Abstract
We introduce in this chapter the paradigm of multilevel simulation whose aim is to dramatically reduce the bias in a Monte Carlo simulation when the (probability distribution of the) random variable under consideration cannot be simulated at a reasonable cost but can be approximated by simulable random variables with a controlled complexity. As typical examples let us cite the discretization scheme of a stochastic process or nested Monte Carlo simulations. The paradigm relies on the existence of both a strong rate and an expansion of the weak error convergence of approximating random variables. We propose an in-depth analysis of both weighted and regular multilevel methods in a an abstract framework, in presence of a higher or firrst rider expansion of the weak error. Various applications are detailed like the pricing of path-dependent or forward start options, quantile computations in actuarial sciences (SCR). When the strong convergence rate is fast enough (like the Milstein scheme for diffusion), multilevel simulation behaves like an unbiased simulation. We conclude by as section about randomized multilevel quantization.
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Notes
- 1.
Other choices for the correlation structure of the families \(Y^{(i)}_{\underline{n}, h}\) could a priori be considered, e.g. some negative correlations between successive levels, but this would cause huge simulation problems since the control of the correlation between two families seems difficult to monitor a priori.
- 2.
c is used here to avoid confusion with the exponent \(\alpha \) of the weak error expansion.
- 3.
Note that in the nested Monte Carlo framework, \(\kappa _i = n_i\), so that \(a_{_N} = (N-1)^{\frac{\beta }{2}}N^{\frac{1-\beta }{2}}\).
- 4.
More generally, a compound option is an option on (the premium of) an option.
- 5.
Due to a notational conflict with the strike prices \(K_1\), \(K_2\), etc, we temporarily modify our standard notations for denoting inner simulations and the bias parameter set \(\mathcal {H}\).
- 6.
As the \(L^1\)-norm is dominated by the \(L^2\)-norm, \(\sum _{n\ge 1} \Vert Z_n\Vert _{_1}<+\infty \) so that \(\mathbb {E}\big (\sum _{n\ge 1} |Z_n|\big )<+\infty \), which in turn implies that \(\sum _{n\ge 1} |Z_n|\) is \({\mathbb P}\)-a.s. absolutely convergent.
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Pagès, G. (2018). Biased Monte Carlo Simulation, Multilevel Paradigm. In: Numerical Probability. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-90276-0_9
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DOI: https://doi.org/10.1007/978-3-319-90276-0_9
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