Related papers: Limit experiments of GARCH
These lecture notes introduce the statistical analysis of continuous-time generative models built from Markov dynamics. We begin with the stochastic-calculus foundations of score-based diffusion models, including time reversal, score…
We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of…
Study of instantaneous dependence among several variable is important in many of the high-dimensional sciences. Multivariate GARCH models are as a standard approach for modelling time-varying covariance matrix such phenomena. Cholesky GARCH…
Recent progresses in single particle tracking have shown evidences of non-Gaussian distribution of displacements in living cells, both near the cellular membrane and inside the cytoskeleton. A similar behavior has also been observed in…
We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved…
Leggett-Garg inequalities (LGI) are constrains on certain combinations of temporal correlations obtained by measuring one and the same system at two different instants of time. The usual derivations of LGI assume \emph {macroscopic realism…
We study a scenario under which variable step random walks give anomalous statistics. We begin by analyzing the Martingale Central Limit Theorem to find a sufficient condition for the limit distribution to be non-Gaussian. We note that the…
This paper considers the statistical inference of the class of asymmetric power-transformed $\operatorname{GARCH}(1,1)$ models in presence of possible explosiveness. We study the explosive behavior of volatility when the strict stationarity…
In biological, glassy, and active systems, various tracers exhibit Laplace-like, i.e., exponential, spreading of the diffusing packet of particles. The limitations of the central limit theorem in fully capturing the behaviors of such…
Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are…
An anomalous diffusion model for ion channel gating is put forward. This scheme is able to describe non-exponential, power-law like distributions of residence time intervals in several types of ion channels. Our method presents a…
The standard small-time functional central limit theorem of semimartingales has been established in (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications.…
We present a parametric study of a nonlinear diffusion equation which generalises Leith's model of a turbulent cascade to an arbitrary cascade having a single conserved quantity. There are three stationary regimes depending on whether the…
Scaling analysis of the magnitude series (volatile series) has been proposed recently to identify possible nonlinear/multifractal signatures in the given data [1-3]. In this letter, correlations of volatile series generated from stationary…
Non-Gaussian diffusion has been intensively studied in recent years, which reflects the dynamic heterogeneity in the disordered media. The recent study on the non-Gaussian diffusion in a static disordered landscape suggests novel phenomena…
It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time.…
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow…
Previous research has shown that for stock indices, the most likely time until a return of a particular size has been observed is longer for gains than for losses. We establish that this so-called gain/loss asymmetry is present also for…
It will be discussed the statistics of the extreme values in time series characterized by finite-term correlations with non-exponential decay. Precisely, it will be considered the results of numerical analyses concerning the return…
We compare our results on empirical analysis of financial data with simulations of two stochastic models of the dynamics of stock market prices. The two models are (i) the truncated L\'evy flight recently introduced by us and (ii) the…