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In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $frac13<H<frac12$. This is a first-order time-discrete numerical approximation scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for It^o SDEs to the case $H>frac12$. The current contribution generalizes the modified Euler scheme to the rough case $frac13<H<frac12$. Namely, we show a convergence rate of order $n^{frac12-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.
We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, dots, B^{m})$ with Hurst parameter $H in (frac 12,1)$. It is well-known that for ordinary differential equa
We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $Hin (0,1)$. We establish strong well-posedness under a variety of as
In this note we consider generalized diffusion equations in which the diffusivity coefficient is not necessarily constant in time, but instead it solves a nonlinear fractional differential equation involving fractional Riemann-Liouville time-derivati
Sufficient and necessary conditions are presented for the comparison theorem of path dependent $G$-SDEs. Different from the corresponding study in path independent $G$-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case.
In this paper, we build the equivalence between rough differential equations driven by the lifted $G$-Brownian motion and the corresponding Stratonovich type SDE through the Wong-Zakai approximation. The quasi-surely convergence rate of Wong-Zakai ap