content_image/317498.jpg content_image/317497.jpg
which is the first time at which the stochastic process hits the boundary, ∂ Ω, and is absorbed. Define a second stopping time, ˆ t = min( γ x,t Ω , t ), which returns t if the process does not hit the boundary before time t , and returns the first time at which the stochastic process hits the boundary otherwise. For convenience, define: 

In this case the FKF takes the form 

<img src='content_image/317498.jpg'>

<img src='content_image/317497.jpg'>

In the case of Neumann boundary conditions a similar modification can be used, in which the Brownian motion is reflected rather than absorbed (Par- doux and R˘ a¸scanu, 2014). 

## 3. Obtaining FKF estimates via path space importance sampling 

The law of the complementary SDE is rarely available in closed-form. We must resort to numerical simulation of the SDE in order to obtain a Monte Carlo estimate of the integrals in (3) and (4). Until recently this would require the use of a discrete time approximation, such as the Euler- Maruyama method, leading to bias in the FKF estimates. The development of path space importance sampling algorithms (commonly referred to as exact 

