53 research outputs found
Extremal behavior of stochastic integrals driven by regularly varying L\'{e}vy processes
We study the extremal behavior of a stochastic integral driven by a
multivariate L\'{e}vy process that is regularly varying with index .
For predictable integrands with a finite -moment, for some
, we show that the extremal behavior of the stochastic integral is
due to one big jump of the driving L\'{e}vy process and we determine its limit
measure associated with regular variation on the space of c\`{a}dl\`{a}g
functions.Comment: Published at http://dx.doi.org/10.1214/009117906000000548 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regularly Varying Measures on Metric Spaces: Hidden Regular Variation and Hidden Jumps
We develop a framework for regularly varying measures on complete separable
metric spaces with a closed cone removed, extending
material in Hult & Lindskog (2006), Das, Mitra & Resnick (2013). Our framework
provides a flexible way to consider hidden regular variation and allows
simultaneous regular variation properties to exist at different scales and
provides potential for more accurate estimation of probabilities of risk
regions. We apply our framework to iid random variables in
with marginal distributions having regularly varying
tails and to c\`adl\`ag L\'evy processes whose L\'evy measures have regularly
varying tails. In both cases, an infinite number of regular variation
properties coexist distinguished by different scaling functions and state
spaces.Comment: 40 page
Functional large deviations for multivariate regularly varying random walks
We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser.
Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64,
193--208] on large deviations for sums of i.i.d. regularly varying random
variables to partial sum processes of i.i.d. regularly varying vectors. The
results are stated in terms of a heavy-tailed large deviation principle on the
space of c\`{a}dl\`{a}g functions. We illustrate how these results can be
applied to functionals of the partial sum process, including ruin probabilities
for multivariate random walks and long strange segments. These results make
precise the idea of heavy-tailed large deviation heuristics: in an asymptotic
sense, only the largest step contributes to the extremal behavior of a
multivariate random walk.Comment: Published at http://dx.doi.org/10.1214/105051605000000502 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Mack's estimator motivated by large exposure asymptotics in a compound Poisson setting
The distribution-free chain ladder of Mack justified the use of the chain
ladder predictor and enabled Mack to derive an estimator of conditional mean
squared error of prediction for the chain ladder predictor. Classical insurance
loss models, i.e. of compound Poisson type, are not consistent with Mack's
distribution-free chain ladder. However, for a sequence of compound Poisson
loss models indexed by exposure (e.g. number of contracts), we show that the
chain ladder predictor and Mack's estimator of conditional mean squared error
of prediction can be derived by considering large exposure asymptotics. Hence,
quantifying chain ladder prediction uncertainty can be done with Mack's
estimator without relying on the validity of the model assumptions of the
distribution-free chain ladder
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