A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditions
Provides a theoretical convergence result for implicit schemes under local one-sided Lipschitz conditions, which is an incremental extension of existing work.
The paper extends Krylov's results on strong convergence of the Euler-Maruyama scheme for SDEs under local one-sided Lipschitz conditions to an implicit numerical scheme, proving convergence in probability as stepsize tends to zero.
Under a local one-sided Lipschitz condition, Krylov [KR] proved the existence and uniqueness of the strong solutions for stochastic differential equations by using the Euler-Maruyama approximation, where he showed that the sequence of numerical solutions converges to the true solution in probability as the stepsize tends to zero. In this note, we shall extend the results in [KR] and investigate an implicit numerical scheme for these equations under a local one-sided Lipschitz condition.