In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both...In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.展开更多
We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay.First,we give the notions of C-stability and B-consistency,and then establish a ...We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay.First,we give the notions of C-stability and B-consistency,and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations,where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear.Then,we construct the split-step backward Euler method which is a special implicit one-step method,and prove that it is C-stable,B-consistent,and strongly convergent of order 1/2.Finally,we give some numerical experiments to support the obtained results.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No. 12301521)the Natural Science Foundation of Shanxi Province (Grant No. 20210302124081)。
文摘In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.
基金supported by the Key Research Program of Higher Education Institutions of Henan Province(No.24B110019)Basic Research Projects of Key Scientific Research Projects Plan in Henan Higher Education Institutions(No.24ZX008)NSF of China(Nos.12171441 and 12301502).
文摘We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay.First,we give the notions of C-stability and B-consistency,and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations,where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear.Then,we construct the split-step backward Euler method which is a special implicit one-step method,and prove that it is C-stable,B-consistent,and strongly convergent of order 1/2.Finally,we give some numerical experiments to support the obtained results.
