| 1. |
- Andersson, Patrik
(författare)
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CARD COUNTING IN CONTINUOUS TIME
- 2012
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 49:1, s. 184-198
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. The sampling without replacement makes it possible to take advantage of the changes in the expected value as the deck is depleted, making large bets when the game is advantageous. Using such a strategy, which is easy to implement, is known as card counting. We consider the case of a large number of decks, making an approximation to continuous time possible. A limit law of the return process is found and the optimal card counting strategy is derived. This continuous-time strategy is shown to be a natural analog of the discrete-time strategy where the so-called effects of removal are replaced by the infinitesimal generator of the card process.
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| 2. |
- Björnberg, Jakob E., et al.
(författare)
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A Stochastic Model for Virus Growth in a Cell Population
- 2014
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 51:3, s. 599-612
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Tidskriftsartikel (refereegranskat)abstract
- In this work we introduce a stochastic model for the spread of a virus in a cell population where the virus has two ways of spreading: either by allowing its host cell to live and duplicate, or by multiplying in large numbers within the host cell, causing the host cell to burst and thereby let the virus enter new uninfected cells. The model is a kind of interacting Markov branching process. We focus in particular on the probability that the virus population survives and how this depends on a certain parameter A which quantifies the 'aggressiveness' of the virus. Our main goal is to determine the optimal balance between aggressive growth and long-term success. Our analysis shows that the optimal strategy of the virus (in terms of survival) is obtained when the virus has no effect on the host cell's life cycle, corresponding to lambda = 0. This is in agreement with experimental data about real viruses.
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| 3. |
- Björnberg, Jakob E., et al.
(författare)
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Coexistence and noncoexistence of Markovian viruses and their hosts
- 2014
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Ingår i: Journal of Applied Probability. - 0021-9002 .- 1475-6072. ; 51:1, s. 191-208
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Tidskriftsartikel (refereegranskat)abstract
- Examining possibilities for the coexistence of two competing populations is a classic problem which dates back to the earliest 'predator-prey' models. In this paper we study this problem in the context of a model introduced in Bjornberg et al. (2012) for the spread of a virus infection in a population of healthy cells. The infected cells may be seen as a population of 'predators' and the healthy cells as a population of 'prey'. We show that, depending on the parameters defining the model, there may or may not be coexistence of the two populations, and we give precise criteria for this.
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| 4. |
- Britton, Tom, et al.
(författare)
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A dynamic network in a dynamic population: asymptotic properties
- 2011
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 1475-6072 .- 0021-9002. ; 48:4, s. 1163-1178
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Tidskriftsartikel (refereegranskat)abstract
- We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model, we derive a criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming that the node population grows to infinity. We also obtain an explicit expression for the degree correlation rho (of neighbouring nodes) which shows that rho is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.
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| 5. |
- Britton, Tom, et al.
(författare)
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Maximizing the size of the giant
- 2012
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 49:4, s. 1156-1165
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Tidskriftsartikel (refereegranskat)abstract
- Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring 'on' the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X-i with E(X-i) = mu, and where there is an edge between i and j with probability 1 - e(-XiXj/(mu n)), independently of other edges. The second model is the thinned configuration model in which then vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = mu. The graph of interest is obtained by deleting edges independently with probability 1 - p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1 - q, where q depends on X or D and p. We investigate for which distributions X and D with given mu and p, 1 - q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.
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| 6. |
- Christensen, Sören, 1982
(författare)
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Phase-type distributions and optimal stopping for autoregressive processes
- 2012
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 49:1, s. 22-39
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Tidskriftsartikel (refereegranskat)abstract
- Autoregressive processes are intensively studied in statistics and other fields of applied stochastics. For many applications, the overshoot and the threshold time are of special interest. When the upward innovations are in the class of phase-type distributions, we determine the joint distribution of these two quantities and apply this result to problems of optimal stopping. Using a principle of continuous fit, this leads to explicit solutions. © 2012 Applied Probability Trust.
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| 7. |
- Deijfen, Maria, 1975-
(författare)
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Random networks with preferential growth and vertex death
- 2010
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 47:4, s. 1150-1163
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Tidskriftsartikel (refereegranskat)abstract
- A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function b of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function d of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices of b and d. When b(i) = i + α and d(i) = β, that is, linear preferential attachment for the newborn and random deaths, then pk ∼ k-(2+α). When b(i) = i + 1 and d(i) = β(i + 1), with β < 1, then pk ∼ (1 + β)-k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, when b(i) = i + 1 and d(i) = β(i + 1)γ, with β, γ < 1, then logpk ∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.
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| 8. |
- Ekström, Erik, et al.
(författare)
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Optimal closing of a momentum trade
- 2013
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 50:2, s. 374-387
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Tidskriftsartikel (refereegranskat)abstract
- There is an extensive academic literature that documents that stocks which have performed well in the past often continue to perform well over some holding period-so-called momentum. We study the optimal timing for an asset sale for an agent with a long position in a momentum trade. The asset price is modelled as a geometric Brownian motion with a drift that initially exceeds the discount rate, but with the opposite relation after an unobservable and exponentially distributed time. The problem of optimal selling of the asset is then formulated as an optimal stopping problem under incomplete information. Based on the observations of the asset, the agent wants to detect the unobservable change point as accurately as possible. Using filtering techniques and stochastic analysis, we reduce the problem to a one-dimensional optimal stopping problem, which we solve explicitly. We also show that the optimal boundary at which the investor should liquidate the trade depends monotonically on the model parameters.
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| 9. |
- Ekström, Erik, et al.
(författare)
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Optimal liquidation of a call spread
- 2010
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Ingår i: Journal of Applied Probability. - 0021-9002 .- 1475-6072. ; 47:2, s. 586-593
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Tidskriftsartikel (refereegranskat)abstract
- We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.
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| 10. |
- Grandell, Jan, et al.
(författare)
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Ruin probabilities in a diffusion environment
- 2011
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 48A, s. 39-50
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Tidskriftsartikel (refereegranskat)abstract
- We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramer-Lundberg approximations are derived.
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| 11. |
- Gudmundsson, Thorbjörn, et al.
(författare)
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Markov chain monte carlo for computing rare-event probabilities for a heavy-tailed random walk
- 2014
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Ingår i: Journal of Applied Probability. - : Applied Probability Trust. - 0021-9002 .- 1475-6072. ; 51:2, s. 359-376
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Tidskriftsartikel (refereegranskat)abstract
- In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.
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| 12. |
- Gut, Allan
(författare)
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Limit theorems for a generalized St Petersburg game
- 2010
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 47:3, s. 752-760
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Tidskriftsartikel (refereegranskat)abstract
- The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr((k-1)/alpha)) = pq(k-1), k = 1,2, ..., where p + q = 1, s = l/p, r = 1/q, and 0 < alpha <= 1. For the case in which alpha = 1, we extend Feller's classical weak law and Martin-Lof's theorem on convergence in distribution along the 2"-subsequence. The analog for 0 < alpha < 1 turns out to converge in distribution to an asymmetric stable law with index a. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.
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| 13. |
- Holst, Lars
(författare)
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Probabilistic proofs of euler identities
- 2013
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 50:4, s. 1206-1212
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Tidskriftsartikel (refereegranskat)abstract
- Formulae for zeta(2n) and L-chi 4 (2n + 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where zeta(2) = pi(2)/6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.
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| 14. |
- Hult, Henrik, et al.
(författare)
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Large deviations for point processes based on stationary sequences with heavy tails
- 2010
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 47:1, s. 1-40
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track both of the magnitude of the extreme values of a process and the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provides a fairly complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of certain probabilities related to partial sum processes as well as ruin probabilities.
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| 15. |
- Janson, Svante, et al.
(författare)
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Large cliques in a power-law random graph
- 2010
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Ingår i: Journal of Applied Probability. - 0021-9002 .- 1475-6072. ; 47:4, s. 1124-1135
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the size of the largest clique omega(G(n, alpha)) in a random graph G(n, alpha) on n vertices which has power-law degree distribution with exponent alpha. We show that, for 'flat' degree sequences with alpha > 2, with high probability, the largest clique in G(n, alpha) is of a constant size, while, for the heavy tail distribution, when 0 < alpha < 2, omega(G (n, alpha)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, alpha) a large clique of size (1 - o(1))omega(G(n, alpha)) in polynomial time.
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| 16. |
- Janson, Svante, et al.
(författare)
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The Probability Of The Alabama Paradox
- 2012
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 49:3, s. 773-794
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Tidskriftsartikel (refereegranskat)abstract
- Hamilton's method is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In the USA it has been abandoned due to some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries. In this paper we give, under certain assumptions, a closed formula for the asymptotic probability, as the number of seats tends to infinity, that the Alabama paradox occurs given the vector p(l), ..., p(m) of relative sizes of the states. From the formula we deduce a number of consequences. For example, the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e and on average approximately 0.123.
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| 17. |
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| 18. |
- Lopes, Fabio, 1984-
(författare)
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Invariant Bipartite Random Graphs on Rd
- 2014
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 51:3, s. 769-779
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Tidskriftsartikel (refereegranskat)abstract
- Suppose that red and blue points occur in R d according to two simple point processes with finite intensities λ R and λ B , respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λ R ν̅ = λ B μ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.
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| 19. |
- Lu, Bing
(författare)
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Recovering a piecewise constant volatility from perpetual put option prices
- 2010
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 47:3, s. 680-692
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.
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| 20. |
- Meiners, Matthias
(författare)
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An almost-sure renewal theorem for branching random walks on the line
- 2010
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 47:3, s. 811-825
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Tidskriftsartikel (refereegranskat)abstract
- In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.
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| 21. |
- Trapman, Pieter, et al.
(författare)
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Splitting trees stopped when the first clock rings and Vervaat's transformation
- 2013
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 50:1, s. 208-227
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Tidskriftsartikel (refereegranskat)abstract
- We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T)is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y'M) conditional on {M≠0}. HereM+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y'M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T}, the ages and residual lifetimes of the n alive individuals at timeT are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital
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| 22. |
- Jagers, Peter, 1941, et al.
(författare)
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Population-size-dependent, age-structured branching processes linger around their carrying capacity
- 2011
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Ingår i: Journal of Applied Probability. - 0021-9002. ; 48A, s. 249-260
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Tidskriftsartikel (refereegranskat)abstract
- Dependence of individual reproduction upon the size of the whole population is studied in a general branching process context. The particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large ones. The transition occurs when population size passes a critical threshold, known in ecology as the carrying capacity. We show that populations either die out directly, never coming close to the carrying capacity, or else they grow quickly towards the latter, subsequently lingering around it for a time that is expected to be exponentially long in terms of a carrying capacity tending to infinity.
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| 23. |
- Pihlsgård, Mats, et al.
(författare)
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On the Dynamics of Semimartingales with Two Reflecting Barriers
- 2013
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Ingår i: Journal of Applied Probability. - 1475-6072. ; 50:3, s. 671-685
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Tidskriftsartikel (refereegranskat)abstract
- We consider a semimartingale X which is reflected at an upper barrier T and a lower barrier S, where S and T are also semimartingales such that T is bounded away from S. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective bathers to X, S, and T. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions on X, S, and T, e.g. if we take X, S, and T to be independent martingales (which satisfy some extra technical conditions).
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| 24. |
- Zhang, Wei-Min, et al.
(författare)
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Synthesis of mesoporous silica by a surface charge reversal route
- 2010
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Ingår i: Journal of Colloid and Interface Science. - : Elsevier BV. - 0021-9797 .- 1095-7103. ; 349:2, s. 473-476
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Tidskriftsartikel (refereegranskat)abstract
- Pore size adjustable mesoporous silica was synthesized by adsorption of varying amount of sodium dodecyl benzenesulfonate at the surface of silica activated by zinc ion via a novel surface charge reversal route. The pore size and volume can be adjusted from 5.9 to 13.76 nm and 0.88 to 1.08 cm3·g-1 respectively with increasing the SDBS concentration from 0.77 to 3.08 mmol·L-1. Adsorption of Zn2+ as a function of pH and N2 adsorption/desorption isotherms demonstrated that the metal ions such as Zn2+ could be readily removed with dilute nitric acid without apparent collapse of the pore structure at the proper range of SDBS concentration.
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