1,837 research outputs found
Probability densities and distributions for spiked and general variance Wishart -ensembles
A Wishart matrix is said to be spiked when the underlying covariance matrix
has a single eigenvalue different from unity. As increases through
, a gap forms from the largest eigenvalue to the rest of the spectrum, and
with of order the scaled largest eigenvalues form a well
defined parameter dependent state. Recent works by Bloemendal and Vir\'ag [BV],
and Mo, have quantified this parameter dependent state for real Wishart
matrices from different viewpoints, and the former authors have done similarly
for the spiked Wishart -ensemble. The latter is defined in terms of
certain random bidiagonal matrices. We use a recursive structure to give an
alternative construction of the spiked and more generally the general variance
Wishart -ensemble, and we give the exact form of the joint eigenvalue
PDF for the two matrices in the recurrence. In the case of real quaternion
Wishart matrices () the latter is recognised as having appeared in
earlier studies on symmetrized last passage percolation, allowing the exact
form of the scaled distribution of the largest eigenvalue to be given. This
extends and simplifies earlier work of Wang, and is an alternative derivation
to a result in [BV]. We also use the construction of the spiked Wishart
-ensemble from [BV] to give a simple derivation of the explicit form of
the eigenvalue PDF.Comment: 18 page
Martingale Option Pricing
We show that our generalization of the Black-Scholes partial differential
equation (pde) for nontrivial diffusion coefficients is equivalent to a
Martingale in the risk neutral discounted stock price. Previously, this was
proven for the case of the Gaussian logarithmic returns model by Harrison and
Kreps, but we prove it for much a much larger class of returns models where the
diffusion coefficient depends on both returns x and time t. That option prices
blow up if fat tails in logarithmic returns x are included in the market
dynamics is also explained
Extreme Thouless effect in a minimal model of dynamic social networks
In common descriptions of phase transitions, first order transitions are
characterized by discontinuous jumps in the order parameter and normal
fluctuations, while second order transitions are associated with no jumps and
anomalous fluctuations. Outside this paradigm are systems exhibiting `mixed
order transitions' displaying a mixture of these characteristics. When the jump
is maximal and the fluctuations range over the entire range of allowed values,
the behavior has been coined an `extreme Thouless effect'. Here, we report
findings of such a phenomenon, in the context of dynamic, social networks.
Defined by minimal rules of evolution, it describes a population of extreme
introverts and extroverts, who prefer to have contacts with, respectively, no
one or everyone. From the dynamics, we derive an exact distribution of
microstates in the stationary state. With only two control parameters,
(the number of each subgroup), we study collective variables of
interest, e.g., , the total number of - links and the degree
distributions. Using simulations and mean-field theory, we provide evidence
that this system displays an extreme Thouless effect. Specifically, the
fraction jumps from to (in the
thermodynamic limit) when crosses , while all values appear with
equal probability at .Comment: arXiv admin note: substantial text overlap with arXiv:1408.542
Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets
Arguably the most important problem in quantitative finance is to understand
the nature of stochastic processes that underlie market dynamics. One aspect of
the solution to this problem involves determining characteristics of the
distribution of fluctuations in returns. Empirical studies conducted over the
last decade have reported that they arenon-Gaussian, scale in time, and have
power-law(or fat) tails. However, because they use sliding interval methods of
analysis, these studies implicitly assume that the underlying process has
stationary increments. We explicitly show that this assumption is not valid for
the Euro-Dollar exchange rate between 1999-2004. In addition, we find that
fluctuations in returns of the exchange rate are uncorrelated and scale as
power-laws for certain time intervals during each day. This behavior is
consistent with a diffusive process with a diffusion coefficient that depends
both on the time and the price change. Within scaling regions, we find that
sliding interval methods can generate fat-tailed distributions as an artifact,
and that the type of scaling reported in many previous studies does not exist.Comment: 12 pages, 4 figure
Exact results for the extreme Thouless effect in a model of network dynamics
If a system undergoing phase transitions exhibits some characteristics of
both first and second order, it is said to be of 'mixed order' or to display
the Thouless effect. Such a transition is present in a simple model of a
dynamic social network, in which extreme introverts/extroverts always
cut/add random links. In particular, simulations showed that , the average fraction of cross-links between the two groups
(which serves as an 'order parameter' here), jumps dramatically when crosses the 'critical point' , as in typical
first order transitions. Yet, at criticality, there is no phase co-existence,
but the fluctuations of are much larger than in typical second order
transitions. Indeed, it was conjectured that, in the thermodynamic limit, both
the jump and the fluctuations become maximal, so that the system is said to
display an 'extreme Thouless effect.' While earlier theories are partially
successful, we provide a mean-field like approach that accounts for all known
simulation data and validates the conjecture. Moreover, for the critical system
, an analytic expression for the mesa-like stationary
distribution, , shows that it is essentially flat in a range
, with . Numerical evaluations of
provides excellent agreement with simulation data for .
For large , we find ,
though this behavior begins to set in only for . For accessible
values of , we provide a transcendental equation for an approximate
which is better than 1% down to . We conjecture how this approach
might be used to attack other systems displaying an extreme Thouless effect.Comment: 6 pages, 4 figure
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