A Tail Estimate with Exponential Decay for the Randomized Incremental Construction of Search Structures

Gudmundsson, Joachim; Seybold, Martin P.

doi:10.1137/1.9781611977073.28

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2022

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“…As an important research object, the tail probability of the random variable has received extensive attention and has been widely used in the fields of probability theory, statistics and econometrics (see e.g. [11,12]). Concerning the error of the numerical method, its tail probability P( err N > ) refers to the probability that the error of the numerical solution does not exceed a given precision > 0.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Structure-Preserving Numerical Methods

Hong

Sun

2022

Lecture Notes in Mathematics

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This paper proposes a novel concept of exponential superiority in probability to compare the numerical methods for general stochastic differential equations from the perspective of the tail probability of the error. We take the linear stochastic oscillator as the test equation and consider several concrete numerical methods. By establishing the large deviation principles of the errors of the considered numerical methods, we show that the symplectic methods are exponentially superior to the non-symplectic methods in probability when the computational time T is sufficiently large. This provides a new way to explain the superiority of stochastic symplectic methods over non-symplectic methods in the long-time simulation.

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Section: Introductionmentioning

confidence: 99%