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For two independent Levy processes ξ and η and an exponentially distributed random variable τ with parameter q > 0 , independent of ξ and η , the killed exponential functional is given by V q , ξ , η ≔ ∫ 0 τ e − ξ s − d η s . Interpreting the case q = 0 as τ = ∞ , the random variable V q , ξ , η is a natural generalisation of the exponential functional ∫ 0 ∞ e − ξ s − d η s , the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional V q , ξ , η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫ 0 t e − ξ s − d η s for fixed t ≥ 0 , as well as for integrals of the form ∫ 0 ∞ f ( s ) d η s for deterministic functions f . Furthermore, applying the same techniques to the case q = 0 , new results on the absolute continuity of the improper integral ∫ 0 ∞ e − ξ s − d η s are derived.

For two independent L\'{e}vy processes $\xi$ and $\eta$ and an exponentially distributed random variable $\tau$ with parameter $q>0$ that is independent of $\xi$ and $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of $V_{q,\xi,\eta}$, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case $q=0$ as $\tau=\infty$ leads to the classical exponential functional $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$, allowing to extend many previous results to include killing.

This paper studies risk balancing features in an insurance market by evaluating ruin probabilities for single and multiple components of a multivariate compound Poisson risk process. The dependence of the components of the process is induced by a random bipartite network. In analogy with the non-network scenario, a network ruin parameter is introduced. This random parameter, which depends on the bipartite network, is crucial for the ruin probabilities. Under certain conditions on the network and for light-tailed claim size distributions we obtain Lundberg bounds and, for exponential claim size distributions, exact results for the ruin probabilities. For large sparse networks, the network ruin parameter is approximated by a function of independent Poisson variables. T

We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Levy case studied by Erickson and Maller we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by Alsmeyer and Buckmann.

Let $k>0$ be an integer and $Y$ a standard $\operatorname{Gamma}(k)$ distributed random variable. Let $X$ be an independent positive random variable with a density that is hyperbolically monotone (HM) of order $k$. Then $Y\cdot X$ and $Y/X$ both have distributions that are generalized gamma convolutions ($\mathrm{GGC}$s). This result extends a result of Roynette et al. from 2009 who treated the case $k=1$ but without use of the $\mathrm{HM}$-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.

We characterize the support of the law of the exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\) of two one-dimensional independent Levy processes ξ and η. Further, we study the range of the mapping Φ ξ for a fixed Levy process ξ, which maps the law of η1 to the law of the corresponding exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\). It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.

Moment conditions for multivariate generalized Ornstein–Uhlenbeck (MGOU) processes are derived and the first and second moments are given in terms of the driving Lévy processes. In the second part of the paper a class of multivariate, positive semidefinite processes of MGOU-type is developed and suggested for use as squared volatility process in multivariate financial modeling.

De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes ( V t ) t ≥ 0 which are basically characterized by the fact that for each h > 0 the equidistantly sampled process ( V n h ) n ∈ N 0 satisfies the random recurrence equation V n h = A ( n − 1 ) h , n h V ( n − 1 ) h + B ( n − 1 ) h , n h , n ∈ N , where ( A ( n − 1 ) h , n h , B ( n − 1 ) h , n h ) n ∈ N is an i.i.d. sequence with positive A 0 , h for each h > 0 . We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Levy process ( X , Y ) in R m × m × R m . The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

For a given bivariate Levy process ( U t , L t ) t ≥ 0 , necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation d V t = V t − d U t + d L t are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V 0 and ( U , L ) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.

We introduce a new kind of symbol in the framework of It\^o processes which are bounded on one side. The connection between this symbol and the infinitesimal generator is analyzed. Based on this concept, an integral criterion for invariant distributions of the underlying process is derived. Some applications are mentioned., Comment: 10 pages

An integral criterion for the existence of an invariant measure of an It\^{o} process is developed. This new criterion is based on the probabilistic symbol of the It\^{o} process. In contrast to the standard integral criterion for invariant measures of Markov processes based on the generator, no test functions and hence no information on the domain of the generator is needed., Comment: Published at http://dx.doi.org/10.3150/14-BEJ618 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Exponential functionals of Levy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping $$\Phi $$ , which maps two independent Levy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Levy) characteristics. Also, continuity of $$\Phi $$ is treated, and some results on its range are obtained.

Financial data are as a rule asymmetric, although most econometric models are symmetric. This applies also to continuous-time models for high-frequency and irregularly spaced data. We discuss some asymmetric versions of the continuous-time GARCH model, concentrating then on the GJR-COGARCH. We calculate higher order moments and extend the first jump approximation. These results are prerequisites for moment estimation and pseudo maximum likelihood estimation of the GJR-COGARCH parameters, respectively, which we derive in detail., Comment: 15 pages, 1 figure

We suggest three superpositions of COGARCH (supCOGARCH) volatility processes driven by L\'evy processes or L\'evy bases. We investigate second-order properties, jump behaviour, and prove that they exhibit Pareto-like tails. Corresponding price processes are defined and studied. We find that the supCOGARCH models allow for more flexible autocovariance structures than the COGARCH. Moreover, other than most financial volatility models, the supCOGARCH processes do not exhibit a deterministic relationship between price and volatility jumps. Furthermore, in one supCOGARCH model not all volatility jumps entail a price jump, while in another supCOGARCH model not all price jumps necessarily lead to volatility jumps., Comment: 41 pages, 6 figures

It is known that in many cases distributions of exponential integrals of Levy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolution. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration., Published in at http://dx.doi.org/10.3150/11-BEJ382 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)