110 research outputs found
Two-sided reflected Markov-modulated Brownian motion with applications to fluid queues and dividend payouts
In this paper we study a reflected Markov-modulated Brownian motion with a
two sided reflection in which the drift, diffusion coefficient and the two
boundaries are (jointly) modulated by a finite state space irreducible
continuous time Markov chain. The goal is to compute the stationary
distribution of this Markov process, which in addition to the complication of
having a stochastic boundary can also include jumps at state change epochs of
the underlying Markov chain because of the boundary changes. We give the
general theory and then specialize to the case where the underlying Markov
chain has two states. Moreover, motivated by an application of optimal dividend
strategies, we consider the case where the lower barrier is zero and the upper
barrier is subject to control. In this case we generalized earlier results from
the case of a reflected Brownian motion to the Markov modulated case.Comment: 22 pages, 1 figur
A new formula for some linear stochastic equations with applications
We give a representation of the solution for a stochastic linear equation of
the form where is a
c\'adl\'ag semimartingale and is a c\'adl\'ag adapted process with bounded
variation on finite intervals. As an application we study the case where
and are nondecreasing, jointly have stationary increments and the jumps of
are bounded by 1. Special cases of this process are shot-noise processes,
growth collapse (additive increase, multiplicative decrease) processes and
clearing processes. When and are, in addition, independent L\'evy
processes, the resulting is called a generalized Ornstein-Uhlenbeck
process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP637 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Useful martingales for stochastic storage processes with L\'{e}vy-type input
In this paper we generalize the martingale of Kella and Whitt to the setting
of L\'{e}vy-type processes and show that the (local) martingales obtained are
in fact square integrable martingales which upon dividing by the time index
converge to zero a.s. and in . The reflected L\'{e}vy-type process is
considered as an example.Comment: 15 pages. arXiv admin note: substantial text overlap with
arXiv:1112.475
On the area between a L\'evy process with secondary jump inputs and its reflected version
We study the the stochastic properties of the area under some function of the
difference between (i) a spectrally positive L\'evy process that jumps
to a level whenever it hits zero, and (ii) its reflected version .
Remarkably, even though the analysis of each of these processes is challenging,
we succeed in attaining explicit expressions for their difference. The main
result concerns the Laplace-Stieltjes transform of the integral of (a
function of) the distance between and until hits zero.
This result is extended in a number of directions, including the area between
and and a Gaussian limit theorem. We conclude the paper with an
inventory problem for which our results are particularly useful
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