Mostrando recursos 1 - 20 de 109

  1. Approximating diffusion reflections at elastic boundaries

    Becherer, Dirk; Bilarev, Todor; Frentrup, Peter
    We show a probabilistic functional limit result for one-dimensional diffusion processes that are reflected at an elastic boundary which is a function of the reflection local time. Such processes are constructed as limits of a sequence of diffusions which are discretely reflected by small jumps at an elastic boundary, with reflection local times being approximated by $\varepsilon $-step processes. The construction yields the Laplace transform of the inverse local time for reflection. Processes and approximations of this type play a role in finite fuel problems of singular stochastic control.

  2. Further studies on square-root boundaries for Bessel processes

    Alili, Larbi; Matsumoto, Hiroyuki
    We look at decompositions of perpetuities and apply them to the study of the distributions of hitting times of Bessel processes of two types of square root boundaries. These distributions are linked giving a new proof of some Mellin transforms results obtained by DeLong [6] and Yor [17]. Several random factorizations and characterizations of the studied distributions are established.

  3. Stein’s method for nonconventional sums

    Hafouta, Yeor
    We obtain almost optimal convergence rate in the central limit theorem for (appropriately normalized) “nonconventional" sums of the form $S_N=\sum _{n=1}^N (F(\xi _n,\xi _{2n},...,\xi _{\ell n})-\bar F)$. Here $\{\xi _n: n\geq 0\}$ is a sufficiently fast mixing vector process with some stationarity conditions, $F$ is bounded Hölder continuous function and $\bar F$ is a certain centralizing constant. Extensions to more general functions $F$ will be discusses, as well. Our approach here is based on the so called Stein’s method, and the rates obtained in this paper significantly improve the rates in [7]. Our results hold true, for instance, when $\xi...

  4. Nonconventional random matrix products

    Kifer, Yuri; Sodin, Sasha
    Let $\xi _1,\xi _2,...$ be independent identically distributed random variables and $F:{\mathbb R}^\ell \to SL_d({\mathbb R})$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi _{q_1(n)},\xi _{q_2(n)},...,\xi _{q_\ell (n)})$ where $0\leq q_1

  5. Stable cylindrical Lévy processes and the stochastic Cauchy problem

    Riedle, Markus
    In this work, we consider the stochastic Cauchy problem driven by the canonical $\alpha $-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.

  6. Harnack inequality and derivative formula for stochastic heat equation with fractional noise

    Yan, Litan; Yin, Xiuwei
    In this note, we establish the Harnack inequality and derivative formula for stochastic heat equation driven by fractional noise with Hurst index $H\in (\frac 14,\frac 12)$. As an application, we introduce a strong Feller property.

  7. Large deviations for the maximum of a branching random walk

    Gantert, Nina; Höfelsauer, Thomas
    We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that...

  8. Comparison inequalities for suprema of bounded empirical processes

    Marchina, Antoine
    In this Note we provide comparison moment inequalities for suprema of bounded empirical processes. Our methods are only based on a decomposition in martingale and on comparison results concerning martingales proved by Bentkus and Pinelis.

  9. Cutoff for a stratified random walk on the hypercube

    Ben-Hamou, Anna; Peres, Yuval
    We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3} {2}n\log n$ with window of size $n$, solving a question posed by Chung and Graham (1997).

  10. Local martingales in discrete time

    Prokaj, Vilmos; Ruf, Johannes
    For any discrete-time $\mathsf{P} $–local martingale $S$ there exists a probability measure $\mathsf{Q} \sim \mathsf{P} $ such that $S$ is a $\mathsf{Q} $–martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any $\varepsilon >0$, the measure $\mathsf{Q} $ can be chosen so that $\frac{\mathrm {d} \mathsf {Q}} {\mathrm{d} \mathsf{P} } \leq 1+\varepsilon $.

  11. Optimal stopping and the sufficiency of randomized threshold strategies

    Henderson, Vicky; Hobson, David; Zeng, Matthew
    In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which the value associated to a stopping rule depends on the law of the stopped process. If this value is quasi-convex on the space of attainable laws then it is well known that it is sufficient to restrict attention to the class of threshold strategies. However, if the objective function is not quasi-convex, this may not be the case. We show that,...

  12. Discrete maximal regularity of an implicit Euler–Maruyama scheme with non-uniform time discretisation for a class of stochastic partial differential equations

    Kazashi, Yoshihito
    An implicit Euler–Maruyama method with non-uniform step-size applied to a class of stochastic partial differential equations is studied. A spectral method is used for the spatial discretization and the truncation of the Wiener process. A discrete analogue of maximal $L^2$-regularity of the scheme and the discretised stochastic convolution is established, which has the same form as their continuous counterpart.

  13. Martingale approximations for random fields

    Magda, Peligrad; Zhang, Na
    In this paper we provide necessary and sufficient conditions for the mean square approximation of a random field by an ortho-martingale. The conditions are formulated in terms of projective criteria. Applications are given to linear and nonlinear random fields with independent innovations.

  14. Existence of solution to scalar BSDEs with $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal values

    Hu, Ying; Tang, Shanjian
    In this paper, we study a scalar linearly growing backward stochastic differential equation (BSDE) with an $L\exp{\left (\!\!\sqrt {{2\over \lambda }\log {(1+L)}}\,\right )} $-integrable terminal value. We prove that a BSDE admits a solution if the terminal value satisfies the preceding integrability condition with the positive parameter $\lambda $ being less than a critical value $\lambda _0$, which is weaker than the usual $L^p$ ($p>1$) integrability and stronger than $L\log L$ integrability. We show by a counterexample that the conventionally expected $L\log L$ integrability and even the preceding integrability for $\lambda >\lambda _0$ are not sufficient for the existence of...

  15. Chaos expansion of 2D parabolic Anderson model

    Gu, Yu; Huang, Jingyu
    We prove a chaos expansion for the 2D parabolic Anderson Model in small time, with the expansion coefficients expressed in terms of the annealed density function of the polymer in a white noise environment.

  16. Where does a random process hit a fractal barrier?

    Benjamini, Itai; Shamov, Alexander
    Given a Brownian path $\beta (t)$ on $\mathbb{R} $, starting at $1$, a.s. there is a singular time set $T_{\beta }$, such that the first hitting time of $\beta $ by an independent Brownian motion, starting at $0$, is in $T_{\beta }$ with probability one. A couple of problems regarding hitting measure for random processes are presented.

  17. A moment-generating formula for Erdős-Rényi component sizes

    Ráth, Balázs
    We derive a simple formula characterizing the distribution of the size of the connected component of a fixed vertex in the Erdős-Rényi random graph which allows us to give elementary proofs of some results of [9] and [12] about the susceptibility in the subcritical graph and the CLT [17] for the size of the giant component in the supercritical graph.

  18. On the ladder heights of random walks attracted to stable laws of exponent 1

    Uchiyama, Kôhei
    Let $Z$ be the first ladder height of a one dimensional random walk $S_n=X_1+\cdots + X_n$ with i.i.d. increments $X_j$ which are in the domain of attraction of a stable law of exponent $\alpha $, $0<\alpha \leq 1$. We show that $P[Z>x]$ is slowly varying at infinity if and only if $\lim _{n\to \infty } n^{-1}\sum _1^n P[S_k>0]=0$. By a known result this provides a criterion for $S_{T(R)} /R \stackrel{{\rm P}} \longrightarrow \infty $ as $R\to \infty $, where $T(R)$ is the time when $S_n$ crosses over the level $R$ for the first time. The proof mostly concerns the case...

  19. Hausdorff dimension of the record set of a fractional Brownian motion

    Benigni, Lucas; Cosco, Clément; Shapira, Assaf; Wiese, Kay Jörg
    We prove that the Hausdorff dimension of the record set of a fractional Brownian motion with Hurst parameter $H$ equals $H$.

  20. Order statistics of the moduli of the eigenvalues of product random matrices from polynomial ensembles

    Wang, Yanhui
    Let $X_{1}, \ldots , X_{m_{N}}$ be independent random matrices of order $N$ drawn from the polynomial ensembles of derivative type. For any fixed $n$, we consider the limiting distribution of the $n$th largest modulus of the eigenvalues of $X = \prod _{k=1}^{m_{N}}X_{k}$ as $N \to \infty $ where $m_{N}/N$ converges to some constant $\tau \in [0, \infty )$. In particular, we find that the limiting distributions of spectral radii behave like that of products of independent complex Ginibre matrices.

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