Abstract
We consider the numerical approximation of P[G ∊ Ω], where the d-dimensional random variable G cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations {G ℓ} ℓ∊ N which can be sampled. The cost of standard Monte Carlo estimation scales poorly with accuracy in this setup since it compounds the approximation and sampling cost. A direct application of multilevel Monte Carlo (MLMC) improves this cost scaling slightly, but returns suboptimal computational complexities since estimation of the probability involves a discontinuous functional of G ℓ. We propose a general adaptive framework which is able to return the MLMC complexities seen for smooth or Lipschitz functionals of G ℓ. Our assumptions and numerical analysis are kept general allowing the methods to be used for a wide class of problems. We present numerical experiments on nested simulation for risk estimation, where G = E [X| Y] is approximated by an inner Monte Carlo estimate. Further experiments are given for digital option pricing, involving an approximation of a d-dimensional SDE.
Original language | English |
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Pages (from-to) | 2125-2149 |
Number of pages | 25 |
Journal | SIAM Journal on Numerical Analysis |
Volume | 60 |
Issue number | 4 |
Early online date | 15 Aug 2022 |
DOIs | |
Publication status | Published - 2022 |
Keywords
- multilevel Monte Carlo
- nested simulation
- risk estimation
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics