arXiv subjects
Publications and source records attributed to Gyula Pap.
We study asymptotic behavior of conditional least squares estimators for critical continuous state and continuous time branching processes with immigration based on discrete time (low frequency) observations.
We propose a change detection method for the famous Cox--Ingersoll--Ross model. This model is widely used in financial mathematics and therefore detecting a change in its parameters is of crucial importance. We develop one- and two-sided testing procedures for both drift parameters of the process. The test process is based on estimators that are motivated by the discrete time least-squares estimators, and its asymptotic distribution under the no-change hypothesis is that of a Brownian bridge. We prove the asymptotic weak consistence of the test, and derive the asymptotic properties of the change-point estimator under the alternative hypothesis of change at one point in time.
Recursions for moments of multi-type continuous state and continuous time branching process with immigration are derived. It turns out that the $k$-th (mixed) moments and the $k$-th (mixed) central moments are polynomials of the initial value of the process, and their degree are at most $k$ and $\lfloor k/2 \rfloor$, respectively.
A multi-type continuous state and continuous time branching process with immigration satisfying some moment conditions is identified as a pathwise unique strong solution of certain stochastic differential equation with jumps.
We study local asymptotic properties of likelihood ratios of certain Heston models. We distinguish three cases: subcritical, critical and supercritical models. For the drift parameters, local asymptotic normality is proved in the subcritical case, only local asymptotic quadraticity is shown in the critical case, while in the supercritical case not even local asymptotic quadraticity holds. For certain submodels, local asymptotic normality is proved in the critical case, and local asymptotic mixed normality is shown in the supercritical case. As a consequence, asymptotically optimal (randomized) tests are constructed in cases of local asymptotic normality. Moreover, local asymptotic minimax bound, and hence, asymptotic efficiency in the convolution theorem sense are concluded for the maximum likelihood estimators in cases of local asymptotic mixed normality.
Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness imply strong existence for stochastic differential equations with jumps.
In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(α,β)$ and of the criticality parameter $\varrho:=α+β$ for a $2$-type critical doubly symmetric positively regular Galton-Watson branching process with immigration is described.
Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Iglói (2008) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (2005). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional Lévy processes are given and we point out that certain limit processes in aggregation models are dilatively stable.
For an affine two factor model, we study the asymptotic properties of the maximum likelihood and least squares estimators of some appearing parameters in the so-called subcritical (ergodic) case based on continuous time observations. We prove strong consistency and asymptotic normality of the estimators in question.
In this paper the asymptotic behavior of a critical multi-type branching process with immigration is described when the offspring mean matrix is irreducible, in other words, when the process is indecomposable. It is proved that sequences of appropriately scaled random step functions formed from periodic subsequences of a critical indecomposable multi-type branching process with immigration converge weakly towards a process supported by a ray determined by the Perron vector of the offspring mean matrix. The types can be partitioned into nonempty mutually disjoint subsets (according to communication of types) such that the coordinate processes belonging to the same subset are multiples of the same squared Bessel process, and the coordinate processes belonging to different subsets are independent.
We consider discrete time Heath-Jarrow-Morton type interest rate models, where the interest rate curves are driven by a geometric spatial autoregression field. Strong consistency and asymptotic normality of the maximum likelihood estimators of the parameters are proved for stable no-arbitrage models containing a general stochastic discounting factor, where explicit form of the ML estimators is not available given a non-i.i.d. sample. The results form the basis of further statistical problems in such models.
We study the existence of a unique stationary distribution and ergodicity for a 2-dimensional affine process. The first coordinate is supposed to be a so-called alpha-root process with α\in(1,2]. The existence of a unique stationary distribution for the affine process is proved in case of α\in(1,2]; further, in case of α=2, the ergodicity is also shown.
In this paper the asymptotic behavior of the conditional least squares estimators of the autoregressive parameters $(α,β)$, of the stability parameter $\varrho := α+ β$, and of the mean $μ$ of the innovation $\vare_k$, $k \in \NN$, for an unstable integer-valued autoregressive process $X_k = α\circ X_{k-1} + β\circ X_{k-2} + \vare_k$, $k \in \NN$, is described. The limit distributions and the scaling factors are different according to the following three cases: (i) decomposable, (ii) indecomposable but not positively regular, and (iii) positively regular models.
Multi-type inhomogeneous Galton--Watson process with immigration is investigated, where the offspring mean matrix slowly converges to a critical mean matrix. Under general conditions we obtain limit distribution for the process, where the coordinates of the limit vector are not necessarily independent.
Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of linear constraints, one for each flat of M. Fractional matchings were introduced by Vande Vate, who showed that the set of fractional matchings is a half-integer relaxation of the matroid matching polytope. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to find for any given weights on the lines in L, a maximum weight fractional matching.
First we provide a simple set of sufficient conditions for the weak convergence of scaled affine processes with state space $R_+ \times R^d$. We specialize our result to one-dimensional continuous state branching processes with immigration. As an application, we study the asymptotic behavior of least squares estimators of some parameters of a two-dimensional critical affine diffusion process.
Change in the coefficients or in the mean of the innovation distribution of an INAR(p) process is a sign of disturbance that is important to detect. The methods of this paper can test for change in any one of these quantities separately, or in any collection of them. They are available in forms that make one-sided tests possible, furthermore, they can be used to test for a temporary change. The tests are based on a CUSUM process using conditional least squares estimators of the parameters. Under alternative hypotheses consistency of the tests is proved and the large sample properties of the change-point estimator are also explored.
Under natural assumptions a Feller type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.