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Continuous-time stochastic processes play an important role in the description of random phenomena, it is therefore of prime interest to study particular variables depending on their paths, like stopping time for example. One approach consists in pointing out explicit expressions of the probability distributions, an other approach is rather based on the numerical generation of the random variables. We propose an algorithm in order to generate the first passage time through a given level of a one-dimensional jump diffusion. This process satisfies a stochastic differential equation driven by a Brownian motion and subject to random shocks characterized by an independent Poisson process. Our algorithm belongs to the family of rejection sampling procedures, also called exact simulation in this context: the outcome of the algorithm and the stopping time under consideration are identically distributed. It is based on both the exact simulation of the diffusion at a given time and on the exact simulation of first passage time for continuous diffusions. It is therefore based on an extension of the algorithm introduced by Herrmann and Zucca [16] in the continuous framework. The challenge here is to generate the exact position of a continuous diffusion conditionally to the fact that the given level has not been reached before. We present the construction of the algorithm and give numerical illustrations, conditions on the recurrence of jump diffusions are also discussed.

We consider the path approximation of Bessel processes and develop a new and efficient algorithm. This study is based on a recent work by the authors, on the path approximation of the Brownian motion, and on the construction of specific own techniques. It is part of the family of the so-called $\varepsilon$-strong approximations. More precisely, our approach constructs jointly the sequences of exit times and corresponding exit positions of some well-chosen domains, the construction of these domains being an important step. Based on this procedure, we emphasize an algorithm which is easy to implement. Moreover, we can develop the method for any dimension. We treat separately the integer dimension case and the non integer framework, each situation requiring appropriate techniques. In particular, for both situations, we show the convergence of the scheme and provide the control of the efficiency with respect to the small parameter $\varepsilon$. We expand the theoretical part by a series of numerical developments.

This paper develops a new technique for the path approximation of one-dimensional stochastic processes, more precisely the Brownian motion and families of stochastic differential equations sharply linked to the Brownian motion (usually known as L and G-classes). We are interested here in the $\epsilon$-strong approximation. We propose an explicit and easy to implement procedure that constructs jointly, the sequences of exit times and corresponding exit positions of some well chosen domains. The main results control the number of steps to cover a fixed time interval and the convergence theorems for our scheme. We combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also described in order to complete the theoretical results.

In order to describe or estimate different quantities related to a specific random variable, it is of prime interest to numerically generate such a variate. In specific situations, the exact generation of random variables might be either momentarily unavailable or too expensive in terms of computation time. It therefore needs to be replaced by an approximation procedure. As was previously the case, the ambitious exact simulation of exit times for diffusion processes was unreachable though it concerns many applications in different fields like mathematical finance, neuroscience or reliability. The usual way to describe exit times was to use discretization schemes, that are of course approximation procedures. Recently, Herrmann and Zucca \cite{Herrmann-Zucca-2} proposed a new algorithm, the so-called GDET-algorithm (General Diffusion Exit Time), which permits to simulate exactly the exit time for one-dimensional diffusions. The only drawback of exact simulation methods using an acceptance-rejection sampling is their time consumption. In this paper the authors highlight an acceleration procedure for the GDET-algorithm based on a multi-armed bandit model. The efficiency of this acceleration is pointed out through numerical examples.

In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm was already introduced in both the Brownian context and in the Ornstein-Uhlenbeck context. Here the aim is therefore to generalize this efficient numerical approach in order to obtain an approximation of both the exit time and position for either a general linear diffusion or a growth diffusion. The efficiency of the method is described with particular care through theoretical results and numerical examples.

In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method., Comment: arXiv admin note: text overlap with arXiv:1906.02969

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability... The usual procedure is to use discretiza-tion schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.

Since diffusion processes arise in so many different fields, efficient tech-nics for the simulation of sample paths, like discretization schemes, represent crucial tools in applied probability. Such methods permit to obtain approximations of the first-passage times as a by-product. For efficiency reasons, it is particularly challenging to simulate directly this hitting time by avoiding to construct the whole paths. In the Brownian case, the distribution of the first-passage time is explicitly known and can be easily used for simulation purposes. The authors introduce a new rejection sampling algorithm which permits to perform an exact simulation of the first-passage time for general one-dimensional diffusion processes. The efficiency of the method, which is essentially based on Girsanov's transformation , is described through theoretical results and numerical examples.

The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirich-let problem for Laplace's equation, its implementation is rather easy. The definition of the random walk is based on a new mean value formula for the heat equation. The convergence results and numerical examples permit to emphasize the efficiency and accuracy of the algorithm.

Under some weak conditions, the first-passage time of the Brownian motion to a continuous curved boundary is an almost surely finite stopping time. Its probability density function (pdf) is explicitly known only in few particular cases. Several mathematical studies proposed to approximate the pdf in a quite general framework or even to simulate this hitting time using a discrete time approximation of the Brownian motion. The authors study a new algorithm which permits to simulate the first-passage time using an iterating procedure. The convergence rate presented in this paper suggests that the method is very efficient.

In the nonlinear diffusion framework, stochastic processes of McKean-Vlasov type play an important role. In some cases they correspond to processes attracted by their own probability distribution: the so-called self-stabilizing processes. Such diffusions can be obtained by taking the hydrodymamic limit in a huge system of linear diffusions in interaction. In both cases, for the linear and the nonlinear processes, small-noise asymptotics have been emphasized by specific large deviation phenomenons. The natural question, therefore, is: is it possible to interchange the mean-field limit with the small-noise limit? The aim here is to consider this question by proving that the rate function of the first particle in a mean-field system converges to the rate function of the hydrodynamic limit as the number of particles becomes large.

In this paper we pursue and complete the study of the simulation of the hitting time of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work, the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method was based mainly on the explicit formula for the distribution of the hitting time and on the connexion between the Bessel process and the Euclidean norm of the Brownian motion. This method does not apply anymore for a non integer dimension. In this paper we consider the simulation of the hitting time of Bessel processes with non integer dimension and provide a new algorithm by using the additivity property of the laws of squared Bessel processes. We split each simulation step in two parts: one is using the integer dimension case and the other one considers hitting time of a Bessel process starting from zero.

In this paper we introduce a new method for the simulation of the exit time and position of a $\delta$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with existing methods are performed.

The influence of a time-periodic forcing on stochastic processes can essentially be emphasized in the large time behaviour of their paths. The statistics of transition in a simple Markov chain model permits to quantify this influence. In particular the first Floquet multiplier of the associated generating function can be explicitly computed and related to the equilibrium probability measure of an associated process in higher dimension. An application to the stochastic resonance is presented., Comment: 21 pages

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.

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that exists between Cox-Ingersoll-Ross processes and Bessel processes., Comment: Published in at http://dx.doi.org/10.1214/12-AAP900 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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 $((Z_t,N_t); t\ge 0)$.

We investigate exit times from domains of attraction for the motion of a self-stabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is the effect of including an ensemble-average attraction in addition to the usual state-dependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers' type law for the particle's exit from the potential's domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that the exit law for the self-stabilizing diffusion coincides with the exit law of a potential diffusion without self-stabilization and a drift component perturbed by average attraction. We show that self-stabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different., Comment: Published in at http://dx.doi.org/10.1214/07-AAP489 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Physical notions of stochastic resonance for potential diffusions in periodically changing double-well potentials such as the spectral power amplification have proved to be defective. They are not robust for the passage to their effective dynamics: continuous-time finite-state Markov chains describing the rough features of transitions between different domains of attraction of metastable points. In the framework of one-dimensional diffusions moving in periodically changing double-well potentials we design a new notion of stochastic resonance which refines Freidlin's concept of quasi-periodic motion. It is based on exact exponential rates for the transition probabilities between the domains of attraction which are robust with respect to the reduced Markov chains. The quality of periodic tuning is measured by the probability for transition during fixed time windows depending on a time scale parameter. Maximizing it in this parameter produces the stochastic resonance points., Comment: Published at http://dx.doi.org/10.1214/105051604000000530 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider potential type dynamical systems in finite dimensions with two meta-stable states. They are subject to two sources of perturbation: a slow external periodic perturbation of period $T$ and a small Gaussian random perturbation of intensity $\epsilon$, and, therefore, are mathematically described as weakly time inhomogeneous diffusion processes. A system is in stochastic resonance, provided the small noisy perturbation is tuned in such a way that its random trajectories follow the exterior periodic motion in an optimal fashion, that is, for some optimal intensity $\epsilon (T)$. The physicists' favorite, measures of quality of periodic tuning--and thus stochastic resonance--such as spectral power amplification or signal-to-noise ratio, have proven to be defective. They are not robust w.r.t. effective model reduction, that is, for the passage to a simplified finite state Markov chain model reducing the dynamics to a pure jumping between the meta-stable states of the original system. An entirely probabilistic notion of stochastic resonance based on the transition dynamics between the domains of attraction of the meta-stable states--and thus failing to suffer from this robustness defect--was proposed before in the context of one-dimensional diffusions. It is investigated for higher-dimensional systems here, by using extensions and refinements of the Freidlin--Wentzell theory of large deviations for time homogeneous diffusions. Large deviations principles developed for weakly time inhomogeneous diffusions prove to be key tools for a treatment of the problem of diffusion exit from a domain and thus for the approach of stochastic resonance via transition probabilities between meta-stable sets., Comment: Published at http://dx.doi.org/10.1214/105051606000000385 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)