point then becomes the initial value in the next iteration of the algorithm, and this continues until the entire path is sampled. More formally, consider the sequence of first passage times for some user defined constant θ , which gives the half-width of the interval. Furthermore denote ( ζ (0) = 0 , Y$_{ζ}$ (0) = y ) as the start point of the (trans- formed) complementary SDE. In LEA the pairs ( ζ ( i $^{)}$, Y$_{ζ}$ ( i ) ) are sequentially simulated until the target time, t , has been surpassed. This requires the simu- lation of two random variables, Δ ζ ( i ) = ζ ( i $^{)}$− ζ ( $^{i}$− 1) and Δ Y ( i ) = Y$_{ζ}$ ( i ) − Y$_{ζ}$ ( i − 1) , which bound the path of Y$_{s}$ . Brownian motion is used as the proposal distribution, requiring that we are able to sample Δ ς ( i ) = ς ( i ) − ς ( $^{i}$− 1) and Δ W ( i ) = W$_{ζ}$ ( i ) − W$_{ζ}$ ( i − 1) , where W$_{s}$ is a standard Brownian motion, and with ( ς (0) = 0 , W$_{ς}$ (0) = y ) . The first passage times of Brownian motion from a symmetric interval can be efficiently sampled using the rejection algorithm developed in Devroye (2009), which has a rejection rate of less than 0 . 001. Once the first passage time is sampled, the symmetric property of Brownian 12 ς ( i ) = inf( s ≥ 0 : | W$_{ς}$ ( i ) − W$_{ς}$ ( i − 1) | > θ ) , i ≥ 1 , ζ ( i ) = inf ( s ≥ 0 : ∣ Y$_{ζ}$ ( i ) − Y$_{ζ}$ ( i − 1) ∣ > θ ) , i ≥ 1