We compute the rate of decay of the persistence probabilities of spherical fractional Brownian motion, which was defined by Lévy (1965) and Istas (2005). The rate resembles the Euclidean case treated in Molchan (1999).
As a by-product we consider the coefficients of series representations of functions with algebraic endpoint singularities in terms of re-scaled Gegenbauer polynomials, which partly generalises Sidi (2009).
We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H,$ respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1,$ which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.
The problem is a log-asymptotics of the probability that the Integrated fractional Brownian motion of index 0<H<1 does not exceed a fixed level during long time. For the growing time interval (0,T) the hypothetical log-asymptotics is (H(H-1)+o(1))Log T. In support of the hypothesis, we update our earlier estimates of the probability and give analytical proofs.
Persistence probabilities for MA(1) sequences with uniform innovations
Technical University of Darmstadt（达姆施塔特工业大学）；CNRS, Université d’Angers（法国国家科学研究中心，昂热大学）；International Research Laboratory France-Vietnam in Mathematics and its Applications, CNRS - VAST - VIASM（法越数学及其应用国际研究实验室）
The fractional Brownian motion of index $0 < H < 1$, H-FBM, with d-dimensional time is considered on an expanding set TG, where G is a bounded convex domain that contains 0 at its boundary. The main result: if 0 is a point of smoothness of the boundary, then the log-asymptotics of probability that H-FBM does not exceed a fixed positive level in TG is $(H - d + o(1)) \log T$, $T\to\infty$. Some generalizations of this result to isotropic but not self-similar Gaussian processes with stationary increments are also considered.
We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under non-negativity assumptions of covariance functions and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the single process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in Cheridito (2001) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent $1-\max(1/2,H),$ where $H$ is …
Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity, or the Hurst exponent. However, recent single particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that can not be attributed to a class of self-similar random processes. Inspired by these observati…
With $\{ξ_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $ρ(i):=\mathbb{E}[ ξ_0ξ_i]$ and $\{σ(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum $\sum_{i=1}^\ell σ(i) ξ_i$, $\ell\ge 1$. For summable correlations $ρ$, we show that the persistence exponent is universal. On the contrary, for non-summable $ρ$, even for polynomial weight functions $σ(i)\sim i^p$ the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter $H$) and on the poly…
We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes.
For each prime $p$, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated $p$-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of $p$-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.
Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f:[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the fractional Brownian motion $B^{H}$ with the deterministic drift $f$. It aims to highlight the role of the regularity properties of the drift $f$ as well as that of the dimension of $E$ in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$ for $F$ a subset of $\mathbb{R}^{d}$ and also fo…
We establish exact formulae for the persistence probabilities of an AR(1) sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results provide factorizations of general AR(1) models, one for negative drifts with continuous innovations, and one for positive drifts with continuous and symmetric innovations. The second factorization exte…
This paper investigates ruin probabilities for a two-dimensional fractional Brownian risk model with a proportional reinsurance scheme. We focus on joint and simultaneous ruin probabilities in a finite-time horizon. The risk processes of both insurance and reinsurance companies are composed of a large number of i.i.d. sub-risk processes, representing independent businesses. We derive the asymptotics as the initial capital tends to infinity.
We consider a stochastic flow $ϕ_t(x,ω)$ in $\mathbb{R}^n$ with initial point $ϕ_0(x,ω)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|ϕ_t(x)\|)}{\|ϕ_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonemp…
For AR(1)-processes $X_n=ρX_{n-1}+ξ_n$, $n\in\mathbb{N}$, where $ρ\in\mathbb{R}$ and $(ξ_i)_{i\in\mathbb{N}}$ is an i.i.d. sequence of random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots, X_N\ge 0)$ for $N\to\infty$. For a wide class of Markov processes a recent result [Aurzada, Mukherjee, Zeitouni; arXiv:1703.06447; 2017] shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the pa…
We analyze the classical Brownian risk models discussing the approximation of ruin probabilities (classical, γ-reflected, Parisian and cumulative Parisian) for the case that ruin can occur only on specific discrete grids. A practical and natural grid of points is for instance G(1) = {0,1,2,...}, which allows us to study the probability of the ruin on the first day, second day, and so one. For such a discrete setting, there are no explicit formulas for the ruin probabilities mentioned above. In this contribution we derive accurate approximations of ruin probabilities for uniform grids by lettin…
We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index $1/2$ and spatial index $H\le1/2$. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. B…
The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathbb{Q}_p$, induces an analogue of the Wiener process in $\mathbb{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.
The (fractional) Brownian sheet is a simplest example of a Gaussian random field X whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self-similar Gaussian processes. Let Y be the homogeneous Gaussian field obtained by applying to X the Lamperti transform, which involves the exponential change of time and the amplitude normalization to have unit variance. Under some assumptions, we prove the existence of the persistence exponents for both fields, X and Y, and find the relation between them. The exponent for X is the limit of ln1/P{X(t)<1, t in…
We propose a new algorithm to generate a fractional Brownian motion, with a given Hurst parameter, 1/2<H<1 using the correlated Bernoulli random variables with parameter p; having a certain density. This density is constructed using the link between the correlation of multivariate Gaussian random variables and the correlation of their dichotomized binary variables and the relation between the correlation coefficient and the persistence parameter. We prove that the normalized sum of trajectories of this proposed random walk yields a Gaussian process whose scaling limit is the desired fractional…
Generalized intersection exponents and local cut points for three-dimensional Brownian loop soup
Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the p…
