On moment stability of numerical schemes for stochastic delay differential equations
by
Hagen Gilsing
Humboldt University at Berlin

For linear, scalar test equations

dX(t) = ( \sum_{i=0, .., n}a(i) X(t-\tau(i)) ) dt +( \sum_{i=0, .., n}b(i) X(t-\tau(i)) ) dW(t), t in (0, \infty),

X(t) = \xi(t), t in [-\tau, 0]

\tau(i) subset [-\tau, 0], i=0, 1, ..., \tau in (0, \infty)

we extend existing estimations of stability regions of p-th moments (p in N) of some numerical algorithms by characterizing the exact stability regions of the p-th moment for the above test equation and numerical algorithms

Y(n+1) = \sum_{i=0, .., n}\alpha(i, n)Y(n-k(i)), n in N

Y(n) = \xi(n/k), n in -N(k)

with suitable (random) coefficients \alpha(i, n), i in N(n), indices k(i), i in N(n), and p in N. The dimension of the resulting stability criterion depends on the discretization of the SDDE and is often very large. A reduction of the dimension can be done at computational costs. This method can be used to determinate the equations governing stability regions of the p-th moments of the above numerical solutions.

Date received: June 12, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calt-18.