We consider a dynamical system in R driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Levy noise of small intensity and such that the heaviest tail of its Levy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.
In this paper we consider a finite state time discrete Markov chain that mimics the behaviour of solutions of the stochastic differential equation $dX=-U'(X)dt+εdL$, where $U$ is a multi-well potential with $n\geq 2$ local minima and L is a symmetric α-stable Lévy process (Lévy flights process). We investigate the spectrum of the generator of this Markov chain in the limit $ε\to 0$ and localize the top n eigenvalues $λ^ε_1,\dots, λ^ε_n$. These eigenvalues turn out to be of the same algebraic order $O(ε^α)$ and are well separated from the rest of the spectrum by a spectral gap. We also determin…
We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.
We study the almost sure behaviour of suitably normalised multivariate Levy processes as t goes to zero. Among other results we find necessary and sufficient conditions for a law of a very slowly varying function which includes a general law of the iterated logarithm in this setting. We also look at the corresponding cluster set problem.
We consider high-frequency sampled continuous-time autoregressive moving average (CARMA) models driven by finite-variance zero-mean Lévy processes. An L^2-consistent estimator for the increments of the driving Lévy process without order selection in advance is proposed if the CARMA model is invertible. In the second part we analyse the high-frequency behaviour of approximating Riemann sum processes, which represent a natural way to simulate continuous-time moving average processes on a discrete grid. We shall compare their autocovariance structure with the one of sampled CARMA processes, where…
Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu, rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) :=E(e^(-theta T_x - mu K_x ρL_x) 1_(T_x<+infinity)), where theta>=0, mu>=0, rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has finite exponential moments we exhibit an asymptotic expansion for F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At la…
In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation $\ddot x^\varepsilon_t +|\dot x^\varepsilon_t|^β=\dot Z^\varepsilon_{\varepsilon t}$, $β\in\mathbb R$, driven by symmetric non-Gaussian Lévy processes $Z^\varepsilon$. This equation describes the dynamics of a one-degree-of-freedom mechanical system subject to non-linear friction and noisy vibrations. For a compound Poisson noise, the process $x^\varepsilon$ on the macroscopic time scale $t/\varepsilon$ has a natural interpretation as a non-linear filter which responds to each single j…
We consider a class of Lévy-type processes derived via a Doob-transform from Lévy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise lower and upper envelopes for the almost sure long time behaviour of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.
In this paper we show that a path-wise solution to the following integral equation $$ Y_t = \int_0^t f(Y_t) dX_t \qquad Y_0=a \in \R^d $$ exists under the assumption that X_t is a Lévy process of finite p-variation for some $p \geq1$ and that f is an $α$-Lipschitz function for some alpha>p. There are two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric the other forward. The geometric solution is obtained by adding fictitious time and solving an associated integral equation. The forward solution is derived from the geometric soluti…
This article studies the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by infinite-dimensional multiplicative Lévy noise with a regularly varying component at intensity $ε>0$. The main results establish the precise asymptotics of the first exit times and locus of the solution $X^ε$ from the domain of attraction of a deterministic stable state, in the limit as $ε\rightarrow 0$. In contrast to the exponential growth for respective Gaussian perturbations the exit times grow essentially as a power function of the noise intensit…
In this paper we obtain new limit theorems for variational functionals of high frequency observations of stationary increments Lévy driven moving averages. We will see that the asymptotic behaviour of such functionals heavily depends on the kernel, the driving Lévy process and the properties of the functional under consideration. We show the "law of large numbers" for our class of statistics, which consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we also prove the associated weak limit theory, which again consists of three differe…
We examine the small expiry behaviour of European call options in stock price models of exponential Lévy type. In most cases of interest, we are able to identify the exact small expiry asymptotics. In "complete generality" we are able to show that the time value of the call option has O(τ) decay as τ(time to expiry) goes to zero. Using our results on the behaviour of call options close to expiry we show that implied volatility explodes as $τ\to0^+$ in "most" exponential Lévy models. Attention is restricted to calls and implied volatilities that are not at-the-money.
We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors $K^i$ each of which supports a unique ergodic probability measure $P^i$, which includes in particular the class of Morse-Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavy-tailed Lévy type noise of small amplitude $\varepsilon>0$. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem …
The reflected process of a random walk or Lévy process arises in many areas of applied probability, and a question of particular interest is how the tail of the distribution of the heights of the excursions away from zero behaves asymptotically. The Lévy analogue of this is the tail behaviour of the characteristic measure of the height of an excursion. Apparently the only case where this is known is when Cramér's condition hold. Here we establish the asymptotic behaviour for a large class of Lévy processes which have exponential moments but do not satisfy Cramér's condition. Our proof also app…
Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional Lévy process $(L_t)_{t \geq 0}$. We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples…
In this paper, we study the asymptotic behaviour of one-dimensional integrated Ornstein-Uhlenbeck processes driven by $α$-stable Lévy processes of small amplitude. We prove that the integrated Ornstein-Uhlenbeck process converges weakly to the underlying $α$-stable Lévy process in the Skorokhod $M_1$-topology which secures the weak convergence of first passage times. This result follows from a more general result about approximations of an arbitrary Lévy process by continuous integrated Ornstein-Uhlenbeck processes in the $M_1$-topology.
We study the convergence in probability in the non-standard $M_1$ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type $\int_0^t F_γ(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven by a Lévy process $L$. In Banach spaces we introduce strong, weak and product modes of $M_1$-convergence, prove a criterion for the $M_1$-convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behaviour of the corresponding oscillation functions, and establish crit…
This paper considers a continuous time analogue of the classical autoregressive moving average processes, Lévy-driven CARMA processes. First we describe limiting properties of the periodogram by means of the so-called truncated Fourier transform if observations are available continuously. The obtained results are in accordance with their counterparts from the discrete-time case. Then we discuss the numerical approximation of the truncated Fourier transform based on non-equidistant high frequency data. In order to ensure convergence of the numerical approximation to the true value of the trunca…
We study the asymptotic behaviour of the tail of the distribution of the first passage time of a Lévy process over a one-sided moving boundary. Our main result states that if the boundary behaves as $t^γ$ for large $t$ for some $γ<1/2$ then the probability that the process stays below the boundary behaves asymptotically as in the case of a constant boundary. We do not have to assume Spitzer's condition in contrast to all previously known results. Both positive ($+t^γ$) and negative ($-t^γ$) boundaries are considered. These results extend the findings of Greenwood and Novikov (1986) and are als…
This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the {\it stabilities} of the times, $\Tstarb(r)$ and $\Tbarb(r)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in\R^2: y\le rt^b, t\ge 0\}$ (one-sided exit), or $\{(t,y)\in\R^2: |y|\le rt^b, t\ge 0\}$ (two-sided exit), $0\le b<1$, as $r\dto 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $…
In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, t…
We consider a Glauber dynamics associated with the Ising model on a large two-dimensional box with with minus boundary conditions and in the limit of a vanishing positive external magnetic field. The volume of this box increases quadratically in the inverse of the magnetic field. We show that at subcritical temperature and for a large class of starting measures, including measures that are supported by configurations with macroscopic plus-spin droplets, the system rapidly relaxes to some metastable equilibrium ---with typical configurations made of microscopic plus-phase droplets in a sea of m…
This paper continues the study of metastable behaviour in disordered mean field models initiated in [2], [3]. We consider the generalized Hopfield model with finitely many independent patterns $ξ_1,...,ξ_p$ where the patterns have i.i.d. components and follow discrete distributions on $[-1,1]$. We show that metastable behaviour occurs and provide sharp asymptotics on metastable exit times and the corresponding capacities. We apply the potential theoretic approach developed by Bovier et al. in the space of appropriate order parameters and use an analysis of the discrete Laplacian to obtain lowe…
