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

Joint Distributions of the Numbers of Visits for Finite-State Markov Chains

AbstractFor a discrete-time Markov chain with finite state space {1, …, r} we consider the joint distribution of the numbers of visits in states 1, …, r−1 during the firstNsteps or before theNth visit tor. From the explicit expressions for the corresponding generating functions we obtain the limiting multivariate distributions asN→∞ when staterbecomes asymptotically absorbing and forj=1, …, r−1 the probability of a transition fromrtojis of order 1/N