Journal: Electronic Journal of Probability

Abbreviation

Electron. J. Probab.

Publisher

University of Washington

Journal Volumes

ISSN

1083-6489

Description

Filters

2020 - 20214
Reset filters

Search Results

Publications1 - 4 of 4
  • Fluctuation around the circular law for random matrices with real entries
    Item type: Journal Article
    Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2021)
    Electronic Journal of Probability
    We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.
  • Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense
    Item type: Journal Article
    Gwynne, Ewain; Holden, Nina; Sun, Xin (2021)
    Electronic Journal of Probability
    Recent works have shown that random triangulations decorated by critical (p=1∕2) Bernoulli site percolation converge in the scaling limit to a √8∕3-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE6 in two different ways: - The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov–Hausdorff topology. - There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE6 -decorated √8∕3 -LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called peanosphere convergence. We prove that one in fact has joint convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to √8∕3-LQG decorated by SLE6 in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into C via the so-called Cardy embedding converge to √8∕3-LQG.
  • Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
    Item type: Journal Article
    Hutzenthaler, Martin; Jentzen, Arnulf; von Wurstemberger, Philippe (2020)
    Electronic Journal of Probability
    Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the dimension and the reciprocal of the prescribed approximation accuracy. We thereby establish, for the first time, that the numerical approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.
  • A unified approach to well-posedness of type-I backward stochastic Volterra integral equations
    Item type: Journal Article
    Hernández, Camilo; Possamaï, Dylan (2021)
    Electronic Journal of Probability
    We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semi-linear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.
Publications1 - 4 of 4