An optimal stopping problem with finite horizon for sums of I.I.D. random variables

AbstractThe problem of selling a commodity optimally at one of n successive time instants leads to the optimal stopping problem for the finite sequence ((n−j)lSj)1⩽j⩽n, where Sj=U1 + … + Uj, U1, U2,… are i.i.d., E(U1) = 0 and E(U21) = 1. The optimal stopping time πn is seen to be of the form τn = inf{j|j = n or j < n, Sj⩾clj,n}, where c1j,1>…>cln−1,n = 0 satisfyn−12 cj,nl → αl(1 − t)11, if n → ∞, j/n →t ṫ[0,1]. αl > 0 is the solution of the equation d2l+2dx2l+2(Ф/φ)(α) = (α + α−1)d2l+2dx2l+2(Ф/φ)(α). For the value vln we have n−32vnl → vl. vl is explicitly computed. In the normal case we also obtain results on the speed of convergence of n−12cj,nl and n−32vnl

On the small-time behavior of subordinators

We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ363 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Perishable inventories with random input:a unifying survey with extensions

This paper is devoted to the theory of perishable inventory systems. In such systems items arrive and stay ‘on the shelf’ until they are either taken by a demand or become outdated. Our aim is to survey, extend and enrich the probabilistic analysis of a large class of such systems. A unifying principle is to consider the so-called virtual outdating process V , where V(t) is the time that would pass from t until the next outdating if no new demands arrived after t. The steady-state density of V is determined for a wide range of perishable inventory systems. Key performance measures like the rate of outdatings, the rate of unsatisfied demands and the distribution of the number of items on the shelf are subsequently expressed in that density. Some of the main ingredients of our analysis are level crossing theory and the observation that the V process can be interpreted as the workload process of a specific single server queueing system.</p