Estimating Multidimensional Density Functions Using the Malliavin-Thalmaier Formula

The Malliavin-Thalmaier formula was introduced in [8] as an alternative expression for the density of a multivariate smooth random variable in Wiener space. In comparison with classical integration by parts formulae, this alternative formulation requires the application of the integration by parts formula only once to obtain an expression that can be simulated. Therefore this expression is free from the curse of dimensionality. Unfortunately, when this formula is applied directly in computer simulation, it exhibits unstable behavior. We propose an approximation to the Malliavin-Thalmaier formula in the spirit of the theory of kernel density estimation to solve this problem. In the first part of this paper, we obtain a central limit theorem for the estimation error. And in the latter part, we apply the Malliavin-Thalmaier formula for the calculation of Greeks in finance.