Different ways of reducing the dimension were proposed.
a- The first consists in using Principal Component
Analysis in order to ''aggregate'' a large number of random
variables in a small number of principal directions. Also it is
used in practice, it does not really solve the problem of the
dimension, but only surrounds it, replacing the initial model by a
more tractable, but
different, one.
b- Instead of relying on PCA, dimension could be
reduced by considering models where a few number of economic
fundamentals
could explain the behavior of many financial assets.
c- In some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT].
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