arXiv subjects
Publications and source records attributed to Tarek Hamdi.
We show how the approach used in `N. Demni, T. Hmidi. Spectral Distribution of the Free unitary Brownian motion: another approach. Sem. Probab. XLIV. 2012. 191-206.' applies to describe the large-size limit of the marginal distribution of the Brownian motion on the complex linear group.
The additive monotone (resp. boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (resp. boolean) independent and stationary increments which are distributed according to the arcsine law (resp. Bernoulli law) . We introduce the monotone and booleen unitary Brownian motions and we derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, whereas in the boolean case the multiplicative analog of the Bernoulli distribution is discrete. Finally, we use quantum stochastic calculus to provide a realization of these processes as the stochastic exponential of the correspending additive Brownian motions.
We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli randoms walks. We prove a Clark-Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.
In this paper, we are interested in the free Jacobi process starting at the unit of the compressed probability space where it takes values and associated with the parameter values $λ=1, θ=1/2$. Firstly, we derive a time-dependent recurrence equation for the moments of the process (valid for any starting point and all parameter values). Secondly, we transform this equation to a nonlinear partial differential one for the moment generating function that we solve when $λ= 1, θ=1/2$. The obtained solution together with tricky computations lead to an explicit expression of the moments which shows that the free Jacobi process is distributed at any time $t$ as $(1/4)(2+Y_{2t}+Y_{2t}^{\star})$ where $Y$ is a free unitary Brownian motion. This expression is recovered relying on enumeration techniques after proving that if $a$ is a symmetric Bernoulli random variable which is free from $\{Y, Y^{\star}\}$, then the distributions of $Y_{2t}$ and that of $aY_taY_t^{\star}$ coincide. We close the exposition by investigating the spectral distribution associated with general sets of parameter values.