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Keywords = doléans-dade exponential

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13 pages, 288 KiB  
Article
On De la Peña Type Inequalities for Point Processes
by Naiqi Liu, Vladimir V. Ulyanov and Hanchao Wang
Mathematics 2022, 10(12), 2114; https://doi.org/10.3390/math10122114 - 17 Jun 2022
Cited by 1 | Viewed by 1841
Abstract
There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain [...] Read more.
There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain de la Peña’s inequalities for a stochastic integral of multivariate point processes. The proof is primarily based on Doléans–Dade exponential formula and the optional stopping theorem. As an application, we obtain an exponential inequality for block counting process in Λ−coalescent. Full article
(This article belongs to the Special Issue Limit Theorems of Probability Theory)
23 pages, 1796 KiB  
Article
Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations
by Jürgen Geiser
Symmetry 2020, 12(3), 383; https://doi.org/10.3390/sym12030383 - 3 Mar 2020
Cited by 2 | Viewed by 3672
Abstract
In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic [...] Read more.
In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods. Full article
(This article belongs to the Special Issue Symmetry in Splitting Methods for Partial and Stochastics Differential Equations: Theory and Applications)
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