Strong Euler-Maruyama’s approximation

Let {(\Omega, \mathcal{F}, \mathbb{P})} be a probability space, on which {\mathcal{F}_t} is filtration satisfying general conditions. {W_t} is a standard Brownian motion. We consider strong approximation of Euler-Maruyama’s method on stochastic differential equation

\displaystyle d X(t) = b(t, X(t)) dt + \sigma(t, X(t)) dW(t), \quad X(0) = x_0.

Under general assumptions, the above SDE has unique strong solution. Strong EM approximation is stated as follows: Suppose {\mathcal{T} = \{\tau_0 = 0, \tau_1, \ldots, \tau_N = T\}} is a partition of {[0,T]}. With notion {\Delta_n = \tau_{n+1} - \tau_n}, we have EM approximation by

\displaystyle Y_{n+1} = Y_n + b(\tau_n, Y_n) \Delta_n + \sigma(\tau_n, Y_n) (W(\tau_{n+1}) - W(\tau_n)), \quad Y_0 = x_0.

Let {\delta = \max_{n= 0}^{N-1} \Delta_n}. It’s continuous interpolation {Y^\delta} is given by

\displaystyle Y^\delta(t) = Y_{n} + b(\tau_n, Y_n) (t- \tau_n) + \sigma(\tau_n, Y_n) (W(t) - W(\tau_n)), \quad \hbox{ for } t\in [\tau_n, \tau_{n+1}).

A classical result on strong EM method under appropriate conditions is that, see for example Theorem 2.7.3 of the book [Mao07],

\displaystyle \mathbb{E} \Big[ \sup_{0\le t\le T} |X(t) - Y^\delta(t)| \Big] \le K \delta^{1/2}.

However, the above inequality fails for the piecewise constant interpolation of EM approximation {\{Y_n\}}. Otherwise, we have following simple conunter-example. Consider EM approximation of {W_t} on {[0,1]} by equal step size {\delta = 1/N}. Then, we have

\displaystyle \mathbb{E}\Big[ \sup_{0\le t \le 1} |W(t) - W([Nt]/N)| \Big]= \mathbb{E} \Big[ \sup_{1\le n \le N} \sup_{(n-1)/N \le t < n/N} |W(t) - W(\frac{n-1}{N})| \Big].

Note that, {\bar W(t) = \sqrt N W(t/N)} is a standard BM on time-scaled filtration. So one can reduce the above equality as

\displaystyle \mathbb{E}\Big[ \sup_{0\le t \le 1} |W(t) - W([Nt]/N)| \Big]= \frac 1 {\sqrt N} \mathbb{E} \Big[ \sup_{1\le n \le N} X_n \Big],

where {\{X_n\}} are i.i.d random variables defined by

\displaystyle X_n = \sup_{n-1 \le t < n} | \bar W(t) - \bar W(n-1) |.

Since, {X_n}‘s are unbounded iid random variables, {\mathbb{E} \Big[ \sup_{1\le n \le N} X_n \Big]} goes to infinity as {N\rightarrow \infty}. This shows that

\displaystyle \mathbb{E}\Big[ \sup_{0\le t \le 1} |W(t) - W([Nt]/N)| \Big] > O(N^{-1/2}).

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