Transformation invariant stochastic catastrophe theory

Wagenmakers, Eric Jan; Molenaar, Peter; Grasman, Raoul P. P. P.; Hartelman, P.A.I.; Maas, Han L. J. van der

doi:10.1016/j.physd.2005.08.014

Cited by 71 publications

(57 citation statements)

References 34 publications

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“…When the diffusion function is constant, σ yt = σ, and the current measurement scale is not to be nonlinearly transformed, the stochastic potential function is proportional to the deterministic potential function, and the probability distribution function corresponding to the solution from Eq. (8) converges to a probability distribution function of a limiting stationary stochastic process because the dynamics of y t are assumed to be much faster than changes in x i,t (Cobb, 1981;Cobb and Zacks, 1985;Wagenmakers et al, 2005). The probability density that describes the distribution of the system's states at any t is then…”

Section: Stochastic Dynamicsmentioning

confidence: 99%

“…Thus, a comparison of the cusp catastrophe model to the logistic function serves as a good indicator of the presence of bifurcations in the data. While these two models are not nested, Wagenmakers et al (2005) suggested comparing them via information criteria, where a stronger Bayesian Information Criterion (BIC) should be required for the decision.…”

Section: Statistical Evaluation Of the Fitmentioning

confidence: 99%

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Realizing stock market crashes: stochastic cusp catastrophe model of returns under time-varying volatility

Baruník

Kukačka

2014

Quantitative Finance

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. This paper develops a two-step estimation methodology that allows us to apply catastrophe theory to stock market returns with time-varying volatility and to model stock market crashes. In the first step, we utilize high-frequency data to estimate daily realized volatility from returns. Then, we use stochastic cusp catastrophe on data normalized by the estimated volatility in the second step to study possible discontinuities in the markets. We support our methodology though simulations in which we discuss the importance of stochastic noise and volatility in a deterministic cusp catastrophe model. The methodology is empirically tested on nearly 27 years of U.S. stock market returns covering several important recessions and crisis periods. While we find that the stock markets showed signs of bifurcation in the first half of the period, catastrophe theory was not able to confirm this behavior in the second half. Translating the results, we find that the U.S. stock market's downturns were more likely to be driven by the endogenous market forces during the first half of the studies period, while during the second half of the period, the exogenous forces seem to be driving the market's instability,. The results suggest that the propose methodology provides an important shift in the application of catastrophe theory to stock markets. Terms of use: Documents in AbstractThis paper develops a two-step estimation methodology that allows us to apply catastrophe theory to stock market returns with time-varying volatility and to model stock market crashes. In the first step, we utilize high-frequency data to estimate daily realized volatility from returns. Then, we use stochastic cusp catastrophe on data normalized by the estimated volatility in the second step to study possible discontinuities in the markets. We support our methodology through simulations in which we discuss the importance of stochastic noise and volatility in a deterministic cusp catastrophe model. The methodology is empirically tested on nearly 27 years of U.S. stock market returns covering several important recessions and crisis periods. While we find that the stock markets showed signs of bifurcation in the first half of the period, catastrophe theory was not able to confirm this behavior in the second half. Translating the results, we find that the U.S. stock market's downturns were more likely to be driven by the endogenous market forces during the first half of the studied period, while during ...

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Section: Stochastic Dynamicsmentioning

confidence: 99%

Section: Statistical Evaluation Of the Fitmentioning

confidence: 99%

Realizing stock market crashes: stochastic cusp catastrophe model of returns under time-varying volatility

Baruník

Kukačka

2014

Quantitative Finance

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“…This issue is closely related to the Catastrophe Theory of Thom from the 1960's, itself an extension of the work on singularity theory of Whitney, in which the principal object of study is the discontinuous or qualitative change in the properties of a system: " [W]hile Newtonian theory only considers smooth, continuous processes, catastrophe theory provides a method for the study of all jump transitions, discontinuities, and sudden qualitative changes" [42]. However, in its original formulation Catastrophe Theory only addressed deterministic systems, it was not until Cobb [43] and more recently Wagenmakers et al [44] that these ideas could be translated to the stochastic systems such as those considered here. With this notion of Catastrophe Theory instabilities this work examines stochastic systems for which qualitative (disruptive) regime shifts occur as a function of the smooth variation in parameters.…”

Section: Quantal Equilibrium Paths: Perturbed Cooperation and The Primentioning

confidence: 99%

Strategic Islands in Economic Games: Isolating Economies From Better Outcomes

Harré

Bossomaier

2014

Entropy

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Many of the issues we face as a society are made more problematic by the rapidly changing context in which important decisions are made. For example buying a petrol powered car is most advantageous when there are many petrol pumps providing cheap petrol whereas buying an electric car is most advantageous when there are many electrical recharge points or high capacity batteries available. Such collective decision-making is often studied using economic game theory where the focus is on how individuals might reach an agreement regarding the supply and demand for the different energy types. But even if the two parties find a mutually agreeable strategy, as technology and costs change over time, for example through cheaper and more efficient batteries and a more accurate pricing of the total cost of oil consumption, so too do the incentives for the choices buyers and sellers make, the result of which can be the stranding of an industry or even a whole economy on an island of inefficient outcomes. In this article we consider the issue of how changes in the underlying incentives can move us from an optimal economy to a sub-optimal economy while at the same time making it impossible to collectively navigate our way to a better strategy without forcing us to pass through a socially undesirable "tipping point". We show that different perturbations to underlying incentives results in the creation or destruction of "strategic islands" isolated by disruptive transitions between strategies. The significant result in this work is the illustration that an economy that remains strategically stationary can over time become stranded in a suboptimal outcome from which there is no easy way to put the economy on a path to better outcomes without going through an economic tipping point.

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“…As pointed out by Hartelman and co-workers [11,14], it is possible to construct a 'coordinate-free' classification for one-dimensional continuous-time diffusions . By defining stochastic analogues of concepts used in catastrophe theory, they arrived at a classification that is, unlike P-equivalence, invariant under monotonically increasing transformations of the real line, or more precisely, a classification that is invariant up to transformations homotopic to the identity mapping.…”

Section: Phenomenological Bifurcationsmentioning

confidence: 99%

“…For one-dimensional continuous time diffusions, a classification that is invariant under transformations has been proposed by Hartelman et al [11,14]. Inspired by their approach, we propose in this paper a classification for (strictly) stationary stochastic processes that are governed by smooth everywhere positive transition density functions.…”

Section: Introductionmentioning

confidence: 99%