The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. ,  and ). The numerical solutions under these conditions have also been discussed intensively (see e.g. , , , , , , ,  and ). Recently, Mao and Rassias  and  established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.
- brownian motion
- stochastic differential delay equation
- itô’s formula