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We deal with the problem of the numerical evaluation by Monte Carlo methods of the expectation of a function of the position at a fixed time of a Brownian motion killed when it reaches time-dependent barriers, a problem of interest in Finance, e.g. for the pricing of barrier options. Two stochastic Euler schemes are compared in this paper: the "discrete" scheme and the "continuous" one. The latter has been already introduced in literature and makes use of some sharp large deviation estimates of the exit probability for a Brownian bridge in a small time interval. We show that, in the weak sense, the "continuous'1 scheme is much faster than the "discrete" one, by proving an inequality which provides a comparison between the rates of convergence.

monte carlo methods and applications

Sharp estimates for the hitting probability on time-dependent barriers for a Brownian Motion. Weak approximation of a Brownian motion killed on time-dependent barriers