We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue of Schrödinger generators. The continuous or discrete time models studied in this work consists of $N$ interacting particles evolving in an environment with soft obstacles related to a potential function $V$. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine...

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue of Schrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We...

The paper deals with the particle filter in state estimation of a discrete-time nonlinear non-Gaussian system. The goal of the paper is to design a sample size adaptation technique to guarantee a quality of a filtering estimate produced by the particle filter which is an approximation of the true filtering estimate. The quality is given by a difference between the approximate filtering estimate and the true filtering estimate. The estimate may be a point estimate or a probability density function...

We propose a general partition-based strategy to estimate conditional density with candidate densities that are piecewise constant with respect to the covariate. Capitalizing on a general penalized maximum likelihood model selection result, we prove, on two specific examples, that the penalty of each model can be chosen roughly proportional to its dimension. We first study a classical strategy in which the densities are chosen piecewise conditional according to the variable. We then consider Gaussian...

This paper deals with the problem of estimating a regression function f, in a random design framework. We build and study two adaptive estimators based on model selection, applied with warped bases. We start with a collection of finite dimensional linear spaces, spanned by orthonormal bases. Instead of expanding directly the target function f on these bases, we rather consider the expansion of h = f ∘ G-1, where G is the cumulative distribution function of the design, following Kerkyacharian and...

In this article we tackle the problem of inverse non linear ill-posed problems from a statistical point of view. We discuss the problem of estimating an indirectly observed function, without prior knowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show...

Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in...

Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in...

Let $F$ be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that $F_u\left(x\right)$, where $F_u$ is the d.f of the excesses over $u$, converges, when $u$ tends to $s_{+}\left(F\right)$, the end-point of $F$, to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma \>-1$, a function $\Lambda$ which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to $0$ faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.

Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that Fu(x), where Fu is the d.f of the excesses over u, converges, when u tends to s+(F), the end-point of F, to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function Λ which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to 0 faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.

For a proper assessment of risks associated with the trading of derivatives, the performance of hedging strategies should be evaluated not only in the context of the idealized model that has served as the basis of strategy development, but also in the context of other models. In this paper we consider the class of so-called interval models as a possible testing ground. In the context of such models the fair price of a derivative contract is not uniquely determined and we characterize the interval...

The periodic autoregressive process with non-vanishing mean and with exogenous variables is investigated in the paper. It is assumed that the model has also periodic variances. The statistical analysis is based on the Bayes approach with a vague prior density. Estimators of the parameters and asymptotic tests of hypotheses are derived.