The Euler-Maruyama approximation for the absorption time of the CEV diffusion

A standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme

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

Numerical simulations of the spiking activity and the related first exit time of stochastic neural systems

The aim of this thesis was to study, using numerical simulation techniques, the possible effects of an additive noise on the firing properties of stochastic neural models, and the related first exit time problems. The research is divided into three main investigations. First, using SDELab, mathematical software for solving stochastic differential equations within MATLAB, we examine the influence of an additive noise on the output spike trains for the space-clamped Hodgkin Huxley (HH) model and the spatially-extended FitzHugh Nagumo (FHN) system. We find that a suitable amount of additive noise can enhance the regularity of the repetitive spiking of the space-clamped HH model. Meanwhile, we find the FHN system to be sensitive to noise, requiring that very small values of noise are chosen, in order to produce regular spikes. Second, under additive noise, we use fixed and exponential time-stepping Euler algorithms, with boundary tests, to calculate the mean first exit times (MFET) for one-dimensional neural diffusion models, represented by a stochastic space-clamped FHN system and the Ornstein-Uhlenbeck (OU) model. The strategies and theory behind these numerical methods and their convergence rates in the MFET are also considered. We find that, for different values of noise, these methods with boundary tests can improve the rate of convergence from order one half to order one, which coincides with previous studies. Finally, we look at spatially-extended systems, represented by the Barkley system with additive noise that is white in time and correlated in space, calculating mean nucleation times and mean lifetimes of traveling waves, using an efficient numerical simulation. A simple model of the dynamics of the underlying Barkley model is introduced, in order to compute the mean lifetimes, particulary for interacting waves. The reduced model is easy to use and allows us to explore the full dynamics of the kinks and antikinks, in particular over long periods. One application of the reduced model is to calculate the mean number of kinks at a given time and use this to obtain the probability that the system is excitable at a given position. With these three investigations into the effects of additive noise on stochastic neural models, we have demonstrated some of the interesting results that can be achieved using numerical techniques. We hope to extend this work, in the future, to include the effects of multiplicative noise.EThOS - Electronic Theses Online ServiceGBUnited Kingdo