Convergence rate of weak Local Linearization schemes for stochastic differential equations with additive noise

Abstract

There exists a diversity of weak Local Linearization (LL) schemes for the integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is employed in the numerical implementation of the weak Local Linear discretizations. On the contrary to the Local Linear discretization, the rate of convergence of the LL schemes has not been considered up to now. In this work, a general theorem about this issue is derived and further is applied to a number of specific schemes. As application, the convergence rate of weak LL schemes for equations with jumps is also presented.

Parametric inference for hypoelliptic ergodic diffusions with full observations

“…Note that the approximated diffusion term now depends on the parameters of the drift term. It is proven that the approximated solutionZ converges weakly to the true solution Z with order 2 (see Theorem 2 in Jimenez and Carbonell (2015)). Now we want to study component-wise the moments of the obtained discretization, build on the observations of the process (Z t ) t≥0 .…”

“…Local Linearization refers to the family of approximation schemes studied by different authors (Biscay et al, 1996, Ozaki, 2012, Jimenez and Carbonell, 2015. The idea consists in approximating the solution of a general SDE by the solution of an autonomous linear SDE, which can be solved explicitly.…”

“…The particular case of Hamiltonian systems with b(X t , Y t ; σ) ≡ σ and a 2 (X t , Y t ; θ) = g 1 (X t ; θ)X t + g 2 (X t ; θ)Y t is considered in Ozaki (1989), where the link between the continuous-time solution of (2) and the corresponding discrete model is obtained with the so-called local linearization scheme. The idea of this scheme is the following: for a SDE with a non-constant drift and a constant variance, its solution can be interval-wise approximated by the solution of a system with a linear drift (see Biscay et al (1996), Ozaki (2012), Jimenez and Carbonell (2015)). This scheme allows to construct a quasi Maximum Likelihood Estimator.…”