Solving Dirichlet Problems Numerically Using the Feynman–Kac Representation

Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

“…Stopped diffusion is a good example that adaptive time steps improve the convergence rate, see Buchmann and Petersen [9] and Moon et al [14]. A priori error estimates of the time discretization error in (3) were first derived by Talay and Tubaro [41].…”

“…The function measures the density of the time discretization error, and n = t n X is defined by (9) for the stochastic time stepping algorithm or n =¯ t n X from (16) for the deterministic time stepping algorithm. Here, the function sign denotes sign x = 1 for x ≤ 0 and sign x = −1 for x < 0.…”

Simulation of stopped diffusions

“…To overcome this problem, instead of the unbounded increments DW k $ Nð0; hÞ, bounded approximations can be used [10,11], or a quantization approach is adequate, see [12] and references therein.…”

A Monte Carlo approach to quantifying model error in Bayesian parameter estimation

“…Let {X t = (X 1 t , X 2 t ), t ≥ 0} be a weak solution to the two-dimensional stochastic differential equation where W 1 t and W 2 t are independent standard Brownian motions. The Feynman-Kac theorem (Buchmann and Petersen, 2003) implies that the forward solution to (23) has the probabilistic representation…”