In this paper,we consider the following stochastic differential equation for(X_(t))_(t)≥0 on R^(d)and its Euler-Maruyama(EM)approximation(Y_(tn)_(n)∈Z^(+)):dX_(t)=b(X_(t))dt+σ(X_(t))dB_(t),Y_(t)_(n+1)=Y_)(t)_(n_(+1...In this paper,we consider the following stochastic differential equation for(X_(t))_(t)≥0 on R^(d)and its Euler-Maruyama(EM)approximation(Y_(tn)_(n)∈Z^(+)):dX_(t)=b(X_(t))dt+σ(X_(t))dB_(t),Y_(t)_(n+1)=Y_)(t)_(n_(+1ηn+1b(Y_(t)_(n)+σ(Y_(t)_(n)(B_(t)_(n)+1-B_(t)_(n),where b:R^(d)↦R^(d),σ:R^(d)→R^(d)×d are measurable,Bt is the d-dimensional Brownian motion,t_(0):=0,and t_(n):=∑_(k=1)^(n)η_(k)for constantsη_(k)>0 satisfying lim_(k)→∞η_(k)=0 and∑_(k=1)^(∞)ηk=∞.We investigate the convergence rates of Y_(t_(n))under both additive and multiplicative noise settings for different smoothness levels of b.When the noise is additive and partial dissipation conditions hold,we obtain explicit convergence rates of W_(p)(L(Y_(t_(n))),L(X_(t_(n))))+W_(p)(L(Y_(t_(n)))),μ)→0 as n→∞,where W_(p) is the Lp-Wasserstein distance for p∈[0,1],L(ξ)denotes the distribution ofξ,andμis the unique invariant probability measure of(X_(t))_(t)≥0.When the noise is multiplicative and global dissipation conditions hold,the convergence rate of Wp(L(Y_(t_(n))),L(X_(t_(n))))for p≥2 is studied.Compared with the existing results where b is usually C^(1) or C^(2) smooth,our estimates apply to Hölder continuous drift and clearly demonstrate the dependence of the convergence rate on the smoothness of b.展开更多
基金supported by the National Key R&D Program of China(Grant Nos.2022YFA1006000 and 2020YFA0712900)National Natural Science Foundation of China(Grant No.11921001)+2 种基金supported by National Natural Science Foundation of China(Grant No.12071499)the Science and Technology Development Fund of Macao(Grant No.S.A.R.FDCT 0074/2023/RIA2)the University of Macao(Grant Nos.MYRG2020-00039-FST and MYRG-GRG2023-00088-FST).
文摘In this paper,we consider the following stochastic differential equation for(X_(t))_(t)≥0 on R^(d)and its Euler-Maruyama(EM)approximation(Y_(tn)_(n)∈Z^(+)):dX_(t)=b(X_(t))dt+σ(X_(t))dB_(t),Y_(t)_(n+1)=Y_)(t)_(n_(+1ηn+1b(Y_(t)_(n)+σ(Y_(t)_(n)(B_(t)_(n)+1-B_(t)_(n),where b:R^(d)↦R^(d),σ:R^(d)→R^(d)×d are measurable,Bt is the d-dimensional Brownian motion,t_(0):=0,and t_(n):=∑_(k=1)^(n)η_(k)for constantsη_(k)>0 satisfying lim_(k)→∞η_(k)=0 and∑_(k=1)^(∞)ηk=∞.We investigate the convergence rates of Y_(t_(n))under both additive and multiplicative noise settings for different smoothness levels of b.When the noise is additive and partial dissipation conditions hold,we obtain explicit convergence rates of W_(p)(L(Y_(t_(n))),L(X_(t_(n))))+W_(p)(L(Y_(t_(n)))),μ)→0 as n→∞,where W_(p) is the Lp-Wasserstein distance for p∈[0,1],L(ξ)denotes the distribution ofξ,andμis the unique invariant probability measure of(X_(t))_(t)≥0.When the noise is multiplicative and global dissipation conditions hold,the convergence rate of Wp(L(Y_(t_(n))),L(X_(t_(n))))for p≥2 is studied.Compared with the existing results where b is usually C^(1) or C^(2) smooth,our estimates apply to Hölder continuous drift and clearly demonstrate the dependence of the convergence rate on the smoothness of b.
