24 research outputs found
On the optimal dividend problem for a spectrally negative L\'{e}vy process
In this paper we consider the optimal dividend problem for an insurance
company whose risk process evolves as a spectrally negative L\'{e}vy process in
the absence of dividend payments. The classical dividend problem for an
insurance company consists in finding a dividend payment policy that maximizes
the total expected discounted dividends. Related is the problem where we impose
the restriction that ruin be prevented: the beneficiaries of the dividends must
then keep the insurance company solvent by bail-out loans. Drawing on the
fluctuation theory of spectrally negative L\'{e}vy processes we give an
explicit analytical description of the optimal strategy in the set of barrier
strategies and the corresponding value function, for either of the problems.
Subsequently we investigate when the dividend policy that is optimal among all
admissible ones takes the form of a barrier strategy.Comment: Published at http://dx.doi.org/10.1214/105051606000000709 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results
Consider two insurance companies (or two branches of the same company) that
divide between them both claims and premia in some specified proportions. We
model the occurrence of claims according to a renewal process. One ruin problem
considered is that of the corresponding two-dimensional risk process first
leaving the positive quadrant; another is that of entering the negative
quadrant. When the claims arrive according to a Poisson process, we obtain a
closed form expression for the ultimate ruin probability. In the general case,
we analyze the asymptotics of the ruin probability when the initial reserves of
both companies tend to infinity under a Cram\'{e}r light-tail assumption on the
claim size distribution.Comment: Published in at http://dx.doi.org/10.1214/08-AAP529 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the drawdown of completely asymmetric Levy processes
The {\em drawdown} process of a completely asymmetric L\'{e}vy process
is equal to reflected at its running supremum : . In this paper we explicitly express in terms of the scale function and the
L\'{e}vy measure of the law of the sextuple of the first-passage time of
over the level , the time of the last supremum of
prior to , the infimum \unl X_{\tau_a} and supremum \ovl
X_{\tau_a} of at and the undershoot and
overshoot of at . As application we obtain explicit
expressions for the laws of a number of functionals of drawdowns and rallies in
a completely asymmetric exponential L\'{e}vy model.Comment: applications added, 26 pages, 3 figures, to appear in SP
On the drawdown of completely asymmetric Levy processes
The {\em drawdown} process of a completely asymmetric L\'{e}vy process is equal to reflected at its running supremum : . In this paper we explicitly express in terms of the scale function and the L\'{e}vy measure of the law of the sextuple of the first-passage time of over the level , the time of the last supremum of prior to , the infimum \unl X_{\tau_a} and supremum \ovl X_{\tau_a} of at and the undershoot and overshoot of at . As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential L\'{e}vy model.
Russian and American options under exponential phase-type Lévy models
Consider the American put and Russian option (Ann. Appl. Probab. 3(1993)603; Theory Probab. Appl. 39 (1994)103; Ann. Appl. Probab. 3(1993)641) with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phase-type jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phase-type case via martingale stopping and Wiener-Hopf factorization. The same type of approach is also applied to the more general class of regime switching Lévy processes with phase-type jumps