arXiv subjects

José Luis Pérez

Publications and source records attributed to José Luis Pérez.

25 records · Page 2Linked to original sources

arXiv2018

Optimality of multi-refraction dividend strategies in the dual model

We consider the multi-refraction strategies in two equivalent versions of the optimal dividend problem in the dual (spectrally positive Lévy) model. The first problem is a variant of the bail-out case where both dividend payments and capital injections must be absolutely continuous with respect to the Lebesgue measure. The second is an extension of Avanzi et al. [4] where a strategy is a combination of two absolutely continuous dividend payments with different upper bounds and different transaction costs. In both problems, it is shown to be optimal to refract the process at two thresholds, with the optimally controlled process being the multi-refracted Lévy process recently studied by Czarna et al. [9]. The optimal strategy and the value function are succinctly written in terms of a version of the scale function. Numerical results are also given.

math.PR↗

arXiv2018

Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(λ_{1}^{(n)}(t),\dots, λ_{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $μ^{(n)}=(μ_{t}^{(n)}; t\geq0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $μ^{(n)}$ converges in probability to a deterministic limit $μ$, in the topology of uniform convergence over compact sets. We show that the process $μ$ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers' equation. Our results extend those of Rogers and Shi for the free Brownian motion and Pardo et al. for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

math.PR↗

arXiv2017

American options under periodic exercise opportunities

In this paper, we study a version of the perpetual American call/put option where exercise opportunities arrive only periodically. Focusing on the exponential Lévy models with i.i.d. exponentially-distributed exercise intervals, we show the optimality of a barrier strategy that exercises at the first exercise opportunity at which the asset price is above/below a given barrier. Explicit solutions are obtained for the cases the underlying Lévy process has only one-sided jumps.

math.PR↗

arXiv2017

Spectrally negative Lévy processes with Parisian reflection below and classical reflection above

We consider a company that receives capital injections so as to avoid ruin. Differently from the classical bail-out settings where the underlying process is restricted to stay at or above zero, we study the case bail-out can only be made at independent Poisson times. Namely, we study a version of the reflected process that is pushed up to zero only on Poisson observation times at which the process is below zero. We also study the case with additional classical reflection above so as to model a company that pays dividends according to a barrier strategy. Focusing on the spectrally negative Lévy case, we compute, using the scale function, various fluctuation identities including capital injections and dividends.

math.PR↗

arXiv2016

Refracted Continuous-State Branching Processes: Self-regulating populations

We construct a modified continuous-state branching process whose Malthusian parameter is replaced by another when passing below a certain level. The construction is obtained via a Lamperti-like transform applied to a refracted Lévy process. Infinitesimal generator, probability of vanishing at infinity, of explosion and some path properties are also provided.

math.PR↗

arXiv2016

On the Free Fractional Wishart Process

We investigate the process of eigenvalues of a fractional Wishart process defined as N=B*B, where B is a matrix fractional Brownian motion recently studied by Nualart and Pérez-Abreu. Using stochastic calculus with respect to the Young integral we show that the eigenvalues do not collide at any time with probability one. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ we derive a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the free fractional Wishart process which constitutes the family of fractional dilations of the free Poisson distribution.

math.PR↗

arXiv2016

A simple construction of positive loops of Legendrians

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu's result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.

math.SG↗