistic case of a quartic potential many additional features arise since the contraction factor is distance dependent. By tuning the barrier-height parameter in the quartic potential, noise-induced transition rates with the characteristics of intermittency are found.     

The correlation length of commodity markets 1. Empirical evidence

It is a common belief nowadays that the world economy is fairly well “integrated”. Yet, this belief often turns out to be in contradiction with empirical evidence. As a matter of fact the way distant markets interact is a question that has largely been ignored by economists. In this series of two papers we examine the role that space, that is to say geographical distance, plays in the economics of commodity markets. The first of these papers presents the empirical evidence while the second develops a theoretical framework. The empirical enquiry discloses several noteworthy features, e.g. (i) with respect to spatial interaction there is a sharp contrast between stock markets and commodity markets. While there is almost perfect spatial arbitrage in the first case, this is not true for commodity markets. (ii) In spite of their chaotic behavior in the course of time commodity prices display well defined spatial patterns, (iii) as in statistical physics and fluid dynamics interactions can be described in terms of correlation length. The correlation length of a set of markets is seen to increase along with the number of transactions; it also increases when transport costs decline as was the case during the “transportation revolution” of the mid-nineteenth century. Using the notion of correlation length one is able to give a quantitative meaning to the otherwise ill-defined concept of market integration. Received 17 May 1999 and Received in final form 31 May 1999     

Effects of nonlinear mode coupling on the chaotic dynamics of multimode lasers

The dynamics of individual modes of a cw multimode dye laser in a 2-mirror configuration has been studied theoretically and experimentally. Numerical solutions of the laser rate equations including nonlinear mode coupling exhibit chaotic and stochastic behavior in the regime of high and low power, respectively. Chaotic behavior due to mode coupling has been observed in the operating regime well above threshold. The mode dynamics in this case is characterized by a chaotic attractor with low dimension between 2 and 3. With decreasing laser power the dimension increases, suggesting stochastic behavior due to quantum noise in the limit of the laser threshold.On leave at 1. Institut für Experimentalphysik, Universität Hamburg, Fed. Rep. Germany     

Creation, Dissipation and Recycling of Resources in Non-Equilibrium Systems

In this article, we define stochastic dynamics for a system coupled to reservoirs. The rules for forward and backward transitions are related by a generalized detailed balance identity involving the system and its reservoirs. We compare the variation of information and of entropy. We define the Carnot dissipation and prove that it can be expressed in terms of cyclic transformations. Lower bounds for partial dissipations are also studied, as well as the effect of switching off certain reservoirs. We also study the near degeneracy of the stochastic matrix, relate it to phase transitions and we show that the reduced dynamics on the set of phases is a permutation. Finally, we relate these concepts to heat, work and more generally to the dissipation and creation of resources, in general systems.     

Multiscaling and clustering of volatility

The dynamics of prices in stock markets has been studied intensively both experimentally (data analysis) and theoretically (models). Nevertheless, while the distribution of returns of the most important indices is known to be a truncated Lévy, the behaviour of volatility correlations is still poorly understood. What is well known is that absolute returns have memory on a long time range, this phenomenon is known in financial literature as clustering of volatility. In this paper we show that volatility correlations are power laws with a non-unique scaling exponent. This kind of multiscale phenomenology is known to be relevant in fully developed turbulence and in disordered systems and it is pointed out here for the first time for a financial series. In our study we consider the New York Stock Exchange (NYSE) daily index, from January 1966 to June 1998, for a total of 8180 working days.     

Noise-induced switching in two adaptively coupled excitable systems

We demonstrate that the interplay of noise and plasticity gives rise to slow stochastic fluctuations in a system of two adaptively coupled active rotators with excitable local dynamics. Depending on the adaptation rate, two qualitatively different types of switching behavior are observed. For slower adaptation, one finds alternation between two modes of noise-induced oscillations, whereby the modes are distinguished by the different order of spiking between the units. In case of faster adaptation, the system switches between the metastable states derived from coexisting attractors of the corresponding deterministic system, whereby the phases exhibit a bursting-like behavior. The qualitative features of the switching dynamics are analyzed within the framework of fast-slow analysis.     

Modeling electricity spot prices using mean-reverting multifractal processes

We discuss stochastic modeling of volatility persistence and anti-correlations in electricity spot prices, and for this purpose we present two mean-reverting versions of the multifractal random walk (MRW). In the first model the anti-correlations are modeled in the same way as in an Ornstein–Uhlenbeck process, i.e. via a drift (damping) term, and in the second model the anti-correlations are included by letting the innovations in the MRW model be fractional Gaussian noise with H<1/2$H<1/2$. For both models we present approximate maximum likelihood methods, and we apply these methods to estimate the parameters for the spot prices in the Nordic electricity market. The maximum likelihood estimates show that electricity spot prices are characterized by scaling exponents that are significantly different from the corresponding exponents in stock markets, confirming the exceptional nature of the electricity market. In order to compare the damped MRW model with the fractional MRW model we use ensemble simulations and wavelet-based variograms, and we observe that certain features of the spot prices are better described by the damped MRW model. The characteristic correlation time is estimated to approximately half a year.