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