BACKWARD SDES, MARTINGALE PROBLEMS AND APPLICATIONS TO MATHEMATICAL FINANCE

The aim of this talk consists in introducing a formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales, coupled with a forward process. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution u of a semilinear PDE of parabolic type coupled with a function v which is associated with the gradient ∇u, when u is of class C1 in space. When u is only a viscosity solution of the PDE, the link associating v to u is not completely clear: sometimes in the literature it is called the identification problem. We introduce in particular the notion of a decoupled mild solution of a PDE, a IPDE, a path-dependent PDE or more generally a deterministic problem associated with a BSDE. The idea is to introduce a suitable analysis to investigate the equivalent of the identification problem first in a general Markovian setting with a class of examples. An interesting application concerns the hedging problem under basis risk of a contingent claim g(XT,ST ), 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. Extensions to non-Markovian (path-dependent) cases are discussed.