Project Details
Projekt
Validating numerical solutions of high-dimensional backward SDEs arising from finance
Subject Area
Mathematics
Term
from 2008 to 2012
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 79152879
Backward stochastic differential equations (BSDEs) are a powerful tool to solve problems arising in mathematical finance, e.g. in the pricing of financial derivatives, the hedging of financial risks, and optimal investment problems. Moreover, they yield stochastic representation formulas for semi-linear parabolic Cauchy problems. Therefore the numerical solvability of BSDEs is a problem of high practical relevance, and it is particularly challenging, if a BSDE depends on a high-dimensional system of random sources. In recent years several Monte-Carlo-algorithms for BSDEs based on stochastic meshes or on quantization techniques have been developed. A serious drawback of these algorithms is that they produce point estimators for the solution only, whose quality is not validated. The aim of this project is to add upper (resp. lower) biased terms to these algorithms, which theoretically vanish in the limit. In the practically relevant pre-limit situations the difference between the corresponding ‘upper’ and ‘lower’ solutions may serve as indicator of the success of the numerical procedure. In particular, the absolute size of the biased terms can be monitored in the single discretization steps, which allows for the development of adaptive algorithms that apply more expensive estimators in critical steps. Apart from an error analysis of the additional biased terms for generic estimators of conditional expectations, a more detailed one for least-squares Monte-Carlo estimators is planned.
DFG Programme
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