Marion, G., Mao, X. and Renshaw, E.
International Mathematical Journal 1, 9-22.
stochastic differential equations
|| Stochastic differential equations provide a useful
means of introducing stochasticity into models
across a broad range of systems from chemistry to
population biology. However, in many applications
the resulting equations have so far proved
intractable to direct analytical solution.
Numerical approximations, such as the Euler scheme
are therefore a vital tool in exploring model
behaviour. Unfortunately, current results
concerning the convergence of such schemes impose
conditions on the drift and diffusion coefficients
of the stochastic differential equation, namely
the linear growth and global Lipschitz conditions,
which are often not met by systems of interest.
In this paper we relax these conditions and prove
that numerical solutions based on the Euler scheme
will converge to the true solution of a broad
class of stochastic differential equations.
The results are illustrated by application to a
stochastic Lotka-Volterra model and a model of
chemical auto-catalysis, neither of which satisfy
either the linear growth nor the global Lipschitz