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

Abstract

There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well.

Locally Linearized Runge Kutta method of Dormand and Prince

“…In this case, because of the flexibility in the numerical implementation of the LLRK methods mentioned in the introduction, the local linearization of the embedded Runge Kutta formulas of Dormand and Prince can be easily formulated in terms of the Krylov-type methods for exponential matrices. In effect, this can be done just by replacing the Padé formula in (10) and (12) by the Krylov-Padé formula as performed in [5,16,17] for the local linearizations schemes for ordinary, random and stochastic differential equations. In this way, the Locally Linearized formulas of Dormand and Prince could be applied to high dimensional ODEs with a reasonable computational cost [23].…”

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

“…, Lemma 11 in [9] implies that P(t n , y n ; h n ) − P(t n , y n ; h n ) M (1 + | y n |) for all n, which implies that |P(t n , y n ; h n )| 2 < ∞ and P(t n , y n ; h n ) 2 < ∞ for all n.…”

“…where vec( P(t n , y n ; h n )) = k p,q mn,kn (h n , I ⊗ C ⊺ β (t n , y n ), vec(L ⊺ 1 )), k p,q mn,kn denotes the (m n , p, q, k n )−Krylov-Padé approximation defined as in [9], and I is the identity matrix of dimension 2d + 2.…”

On the oscillatory behavior of coupled stochastic harmonic oscillators driven by random forces

“…The Locally Linearized integrator x n+1 converges, strongly with order 1, to the solution x(t n+1 ) of (1) at t n+1 as h goes to zero ( [10], [5]).…”