We consider a kind of non-autonomous mixed stochastic differential equations driven by standard Brownian motions and fractional Brownian motions with Hurst index H ∈(1/2,1). In the sense of stochastic Besov norm with...We consider a kind of non-autonomous mixed stochastic differential equations driven by standard Brownian motions and fractional Brownian motions with Hurst index H ∈(1/2,1). In the sense of stochastic Besov norm with index γ, we prove that the rate of convergence for Euler approximation is O(δ^(2H-2γ)), here δ is the mesh of the partition of [0,T].展开更多
Let Xt（x） be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt （x） be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that f...Let Xt（x） be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt （x） be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that for any R〉0 and γ∈（0, 1/2）, sup t∈[0,1],｜x｜≤R｜X nt （x,ω）-Xt （x,ω）｜≤ξR,γ（ω）2-nγ, n≥1, q.e., whereξR,γ（ω） is quasi-everywhere finite.展开更多
We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obt...We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.展开更多
This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this prob...This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod’s representation for jumping processes. Furthermore, we establish the strong convergence of Euler–Maruyama’s approximations, and estimate the order of error. The continuous dependence on initial values of Euler–Maruyama’s approximations is also investigated in the end.展开更多
基金This work was supported by Project of Department of Education of Guangdong Province(No.2018KTSCx072).
文摘We consider a kind of non-autonomous mixed stochastic differential equations driven by standard Brownian motions and fractional Brownian motions with Hurst index H ∈(1/2,1). In the sense of stochastic Besov norm with index γ, we prove that the rate of convergence for Euler approximation is O(δ^(2H-2γ)), here δ is the mesh of the partition of [0,T].
文摘Let Xt（x） be the solution of stochastic differential equations with smooth and bounded derivatives coefficients. Let Xnt （x） be the Euler discretization scheme of SDEs with step 2-n . In this note, we prove that for any R〉0 and γ∈（0, 1/2）, sup t∈[0,1],｜x｜≤R｜X nt （x,ω）-Xt （x,ω）｜≤ξR,γ（ω）2-nγ, n≥1, q.e., whereξR,γ（ω） is quasi-everywhere finite.
基金Supported by National Natural Science Foundation of China (Grant Nos.10901065 and 11271013)Fundamental Research Funds for the Central Universities,Huazhong University of Science and Technology (Grant No.2012QN028)
文摘We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.
基金Supported in part by NNSFs of China(Grant Nos.11771327,11431014,11831014)。
文摘This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod’s representation for jumping processes. Furthermore, we establish the strong convergence of Euler–Maruyama’s approximations, and estimate the order of error. The continuous dependence on initial values of Euler–Maruyama’s approximations is also investigated in the end.
