QMC methods in quantitative finance, tradition and perspectives
Sprache des Vortragstitels:
Quasi-Monte Carlo (QMC) methods have been developed in the second half
of the 20th century with the primary goal of being able to compute integrals
over $d$-dimensional domains for modest dimensions $5<d<15$. Notable
contributions have been made by Hlawka, Korobov, Sobol, Niederreiter and others.
Towards the end of the century it has been observed that those methods can be
put to good use for integration problems in much higher dimensions, with
$d$ being in the hundreds or thousands. This was in particular true for
many problems from financial mathematics, i.e., for evaluation of financial
derivatives, but also, more generally, for numerical treatment of
stochastic differential equations.
The success of QMC in these areas could not be explained by the theory
available at that time, and since then researchers have been striving to
In our talk we will review the basics of quasi-Monte Carlo, e.g., the notion of
low-discrepancy sequences and Koksma-Hlawka-type inequalities.
Subsequently we shall have a look at some by now classical explanations for
the effectiveness of QMC for financial problems, in particular the concepts
of effective dimension and weighted Koksma-Hlawka-type inequalities.
Then we will present some of our own contributions to those topics, whereby
we shall concentrate on the recently developed concepts of Hermite spaces and
fast efficient orthogonal transforms. We shall briefly consider tractability
of integration in certain Hermite spaces.
Numerical examples will serve to illustrate theoretical findings.