Narrow Results

We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric α-stable (SαS) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with SαS innovations. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks.

We generate paths of the LFSM process by using Riemann-sum approximations of its SαS stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error.

We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustrate our method. We include MATLAB implementations of these simulation algorithms and indicate how the practitioner can use them.

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}_t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t).

We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents.

The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

We calculate numerically the optimal allocation and consumption strategies for Merton's optimal portfolio management problem when the risky asset is modelled by a geometric normal inverse Gaussian Lévy process. We compare the computed strategies to the ones given by the standard asset model of geometric Brownian motion. To have realistic parameters in our studies, we choose Norsk Hydro quoted on the New York Stock Exchange as the risky asset. We find that an investor believing in the normal inverse Gaussian model puts a greater fraction of wealth into the risky asset. We also investigate the limiting investment rate when the volatility increases. We observe different behaviour in the two models depending on which parameters we vary in the normal inverse Gaussian distribution.

We model spot prices in energy markets with exponential non-Gaussian Ornstein–Uhlenbeck processes. We generalize the classical geometric Brownian motion and Schwartz' mean-reversion model by introducing Lévy processes as the driving noise rather than Brownian motion. Instead of modelling the spot price dynamics as the solution of a stochastic differential equation with jumps, it is advantageous from a statistical point of view to model the price process directly. Imposing the normal inverse Gaussian distribution as the statistical model for the Lévy increments, we obtain a superior fit compared to the Gaussian model when applied to spot price data from the oil and gas markets. We also discuss the problem of pricing forwards and options and outline how to find the market price of risk in an incomplete market.

In this paper, we propose a novel, analytically tractable, one-factor stochastic model for the dynamics of credit default swap (CDS) spreads and their returns, which we refer to as the spread-return mean-reverting (SRMR) model. The SRMR model can be seen as a hybrid of the Black–Karasinski model on spreads and the Ornstein–Uhlenbeck model on spread returns, and is able to capture empirically observed properties of CDS spreads and returns, including spread mean-reversion, heavy tails of the return distribution, and return autocorrelations. Although developed for modeling CDS spreads, the SRMR model has applications for many other stochastic processes with similar empirical properties, including more general rate processes.

Let $\mathrm{\Gamma }$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope ${\mathrm{\Gamma }}^{\ast }{B}_1^N$ in ${ℝ}^n$ (the absolute convex hull of rows of $\mathrm{\Gamma }$). In particular, we show that

We study the geometry of centrally symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in ${ℝ}^n$. We show that under minimal assumptions on $X$, for $N\gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector — namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors, we recover the estimates that were obtained previously, and thanks to the minimal assumptions on $X$, we derive estimates in cases that were out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when $X$ is $q$-stable or when $X$ has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing — noise blind sparse recovery.

The central result of this paper is the existence of limiting distributions for two classes of critical homogeneous-in-space branching processes with heavy tails spatial dynamics in dimension d = 2. In dimension d ≥ 3, the same results are true without any special assumptions on the underlying (non-degenerated) stochastic dynamics.

Stochastic resonance is an amplification and synchronization effect of weak periodic signals in nonlinear systems through a small noise perturbation. In this paper we study the dynamics of stochastic resonance in a bistable system driven by multiplicative Lévy noise with heavy tails, e.g., $\alpha$-stable Lévy noise. We determine the optimal tuning with respect to a probabilistic synchronization measure for both the jump-diffusion and the reduced two-state Markov chain. These results extend the theory of stochastic resonance to the case of heavy tail Lévy perturbations.

Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e. the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.

One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+𝜃v{v}^T$ with $A={(a_{ij})}_{1\le i,j\le n}\in {ℝ}^{n\times n}$ symmetric, ${(a_{ij})}_{1\le i\le j\le n}$ i.i.d. with centered standard normal distributions, and $𝜃>0,v\in {𝕊}^{n-1}.$ It is well known ${\lambda }_1(P),$ the largest eigenvalue of $P,$ has a phase transition at ${𝜃}_0=1$: when $𝜃\le 1,$${\lambda }_1(P)\underset{}{\overset{\mathrm{\text{a.s.}}}{\to }}2,$ whereas for $𝜃>1,$${\lambda }_1(P)\underset{}{\overset{\mathrm{\text{a.s.}}}{\to }}𝜃+{𝜃}^{-1}.$ Under more general conditions, the limiting behavior of ${\lambda }_1(P),$ appropriately normalized, has also been established: it is normal if $\left|\right|v|{|}_{\infty }=o(1),$ or the convolution of the law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions $a_{11},$ heavy-tailed with index $\alpha \in (0,4),$ the fluctuations are shown to be universal and dependent on $𝜃$ but not on $v,$ whereas a subfamily of the edge case $\alpha =4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a continuous phase transition at ${𝜃}_0=1,{𝜃}_0\in [1,\frac{128}{89}],$ respectively. These results build on our previous work which analyzes the asymptotic behavior of ${\lambda }_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.