Advances in Applied Probability

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ISSN / EISSN : 0001-8678 / 1475-6064
Published by: Cambridge University Press (CUP) (10.1017)
Total articles ≅ 7,191
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, , Alexey Piunovskiy,
Advances in Applied Probability, Volume 53, pp 301-334; https://doi.org/10.1017/apr.2020.64

Abstract:
We consider a gradual-impulse control problem of continuous-time Markov decision processes, where the system performance is measured by the expectation of the exponential utility of the total cost. We show, under natural conditions on the system primitives, the existence of a deterministic stationary optimal policy out of a more general class of policies that allow multiple simultaneous impulses, randomized selection of impulses with random effects, and accumulation of jumps. After characterizing the value function using the optimality equation, we reduce the gradual-impulse control problem to an equivalent simple discrete-time Markov decision process, whose action space is the union of the sets of gradual and impulsive actions.
Mathieu Rosenbaum,
Published: 1 June 2021
by 10.1017
Advances in Applied Probability, Volume 53, pp 425-462; https://doi.org/10.1017/apr.2020.60

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, Ronald W. Wolff
Published: 1 June 2021
by 10.1017
Advances in Applied Probability, Volume 53, pp 463-483; https://doi.org/10.1017/apr.2020.63

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Advances in Applied Probability, Volume 53, pp 484-509; https://doi.org/10.1017/apr.2020.66

Abstract:
The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob. 22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.
Konstantinos Karatapanis
Advances in Applied Probability, Volume 53, pp 575-607; https://doi.org/10.1017/apr.2020.10

Abstract:
We consider stochastic differential equations of the form $dX_t = |f(X_t)|/t^{\gamma} dt+1/t^{\gamma} dB_t$ , where f(x) behaves comparably to $|x|^k$ in a neighborhood of the origin, for $k\in [1,\infty)$ . We show that there exists a threshold value $ \,{:}\,{\raise-1.5pt{=}}\, \tilde{\gamma}$ for $\gamma$ , depending on k, such that if $\gamma \in (1/2, \tilde{\gamma})$ , then $\mathbb{P}(X_t\rightarrow 0) = 0$ , and for the rest of the permissible values of $\gamma$ , $\mathbb{P}(X_t\rightarrow 0)>0$ . These results extend to discrete processes that satisfy $X_{n+1}-X_n = f(X_n)/n^\gamma +Y_n/n^\gamma$ . Here, $Y_{n+1}$ are martingale differences that are almost surely bounded. This result shows that for a function F whose second derivative at degenerate saddle points is of polynomial order, it is always possible to escape saddle points via the iteration $X_{n+1}-X_n =F'(X_n)/n^\gamma +Y_n/n^\gamma$ for a suitable choice of $\gamma$ .
Quentin Le Gall, Bartłomiej Błaszczyszyn, Élie Cali, Taoufik En-Najjary
Advances in Applied Probability, Volume 53, pp 510-536; https://doi.org/10.1017/apr.2020.69

Abstract:
In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.
, Jean-François Delmas
Advances in Applied Probability, Volume 53, pp 537-574; https://doi.org/10.1017/apr.2020.70

Abstract:
We consider a model of a stationary population with random size given by a continuous-state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure for the genealogical tree of n individuals randomly chosen among the extant population at a given time. Then we prove the convergence of the renormalized total length of this genealogical tree as n goes to infinity; see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant-size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time, which was defined by Aldous and Popovic (2005) in the critical case.
Advances in Applied Probability, Volume 53, pp 370-399; https://doi.org/10.1017/apr.2020.61

Abstract:
We derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an $\alpha$ -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters $\alpha\in (0,1)$ from the subordinator and $r>0$ from the negative binomial process. The large-sample distribution of the number of species is derived as sample size $n\to\infty$ . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.
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