Jg. Gaines et Tj. Lyons, VARIABLE STEP-SIZE CONTROL IN THE NUMERICAL-SOLUTION OF STOCHASTIC DIFFERENTIAL-EQUATIONS, SIAM journal on applied mathematics, 57(5), 1997, pp. 1455-1484
We introduce a variable step size method for the numerical approximati
on of pathwise solutions to stochastic differential equations (SDEs).
The method, which is dependent on a representation of Brownian paths a
s binary trees, involves estimation of local errors and of their contr
ibution to the global error. We advocate controlling the variance of t
he one-step errors, conditional on knowledge of the Brownian path, in
such a way that after propagation along the trajectory the error over
each step will provide an equal contribution to the variance of the gl
obal error. Discretization schemes can be chosen that reduce the mean
local error so that it is negligible beside the standard deviation. We
show that to obtain convergence of variable step size methods for SDE
s; in general it is not sufficient to evaluate the Brownian path only
at the points in time where one tries to approximate the solution. We
prove that convergence of such methods is guaranteed if the Levy area
is approximated well enough by further subdivision of the Brownian pat
h and the discretization scheme employed uses appropriately both the a
pproximate Levy areas and increments of the Brownian path.