content_image/323206.jpg content_image/323205.jpg
motion means that 

<img src='content_image/323206.jpg'>

The value of θ does not need to remain constant, and we could instead consider selecting a sequence of constants, θ ( i $^{)}$. In particular, when the SDE is bounded as in the Dirichlet problem, then we can exactly simulate the absorption time of the path by adaptively choosing θ ( i ) so that one side of the interval coincides with the nearest boundary. It is also beneficial to include an upper limit, ˜ θ = √$_{t}$ − ς ( $^{i}$− $^{1)}$, where t is the target time, in order to avoid the extra computational expense incurred when sampled first passage times are significantly larger than the target time. When a sampled ς ( i ) exceeds the target time, the terminal value, W$_{t}$ , can be sampled using the law of a Bessel bridge with minimum (or maximum) W$_{ς}$ ( i ) . 

Denoting Q$_{[}$$_{ς}$ ( i − $^{1)}$,ς( i $^{)}$∧ t ] as the law of the transformed SDE, and W$_{[}$$_{ς}$ ( i − $^{1)}$,ς( i $^{)}$∧ t ] as the law of a standard Brownian motion, the Radon-Nikod´ ym derivative of Q$_{[}$$_{ς}$ ( i − $^{1)}$,ς( i $^{)}$∧ t ] with respect to W$_{[}$$_{ς}$ ( i − $^{1)}$,ς( i $^{)}$∧ t ] is 

<img src='content_image/323205.jpg'>

where A ( v ) = ∫ v 0 α ( z ) dz , and φ ( v ) = 1 2 ( α ( v ) 2 + α ′ ( v )). It was noted in Chen and Huang (2013) that the continuity of φ ( v ) ensures that the Radon- Nikod´ ym derivative is bounded, allowing the construction of a valid accep- 

