Publications

In this paper, we consider limit laws for the model, which is a generalisation of the random energy model (REM) to the case when the energy levels have the mixture distribution. More precisely, the distribution of the energy levels is assumed to be a mixture of two normal distributions, one of which is standard normal, while the second has the mean \(\sqrt{n}a\) with some \(a\in \R,\) and the variance \(\sigma \ne 1\). The phase space \((a,\sigma) \subset \R \times \R_+\) is divided onto several domains, where after appropriate normalisation, the partition function converges in law to the stable distribution. These domains are separated by the critical surfaces, corresponding to transitions from the normal distribution to \(\alpha-\)stable with \(\alpha \in (1,2)\), after to 1-stable, and finally to \(\alpha-\)stable with \(\alpha \in (0,1).\) The corresponding phase diagram is the central result of this paper.

In this paper we study the problem of statistical inference for a continuous-time moving average  L\'evy process of the form 

\[ Z_{t}=\int_{\R}\mathcal{K}(t-s)\, dL_{s},\quad t\in\mathbb{R}, \]  with a deterministic kernel \(\K\) and a  L{\'e}vy process \(L\). Especially the  estimation of  the L\'evy measure \(\nu\) of  $L$ from low-frequency observations of the process $Z$ is considered. We construct a consistent estimator, derive its convergence rates and  illustrate its performance  by a numerical example.  On the mathematical level, we  establish  some new results on exponential mixing  for  continuous-time moving average  L\'evy processes.    

In this paper, we consider a multidimensional time-changed stochastic process in the context of asset-pricing modeling. The proposed model is constructed from stable processes, and its construction is based on two popular concepts:multivariate subordination and Lévy copulas. From a theoretical point of view, our main result is Theorem 1, which yields a simulation method from the considered class of processes. Our empirical study shows that the model represents the correlation between asset returns quite well. Moreover, we provide some evidence that this model is more appropriate for describing stock prices than classical time-changed Brownian motion, at least if the cumulative amount of transactions is used for a stochastic time change.

Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.

We consider a control problem over an infinite time horizon with a linear stochastic system with an unstable asymptotically unbounded state matrix. We extend the notion of anti-stability of a matrix to the case of non-exponential anti-stability, and introduce an antistability rate function as a characteristic of the rate of growth for the norm of the corresponding fundamental matrix. We show that the linear stable feedback control law is optimal with respect to the criterion of the adjusted extended long-run average. The designed criterion explicitly includes information about the rate of anti-stability and the parameters of the disturbances. We also analyze optimality conditions.

The aim of this work is to investigate the optimal vaccine sharing between two SIR centres in the presence of migration fluxes of susceptibles and infected individuals during the mumps outbreak. Optimality of the vaccine allocation means the minimisation of the total number of lost working days during the whole period of epidemic outbreak $[0,t_f]$, which can be described by the functional $Q=\int_0^{t_f}I(t){\rm d}t$ where $I(t)$ stands for the number of infectives at time $t$. We explain the behaviour of the optimal allocation, which depends on the model parameters and the amount of vaccine available $V$.

In this paper, we consider some properties of the model constructed from the one-dimensional stable processes by changing the time to a non-decreasing Lévy process. Our first result reveals a relation between this class of processes and the class of time-changed Brownian motions. Moreover, we describe the CGMY (Carr-Geman-Madan-Yor) model as subordinated stable process, and show the representation of the Lévy density of the corresponding subordinator via the Mellin–Barnes integral.

We consider a new method of semiparametric statistical estimation for the continuous-time moving-average Lévy processes. We derive the convergence rates of the proposed estimators and show that these rates are optimal in minimax sense.

We study 2D Navier–Stokes equations with a constraint forcing the conservation of the energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier–Stokes equation on R2 and T2, by a fixed point argument. We also show that the solution of the constrained equation converges to the solution of the Euler equation as the viscosity ν vanishes.

This is an advanced guide to optimal stopping and control, focusing on advanced Monte Carlo simulation and its application to finance. Written for quantitative finance practitioners and researchers in academia, the book looks at the classical simulation based algorithms before introducing some of the new, cutting edge approaches under development.

A rigorous connection between large deviations theory and Gamma-convergence is established. Applications include representations formulas for rate functions, a contraction principle for measurable maps, a large deviations principle for coupled systems and a second order Sanov theorem.

The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, a Gaussian convergence can be established for the procedure. Here, we are interested in the local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.

We consider the problem of modeling anomalous diffusions with the Ornstein–Uhlenbeck process with time-varying coefficients. An anomalous diffusion is defined as a process whose mean-squared displacement non-linearly grows in time which is nonlinearly growing in time. We classify diffusions into types (subdiffusion, normal diffusion, or superdiffusion) depending on the parameters of the underlying process. We solve the problem of finding the coefficients of dynamics equations for the Ornstein–Uhlenbeck process to reproduce a given mean-squared displacement function.

We consider a general continuous mean-variance problem where the cost functional has an integral and a terminal-time component. We transform the problem into a superposition of a static and a dynamic optimization problem. The value function of the latter can be considered as the solution to a degenerate HJB equation either in viscosity or in Sobolev sense (after regularization) under suitable assumptions and with implications with regards to the optimality of strategies.

In this paper, we define and study a new notion of stability for the k-means clustering scheme building upon the notion of quantization of a probability measure. We connect this notion of stability to a geometric feature of the underlying distribution of the data, named absolute margin condition, inspired by recent works on the subject.

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 one-sided 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.

Recently developed toy models for the mean-field games of corruption and botnet defence in cyber-security with three or four states of agents are extended to a more general mean-field-game model with 2d states, d ∈ N. In order to tackle new technical difficulties arising from a larger state-space we introduce new asymptotic regimes, namely small discount and small interaction asymptotics. Moreover, the link between stationary and time-dependent solutions is established rigorously leading to a performance of the turnpike theory in a mean-field-game setting.