Contrast function estimation for the drift parameter of ergodic jump diffusion process

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $\theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $\Delta$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an estimator of $\theta$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $\Delta$n $\rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$\Delta$ 3 n $\rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $\theta$ is feasible under the condition that n$\Delta$ k n $\rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{\'e}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods

Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

The problem of integrated volatility estimation for the solution X of a stochastic differential equation with L{\'e}vy-type jumps is considered under discrete high-frequency observations in both short and long time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results the condition $\beta$ > 1 2(2--$\alpha$) on $\beta$ (that is such that (1/n) $\beta$ is the threshold of the truncated quadratic variation) and on the degree of jump activity $\alpha$ was needed to have the original truncated realized volatility well-performed (see [22], [13]). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple ($\alpha$, $\beta$). L{\'e}vy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data

Parcela - uvod u planersko pismo

The purpose of this paper is to bring to life universal spatial planning rules whose marks are symbols reminiscent of letters for writing words, and are in practice pictures of drawings. The goal of the research is to determine general terms for the development of a hierarchy of use with four levels of land plots. The research methodology includes an analysis of the way spatial planning takes place at present in theory and in practice as well as experience in preparing and implementing spatial planning plans. The general terms in order from superior to subordinate units of use are: Planning area, Planning zone, Planning block and Planning parcel.Svrha ovog rada je oživotvoriti univerzalna pravila planiranja prostora čije su oznake simboli koji podsjećaju na slova za pisanje teksta, a praktično su slika crteža. Cilj istraživanja je utvrditi opće termine za razradu hijerarhije namjene na četiri razine parcela. Metodologija istraživanja obuhvaća analizu dosadašnjeg načina planiranja prostora u teoriji i praksi, te iskustvo u izradi i provedbi planova prostornog uređenja. Opći termini od nadređene do podređene jedinice namjene su: Planersko područje, Planerska zona, Planerski blok i Planerska čestica

Parcela - uvod u planersko pismo

The purpose of this paper is to bring to life universal spatial planning rules whose marks are symbols reminiscent of letters for writing words, and are in practice pictures of drawings. The goal of the research is to determine general terms for the development of a hierarchy of use with four levels of land plots. The research methodology includes an analysis of the way spatial planning takes place at present in theory and in practice as well as experience in preparing and implementing spatial planning plans. The general terms in order from superior to subordinate units of use are: Planning area, Planning zone, Planning block and Planning parcel.Svrha ovog rada je oživotvoriti univerzalna pravila planiranja prostora čije su oznake simboli koji podsjećaju na slova za pisanje teksta, a praktično su slika crteža. Cilj istraživanja je utvrditi opće termine za razradu hijerarhije namjene na četiri razine parcela. Metodologija istraživanja obuhvaća analizu dosadašnjeg načina planiranja prostora u teoriji i praksi, te iskustvo u izradi i provedbi planova prostornog uređenja. Opći termini od nadređene do podređene jedinice namjene su: Planersko područje, Planerska zona, Planerski blok i Planerska čestica

Wheat Productivity and Plough Land Inequality in Rural Croatia

The unequal distribution of plough land could be according to a prior naive theorizing be a source of inefficiency in wheat production. The paper investigates whether, plough land inequality due to specific less or more egalitarian land distribution, and is a source of possible inefficiency measured by wheat productivity within Croatia's counties. We analyze these issues by using cross-county data on inequality in operational holdings of plough land from Agricultural Survey in 2003. After constructing the Gini coefficient for plough land holdings, and other relevant exogenous variable which cover necessary inputs condition as a average holding size per ha, labor, capital (represented by alternative variables summed by number of combine harvester and tractor), among counties, an estimation of an production function, is done by OLS estimations of wheat output.Wheat Productivity, Production Function, Plough Land Inequality, Croatia

Optimization model for family house plot elements – the Istria case

Na današnjem stupnju prostornog razvoja, postavljeni su kriteriji optimalizacije elemenata parcele koji uključuju primjenu ambijentalnih značajki, strukturu parcele, formule prosjeka i jedinstvo parametara. Optimalno je najbolje iskorištena površina na parceli obiteljske kuće za život njenih ukućana. Istraživanja ukazuju da brojčani iskazi količina i površina pojedinih strukturnih elemenata, potrebnih za kompletiranje parcele obiteljske kuće, nisu jednaki. Prema kriterijima optimalizacije, utvrđene su srednje površine svih strukturnih elemenata unutar limitirane, minimalne i maksimalne površine parcele.At today’s level of spatial planning the criteria for the plot elements optimization are set including the ambient features’ application, the plot structure, formulas average and unity parameters. The optimal model refers to the best utilization of a family house plot from the aspect of the house inhabitants. Research indicates that numerical expressions of quantities and the surfaces of particular structural elements necessary to complete a family house plot are unequal. According to the optimization criteria, average surfaces for structural elements are determined within limited, minimal and maximal plot surfaces

Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

The problem of integrated volatility estimation for the solution X of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations in both short and long time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results the condition β > 1 2(2−α) on β (that is such that (1/n) β is the threshold of the truncated quadratic variation) and on the degree of jump activity α was needed to have the original truncated realized volatility well-performed (see [22], [13]). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple (α, β). Lévy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data

Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes

We study the problem of the nonparametric estimation for the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$ with possibly unbounded drift. From the continuous observation of the sampling path on $[0, T]$, we study the rate of estimation of $\pi(x)$ as $T$ goes to infinity. One finding is that, for $d \ge 3$, the rate of estimation depends on the smoothness $\beta = (\beta_1, ... , \beta_d)$ of $\pi$. In particular, having ordered the smoothness such that $\beta_1 \le ... \le \beta_d$, it depends on the fact that $\beta_2 < \beta_3$ or $\beta_2 = \beta_3$. We show that kernel density estimators achieve the rate $(\frac{\log T}{T})^\gamma$ in the first case and $(\frac{1}{T})^\gamma$ in the second, for an explicit exponent $\gamma$ depending on the dimension and on $\bar{\beta}_3$, the harmonic mean of the smoothness over the $d$ directions after having removed $\beta_1$ and $\beta_2$, the smallest ones. Moreover, we obtain a minimax lower bound on the $\mathbf{L}^2$-risk for the pointwise estimation with the same rates $(\frac{\log T}{T})^\gamma$ or $(\frac{1}{T})^\gamma$, depending on the value of $\beta_2$ and $\beta_3$

Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime

Let $(X_t)_{t \ge 0}$ be solution of a one-dimensional stochastic differential equation. Our aim is to study the convergence rate for the estimation of the invariant density in intermediate regime, assuming that a discrete observation of the process $(X_t)_{t \in [0, T]}$ is available, when $T$ tends to $\infty$. We find the convergence rates associated to the kernel density estimator we proposed and a condition on the discretization step $\Delta_n$ which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is $n^{- \frac{2 \beta}{2 \beta + 1}}$, where $\beta$ is the smoothness of the invariant density. After that, we complement the upper bounds previously found with a lower bound over the set of all the possible estimator, which provides the same convergence rate: it means it is not possible to propose a different estimator which achieves better convergence rates. This is obtained by the two hypothesis method; the most challenging part consists in bounding the Hellinger distance between the laws of the two models. The key point is a Malliavin representation for a score function, which allows us to bound the Hellinger distance through a quantity depending on the Malliavin weight