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

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