content_image/297377.jpg content_image/297378.jpg
of a unique solution (see Evans (1998), for example). The FKF allow us to represent the local solution of the PDE, u ( x x, t ), as the expectation of the initial function, f ( x ), with respect to the law of a complementary SDE satisfying 

where W W$_{s}$ ∈ R d is a vector of independent Brownian motions at time s , X x s ∈ R d is the state of the SDE at time s and initialised at x , i.e. X x 0 = x , and σ satisfies aaa$_{ij}$ ( x x, s ) = ∑ $_{k}$σ$_{ik}$ ( x x, s ) σ$_{kj}$ ( x x, s ). In this specific case, t 

<img src='content_image/297377.jpg'>

<img src='content_image/297378.jpg'>

Note that we have further assumed that bbb ( x x, s ) and σ ( x x, s ) are sufficiently regular to ensure a unique, non-explosive, weak solution to (2) (see Karatzas and Shreve (1991), for example). 

In nontrivial cases (3) will need to be estimated using Monte Carlo meth- ods. In comparison to deterministic solvers we have replaced the determin- istic discretization error with Monte Carlo error, and so the error analysis of (3) can be performed with the Central Limit Theorem (CLT), for example (Graham et al., 1996). This probabilistic approach is more naturally suited for performing statistical inference. 

## 2.1. Bounded Domains 

In many situations the PDE is bounded on the domain Ω. Common 

