On Sampling of stationary increment processes

Under a complex technical condition, similar to such used in extreme value theory, we find the rate q(\epsilon)^{-1} at which a stochastic process with stationary increments \xi should be sampled, for the sampled process \xi(\lfloor\cdot /q(\epsilon)\rfloor q(\epsilon)) to deviate from \xi by at most \epsilon, with a given probability, asymptotically as \epsilon \downarrow0. The canonical application is to discretization errors in computer simulation of stochastic processes.Comment: Published at http://dx.doi.org/10.1214/105051604000000468 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On overload in a storage model, with a self-similar and infinitely divisible input

Let {X(t)}_{t\ge0} be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index H>0. Pick constants \gamma >H and c>0. Let \nu be the L\'evy measure on R^{[0,\infty)} of X, and suppose that R(u)\equiv\nu({y\inR^{[0,\infty)}:supt\ge 0y(t)/(1+ct^{\gamma})>u}) is suitably ``heavy tailed'' as u\to\infty (e.g., subexponential with positive decrease). For the ``storage process'' Y(t)\equiv sup_{s\ge t}(X(s)-X(t)-c(s-t)^{\gamma}), we show that P{sup_{s\in[0,t(u)]}Y(s)>u}\sim P{Y(\hat t(u))>u} as u\to\infty, when 0\le \hat t(u)\le t(u) do not grow too fast with u [e.g., t(u)=o(u^{1/\gamma})]

On the infimum attained by a reflected L\'evy process

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of \prob{M(T_u)> u} (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios

Implied volatility of basket options at extreme strikes

In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a model-free tail-wing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function

Determinant and Weyl anomaly of Dirac operator: a holographic derivation

We present a holographic formula relating functional determinants: the fermion determinant in the one-loop effective action of bulk spinors in an asymptotically locally AdS background, and the determinant of the two-point function of the dual operator at the conformal boundary. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk partition function to a subleading large-N contribution in the boundary partition function. We use this holographic picture to address questions in spectral theory and conformal geometry. As an instance, we compute the type-A Weyl anomaly and the determinant of the iterated Dirac operator on round spheres, express the latter in terms of Barnes' multiple gamma function and gain insight into a conjecture by B\"ar and Schopka.Comment: 11 pages; new comments and references added, typos correcte

Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations

The new digital economy has renewed interest in how digital agents can innovate. This follows the legacy of John von Neumann dynamical systems theory on complex biological systems as computation. The Gödel-Turing-Post (GTP) logic is shown to be necessary to generate innovation based structure changing Type 4 dynamics of the Wolfram-Chomsky schema. Two syntactic procedures of GTP logic permit digital agents to exit from listable sets of digital technologies to produce novelty and surprises. The first is meta-analyses or offline simulations. The second is a fixed point with a two place encoding of negation or opposition, referred to as the Gödel sentence. It is postulated that in phenomena ranging from the genome to human proteanism, the Gödel sentence is a ubiquitous syntactic construction without which escape from hostile agents qua the Liar is impossible and digital agents become entrained within fixed repertoires. The only recursive best response function of a 2-person adversarial game that can implement strategic innovation in lock-step formation of an arms race is the productive function of the Emil Post [58] set theoretic proof of the Gödel incompleteness result. This overturns the view of game theorists that surprise and innovation cannot be a Nash equilibrium of a game

Cerebral activations related to ballistic, stepwise interrupted and gradually modulated movements in parkinson patients

Patients with Parkinson's disease (PD) experience impaired initiation and inhibition of movements such as difficulty to start/stop walking. At single-joint level this is accompanied by reduced inhibition of antagonist muscle activity. While normal basal ganglia (BG) contributions to motor control include selecting appropriate muscles by inhibiting others, it is unclear how PD-related changes in BG function cause impaired movement initiation and inhibition at single-joint level. To further elucidate these changes we studied 4 right-hand movement tasks with fMRI, by dissociating activations related to abrupt movement initiation, inhibition and gradual movement modulation. Initiation and inhibition were inferred from ballistic and stepwise interrupted movement, respectively, while smooth wrist circumduction enabled the assessment of gradually modulated movement. Task-related activations were compared between PD patients (N = 12) and healthy subjects (N = 18). In healthy subjects, movement initiation was characterized by antero-ventral striatum, substantia nigra (SN) and premotor activations while inhibition was dominated by subthalamic nucleus (STN) and pallidal activations, in line with the known role of these areas in simple movement. Gradual movement mainly involved antero-dorsal putamen and pallidum. Compared to healthy subjects, patients showed reduced striatal/SN and increased pallidal activation for initiation, whereas for inhibition STN activation was reduced and striatal-thalamo-cortical activation increased. For gradual movement patients showed reduced pallidal and increased thalamo-cortical activation. We conclude that PD-related changes during movement initiation fit the (rather static) model of alterations in direct and indirect BG pathways. Reduced STN activation and regional cortical increased activation in PD during inhibition and gradual movement modulation are better explained by a dynamic model that also takes into account enhanced responsiveness to external stimuli in this disease and the effects of hyper-fluctuating cortical inputs to the striatum and STN in particular

Rabies screen reveals GPe control of cocaine-triggered plasticity.

Identification of neural circuit changes that contribute to behavioural plasticity has routinely been conducted on candidate circuits that were preselected on the basis of previous results. Here we present an unbiased method for identifying experience-triggered circuit-level changes in neuronal ensembles in mice. Using rabies virus monosynaptic tracing, we mapped cocaine-induced global changes in inputs onto neurons in the ventral tegmental area. Cocaine increased rabies-labelled inputs from the globus pallidus externus (GPe), a basal ganglia nucleus not previously known to participate in behavioural plasticity triggered by drugs of abuse. We demonstrated that cocaine increased GPe neuron activity, which accounted for the increase in GPe labelling. Inhibition of GPe activity revealed that it contributes to two forms of cocaine-triggered behavioural plasticity, at least in part by disinhibiting dopamine neurons in the ventral tegmental area. These results suggest that rabies-based unbiased screening of changes in input populations can identify previously unappreciated circuit elements that critically support behavioural adaptations