scholarly journals Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

2021 ◽  
Author(s):  
Sixian Jin ◽  
Kei Kobayashi
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Approximation

scholarly journals On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

2017 ◽  
Vol 473 (2207) ◽  
pp. 20170104 ◽  
Author(s):  
Máté Gerencsér ◽  
Arnulf Jentzen ◽  
Diyora Salimova
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Recent Article ◽  
Strong Approximation ◽  
Convergence Rates ◽  
Speed Of Convergence ◽  
Slow Convergence ◽  
Two Space Dimensions ◽  
The Given

In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14 , 1477–1500 ( doi:10.4310/CMS.2016.v14.n6.a1 )), it has been established that, for every arbitrarily slow convergence speed and every natural number d ∈{4,5,…}, there exist d -dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two ( d =2) and three ( d =3) space dimensions.

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