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In this paper, we investigate suffcient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case [1] and nonlinear expectation framework [8].
An extended Polya urn Model with two colors, black and white, is studied with some SLLN and CLT on the proportion of white balls.
Shafer and Vovk introduce in their book Game-theoretic foundations for probability and finance the notion of instant enforcement. In this paper we introduce an outer measure on the space of continuous paths which assigns zero value exactly to those s ets (properties) of pairs of time $t$ and elementary event $omega$ which are instantly blockable. Next, for the introduced measure we prove BDG inequalities and use them to define It^o-type integral. Additionally, we prove few properties for the quadratic variation of model-free continuous paths which hold with instant enforcement.
We construct an exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of t he density but it is relevant for the evolution of the current. In particular because of that, the Ficks law is violated in the diffusive limit. Switching on a weakly external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by the Einstein relation. We show that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.
The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about non-central moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.
262 - Xiaojuan Li 2021
In this paper, we first find a type of viscosity solution of $G$-heat equation under degenerate case, and then obtain the related $G$-capacity $c({B_{T}in A})$ for any Borel set $A$. Furthermore, we prove that $I_{A}(B_{T})$ is not quasi-continuous when it is not a constant function.
Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variab les on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passing times.
The aim of this paper is to prove a Large Deviation Principle (LDP) for cumulative processes also known as coumpound renewal processes. These processes cumulate independent random variables occuring in time interval given by a renewal process. Our re sult extends the one obtained in Lefevere et al. (2011) in the sense that we impose no specific dependency between the cumulated random variables and the renewal process. The proof is inspired from Lefevere et al. (2011) but deals with additional difficulties due to the general framework that is considered here. In the companion paper Cattiaux-Costa-Colombani (2021) we apply this principle to Hawkes processes with inhibition. Under some assumptions Hawkes processes are indeed cumulative processes, but they do not enter the framework of Lefevere et al. (2011).
161 - Maxime Egea 2021
We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $pi$ on R d , based on (over-damped) Langevin diffusions. This method both inspired by [PP18] and [GMS + 20] relies on a multilevel occupation measur e, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with respective steps $gamma$r = $gamma$02 --r , r = 0,. .. , R. For a given diffusion, we first state a result under general assumptions which guarantees an $epsilon$-approximation (in a L 2-sense) with a cost proportional to $epsilon$ --2 (i.e. proportional to a Monte-Carlo method without bias) or $epsilon$ --2 | log $epsilon$| 3 under less contractive assumptions. This general result is then applied to over-damped Langevin diffusions in a strongly convex setting, with a study of the dependence in the dimension d and in the spectrum of the Hessian matrix D 2 U of the potential U : R d $rightarrow$ R involved in the Gibbs distribution. This leads to strategies with cost in O(d$epsilon$ --2 log 3 (d$epsilon$ --2)) and in O(d$epsilon$ --2) under an additional condition on the third derivatives of U. In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by ($lambda$ U $lor$1) 2 $lambda$ 3 U d$epsilon$ --2 (where$lambda$U and $lambda$ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U). In our numerical illustrations, we show that our theoretical bounds are confirmed in practice and finally propose an opening to some theoretical or numerical strategies in order to increase the robustness of the procedure when the largest and smallest eigenvalues of D 2 U are respectively too large or too small.
124 - Mahmoud Khabou 2021
In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavins calculus defined on an underlying Poisson embedding. The upper bound is then used to infer the speed of convergence of Central Limit Theorems for the multivariate compound Hawkes process with exponential kernels as the observation time T goes to infinity.
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