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The complementary SDE is 

<img src='content_image/323619.jpg'>

This reflects a challenging problem as the complementary SDE is nonlinear and we need exact first passage information for the boundary. Note that both of the major restrictions of our EA approach are met. The complementary SDE has unit volatility, and the potential function A ( xxx ) is given by k ( xxx ). 

Since no analytical solution to this PDE exists, we compare an estimate obtained via EA with estimates obtained using the Euler-Murayama scheme with decreasing step sizes. It is known that in the limit of zero step size, the weak approximation error of the Euler-Murayama approach reduces to 0 (Gobet, 2000). Hence, the Euler-Murayama estimates should converge to (within Monte Carlo error of) the EA estimate as the discretisation interval is reduced. The comparison is shown in Figure 1 for t = 2, xxx = (0 . 2 , 0 . 2) T and xxx = (0 . 8 , 0 . 8) $^{T}$, where 10 6 simulations were used for each estimate, and we see that this is indeed the case. The EA estimates took 18 seconds and 13 seconds respectively to obtain, roughly the same as the Euler-Maruyama estimate with 1024 steps, and 512 steps respectively per unit interval. The discretisation error at this level of refinement is larger than the Monte Carlo error. These comparisons demonstrate that the EA approach is more com- putationally efficient in this example. 

