Abstract
This chapter provides a (partial) answer to the following question: can we simulate the continuous – or genuine – Euler scheme? To this end, we first investigate the Brownian bridge and its avatar for diffusion processes which. It allows to simulate in an exat way some functionals of the genuine Euler scheme involving its maximum or its minimum over a given time interval and provide sharper approximations of functionals involving time integrals. Several first order weak error are stated with precise references. Applications to several families of path-dependent European options (Asian, lookback, barrier) are given, including some variance reduction methods for barrier options.
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- 1.
We need to define F on càdlàg functions in view of the stepwise constant Euler scheme, not to speak of jump diffusion driven by Lévy processes.
- 2.
When \(\alpha \) is continuous or stepwise constant and càdlàg, \(\tau _{_D}(\alpha ):=\inf \{s\!\in [0,T]: \alpha (s)\!\notin \, D\}\).
- 3.
...Of course one needs to compute the empirical variance (approximately) given by
$$ \frac{1}{M} \sum _{m=1}^M f (\Xi ^{(m)})^2-\left( \frac{1}{M} \sum _{m=1}^M f(\Xi ^{(m)})\right) ^2 $$in order to design a confidence interval, without which the method is simply nonsense....
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Pagès, G. (2018). The Diffusion Bridge Method: Application to Path-Dependent Options (II). In: Numerical Probability. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-90276-0_8
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DOI: https://doi.org/10.1007/978-3-319-90276-0_8
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