Publications
A random flight on a plane with nonisotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and nonisotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.
We introduce a longevity feature to the classical optimal dividend problem by adding a constraint on the time of ruin of the firm. We extend the results in [HJ15], now in context of onesided Lévy risk models. We consider de Finetti's problem in both scenarios with and without fix transaction costs, e.g. taxes. We also study the constrained analog to the so called Dual model. To characterize the solution to the aforementioned models we introduce the dual problem and show that the complementary slackness conditions are satisfied and therefore there is no duality gap. As a consequence the optimal value function can be obtained as the pointwise infimum of auxiliary value functions indexed by Lagrange multipliers. Finally, we illustrate our findings with a series of numerical examples.
For a probability distribution P on an at most countable alphabet A, this article gives finite sample bounds for the expected occupancy counts EKn,r and probabilities EMn,r. Both upper and lower bounds are given in terms of the counting function ν of P. Special attention is given to the case where ν is bounded by a regularly varying function. In this case, it is shown that our general results lead to an optimalrate control of the expected occupancy counts and probabilities with explicit constants. Our results are also put in perspective with Turing’s formula and recent concentration bounds to deduce bounds in probability. At the end of the paper, we discuss an extension of the occupancy problem to arbitrary distributions in a metric space.
The paper considers a family of probability distributions depending on a pa rameter. The goal is to derive the generalized versions of Cram´erRao and Bhattacharyya inequalities for the weighted covariance matrix and of the Kullback inequality for the weighted Kullback distance, which are important objects themselves [9, 23, 28]. The asymp totic forms of these inequalities for a particular family of probability distributions and for a particular class of continuous weight functions are given.
Existence, regularity and uniqueness of the solution to a HamiltonJacobiBellman (HJB) equation were studied recently in [H. A. MorenoFranco, Appl. Math. Opt. 2016], when the Lévy measure associated with the integral part of the elliptic integrodifferential operator, is finite. Here we extend the results obtained in this paper, in the case that the Lévy measure associated with this operator has bounded variation. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a jumpdiffusion. For a particular case, by means of εpenalized controls we show that the value function associated to this class of problems agrees with the solution to our HJB equation.
In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of lighttail and truncated heavytail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions of the sums due to various relations between these parameters.
In this work we study the optimal execution problem with multiplicative price impact in algorithm trading, when an agent holds an initial position of shares of a financial asset. The intersellingdecision times are modelled by the arrival times of a Poisson process. The criterion to be optimised consists in maximising the expected net present value of gains of the agent, and it is proved that an optimal strategy has a barrier form, depending only on the number of shares left and the level of asset price.
In this paper, we analyze a L{\'e}vy model based on two popular concepts  subordination and L{\'e}vy copulas. More precisely, we consider a twodimensional L{\'e}vy process such that each component is a timechanged (subordinated) Brownian motion and the dependence between subordinators is described via some L{\'e}vy copula. The main result of this paper is the series representation for our model, which can be efficiently used for simulation purposes.
We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients.
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.
We consider the diffusion process and its approximation by Markov chain with nonlinear unbounded trends. The usual parametrix method is not applicable because these models have unbounded trends. We describe a procedure that allows to exclude nonlinear unbounded trend and move to stochastic differential equation with bounded drift and diffusion coefficients. A similar procedure is considered for a Markov chain.

We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hörmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multiscale behavior of the process.
In this paper we develop an asymptotic theory for the QuasiMaximum Likelihood Estimator (QMLE) of the parametric GARCHinMean model. The asymptotics is based on a study of the volatility as a process of the model parameters. The proof makes use of stochastic recurrence equations for this random function and uses exponential inequalities to localize the problem. Our results show why the asymptotics for this specification is quite complex although it is a rather standard parametric model. Nevertheless, our theory does not yet treat all standard specifications of the mean function.