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Pierre Vallois

Publications and source records attributed to Pierre Vallois.

At least 19 recordsLinked to original sources
arXiv2025

Modelling the effects of biological intervention in a dynamical gene network

Cellular response to environmental and internal signals can be modeled by dynamical gene regulatory networks (GRN). In the literature, three main classes of gene network models can be distinguished: (i) non-quantitative (or data-based) models which do not describe the probability distribution of gene expressions; (ii) quantitative models which fully describe the probability distribution of all genes coexpression; and (iii) mechanistic models which allow for a causal interpretation of gene interactions. We propose two rigorous frameworks to model gene alteration in a dynamical GRN, depending on whether the network model is quantitative or mechanistic. We explain how these models can be used for design of experiment, or, if additional alteration data are available, for validation purposes or to improve the parameter estimation of the original model. We apply these methods to the Gaussian graphical model, which is quantitative but non-mechanistic, and to mechanistic models of Bayesian networks and penalized linear regression.

q-bio.MN↗
arXiv2024

Drawdowns of diffusions

In this paper we give excursion theoretical proofs of Lehoczky's formula (in an extended form allowing a lower bound for the underlying diffusion) for the joint distribution of the first drawdown time and the maximum before this time, and of Malyutin's formula for the joint distribution of the first hitting time and the maximum drawdown before this time. It is remarkable -- but there is a clean explanation -- that the excursion theoretical approach which we developed first for Lehoczky's formula provides also a proof for Malyutin's formula. Moreover, we discuss some generalizations and analyze the pure jump process describing the maximum before the first drawdown time when the size of the drawdown is varying

math.PR↗
arXiv2021

Asymptotic error distribution for the Riemann approximation of integrals driven by fractional Brownian motion

We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a `controlled' process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals

math.PR↗
arXiv2021

On a first hit distribution of the running maximum of Brownian motion

Let $(S_t)_{t\geq 0}$ be the running maximum of a standard Brownian motion $(B_t)_{t\geq 0}$ and $T_m:=\inf\{t; \, mS_t 0$. In this note we calculate the joint distribution of $T_m$ and $B_{T_m}$. The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.

math.PR↗
arXiv2019

Fractional extreme distributions

We consider three classes of linear differential equations on distribution functions, with a fractional order $α\in [0,1].$ The integer case $α=1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $α-$stable subordinator. From the analytical viewpoint, this law is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case. By the stochastic representation, we can derive several analytical properties for the latter special functions, extending known features of the classical Mittag-Leffler function, and dealing with monotonicity, complete monotonicity, infinite divisibility, asymptotic behaviour at infinity, uniform hyperbolic bounds.

math.PR↗
arXiv2015

Probability that the maximum of the reflected Brownian motion over a finite interval $[0,t]$ is achieved by its last zero before $t$

We calculate the probability $p_c$ that the maximum of a reflected Brownian motion $U$ is achieved on a complete excursion, i.e. $p_c:=P\big(\overline{U}(t)=U^*(t)\big)$ where $\overline{U}(t)$ (respectively $U^*(t)$) is the maximum of the process $U$ over the time interval $[0,t]$ (resp. $\big[0,g(t)\big]$ where $g(t)$ is the last zero of $U$ before $t$).

math.PR↗
arXiv2012

Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes

A classical random walk $(S_t, t\in\mathbb{N})$ is defined by $S_t:=\displaystyle\sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in\mathbb{N}}$ are a one-order Markov chain, a short memory is introduced in the dynamics of $(S_t)$. This so-called "persistent" random walk is nolonger Markovian and, under suitable conditions, the rescaled process converges towards the integrated telegraph noise (ITN) as the time-scale and space-scale parameters tend to zero (see Herrmann and Vallois, 2010; Tapiero-Vallois, Tapiero-Vallois2}). The ITN process is effectively non-Markovian too. The aim is to consider persistent random walks $(S_t)$ whose increments are Markov chains with variable order which can be infinite. This variable memory is enlighted by a one-to-one correspondence between $(X_n)$ and a suitable Variable Length Markov Chain (VLMC), since for a VLMC the dependency from the past can be unbounded. The key fact is to consider the non Markovian letter process $(X_n)$ as the margin of a couple $(X_n,M_n)_{n\ge 0}$ where $(M_n)_{n\ge 0}$ stands for the memory of the process $(X_n)$. We prove that, under a suitable rescaling, $(S_n,X_n,M_n)$ converges in distribution towards a time continuous process $(S^0(t),X(t),M(t))$. The process $(S^0(t))$ is a semi-Markov and Piecewise Deterministic Markov Process whose paths are piecewise linear.

math.PR↗
arXiv2008

From persistent random walks to the telegraph noise

We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process $(Z_t)$ which can be easely expressed in terms of a counting process $(N_t)$. In a particular case the counting process is a Poisson process, and $(Z_t)$ permits to represent the solution of the telegraph equation. We study in detail the Markov process $((Z_t,N_t); t\ge 0)$.

math.PR↗
arXiv2008

On subexponentiality of the Lévy measure of the diffusion inverse local time; with applications to penalizations

For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the Lévy measure of the inverse local time is subexponential this distribution behaves asymtotically as a multiple of the Lévy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on $\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.

math.PR↗
arXiv2007

Approximation via regularization of the local time of semimartingales and Brownian motion

Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartingales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in $L^2$, and a.s. convergence of some of our schemes.

math.PR↗
arXiv2007

Quelques approximations du temps local brownien

We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $ \frac{2}ε\int_0^{t} X_{(u+ε)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}ε\int_0^{t} X_{(u+ε) \wedge t}^- \indi_{\{X_u>0\}} du$ and $\frac{4}ε\int_0^{t} X_u^- \indi_{\{X_{(u+ε) \wedge t} > 0\}} du$ converge in the ucp sense to $L_t^0$, as $ε\to 0$. We show that $ \frac{1}ε\int_0^t (\indi_{\{x<X_{s+ε}\}} - \indi_{\{x<X_{s}\}}) (X_{s+ε}-X_{s})ds$ goes to $L_t^x$ in $L^2(Ω)$ as $ε\to 0$, and that the rate of convergence is of order $ε^α$, for any $α< {1/4}$.

math.PR↗
arXiv2006

On the excursion theory for linear diffusions

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein's representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss Ornstein-Uhlenbeck processes.

math.PR↗
arXiv2006

Elements of Stochastic Calculus via Regularisation

This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure Itô and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite quadratic variation processes, Dirichlet and weak Dirichlet processes.

math.PR↗
arXiv2005

On Maximum Increase and Decrease of Brownian Motion

The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient between these at a fixed time and this is seen to be -0.47936... .

math.PR↗
arXiv2005

Limiting laws for long Brownian Bridges perturbed by their one-sided maximum, III

Results of penalization of a one-dimensional Brownian motion $(X_t) $, by its one-sided maximum $\dis (S_t=\sup_{0 \leq u \leq t}X_u)$, which were recently obtained by the authors are improved with the consideration-in the present paper- of the asymptotic behaviour of the likewise penalized Brownian bridges of length $t$, as $t\to \infty$, or penalizations by functions of $(S_t,X_t)$, and also the study of the speed of convergence, as $t\to \infty$, of the penalized distributions at time $t$.

math.PR↗