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Variance reduction methods for simulation of densities on Wiener space. (English) Zbl 1019.60055
The authors consider Monte Carlo simulation methods for the estimation of quantities of the type \(E[f(x)]\). Here \(x\) is a random variable, typically the final value of a diffusion, i.e. \(x = X_1\) with \[ X_t = X_0 + \int_0^t b(X_s) ds + \int_0^t \sigma(X_s) dW_s \] for \(t \in [0,1]\). Most results concerning error analysis of the Monte Carlo simulation methods assume that \(f\) is a rather smooth function. The authors are interested in the case where \(f\) is a generalised function, e.g. a Dirac delta function (then \(E[f(x)]\) is the density of \(x\)) or an indicator function (then \(E[f(x)]\) is the distribution function of \(x\)). However, the generalised function needs to be regularised and thus an additional error is introduced. Variance reduction methods are widely used to control the statistical error in Monte Carlo methods, here they are furthermore useful for controlling that additional regularisation error. The authors analyse the problem using Malliavin calculus for Wiener space. They give a control variate method for the variance reduction, analyse the mean-square errors, compare with a standard density estimation method and finally provide a numerical example.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
65C30 Numerical solutions to stochastic differential and integral equations
68U20 Simulation (MSC2010)
60H07 Stochastic calculus of variations and the Malliavin calculus
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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