17 research outputs found
The tail of the stationary distribution of a random coefficient AR(q) model
We investigate a stationary random coefficient autoregressive process.
Using renewal type arguments tailor-made for such processes, we show that the
stationary distribution has a power-law tail. When the model is normal, we show
that the model is in distribution equivalent to an autoregressive process with
ARCH errors. Hence, we obtain the tail behavior of any such model of arbitrary
order
Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions
We investigate optimal consumption problems for a Black-Scholes market under
uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic
utility functions. We find the solutions in terms of a dynamic strategy in
explicit form, which can be compared and interpreted. This paper continues our
previous work, where we solved similar problems for power utility functions
Risk in a large claims insurance market with bipartite graph structure
We model the influence of sharing large exogeneous losses to the reinsurance
market by a bipartite graph. Using Pareto-tailed claims and multivariate
regular variation we obtain asymptotic results for the Value-at-Risk and the
Conditional Tail Expectation. We show that the dependence on the network
structure plays a fundamental role in their asymptotic behaviour. As is
well-known in a non-network setting, if the Pareto exponent is larger than 1,
then for the individual agent (reinsurance company) diversification is
beneficial, whereas when it is less than 1, concentration on a few objects is
the better strategy. An additional aspect of this paper is the amount of
uninsured losses which have to be convered by society. In the situation of
networks of agents, in our setting diversification is never detrimental
concerning the amount of uninsured losses. If the Pareto-tailed claims have
finite mean, diversification turns out to be never detrimental, both for
society and for individual agents. In contrast, if the Pareto-tailed claims
have infinite mean, a conflicting situation may arise between the incentives of
individual agents and the interest of some regulator to keep risk for society
small. We explain the influence of the network structure on diversification
effects in different network scenarios
Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes
We formulate the insurance risk process in a general Levy process setting,
and give general theorems for the ruin probability and the asymptotic
distribution of the overshoot of the process above a high level, when the
process drifts to -\infty a.s. and the positive tail of the Levy measure, or of
the ladder height measure, is subexponential or, more generally, convolution
equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64
(1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226]
for ruin probabilities and the overshoot in random walk and compound Poisson
models are shown to have analogues in the general setup. The identities we
derive open the way to further investigation of general renewal-type properties
of Levy processes.Comment: Published at http://dx.doi.org/10.1214/105051604000000927 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
PARAMETER-ESTIMATION FOR ARMA MODELS WITH INFINITE VARIANCE INNOVATIONS
We consider a standard ARMA process of the form phi(B)X(t) = B(B)Z(t), where the innovations Z(t) belong to the domain of attraction of a stable law, so that neither the Z(t) nor the X(t) have a finite variance. Our aim is to estimate the coefficients of phi and theta. Since maximum likelihood estimation is not a viable possibility (due to the unknown form of the marginal density of the innovation sequence), we adopt the so-called Whittle estimator, based on the sample periodogram of the X sequence. Despite the fact that the periodogram does not, a priori, seem like a logical object to study in this non-L(2) situation, we show that our estimators are consistent, obtain their asymptotic distributions and show that they converge to the true values faster than in the usual L(2) case
Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions
We investigate optimal consumption problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the solutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous work, where we solved similar problems for power utility functions
Optimal consumption and investment with bounded downside risk for power utility functions
We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.