Hitting time for Bessel processes - walk on moving spheres algorithm (WoMS)

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

Stochastic equation of fragmentation and branching processes related to avalanches

We give a stochastic model for the fragmentation phase of a snow avalanche. We construct a fragmentation-branching process related to the avalanches, on the set of all fragmentation sizes introduced by J. Bertoin. A fractal property of this process is emphasized. We also establish a specific stochastic equation of fragmentation. It turns out that specific branching Markov processes on finite configurations of particles with sizes bigger than a strictly positive threshold are convenient for describing the continuous time evolution of the number of the resulting fragments. The results are obtained by combining analytic and probabilistic potential theoretical tools.Comment: 17 page

Simulation of diffusions by means of importance sampling paradigm

The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is to combine random walk on squares or rectangles methods with importance sampling techniques. The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows one to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to performing variance reduction techniques and simulating rare events.Comment: Published in at http://dx.doi.org/10.1214/09-AAP659 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Initial-boundary value problem for the heat equation - A stochastic algorithm

International audienceThe initial-boundary value problem for the heat equation is solved by using an 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 Dirichlet problem for Laplace’s equation, its implementation is rather easy. The construction of this algorithm can be considered as a natural consequence of previous works the authors completed on the hitting time approximation for Bessel processes and Brownian motion [Ann. Appl. Probab. 23 (2013) 2259–2289, Math. Comput. Simulation 135 (2017) 28–38, Bernoulli 23 (2017) 3744–3771]. A similar procedure was introduced previously in the paper [Random Processes for Classical Equations of Mathematical Physics (1989) Kluwer Academic].The definition of the random walk is based on a particular mean value formula for the heat equation. We present here a probabilistic view of this formula.The aim of the paper is to prove convergence results for this algorithm and to illustrate them by numerical examples. These examples permit to emphasize the efficiency and accuracy of the algorithm

The walk on moving spheres: a new tool for simulating Brownian motion's exit time from a domain

International audienceIn 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

A random walk on rectangles algorithm

http://www.springerlink.com/content/1573-7713/In this article, we introduce an algorithm that simulates efficiently the first exit time and position from a rectangle (or a parallelepiped) for a Brownian motion that starts at any point inside. This method provides an exact way to simulate the first exit time and position from any polygonal domain and then to solve some Dirichlet problems, whatever the dimension. This method can be used as a replacement or complement of the method of the random walk on spheres and can be easily adapted to deal with Neumann boundary conditions or Brownian motion with a constant drift

Simulation of exit times and positions for Brownian motions and Diffusions

International audienceWe present in this note some variations of the Monte Carlo method for the random walk on spheres which allow to solve many elliptic and parabolic problems involving the Laplace operator or second-order differential operators. In these methods, the spheres are replaced by rectangles or parallelepipeds. Our first method constructs the exit time and the exit position of a rectangle for a Brownian motion. The second method exhibits a variance reduction technique. The main point is to reduce the problem only to the use of some distributions related to the standard one-dimensional Brownian motion