Stochastic Processes and their Applications最新文献

The Kac polynomial $f_n\left(x\right)=\sum _{i=0}^n{\xi }_i{x}^i$ with independent coefficients of variance 1 is one of the most studied models of random polynomials.

It is well-known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work (Michelen, 2020), Michelen showed that the holes at $\pm 1$ are of order $1/n$. We show that in fact, all the hole radii are of the same order. The same phenomenon is established for the derivatives of the Kac polynomial as well.

Given $d\ge 1$, we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as $e^{\sqrt{2d}X}d\mu$ where $X$ is a $log$-correlated Gaussian field and $\mu$ is a locally finite measure on $R^d$. Our construction generalizes the one performed in the case where $\mu$ is the Lebesgue measure. It requires that the measure $\mu$ is sufficiently spread out, namely that for $\mu$ almost every $x$ we have ${\int }_{B(x,1)}\frac{\mu \left(dy\right)}{{|x-y|}^d{e}^{\rho \left(log\frac{1}{|x-y|}\right)}}<\infty ,$ where $\rho :R_{+}\to R_{+}$ can be chosen to be any lower envelope function for the 3-Bessel process (this includes $\rho \left(x\right)=x^{\alpha }$ with $\alpha \in (0,1/2)$). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure $\mu$ is in a sense optimal.

We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on $R_{+}^m$, in cases where their domain is large enough. In general the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson–Watanabe superprocesses. From a mathematical finance point of view, the polynomial framework is especially attractive since it allows to transfer many famous finite dimensional models and their tractability properties to an infinite dimensional measure-valued setting.

We show a deviation inequality inequalities for multi-indexed martingale We then provide applications to kernel regression for random fields and rates in the law of large numbers for orthomartingale difference random fields.

We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. In order to apply the general theory, it is necessary to estimate functionals that are a randomly weighted sum of products of multiple integrals of the fractional Brownian motion, in expanding the quadratic variation and identifying the limit random symbols. To overcome the difficulty, we utilized the theory of exponents of functionals, which was introduced by the authors in Yamagishi and Yoshida (2023).

We study scaling limits of nonlinear functions $G$ of random grain model $X$ on $R^d$ with long-range dependence and marginal Poisson distribution. Following Kaj et al. (2007) we assume that the intensity $M$ of the underlying Poisson process of grains increases together with the scaling parameter $\lambda$ as $M={\lambda }^{\gamma }$, for some $\gamma >0$. The results are applicable to the Boolean model and exponential $G$ and rely on an expansion of $G$ in Charlier polynomials and a generalization of Mehler’s formula. Application to solution of Burgers’ equation with initial aggregated random grain data is discussed.

We study parametric inference for ergodic diffusion processes with a degenerate diffusion matrix. Existing research focuses on a particular class of hypo-elliptic Stochastic Differential Equations (SDEs), with components split into ‘rough’/‘smooth’ and noise from rough components propagating directly onto smooth ones, but some critical model classes arising in applications have yet to be explored. We aim to cover this gap, thus analyse the highly degenerate class of SDEs, where components split into further sub-groups. Such models include e.g. the notable case of generalised Langevin equations. We propose a tailored time-discretisation scheme and provide asymptotic results supporting our scheme in the context of high-frequency, full observations. The proposed discretisation scheme is applicable in much more general data regimes and is shown to overcome biases via simulation studies also in the practical case when only a smooth component is observed. Joint consideration of our study for highly degenerate SDEs and existing research provides a general ‘recipe’ for the development of time-discretisation schemes to be used within statistical methods for general classes of hypo-elliptic SDEs.

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator, derived in Altmeyer and Reiß (2021) for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.

We establish sufficient conditions for the existence, and derive explicit formulas for the $\kappa$’th moments, $\kappa \ge 1$, of Markov modulated generalized Ornstein–Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process.

Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.