Let R be a symmetric a-stable Riemann-Liouville process with Hurst parameter H > 0. Consider ||.|| a translation invariant, b-self-similar, and p-pseudo-additive functional semi-norm. We show that if H > (b + 1/p) and c = (H - b - 1/p), then x log P [ log ||R|| < c log x ] -> - k < 0, when x -> 0, with k finite in the Gaussian case a = 2. If a < 2, we prove that k is finite when R is continuous and H > (b + 1/p + 1/a). We also show that under the above assumptions, x log P [ log ||X|| < c log x ] -> - k < 0 when x -> 0, where k is finite and X is the linear a-stable fractional motion with Hurst parameter 0 < H < 1 (if a = 2, then X is the classical fractional Brownian motion). These general results recover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and Non-Gaussian frameworks.
We investigate the sharp large deviation properties of the energy and the maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst index greater than one half.
In this note, we prove the large deviation principle for the 2D-fractional stochastic Navier-Stokes equation on the torus under the dissipation order $ α\in [\frac43, 2]$.
The term "moderate deviations" is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about "non-central moderate deviations" when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.
We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time cha
We study small noise large deviation asymptotics for stochastic differential equations with a multiplicative noise given as a fractional Brownian motion $B^H$ with Hurst parameter $H>\frac12$. The solutions of the stochastic differential equations are defined pathwise under appropriate conditions on the coefficients. The ingredients in the proof of the large deviation principle, which include a variational representation for nonnegative functionals of fractional Brownian motions and a general sufficient condition for a LDP for a collection of functionals of a fractional Brownian motions, have …
We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein--Uhlenbeck process driven by mixed fractional Brownian motion.
Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.
In this paper, we study the Moderate Deviation Principle for a perturbed stochastic heat equation in the whole space $\rr^d, d\ge1$. This equation is driven by a Gaussian noise, white in time and correlated in space, and the differential operator is a fractional derivative operator. The weak convergence method plays an important role.
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of…
For a fractional Brownian motion $B^H$ with Hurst parameter $H\in]{1/4},{1/2}[\cup]{1/2},1[$, multiple indefinite integrals on a simplex are constructed and the regularity of their sample paths are studied. Then, it is proved that the family of probability laws of the processes obtained by replacing $B^H$ by $ε^{1/2} B^H$ satisfies a large deviation principle in Hölder norm. The definition of the multiple integrals relies upon a representation of the fractional Brownian motion in terms of a stochastic integral with respect to a standard Brownian motion. For the large deviation principle, the a…
In this paper, we investigate two-sided bounds for the small ball probability of a mixed fractional Brownian motion with a general deterministic trend function, in terms of respective small ball probability of a mixed fractional Brownian motion without trend. To maximize the lower bound, we consider various ways to split the trend function between the components of the mixed fractional Brownian motion for the application of Girsanov theorem, and we show that the optimal split is the solution of a Fredholm integral equation. We find that the upper bound for the probability is also a function of…
In this paper we study the Large Deviation Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space $\mathbb{R}^d$, with arbitrary dimension $d\geq 1$, under random influence which is a Gaussian noise, white in time and correlated in space. The differential operator is a fractional derivative operator. We prove a large deviations principle for our equation, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion. This approach reduces the proof of LDP to establishin…
We prove large deviation principles for $\int_0^t γ(X_s)ds$, where $X$ is a $d$-dimensional self-similar Gaussian process and $γ(x)$ takes the form of the Dirac delta function $δ(x)$, $|x|^{-β}$ with $β\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-β_i}$ with $β_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.
Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $α$-stable motions called local time fractional stable motions. When $α=2$, these processes are precisely fractional Brownian motions with $1/2<H<1$. Motivated by random walks in alternating scenery, we find a "complementary" family of symmetric $α$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $α=2$, one gets fractional Brownian motions with $0<H<1/2$.
Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics
National Research University Higher School of Economics（国立高等经济学院）；Universidade de Lisboa（里斯本大学）；Instituto Superior Técnico（技术高等学院）；Instituto de matemática pura e aplicada（纯粹与应用数学研究所）
We study stochastic volatility models in which the volatility process is a function of a continuous fractional stochastic process, which is an integral transform of the solution of an SDE satisfying the Yamada-Watanabe condition. We establish a small-noise large deviation principle for the log-price, and, for a special case of our setup, obtain logarithmic call price asymptotics for large strikes.
Stable non-Gaussian self-similar mixed moving averages can be decomposed into several components. Two of these are the periodic and cyclic fractional stable motions which are the subject of this study. We focus on the structure of their integral representations and show that the periodic fractional stable motions have, in fact, a canonical representation. We study several examples and discuss questions of uniqueness, namely how to determine whether two given integral representations of periodic or cyclic fractional stable motions give rise to the same process.
Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the approximations weakly converge as the mesh-size goes to zero. As an application, we improve upon previous approximation schemes for integrals with respect to linear fractional stable motions.
Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0\}$ be the local time process at zero of a strictly stable Lévy process $X=\{X_t, t \ge 0\}$ of index $1<α\leq 2$ independent of $W^H$. The $\a$-stable local time fractional Brownian motion $Z^H=\{Z^H(t), t \ge 0\}$ is defined by $Z^H(t) = W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1 - \frac 1 α)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coup…
Expressions are given for the truncated fractional moments $E X_+^p$ of a general stable law. These involve families of special functions that arose out of the study of multivariate stable densities and probabilities. As a particular case, an expression is given for $E(X-a)_+$ when $α> 1$.
In this paper we prove exponential inequalities (also called Bernstein's inequality) for fractional martingales. As an immediate corollary, we will discuss weak law of large numbers for fractional martingales under divergence assumption on the $β-$variation of the fractional martingale. A non trivial example of application of this convergence result is proposed.
In this paper, we simulate sample paths of a class of symmetric $α$-stable processes using their series expression. We will develop a result in the approximation of shot-noise series. And finally, we will get a convergence rate for the approximation.
Meerschaert and Sabzikar [12], [13] introduced tempered fractional Brownian/stable motion (TFBM/TFSM) by including an exponential tempering factor in the moving average representation of FBM/FSM. The present paper discusses another tempered version of FBM/FSM, termed tempered fractional Brownian/stable motion of second kind (TFBM II/TFSM II).We prove that TFBM/TFSM and TFBM II/TFSM II are different processes. Particularly, large time properties of TFBM II/TFSM II are similar to those of FBM/FSM and are in deep contrast to large time properties of TFBM/TFSM.
This work defines two classes of processes, that we term {\it tempered fractional multistable motion} and {\it tempered multifractional stable motion}. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise Hölder exponent, Hölder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These proc…
In this paper, we define a new and broad family of vector-valued random fields called tempered operator fractional operator-stable random fields (TRF, for short). TRF is typically non-Gaussian and generalizes tempered fractional stable stochastic processes. TRF comprises moving average and harmonizable-type subclasses that are constructed by tempering (matrix-) homogeneous, matrix-valued kernels in time- and Fourier-domain stochastic integrals with respect to vector-valued, strictly operator-stable random measures. We establish the existence and fundamental properties of TRF. Assuming both Gau…
We investigate the large deviation behaviour of a point process sequence based on a stationary symmetric stable non-Gaussian discrete-parameter random field using the framework of Hult and Samorodnitsky (2010). Depending on the ergodic theoretic and group theoretic structures of the underlying nonsingular group action, we observe different large deviation behaviours of this point process sequence. We use our results to study the large deviations of various functionals (e.g., partial sum, maxima, etc.) of stationary symmetric stable fields.
Define the incremental fractional Brownian field $B_{H}(s+τ)-B_{H}(s), H\in (0,1)$, where $B_{H}(s)$ is a standard fractional Brownian motion with Hurst index $H\in(0,1)$. In this paper we derive the exact asymptotic behaviour of the maximum $\max_{(τ,s)\in[0,1]\times[0,T]} (B_{H}(s+τ)-B_{H}(s)) $ for any $H\in (0,1/2)$ complimenting thus the result of Zholud (2008) for the Brownian motion.
We consider stochastic nonlinear Schrodinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter H in (0,1). It is also colored in space and the space correlation operator is assumed to be nuclear. We study the local well-posedness of the equation. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove a sample path large deviations principle and a support result. These results are stated in a space of exploding paths which are Holder continuous in time until blow-up. We treat t…
In this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional s…
It has been shown recently that the optimal fluctuation method -- essentially geometrical optics -- provides a valuable insight into large deviations of Brownian motion. Here we extend the geometrical optics formalism to two-sided, $-\infty<t<\infty$, fractional Brownian motion (fBM) on the line, which is "pushed" to a large deviation regime by imposed constraints. We test the formalism on three examples where exact solutions are available: the two- and three-point probability distributions of the fBm and the distribution of the area under the fBm on a specified time interval. Then we apply th…
Generalizing both Substable FSMs and Indicator FSMs, we introduce alpha-stabilized subordination, a procedure which produces new FSMs (H-sssi symmetric stable processes) from old ones. We extend these processes to isotropic stable fields which have stationary increments in the strong sense, i.e., processes which are invariant under Euclidean rigid motions of the multi-dimensional time parameter. We also prove a Stable Central Limit Theorem which provides an intuitive picture of alpha-stabilized subordination. Finally we show that alpha-stabilized subordination of Linear FSMs produces null-cons…
In this paper we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable expone…
It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ ξ_j $ for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ ξ_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion …
For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form of measure-theoretic duality" between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in "alternating" scenery, thus hinting at a mechanism…
Long memory processes driven by Lévy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here, we study a class of Lévy process whose second-order moments are infinite, the so-called $α$-stable processes. Based on Samorodnitsky and Taqqu (2000), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, follows naturally an i…
Consider the solution $\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\mathcal{Z}(0,x) = δ(x)$. For any real $p>0$, we obtained detailed estimates of the $p$-th moment of $e^{t/12}\mathcal{Z}(2t,0)$, as $t\to\infty$, and from these estimates establish the one-point upper-tail large deviation principle of the Kardar-Parisi-Zhang equation. The deviations have speed $t$ and rate function $Φ_+(y)=\frac{4}{3}y^{3/2}$. Our result confirms the existing physics predictions [Le Doussal, Majumdar, Schehr 16…
We shall establish a large deviation principle for some occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. As a consequence, we obtain the exact rate of exponential ergodicity of the stochastic system under $τ$-topology.
Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely convergent in all the Hölder spaces $C^γ([-M,M]^N)$, where $M>0$ and $γ\in (0, \min\{H_1,\ldots, H_N\})$ are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let $X=\{X(t), t \in \R^N\}$ be an $\R^d$-valued harmonizable fractional stable sheet whose components …
We define and study fractional versions of the well-known Gamma subordinator $Γ:=\{Γ(t),$ $t\geq 0\},$ which are obtained by time-changing $% Γ$ by means of an independent stable subordinator or its inverse. Their densities are proved to satisfy differential equations expressed in terms of fractional versions of the shift operator (with fractional parameter greater or less than one, in the two cases). As a consequence, the fractional generalization of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding…
