  Stochastic differential equations (SDEs) and the Kolmogorov partial
differential equations (PDEs) associated to them have been widely used in
models from engineering, finance, and the natural sciences. In particular, SDEs
and Kolmogorov PDEs, respectively, are highly employed in models for the
approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs,
respectively, can typically not be solved explicitly and it has been and still
is an active topic of research to design and analyze numerical methods which
are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly
all approximation methods for Kolmogorov PDEs in the literature suffer under
the curse of dimensionality or only provide approximations of the solution of
the PDE at a single fixed space-time point. In this paper we derive and propose
a numerical approximation method which aims to overcome both of the above
mentioned drawbacks and intends to deliver a numerical approximation of the
Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse
of dimensionality. Numerical results on examples including the heat equation,
the Black-Scholes model, the stochastic Lorenz equation, and the Heston model
suggest that the proposed approximation algorithm is quite effective in high
dimensions in terms of both accuracy and speed.
