12,918 research outputs found
Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths
We study the Taylor expansion for the solutions of differential equations
driven by -rough paths with . We prove a general theorem concerning the
convergence of the Taylor expansion on a nonempty interval provided that the
vector fields are analytic on a ball centered at the initial point. We also
derive criteria that enable us to study the rate of convergence of the Taylor
expansion. Finally and this is also the main and the most original part of this
paper, we prove Castell expansions and tail estimates with exponential decays
for the remainder terms of the solutions of the stochastic differential
equations driven by continuous centered Gaussian process with finite
variation and fractional Brownian motion with Hurst parameter
.Comment: Final version for publis
Sharpening and generalizations of Shafer's inequality for the arc tangent function
In this paper, we sharpen and generalize Shafer's inequality for the arc
tangent function. From this, some known results are refined
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