The standard Merton-Black-Scholes formula for European Option pricing serves only as approximation to real values of options. More advanced extensions include applications of Lévy processes and are based on characteristic functions, which are more convenient to use than the corresponding probability distributions. We found one of the Lewis (2001) general theoretical formulae for option pricing based on characteristic functions particularly suitable for a statistical approach to option pricing. By...

This paper is an attempt to present and analyse stochastic mortality models. We propose a couple of continuous-time stochastic models that are natural generalizations of the Gompertz law in the sense that they reduce to the Gompertz function when the volatility parameter is zero. We provide a statistical analysis of the available demographic data to show that the models fit historical data well. Finally, we give some practical examples for the multidimensional models.

We study a class of models used with success in the modelling of climatological sequences. These models are based on the notion of renewal. At first, we examine the probabilistic aspects of these models to afterwards study the estimation of their parameters and their asymptotical properties, in particular the consistence and the normality. We will discuss for applications, two particular classes of alternating renewal processes at discrete time. The first class is defined by laws of sojourn time...

We prove a moderate deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space.

In the present paper we prove moderate deviations for a Curie–Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1–30]. The results extend those already obtained for the Curie–Weiss model without external field by Eichelsbacher and Löwe in [P. Eichelsbacher and M. Löwe, Markov Process. Related Fields 10 (2004) 345–366]. The Curie–Weiss model with...

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^n{X}_i/b_n$ – where $\frac{b_n}{n}\downarrow 0$, but $\frac{b_n}{\sqrt{n}}\uparrow \infty$ – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^n{X}_i/b_n$ – where $\frac{b_n}{n}\downarrow 0$, but $\frac{b_n}{\sqrt{n}}\uparrow \infty$ – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

Functionals in geometric probability are often expressed as sums of bounded functions exhibiting exponential stabilization. Methods based on cumulant techniques and exponential modifications of measures show that such functionals satisfy moderate deviation principles. This leads to moderate deviation principles and laws of the iterated logarithm for random packing models as well as for statistics associated with germ-grain models and k nearest neighbor graphs.

In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of ϕ-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given.

The purpose of this paper is to investigate moderate deviations for the Durbin–Watson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. It enables us to provide a moderate deviation...

Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance ${\sigma }_1^2$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance ${\sigma }_2^2$ independent of {Xi,1 ≤ i ≤ n1}. Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-({\mu }_1-{\mu }_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$. This paper shows that ln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, in addition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T\ge x)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))