Adv Appl Probab Research Articles

Hedging goals

Goal-based investing is concerned with reaching a monetary investment goal by a given finite deadline, which differs from mean-variance optimization in modern portfolio theory. In this article, we expand the close connection between goal-based investing and option hedging that was originally discovered in Browne (Adv Appl Probab 31(2):551–577, 1999) by allowing for varying degrees of investor risk aversion using lower partial moments of different orders. Moreover, we show that maximizing the probability of reaching the goal (quantile hedging, cf. Föllmer and Leukert in Finance Stoch 3:251–273, 1999) and minimizing the expected shortfall (efficient hedging, cf. Föllmer and Leukert in Finance Stoch 4:117–146, 2000) yield, in fact, the same optimal investment policy. We furthermore present an innovative and model-free approach to goal-based investing using methods of reinforcement learning. To the best of our knowledge, we offer the first algorithmic approach to goal-based investing that can find optimal solutions in the presence of transaction costs.

A dual skew symmetry for transient reflected Brownian motion in an orthant

We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison (Adv Appl Probab 10:886–905, 1978), which characterizes the exponential stationary distributions in the recurrent case. The duality comes from that the partial differential equation satisfied by the absorption probability is dual to the one associated with the stationary distribution in the recurrent case.

Spectral analysis of multifractional LRD functional time series

Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak-consistent parametric estimator of the long-memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss. The results derived allow, in particular, to develop inference from the discrete sampling of the Gaussian solution to fractional and multifractional pseudodifferential models introduced in Anh et al. (Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434–1459, 2016) and Kelbert (Adv Appl Probab 37(1):1–25, 2005).

A Local Barycentric Version of the Bak\u2013Sneppen Model

We study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let Nge 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution zeta . We show that in case where zeta is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

Hipster random walks

We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (Pemantle’s Min-Plus Binary Tree, 2017. arXiv:1709.07849 [math.PR]), and to the critical random hierarchical lattice studied by Hambly and Jordan (Adv Appl Probab 36(3):824–838, 2004. https://doi.org/10.1239/aap/1093962236 ). We prove distributional convergence for the processes, after rescaling, by showing that their evolutions can be understood as a discrete analogues of certain convection–diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.

Convergence in the p-Contest

We study asymptotic properties of the following Markov system of N ge 3 points in [0, 1]. At each time step, the point farthest from the current centre of mass, multiplied by a constant p>0, is removed and replaced by an independent zeta -distributed point; the problem, inspired by variants of the Bak–Sneppen model of evolution and called a p-contest, was posed in Grinfeld et al. (J Stat Phys 146, 378–407, 2012). We obtain various criteria for the convergences of the system, both for p<1 and p>1. In particular, when p<1 and zeta sim U[0,1], we show that the limiting configuration converges to zero. When p>1, we show that the configuration must converge to either zero or one, and we present an example where both outcomes are possible. Finally, when p>1, N=3 and zeta satisfies certain mild conditions (e.g. zeta sim U[0,1]), we prove that the configuration converges to one a.s. Our paper substantially extends the results of Grinfeld et al. (Adv Appl Probab 47:57–82, 2015) and Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018) where it was assumed that p=1. Unlike the previous models, one can no longer use the Lyapunov function based just on the radius of gyration; when 0<p<1 one has to find a more finely tuned function which turns out to be a supermartingale; the proof of this fact constitutes an unwieldy, albeit necessary, part of the paper.

A stochastic model for cell adhesion to the vascular wall.

Cell dynamics in the vicinity of the vascular wall involves several factors of mechanical or biochemical origins. It is driven by the competition between the drag force of the blood flow and the resistive force generated by the bonds created between the circulating cell and the endothelial wall. Here, we propose a minimal mathematical model for the adhesive interaction between a circulating cell and the blood vessel wall in shear flow when the cell shape is neglected. The bond dynamics in cell adhesion is modeled as a nonlinear Markovian Jump process that takes into account the growth of adhesion complexes. Performing scaling limits in the spirit of Joffe and Metivier (Adv Appl Probab 18(1):20, 1986), Ethier and Kurtz (Markov processes: characterization and convergence, Wiley, New York, 2009), we obtain deterministic and stochastic continuous models, whose analysis allow to identify a threshold shear velocity associated with the transition from cell rolling and firm adhesion. We also give an estimation of the mean stopping time of the cell resulting from this dynamics. We believe these results can have strong implications for the understanding of major biological phenomena such as cell immunity and metastatic development.

Extensions of the Generalized Pólya Process

A new self-exciting counting process is here considered, which extends the generalized Polya process introduced by Cha (Adv Appl Probab 46:1148–1171, 2014). Contrary to Cha’s original model, where the intensity of the process (linearly) increases at each jump time, the extended version allows for more flexibility in the dependence between the point-wise intensity of the process at some time t and the number of already observed jumps. This allows the “extended Polya process” to be appropriate, e.g., for describing successive failures of a system subject to imperfect but effective repairs, where the repair can lower the intensity of the underlying counting process. Probabilistic properties of the new process are studied (construction from a homogeneous pure-birth process, conditions of non explosion, computation of distributions, convergence of a sequence of such processes, ...) and its connection with Generalized Order Statistics is highlighted. Positive dependence properties are next explored. Finally, the maximum likelihood method is considered in a parametric setting and tested on a few simulated data sets, to highlight the practical use of the new process in an application context.

Inclusion–Exclusion Principles for Convex Hulls and the Euler Relation

Consider n points \(X_1,\ldots ,X_n\) in \(\mathbb {R}^d\) and denote their convex hull by \({\Pi }\). We prove a number of inclusion–exclusion identities for the system of convex hulls \({\Pi }_I:=\mathrm{conv}(X_i: i\in I)\), where I ranges over all subsets of \(\{1,\ldots ,n\}\). For instance, denoting by \(c_k(X)\) the number of k-element subcollections of \((X_1,\ldots ,X_n)\) whose convex hull contains a point \(X\in \mathbb {R}^d\), we prove that $$\begin{aligned} c_1(X)-c_2(X)+c_3(X)-\cdots + (-1)^{n-1} c_n(X) = (-1)^{\dim {\Pi }} \end{aligned}$$ for allX in the relative interior of \({\Pi }\). This confirms a conjecture of Cowan (Adv Appl Probab 39(3):630–644, 2007) who proved the above formula for almost allX. We establish similar results for the number of polytopes \({\Pi }_J\) containing a given polytope \({\Pi }_I\) as an r-dimensional face, thus proving another conjecture of Cowan (Discrete Comput Geom 43(2):209–220, 2010). As a consequence, we derive inclusion–exclusion identities for the intrinsic volumes and the face numbers of the polytopes \({\Pi }_I\). The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler–Schlafli–Poincare relation and is of independent interest.

Percolation and local isoperimetric inequalities

In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension $$d > 1$$ , then $$p_c(G) < 1$$ . This gives a partial answer to a question of Benjamini and Schramm (Electron Commun Probab 1(8):71–82 1996). As a consequence of this result and Haggstrom (Adv Appl Probab 32(1):39–66 2000) we derive, that these graphs also undergo a non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and the beach model. Our techniques are also applied to dependent percolation processes with long range correlations. We provide results on the uniqueness of the infinite percolation cluster and quantitative estimates on the size of finite components. Finally we leave some remarks and questions that arise naturally from this work.

Multivariate Higher-Degree Stochastic Increasing Convexity

Building on the seminal work by Shaked and Shanthikumar (Adv Appl Probab 20:427–446, 1988a; Stoch Process Appl 27:1–20, 1988b), Denuit et al. (Eng Inf Sci 13:275–291, 1999; Methodol Comput Appl Probab 2:231–254, 2000; 2001) studied the stochastic s-increasing convexity properties of standard parametric families of distributions. However, the analysis is restricted there to a single parameter. As many standard families of distributions involve several parameters, multivariate higher-order stochastic convexity properties also deserve consideration for applications. This is precisely the topic of the present paper, devoted to stochastic $$(s_1,s_2,\ldots ,s_d)$$ -increasing convexity of distribution families indexed by a vector $$(\theta _1,\theta _2,\ldots ,\theta _d)$$ of parameters. This approach accounts for possible correlation in multivariate mixture models.

Ancestries of a recombining diploid population

We derive the exact one-step transition probabilities of the number of lineages that are ancestral to a random sample from the current generation of a bi-parental population that is evolving under the discrete Wright-Fisher model with n diploid individuals. Our model allows for a per-generation recombination probability of r . When r = 1, our model is equivalent to Chang's (Adv Appl Probab 31:1002-1038, 1999) model for the karyotic pedigree. When r = 0, our model is equivalent to Kingman's (Stoch Process Appl 13:235-248, 1982) discrete coalescent model for the cytoplasmic tree or sub-karyotic tree containing a DNA locus that is free of intra-locus recombination. When 0 < r < 1 our model can be thought to track a sub-karyotic ancestral graph containing a DNA sequence from an autosomal chromosome that has an intra-locus recombination probability r . Thus, our family of models indexed by r ∈ [0, 1] connects Kingman's discrete coalescent to Chang's pedigree in a continuous way as r goes from 0 to 1. For large populations, we also study three properties of the ancestral process corresponding to a given r ∈ (0, 1): the time Tn to a most recent common ancestor (MRCA) of the population, the time Un at which all individuals are either common ancestors of all present day individuals or ancestral to none of them, and the fraction of individuals that are common ancestors at time Un. These results generalize the three main results of Chang's (Adv Appl Probab 31:1002-1038, 1999). When we appropriately rescale time and recombination probability by the population size, our model leads to the continuous time Markov chain called the ancestral recombination graph of Hudson (Theor Popul Biol 23:183-201, 1983) and Griffiths (The two-locus ancestral graph, Institute of Mathematical Statistics 100-117, 1991).

Perfect and nearly perfect sampling of work-conserving queues

In this paper, we explore algorithms for perfect and nearly perfect sampling from the stationary distribution of the waiting times in various Poisson arrival multi-class and multi-server queues with non-preemptive work-conserving service disciplines. The service duration distributions of these classes may be identical or may vary from class to class. The algorithms follow the idea of dominated coupling from the past (Kendall, Adv Appl Probab 32:844–865 2000) and are variations on an algorithm of Sigman (J Appl Prob 48A:37–43, 2011). A coupled first come first serve queue is constructed for each work-conserving queue. When the service duration distributions do not vary, we achieve perfect simulation by finding times when the system is known to be totally idle. When the distributions differ, the totally idle times may be impossible to determine exactly, but we can achieve simulations with a specified error limit \(\epsilon > 0\).

A note on relationships between some univariate stochastic orders and the corresponding joint stochastic orders

In order to take into account any possible dependence between alternatives in optimization problems, bivariate characterizations of some well-know univariate stochastic orders have been defined and studied by Shanthikumar and Yao (Adv Appl Probab 23:642–659, 1991). These characterizations gave rise to new stochastic comparisons, commonly called joint stochastic orders, which are equivalent to the original ones under assumption of independence, but are different whenever the variables to be compared are dependent. In this note we provide sufficient conditions on the survival copula describing the dependence among the compared variables such that the standard stochastic orders imply the corresponding joint stochastic orders, and viceversa. Also, simple conditions for joint stochastic orders between the components of random vectors defined through multivariate frailty models are provided.

Transition kernels and the conditional extreme value model

Classical multivariate extreme value modelling assumes that the joint distribution belongs to a multivariate domain of attraction and this assumption requires that each marginal distribution be individually attracted to a univariate extreme value distribution. The Heffernan and Tawn (J R Stat Soc Ser B (Stat Methodol) 66(3):497–546, 2004) alternative extremal model for multivariate data does not require all the components belong to an extremal domain of attraction but assumes instead the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components is extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation of the probability of certain risk regions. The original focus on conditional distributions has technical drawbacks but is a natural assumption in many contexts. The technical drawbacks are overcome by relying on convergence of measures and the theory of extended regular variation Heffernan and Resnick (Ann Appl Probab 17(2):537–71, 2007); Das and Resnick (Extremes 14(1):29–61, 2000a); Das et al. (Adv Appl Probab 45(1):139–163, 2013). We compare the two approaches and describe in what way relying on variational limit properties of conditional distributions restricts the class of limit approximations.

Error Rates and Improved Algorithms for Rare Event Simulation with Heavy Weibull Tails

Let Y 1,...,Y n be i.i.d. subexponential and S n = Y 1 + ⋯ + Y n . Asmussen and Kroese (Adv Appl Probab 38:545–558, 2006) suggested a simulation estimator for evaluating ${\mathbb P}(S_n>x)$ , combining an exchangeability argument with conditional Monte Carlo. The estimator was later shown by Hartinger and Kortschak (Bl DGVFM 30:363–377, 2009) to have vanishing relative error. For the Weibull and related cases, we calculate the exact error rate and suggest improved estimators. These improvements can be seen as control variate estimators, but are rather motivated by second order subexponential theory which is also at the core of the technical proofs.

Large Deviation Approaches for the Numerical Computation of the Hitting Probability for Gaussian Processes

We state large deviations for small time of a pinned n-conditional Gaussian process, i.e. the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants, by letting all the past monitoring instants to depend on the small parameter going to 0. Differently from what already developed in Caramellino and Pacchiarotti (Adv Appl Probab 40:424–453, 2008), this procedure is able to catch the dependence on the past observations. We apply the results to numerical experiments that involve the fractional Brownian motion, for the computation of the hitting probability through Monte Carlo methods.

Adjustment Coefficient for Risk Processes in Some Dependent Contexts

Following Müller and Pflug (Insur Math Econ 28:381–392, 2001) and Nyrhinen (Adv Appl Probab 30:1008–1026, 1998; J Appl Probab 36:733–746, 1999), we study the adjustment coefficient of ruin theory in a context of temporal dependency. We provide a consistent estimator for this coefficient, and perform some simulations.

A random mapping with preferential attachment

AbstractIn this article we investigate the asymptotic structure of a random mapping model with preferential attachment, T, which maps the set {1,2,…,n} into itself. The model T was introduced in a companion article [Hansen and Jaworski, Random Struct Algorithms 33 (2008), 105–126] and the asymptotic structure of the associated directed graph G which represents the action of T on the set {1,2,…,n} was investigated in [Hansen and Jaworski, Random Struct Algorithms 33 (2008), 105–126] and [Hansen and Jaworski, Adv Appl Probab 40 (2008), 183–205] in the case when the attraction parameter ρ &gt; 0 is fixed as n →∞. In this article we consider the asymptotic structure of G when the attraction parameter ρ  ρ(n) is a function of n as n →∞. We show that there are three distinct regimes during the evolution of G: (i) ρn →∞ as n →∞, (ii) ρn → β &gt; 0 as n →∞, and (iii) ρn → 0 as n →∞. It turns out that the asymptotic structure of G is, in some cases, quite different from the asymptotic structure of well‐known models such as the uniform random mapping model and models with an attracting center. In particular, in regime (ii) we obtain some interesting new limiting distributions which are related to the incomplete gamma function. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Zero-Automatic Networks

We continue the study of zero-automatic queues first introduced in Dao-Thi and Mairesse (Adv Appl Probab 39(2):429---461, 2007). These queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers are the two simplest 0-automatic queues. All stable 0-automatic queues have an explicit "multiplicative" stationary distribution and a Poisson departure process (Dao-Thi and Mairesse, Adv Appl Probab 39(2):429---461, 2007). In this paper, we introduce and study networks of 0-automatic queues. We consider two types of networks, with either a Jackson-like or a Kelly-like routing mechanism. In both cases, and under the stability condition, we prove that the stationary distribution of the buffer contents has a "product-form" and can be explicitly determined. Furthermore, the departure process out of the network is Poisson.