The aim of this talk consists in introducing a new formalism for the deterministic analysis  associated with backward stochastic differential equations  driven by general  martingales, coupled with a forward  process.

 

When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type coupled with a function which is associated with the  , when is of class in space. When is only a viscosity solution of the PDE, the link associating to is not completely clear: sometimes in the literature it is called the identification problem.

 

The idea is to introduce a suitable analysis to investigate the equivalent of the identification problem in a general Markovian setting with a class of examples. An interesting application concerns  the hedging problem under basis risk of a contingent claim , where (resp. ) 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 is a diffusion and we provide explicit expressions when is an exponential of additive processes. Extensions to non-Markovian (path-dependent cases) are discussed.