arXiv subjects

Robert Stelzer

Publications and source records attributed to Robert Stelzer.

28 records · Page 2Linked to original sources

arXiv2012

Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

The class of multivariate Lévy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular ($β$-mixing) under a mild continuity assumption on the driving Lévy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving Lévy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite Lévy measure admitting a density around zero.

math.ST↗

arXiv2012

A Class of Infinitely Divisible Multivariate and Matrix Gamma Distributions and Cone-valued Generalised Gamma Convolutions

Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an Itô-Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a Lévy process is established. A new example of an infinitely divisible positive definite Gamma random matrix is introduced. It has properties which make it appealing for modelling under an infinite divisibility framework. An interesting relation of the moments of the Lévy measure and the Wishart distribution is highlighted which we suppose to be important when considering the limiting distribution of the eigenvalues .

math.PR↗

arXiv2011

CARMA Processes driven by Non-Gaussian Noise

We present an outline of the theory of certain Lévy-driven, multivariate stochastic processes, where the processes are represented by rational transfer functions (Continuous-time AutoRegressive Moving Average or CARMA models) and their applications in non-Gaussian time series modelling. We discuss in detail their definition, their spectral representation, the equivalence to linear state space models and further properties like the second order structure and the tail behaviour under a heavy-tailed input. Furthermore, we study the estimation of the parameters using quasi-maximum likelihood estimates for the auto-regressive and moving average parameters, as well as how to estimate the driving Lévy process.

math.PR↗

arXiv2011

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.

q-fin.PR↗

arXiv2011

Stationarity and Geometric Ergodicity of BEKK Multivariate GARCH Models

Conditions for the existence of strictly stationary multivariate GARCH processes in the so-called BEKK parametrisation, which is the most general form of multivariate GARCH processes typically used in applications, and for their geometric ergodicity are obtained. The conditions are that the driving noise is absolutely continuous with respect to the Lebesgue measure and zero is in the interior of its support and that a certain matrix built from the GARCH coefficients has spectral radius smaller than one. To establish the results semi-polynomial Markov chains are defined and analysed using algebraic geometry.

math.PR↗

arXiv2011

Spectral Representation of Multivariate Regularly Varying Lévy and CARMA processes

A spectral representation for regularly varying Lévy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail giving also new insight in the $L^2$-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying Lévy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.

math.PR↗

arXiv2011

On strong solutions for positive definite jump-diffusions

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics. Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the alpha-th positive semidefinite power of the process itself with 0.5<alpha<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.

math.PR↗

arXiv2010

Multivariate supOU processes

Univariate superpositions of Ornstein--Uhlenbeck-type processes (OU), called supOU processes, provide a class of continuous time processes capable of exhibiting long memory behavior. This paper introduces multivariate supOU processes and gives conditions for their existence and finiteness of moments. Moreover, the second-order moment structure is explicitly calculated, and examples exhibit the possibility of long-range dependence. Our supOU processes are defined via homogeneous and factorizable Lévy bases. We show that the behavior of supOU processes is particularly nice when the mean reversion parameter is restricted to normal matrices and especially to strictly negative definite ones. For finite variation Lévy bases we are able to give conditions for supOU processes to have locally bounded càdlàg paths of finite variation and to show an analogue of the stochastic differential equation of OU-type processes, which has been suggested in \cite barndorffnielsen01 in the univariate case. Finally, as an important special case, we introduce positive semi-definite supOU processes, and we discuss the relevance of multivariate supOU processes in applications.

math.PR↗

arXiv2010

Multivariate COGARCH(1,1) processes

Multivariate $\operatorname {COGARCH}(1,1)$ processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the $\operatorname {COGARCH}(1,1)$ process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving Lévy process, explicit expressions for the first and second-order moments and (asymptotic) second-order stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate $\operatorname {COGARCH}(1,1)$ process and their "squares".

math.ST↗

arXiv2009

On the definition, stationary distribution and second order structure of positive semidefinite Ornstein--Uhlenbeck type processes

Several important properties of positive semidefinite processes of Ornstein--Uhlenbeck type are analysed. It is shown that linear operators of the form $X\mapsto AX+XA^{\mathrm{T}}$ with $A\in M_d(\mathbb{R})$ are the only ones that can be used in the definition provided one demands a natural non-degeneracy condition. Furthermore, we analyse the absolute continuity properties of the stationary distribution (especially when the driving matrix subordinator is the quadratic variation of a $d$-dimensional Lévy process) and study the question of how to choose the driving matrix subordinator in order to obtain a given stationary distribution. Finally, we present results on the first and second order moment structure of matrix subordinators, which is closely related to the moment structure of positive semidefinite Ornstein--Uhlenbeck type processes. The latter results are important for method of moments based estimation.

math.ST↗