Searcharxiv⌕ Search

arXiv subjects

K. Hamza

Publications and source records attributed to K. Hamza.

4 recordsLinked to original sources
arXiv2020

Limit theorems and ergodicity for general bootstrap random walks

Given the increments of a simple symmetric random walk $(X_n)_{n\ge0}$, we characterize all possible ways of recycling these increments into a simple symmetric random walk $(Y_n)_{n\ge0}$ adapted to the filtration of $(X_n)_{n\ge0}$. We study the long term behavior of a suitably normalized two-dimensional process $((X_n,Y_n))_{n\ge0}$. In particular, we provide necessary and sufficient conditions for the process to converge to a two-dimensional Brownian motion (possibly degenerate). We also discuss cases in which the limit is not Gaussian. Finally, we provide a simple necessary and sufficient condition for the ergodicity of the recycling transformation, thus generalizing results from Dubins and Smorodinsky (1992) and Fujita (2008), and solving the discrete version of the open problem of the ergodicity of the general L\'evy transformation (see Mansuy and Yor, 2006).

math.PR↗
arXiv2018

Persistence of Small Noise and Random initial conditions

The effect of small noise in a smooth dynamical system is negligible on any finite time interval. Here we study situations when it persists on intervals increasing to infinity. Such asymptotic regime occurs when the system starts from initial condition, sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to solution of the unperturbed system, started from a certain {\em random} initial condition. In this paper we consider the case of one dimensional diffusions on the positive half line, which often arise as scaling limits in population dynamics.

math.PR↗
arXiv2005

On Solutions of First Order Stochastic Partial Differential Equations

This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order stochastic partial differential equations of the form $\pd Ut = DW$, and $\pd Ut-\pd Ux= DW$, where $D$ is a differential operator and $W(t,x)$ is a continuous but non-differentiable function (field). We give a necessary and sufficient condition for stochastic equations to have solutions as functions. The result is then applied to the equation for a yield curve. Proofs are based on probability arguments.

math.PR↗