Let be a probability space, on which
is filtration satisfying general conditions.
is a standard Brownian motion. We consider strong approximation of Euler-Maruyama’s method on stochastic differential equation
Under general assumptions, the above SDE has unique strong solution. Strong EM approximation is stated as follows: Suppose is a partition of
. With notion
, we have EM approximation by
Let . It’s continuous interpolation
is given by
A classical result on strong EM method under appropriate conditions is that, see for example Theorem 2.7.3 of the book [Mao07],
However, the above inequality fails for the piecewise constant interpolation of EM approximation . Otherwise, we have following simple conunter-example. Consider EM approximation of
on
by equal step size
. Then, we have
Note that, is a standard BM on time-scaled filtration. So one can reduce the above equality as
where are i.i.d random variables defined by
Since, ‘s are unbounded iid random variables,
goes to infinity as
. This shows that