How many inner simulations to compute conditional expectations...
Aurélien Alfonsi, Bernard Lapeyre, Jérôme Lelong
The problem of computing the conditional expectation E[f (Y)|X] with
least-square Monte-Carlo is of general importance and has been widely studied.
To solve this problem, it is usually assumed that one has as many samples of Y
as of X. However, when samples are generated by computer simulation and the
conditional law of Y given X can be simulated, it may be relevant to sample K
$\in$ N values of Y for each sample of X. The present work determines the
optimal value of K for a given computational budget, as well as a way to
estimate it. The main take away message is that the computational gain can be
all the more important that the computational cost of sampling Y given X is
small with respect to the computational cost of sampling X. Numerical
illustrations on the optimal choice of K and on the computational gain are
given on different examples including one inspired by risk management.