We model the growth of a cell population by a piecewise deter-ministic Markov branching tree. Each cell splits into two offsprings at a division rate B(x) that depends on its size x. The size of each cell grows exponentially in time, at a rate that varies for each individual. We show that the mean empirical measure of the model satisfies a… (More)
Multifractal analysis of multiplicative random cascades is revis-ited within the framework of mixed asymptotics. In this new framework , statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single… (More)
We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative… (More)
Many organisms coordinate cell growth and division through size control mechanisms: cells must reach a critical size to trigger a cell cycle event. Bacterial division is often assumed to be controlled in this way, but experimental evidence to support this assumption is still lacking. Theoretical arguments show that size control is required to maintain size… (More)
In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0, T ] when T → ∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the… (More)
In this note, we present results on the behavior for the partition function of multiplicative cascades in the case where the total time of observation is large, compared to the scale of decay for the correlation of the cascade process.