An almost sure conditional convergence result and an application to a generalized Polya urn

We prove an almost sure conditional convergence result toward a Gaussian kernel and we apply it to a two-colors randomly reinforced urn

Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment

The objective of the present paper is to establish exponential large deviation inequalities, and to use them to show exponential concentration inequalities for the free energy of a polymer in general random environment, its rate of convergence, and an expression of its limit value in terms of those of some multiplicative cascades.Comment: 25 page

Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series

We consider some versions and generalizations of the approach to expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series. The expansions of iterated stochastic integrals with respect to martingale Poisson measures and with respect to martingales were obtained. For the iterated stochastic integrals with respect to martingales we have proved two theorems. The first theorem is the generalization of expansion for iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series. The second one is the modification of the first theorem for the case of complete orthonormal with weight $r(t_1)\ldots r(t_k)\ge 0$ systems of functions in the space $L_2([t, T]^k)$ (in the first theorem $r(t_1)\ldots r(t_k)\equiv 1$). Mean-square convergence of the considered expansions is proved. The example of expansion of iterated (double) stochastic integrals with respect to martingales with using the system of Bessel functions is considered.Comment: 37 pages. Minor changes. arXiv admin note: text overlap with arXiv:1712.09746, arXiv:1801.05654, arXiv:1801.01564, arXiv:1712.08991, arXiv:1801.00231, arXiv:1801.03195, arXiv:1712.0951

Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time.For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.Comment: Accepted by Bernoull

Dirichlet forms methods, an application to the propagation of the error due to the Euler scheme

We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme in terms of a Dirichlet form on the Wiener space, what allows to propagate this error thanks to functional calculus.Comment: 15

On martingale tail sums in affine two-color urn models with multiple drawings

In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.Comment: 17 page

Statistical estimation of the Oscillating Brownian Motion

We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors' estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.Comment: 31 pages, 1 figur