Mostrando recursos 1 - 20 de 33

  1. On the unique crossing conjecture of Diaconis and Perlman on convolutions of gamma random variables

    Yu, Yaming
    Diaconis and Perlman [In Topics in Statistical Dependence (Somerset, PA, 1987) (1990) 147–166, IMS] conjecture that the distribution functions of two weighted sums of i.i.d. gamma random variables cross exactly once if one weight vector majorizes the other. We disprove this conjecture when the shape parameter of the gamma variates is $\alpha<1$ and prove it when $\alpha\geq1$.

  2. The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

    Jahnel, Benedikt; Külske, Christof
    We consider the continuum Widom–Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs–non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. ¶ We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from...

  3. Synchronization of reinforced stochastic processes with a network-based interaction

    Aletti, Giacomo; Crimaldi, Irene; Ghiglietti, Andrea
    Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement that recently has been introduced in [Crimaldi et al. (2016)]. Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependence on the number of times that events of the same type occurred...

  4. Financial markets with a large trader

    Blümmel, Tilmann; Rheinländer, Thorsten
    We construct a large trader model using tools from nonlinear stochastic integration theory and an impact function. It encompasses many well-known models from the literature. In particular, the model allows price changes to depend on the size as well as on the speed and timing of the large trader’s transactions. Moreover, a volume impact limit order book can be studied in this framework. Relaxing a condition about existence of a universal martingale measure governing all resulting small trader models, we can show absence of arbitrage for the small trader under mild conditions. Furthermore, a case study on utility maximization from...

  5. Robust bounds in multivariate extremes

    Engelke, Sebastian; Ivanovs, Jevgenijs
    Extreme value theory provides an asymptotically justified framework for estimation of exceedance probabilities in regions where few or no observations are available. For multivariate tail estimation, the strength of extremal dependence is crucial and it is typically modeled by a parametric family of spectral distributions. In this work, we provide asymptotic bounds on exceedance probabilities that are robust against misspecification of the extremal dependence model. They arise from optimizing the statistic of interest over all dependence models within some neighborhood of the reference model. A certain relaxation of these bounds yields surprisingly simple and explicit expressions, which we propose to...

  6. Asymptotic Lyapunov exponents for large random matrices

    Nguyen, Hoi H.
    Suppose that $A_{1},\dots ,A_{N}$ are independent random matrices of size $n$ whose entries are i.i.d. copies of a random variable $\xi $ of mean zero and variance one. It is known from the late 1980s that when $\xi $ is Gaussian then $N^{-1}\log \Vert A_{N}\dots A_{1}\Vert $ converges to $\log \sqrt{n}$ as $N\to \infty $. We will establish similar results for more general matrices with explicit rate of convergence. Our method relies on a simple interplay between additive structures and growth of matrices.

  7. Improved Fréchet–Hoeffding bounds on $d$-copulas and applications in model-free finance

    Lux, Thibaut; Papapantoleon, Antonis
    We derive upper and lower bounds on the expectation of $f(\mathbf{S})$ under dependence uncertainty, that is, when the marginal distributions of the random vector $\mathbf{S}=(S_{1},\ldots,S_{d})$ are known but their dependence structure is partially unknown. We solve the problem by providing improved Fréchet–Hoeffding bounds on the copula of $\mathbf{S}$ that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of $[0,1]^{d}$, the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show...

  8. Optimal dividend and investment problems under Sparre Andersen model

    Bai, Lihua; Ma, Jin; Xing, Xiaojing
    In this paper, we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a “renewal” process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique, we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic...

  9. Large deviations for the exclusion process with a slow bond

    Franco, Tertuliano; Neumann, Adriana
    We consider the one-dimensional symmetric simple exclusion process with a slow bond. In this model, whilst all the transition rates are equal to one, a particular bond, the slow bond, has associated transition rate of value $N^{-1}$, where $N$ is the scaling parameter. This model has been considered in previous works on the subject of hydrodynamic limit and fluctuations. In this paper, assuming uniqueness for weak solutions of hydrodynamic equation associated to the perturbed process, we obtain dynamical large deviations estimates in the diffusive scaling. The main challenge here is the fact that the presence of the slow bond gives...

  10. The dividend problem with a finite horizon

    De Angelis, Tiziano; Ekström, Erik
    We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton–Jacobi–Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund’s value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at $0$ and created at a rate proportional to its local time.

  11. General Edgeworth expansions with applications to profiles of random trees

    Kabluchko, Zakhar; Marynych, Alexander; Sulzbach, Henning
    We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16 (2006) 886–918], Fuchs, Hwang and Neininger [Algorithmica 46 (2006) 367–407], and Drmota and Hwang [Adv. in...

  12. Dynamic approaches for some time-inconsistent optimization problems

    Karnam, Chandrasekhar; Ma, Jin; Zhang, Jianfeng
    In this paper, we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time consistency as well as any method that is based on Pontryagin’s maximum principle. The fundamental obstacle is the dilemma of having to invoke the Dynamic Programming Principle (DPP) in a time-inconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the “dynamic utility” under which the original time-inconsistent problem (under the fixed utility) becomes a time-consistent one. As...

  13. On the instability of matching queues

    Moyal, Pascal; Perry, Ohad
    A matching queue is described via a graph, an arrival process and a matching policy. Specifically, to each node in the graph there is a corresponding arrival process of items, which can either be queued or matched with queued items in neighboring nodes. The matching policy specifies how items are matched whenever more than one matching is possible. Given the matching graph and the matching policy, the stability region of the system is the set of intensities of the arrival processes rendering the underlying Markov process positive recurrent. In a recent paper, a condition on the arrival intensities, which was...

  14. On dynamic deviation measures and continuous-time portfolio optimization

    Pistorius, Martijn; Stadje, Mitja
    In this paper, we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency, we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of stochastic differential equations. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion of dynamic deviation measure, we formulate a dynamic mean-deviation portfolio optimization problem...

  15. Unbiased simulation of stochastic differential equations

    Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar
    We propose an unbiased Monte 3 estimator for $\mathbb{E}[g(X_{t_{1}},\ldots,X_{t_{n}})]$, where $X$ is a diffusion process defined by a multidimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by the Bismut–Elworthy–Li formula from Malliavin calculus, as...

  16. Asymptotic bias of stochastic gradient search

    Tadić, Vladislav B.; Doucet, Arnaud
    The asymptotic behavior of the stochastic gradient algorithm using biased gradient estimates is analyzed. Relying on arguments based on dynamic system theory (chain-recurrence) and differential geometry (Yomdin theorem and Lojasiewicz inequalities), upper bounds on the asymptotic bias of this algorithm are derived. The results hold under mild conditions and cover a broad class of algorithms used in machine learning, signal processing and statistics.

  17. Erratum: First passage percolation on random graphs with finite mean degrees [Ann. Appl. Probab. 20(5) (2010) 1907–1965]

    Bhamidi, Shankar; van der Hofstad, Remco; Hooghiemstra, Gerard
    In this erratum, we correct a mistake in the above paper, where we were using an exchangeability result that is obviously false.

  18. A mean-field stochastic control problem with partial observations

    Buckdahn, Rainer; Li, Juan; Ma, Jin
    In this paper, we are interested in a new type of mean-field, non-Markovian stochastic control problems with partial observations. More precisely, we assume that the coefficients of the controlled dynamics depend not only on the paths of the state, but also on the conditional law of the state, given the observation to date. Our problem is strongly motivated by the recent study of the mean field games and the related McKean–Vlasov stochastic control problem, but with added aspects of path-dependence and partial observation. We shall first investigate the well-posedness of the state-observation dynamics, with combined reference probability measure arguments in...

  19. Ballistic and sub-ballistic motion of interfaces in a field of random obstacles

    Dondl, Patrick W.; Scheutzow, Michael
    We consider a discretized version of the quenched Edwards–Wilkinson model for the propagation of a driven interface through a random field of obstacles. Our model consists of a system of ordinary differential equations on a $d$-dimensional lattice coupled by the discrete Laplacian. At each lattice point, the system is subject to a constant driving force and a random obstacle force impeding free propagation. The obstacle force depends on the current state of the solution, and thus renders the problem nonlinear. For independent and identically distributed obstacle strengths with an exponential moment, we prove ballistic propagation (i.e., propagation with a positive...

  20. Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

    Grigorova, Miryana; Imkeller, Peter; Offen, Elias; Ouknine, Youssef; Quenez, Marie-Claire
    In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some results from optimal stopping theory, some tools from the general theory of processes such as Mertens’ decomposition of optional strong supermartingales, as well as an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE...

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