Narrow Results

We aim at detecting stress in newborns by observing heart rate variability (HRV). The HRV features nonlinearities. Fractal dynamics is a usual way to model them and the Hurst exponent summarizes the fractal information. In our framework, we have observations of short duration, for which usual estimators of the Hurst exponent, like detrended fluctuation analysis (DFA), are not adapted. Moreover, we observe that the Hurst exponent does not vary much between stress and rest phases, but its decomposition in memory and underlying properties of the probability distribution leads to satisfactory diagnostic tools. This decomposition of the Hurst exponent is in addition embedded in a mean-reverting model. The resulting model is a mean-reverting fractional Lévy stable motion (FLSM). We estimate it and use its parameters as diagnostic tools of neonatal stress. Indeed, the value of the speed of reversion parameter is a significant indicator of stress. The evolution of both parameters in which the Hurst exponent is decomposed provides us with significant indicators as well. On the contrary, the Hurst exponent itself does not bear useful information.

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏ_t = -∇U(Y_t) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

Let $d\ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equations: $d{X}_t=d{S}_t+b(s+t,X_t)dt,t\ge 0,X_0=x$, where $(s,x)\in {ℝ}_{+}\times {ℝ}^d$ is the starting point, $b:{ℝ}_{+}\times {ℝ}^d\to {ℝ}^d$ is measurable, and $S={(S_t)}_{t\ge 0}$ is a $d$-dimensional centered $\alpha$-stable process with index $\alpha \in (1,2)$. We show that if the centered $\alpha$-stable process $S$ is non-degenerate and $b\in L_{\mathrm{\text{loc}}}^{\infty }({ℝ}_{+};L^{\infty }({ℝ}^d))+L_{\mathrm{\text{loc}}}^q({ℝ}_{+};L^p({ℝ}^d))$ for some $p,q>0$ with $d/p+\alpha /q<\alpha -1$, then the above SDE has a unique weak solution for every starting point $(s,x)\in {ℝ}_{+}\times {ℝ}^d$.

In this paper, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $\alpha$-stable process. As the main results, we establish a functional central limit theorem (in the case when $d/\alpha >3/2$) with a standard one-dimensional Brownian motion in the limit, and Khintchine’s and Chung’s laws of the iterated logarithm (in the case when $d/\alpha >9/5$).

In this paper, we provide a Monte Carlo method to calculate multiple stochastic integrals which is based on a.s. distributional convergence.

The [GLP & MEMM] pricing model (= [Geometric Lévy Process & Minimal Entropy Martingale Measure] pricing model) has been introduced as a pricing model for the incomplete financial market. This model has many good properties and is applicable to very wide classes of underlying asset price processes including the geometric stable processes. We explain those good properties and see several examples of this model. After that we investigate the calibration problems of [GLP & MEMM] model.

The objective of this paper is to introduce the [GSP & MEMM] (geometric stable processes and minimal entropy martingale measure) pricing model and show its advantages for pricing options with its fat tailed property. As the result of empirical analysis the good fitness of the [GSP & MEMM] pricing model is shown to the currency options.