Mostrando recursos 1 - 20 de 100

  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

    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.

  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

    Gabrysch, Katja; Thörnblad, Erik
    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.

  19. Column normalization of a random measurement matrix

    Mendelson, Shahar
    In this note we answer a question of G. Lecué, by showing that column normalization of a random matrix with iid entries need not lead to good sparse recovery properties, even if the generating random variable has a reasonable moment growth. Specifically, for every $2 \leq p \leq c_1\log d$ we construct a random vector $X \in \mathbb{R} ^d$ with iid, mean-zero, variance $1$ coordinates, that satisfies $\sup _{t \in S^{d-1}} \|\bigl < {X,t} \bigr >\|_{L_q} \leq c_2\sqrt{q} $ for every $2\leq q \leq p$. We show that if $m \leq c_3\sqrt{p} d^{1/p}$ and $\tilde{\Gamma } :\mathbb{R} ^d \to \mathbb{R}...

  20. The lower Snell envelope of smooth functions: an optional decomposition

    Trevino Aguilar, Erick
    In this paper we provide general conditions under which the lower Snell envelope defined with respect to the family $\mathcal{M} $ of equivalent local-martingale probability measures of a semimartingale $S$ admits a decomposition as a stochastic integral with respect to $S$ and an optional process of finite variation. On the other hand, based on properties of predictable stopping times we establish a version of the classical backwards induction algorithm in optimal stopping for the non-linear super-additive expectation associated to $\mathcal{M} $. This result is of independent interest and we show how to apply it in order to systematically construct instances...

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