## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (203.209 recursos)

Electronic Communications in Probability

1. #### 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 $\frac{\mathrm {d} \mathsf {Q}} {\mathrm{d} \mathsf{P} } \leq 1+\varepsilon$.

2. #### 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,...

3. #### 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.

4. #### Martingale approximations for random fields

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.

5. #### 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...

6. #### 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.

7. #### 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.

8. #### 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.

9. #### 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...

10. #### 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 $H$.

11. #### 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.

12. #### The largest root of random Kac polynomials is heavy tailed

Butez, Raphaël
We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviations principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviations principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at...

13. #### The Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibrium

Duong, Manh Hong; Tugaut, Julian
In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.

14. #### How fast planar maps get swallowed by a peeling process

Curien, Nicolas; Marzouk, Cyril
The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [15] that any peeling process will eventually discover the whole map. In this paper we study the probability that the origin is not swallowed by the peeling process until time $n$ and show it decays at least as $n^{-2c/3}$ where $c \approx 0.1283123514178324542367448657387285493314266204833984375...$ is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable Lévy process conditioned to remain positive [12] which appears as a scaling...

15. #### Almost-sure asymptotics for the number of heaps inside a random sequence

Basdevant, A.-L.; Singh, A.
We study the minimum number of heaps required to sort a random sequence using a generalization of Istrate and Bonchis’s algorithm (2015). In a previous paper, the authors proved that the expected number of heaps grows logarithmically. In this note, we improve on the previous result by establishing the almost-sure and $L^1$ convergence.

16. #### Asymptotic results in solvable two-charge models

Dal Borgo, Martina; Hovhannisyan, Emma; Rouault, Alain
In [16], a solvable two charge ensemble of interacting charged particles on the real line in the presence of the harmonic oscillator potential is introduced. It can be seen as a special form of a grand canonical ensemble with the total charge being fixed and unit charge particles being random. Moreover, it serves as an interpolation between the Gaussian orthogonal and the Gaussian symplectic ensembles and maintains the Pfaffian structure of the eigenvalues. A similar solvable ensemble of charged particles on the unit circle was studied in [17, 10]. ¶ In this paper we explore the sharp asymptotic behavior of the number...

17. #### Particle approximation for Lagrangian Stochastic Models with specular boundary condition

Bossy, Mireille; Jabir, Jean-François
In this paper, we prove a particle approximation, in the sense of the propagation of chaos, of a Lagrangian stochastic model submitted to specular boundary condition and satisfying the mean no-permeability condition.

18. #### The greedy walk on an inhomogeneous Poisson process

The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. We prove that the property of visiting all points of the point process satisfies a $0$–$1$ law and determine explicit sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.