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Gyula Pap

Publications and source records attributed to Gyula Pap.

At least 37 records · Page 2Linked to original sources
arXiv2018

Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian Lévy process of not necessarily bounded variation with a Lévy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. We also present some numerical illustrations to confirm our results.

math.ST↗
arXiv2018

On aggregation of multitype Galton-Watson branching processes with immigration

Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton-Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton-Watson processes with immigration as well.

math.PR↗
arXiv2017

Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process based on continuous time observations

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump-diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.

math.ST↗
arXiv2017

On conditional least squares estimation for affine diffusions based on continuous time observations

We study asymptotic properties of conditional least squares estimators for the drift parameters of two-factor affine diffusions based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. For all the drift parameters, in the subcritical and supercritical cases, asymptotic normality and asymptotic mixed normality is proved, while in the critical case, non-standard asymptotic behavior is described.

math.PR↗
arXiv2016

Iterated limits for aggregation of randomized INAR(1) processes with Poisson innovations

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $α\in(0,1)$ and with idiosyncratic Poisson innovations. Assuming that $α$ has a density function of the form $ψ(x)(1 - x)^β$, $x\in(0,1)$, with $\lim_{x\uparrow 1}ψ(x) = ψ_1 \in(0,\infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $β\in(-1,0)$, $β= 0$, $β\in(0,1)$ or $β\in(1,\infty)$, when taking first the limit as $N\to\infty$ and then the time scale $n\to\infty$, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaite and Surgailis (2014) by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.

math.PR↗
arXiv2016

Statistical inference of 2-type critical Galton-Watson processes with immigration

In this paper the asymptotic behavior of the conditional least squares estimators of the offspring mean matrix for a 2-type critical positively regular Galton-Watson branching process with immigration is described.We also study this question for a natural estimator of the spectral radius of the offspring mean matrix, which we call criticality parameter. We discuss the subcritical case as well.

math.ST↗
arXiv2016

Iterated scaling limits for aggregation of random coefficient AR(1) and INAR(1) processes

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary AR(1) and INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $α\in (0, 1)$ and idiosyncratic innovations. Assuming that $α$ has a density function of the form $ψ(x) (1 - x)^β$, $x \in (0, 1)$, with $\lim_{x\uparrow 1} ψ(x) = ψ_1 \in (0, \infty)$, different Brownian limit processes of appropriately centered and scaled aggregated partial sums are shown to exist in case $β=1$ when taking first the limit as $N \to \infty$ and then the time scale $n \to \infty$, or vice versa. This paper completes the one of Pilipauskaitė and Surgailis (2014), and Barczy, Nedényi and Pap (2015), where the iterated limits are given for every other possible value of the parameter $β$ for the two types of models.

math.PR↗
arXiv2015

One-parameter statistical model for linear stochastic differential equation with time delay

Assume that we observe a stochastic process $(X(t))_{t\in[-r,T]}$, which satisfies the linear stochastic delay differential equation \[ \mathrm{d} X(t) = \vartheta \int_{[-r,0]} X(t + u) \, a(\mathrm{d} u) \, \mathrm{d} t + \mathrm{d} W(t) , \qquad t \geq 0 , \] where $a$ is a finite signed measure on $[-r, 0]$. The local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of $v_\vartheta^* < 0$, local asymptotic quadraticity is shown if $v_\vartheta^* = 0$, and, under some additional conditions, local asymptotic mixed normality or periodic local asymptotic mixed normality is valid if $v_\vartheta^* > 0$, where $v_\vartheta^*$ is an appropriately defined quantity. As an application, the asymptotic behaviour of the maximum likelihood estimator $\widehat{\vartheta}_T$ of $\vartheta$ based on $(X(t))_{t\in[-r,T]}$ can be derived as $T \to \infty$.

math.ST↗
arXiv2015

Asymptotic inference for a stochastic differential equation with uniformly distributed time delay

For affine stochastic differential equation with uniformly distributed time delay the local asymptotic properties of the likelihood function are studied. Local asymptotic normality, local asymptotic mixed normality, periodic local asymptotic mixed normality or local asymptotic quadraticity is proved for different values of the parameter. Applications to the asymptotic behaviour of the maximum likelihood estimator of the parameter based on continuous sample are given.

math.ST↗
arXiv2015

On convergence properties of infinitesimal generators of scaled multi-type CBI processes

It is a common method for proving weak convergence of a sequence of time-homogeneous Markov processes towards a time-homogeneous Markov process first to show convergence of the corresponding infinitesimal generators and then to check some additional conditions. The aim of the present paper is to investigate convergence properties of discrete infinitesimal generators of appropriately scaled random step functions formed from a multi-type continuous state and continuous time branching process with immigration. We also present a convergence result for usual infinitesimal generators of the branching processes in question appropriately normalized.

math.PR↗
arXiv2015

Blocking unions of arborescences

Given a digraph $D=(V,A)$ and a positive integer $k$, a subset $B\subseteq A$ is called a \textbf{$k$-union-arborescence}, if it is the disjoint union of $k$ spanning arborescences. When also arc-costs $c:A\to \mathbb{R}$ are given, minimizing the cost of a $k$-union-arborescence is well-known to be tractable. In this paper we take on the following problem: what is the minimum cardinality of a set of arcs the removal of which destroys every minimum $c$-cost $k$-union-arborescence. Actually, the more general weighted problem is also considered, that is, arc weights $w:A\to \mathbb{R}_+$ (unrelated to $c$) are also given, and the goal is to find a minimum weight set of arcs the removal of which destroys every minimum $c$-cost $k$-union-arborescence. An equivalent version of this problem is where the roots of the arborescences are fixed in advance. In an earlier paper [A. Bernáth and Gy. Pap, \emph{Blocking optimal arborescences}, Integer Programming and Combinatorial Optimization, Springer, 2013] we solved this problem for $k=1$. This work reports on other partial results on the problem. We solve the case when both $c$ and $w$ are uniform -- that is, find a minimum size set of arcs that covers all $k$-union-arbosercences. Our algorithm runs in polynomial time for this problem. The solution uses a result of [M. Bárász, J. Becker, and A. Frank, \emph{An algorithm for source location in directed graphs}, Oper. Res. Lett. \textbf{33} (2005)] saying that the family of so-called insolid sets (sets with the property that every proper subset has a larger in-degree) satisfies the Helly-property, and thus can be (efficiently) represented as a subtree hypergraph. We also give an algorithm for the case when only $c$ is uniform but $w$ is not. This algorithm is only polynomial if $k$ is not part of the input.

math.CO↗
arXiv2015

Asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations

We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, while in the critical and supercritical cases, non-standard asymptotic behavior is described.

math.ST↗
arXiv2015

Blocking optimal arborescences

The problem of covering minimum cost common bases of two matroids is NP-complete, even if the two matroids coincide, and the costs are all equal to 1. In this paper we show that the following special case is solvable in polynomial time: given a digraph $D=(V,A)$ with a designated root node $r\in V$ and arc-costs $c:A\to \mathbb{R}$, find a minimum cardinality subset $H$ of the arc set $A$ such that $H$ intersects every minimum $c$-cost $r$-arborescence. By an $r$-arborescence we mean a spanning arborescence of root $r$. The algorithm we give solves a weighted version as well, in which a nonnegative weight function $w:A\to \mathbb{R}_+$ (unrelated to $c$) is also given, and we want to find a subset $H$ of the arc set such that $H$ intersects every minimum $c$-cost $r$-arborescence, and $w(H)=\sum_{a\in H}w(a)$ is minimum. The running time of the algorithm is $O(n^3T(n,m))$, where $n$ and $m$ denote the number of nodes and arcs of the input digraph, and $T(n,m)$ is the time needed for a minimum $s-t$ cut computation in this digraph. A polyhedral description is not given, and seems rather challenging.

math.CO↗
arXiv2015

Asymptotic behavior of critical irreducible multi-type continuous state and continuous time branching processes with immigration

Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.

math.PR↗