BSE’s, BSDE’s and fixed-point problems
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Patrick Cheridito [1] ; Kihun Nam [2]
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[1] Swiss Federal Institute of Technology in Zurich
Swiss Federal Institute of Technology in Zurich
Zürich, Suiza
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[2] Rutgers University
Rutgers University
City of New Brunswick, Estados Unidos
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 1, 2017, págs. 3795-3828
- Idioma: inglés
- Enlaces
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Resumen
In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed-point problem in a space of random vectors. This makes it possible to employ general fixed-point arguments to establish the existence of a solution. For instance, Banach’s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed-point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean–Vlasov-type equations and equations with coefficients of superlinear growth.
