Peter Kloeden

Arnulf Jentzen, Peter Kloeden, Georg Winkel

Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649-667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The key idea was to use suitable linear functionals of the noise process in the numerical scheme which allow a higher approximation order to be obtained. Following this approach, a new simplified version of the scheme in the above named reference is proposed and analyzed in this article. The main advantage of the convergence result given here is the higher convergence order for nonlinear parabolic SPDEs with additive noise, although the used numerical scheme is very simple to simulate and implement.

Peter Kloeden, Andreas Neuenkirch

Many stochastic differential equations that occur in financial modelling do not satisfy the standard assumptions made in convergence proofs of numerical schemes that are given in textbooks, i.e., their coefficients and the corresponding derivatives appearing in the proofs are not uniformly bounded and hence, in particular, not globally Lipschitz. Specific examples are the Heston and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia model with rational coefficient functions. Simple examples show that, for example, the Euler-Maruyama scheme may not converge either in the strong or weak sense when the standard assumptions do not hold. Nevertheless, new convergence results have been obtained recently for many such models in financial mathematics. These are reviewed here. Although weak convergence is of traditional importance in financial mathematics with its emphasis on expectations of functionals of the solutions, strong convergence plays a crucial role in Multi Level Monte Carlo methods, so it and also pathwise convergence will be considered along with methods which preserve the positivity of the solutions.

Peter Kloeden, Tony Shardlow

Mean-field SDEs, also known as McKean-Vlasov equations, are stochastic differential equations where the drift and diffusion depend on the current distribution in addition to the current position. We describe an efficient numerical method for approximating the distribution at time t of the solution to the initial-value problem for one-dimensional mean-field SDEs. The idea is to time march (e.g., using the Euler-Maruyama time-stepping method) an m-point Gauss quadrature rule. With suitable regularity conditions, convergence with first order is proved for Euler-Maruyama time stepping. We also estimate the work needed to achieve a given accuracy in terms of the smoothness of the underlying problem. Numerical experiments are given, which show the effectiveness of this method as well as two second-order time-stepping methods. The methods are also effective for ordinary SDEs in one dimension, as we demonstrate by comparison with the multilevel Monte Carlo method.