If then u † ( x x, t ) is an unbiased estimator of u ( x x, t ) with The necessary proof is given in Theorem 1 of Rhee and Glynn (2014). Note that this proof requires that κ ( x ) be continuous. Whilst this will usually be the case for PDEs on infinite domains, it is not generally true of Dirichlet problems. In the following we set N = 1 (each level of the approximation is estimated using a single sample path from the discretized SDE), and define { h$_{j}$ } ∞ j =0 through h$_{0}$ = t , and the relation h$_{j}$$_{+1}$ = h$_{j}$ ⧸ 2. It is clear from (7) that the efficiency of the debiasing approach depends strongly on the convergence rate of ˜ u$_{h}$ ( x x, t ), and hence how { ˜ u$_{h}$$_{j}$ ( x x, t ) } ∞ j =0 are jointly generated (termed the coupling ). In Rhee and Glynn (2014) the view is taken that a single Brownian motion drives every level of the discretization. This can be achieved by either simulating the entire Brownian path at the finest discretisation level (conditional on the halting value), and then sum- ming the components as necessary to evaluate the lower levels, or through the use of Brownian bridges from the Brownian path obtained at the initial level. Using this coupling, E { ( ˜ u$_{h}$$_{j}$ ( x x, t ) − ˜ u$_{h}$$_{∞}$ ( x x, t ) ) 2 } is O (2 − $^{jp}$), where p 19 E { u † ( x x, t ) $^{2}$} = ∞ ∑ j =0 E { ( ˜ u$_{h}$$_{j}$$_{−}$$_{1}$ ( x x, t ) − ˜ u$_{h}$$_{∞}$ ( x x, t ) ) 2 } − E { ( ˜ u$_{h}$$_{j}$ ( x x, t ) − ˜ u$_{h}$$_{∞}$ ( x x, t ) ) 2 } P ( H ≥ j ) . (7) ∞ ∑ j =1 E { ( ˜ u$_{h}$$_{j}$$_{−}$$_{1}$ ( x x, t ) − ˜ u$_{h}$$_{∞}$ ( x x, t ) ) 2 } P ( H ≥ j ) < ∞