Search results for "60G50"
showing 6 items of 6 documents
Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon
2023
This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an $\alpha$-stable law, $\alpha\in (0,2]$, with emphasis on the case $\alpha=2$. There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when $\alpha \in (0,2)$ (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order $2+\delta$ for some $\delta>0$. We extend these results, inclu…
One-dimensional random walks with self-blocking immigration
2017
We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as $c \sqrt{t} \log t$. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration.
Time-dependent weak rate of convergence for functions of generalized bounded variation
2016
Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$
Mean square rate of convergence for random walk approximation of forward-backward SDEs
2020
AbstractLet (Y,Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk$B^n$from the underlying Brownian motionBby Skorokhod embedding, one can show$L_2$-convergence of the corresponding solutions$(Y^n,Z^n)$to$(Y, Z).$We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in$C^{2,\alpha}$. The proof relies on an approximative representation of$Z^n$and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to t…
Uniform measure density condition and game regularity for tug-of-war games
2018
We show that a uniform measure density condition implies game regularity for all 2 < p < ∞ in a stochastic game called “tug-of-war with noise”. The proof utilizes suitable choices of strategies combined with estimates for the associated stopping times and density estimates for the sum of independent and identically distributed random vectors. peerReviewed
Random walk approximation of BSDEs with H{\"o}lder continuous terminal condition
2018
In this paper, we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space. peerReviewed