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Register a new userUnified approach for solving exit problems for additive-increase and multiplicative-decrease processes
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We analyse an additive-increase and multiplicative-decrease (aka growth-collapse) process that grows linearly in time and that experiences downward jumps at Poisson epochs that are (deterministically) proportional to its present position. This process is used for example in modelling of Transmission Control Protocol (TCP) and can be viewed as a particular example of the so-called shot noise model, a basic tool in modeling earthquakes, avalanches and neuron firings. For this process, and also for its reflect
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In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no-information, rank-dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
In this paper we develop the theory of the so-called $mathbf{W}$ and $mathbf{Z}$ scale matrices for (upwards skip-free) discrete-time and discrete-space Markov additive processes, along the lines of the analogous theory for Markov additive processes in continuous-time. In particular, we provide their probabilistic construction, identify the form of the generating function of $mathbf{W}$ and its connection with the occupation mass formula, which provides the tools for deriving semi-explicit expressions for corresponding exit problems for the upward-skip free process and its reflections, in terms the scale matrices.
We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some combinatorial coalescent processes: a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $nto +infty$. We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous [Ann. Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time $lambda$, converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $(B_t+lambda t - t^2/2, tgeq 0)$, but we also construct a version of the standard augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel., to appear] using an additional Poisson point process.
In this article we determine the Laplace transforms of the main boundary functionals of the oscillating compound Poisson process. These are the first passage time of the level, the joint distribution of the first exit time from the interval and the value of the overshoot through the boundary. Under certain conditions we establish the asymptotic behaviour of the mentioned functionals.
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