BSDEs, càdlàg martingale problems and orthogonalisation under basis risk

Ismail Laachir and Francesco Russo ()
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Ismail Laachir: UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - IP Paris - Institut Polytechnique de Paris, Lab-STICC_UBS_CACS_IAS - Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance - UEB - Université européenne de Bretagne - European University of Brittany - ENIB - École Nationale d'Ingénieurs de Brest - UBS - Université de Bretagne Sud - UBO - Université de Brest - Télécom Bretagne - IBNM - Institut Brestois du Numérique et des Mathématiques - UBO - Université de Brest - ENSTA Bretagne - École Nationale Supérieure de Techniques Avancées Bretagne - IMT - Institut Mines-Télécom [Paris] - CNRS - Centre National de la Recherche Scientifique
Francesco Russo: UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - IP Paris - Institut Polytechnique de Paris, OC - Optimisation et commande - UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - IP Paris - Institut Polytechnique de Paris

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Abstract: The aim of this paper is to introduce a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type. A significant application concerns the hedging problem under basis risk of a contingent claim $g(X_T,S_T)$, where $S$ (resp. $X$) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes $(X,S)$ is a diffusion and we provide explicit expressions when $(X,S)$ is an exponential of additive processes.

Keywords: Backward stochastic differential equations; cadlàg martingales; basis risk; Föllmer-Schweizer decomposition; quadratic hedging; martingale problem (search for similar items in EconPapers)
Date: 2016
Note: View the original document on HAL open archive server: https://inria.hal.science/hal-01086227v2
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Published in SIAM Journal on Financial Mathematics, 2016, 7, pp.308-356. ⟨10.1137/140996239⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01086227

DOI: 10.1137/140996239

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