  The aim of this paper is to introduce a new formalism for the deterministic
analysis associated with backward stochastic differential equations driven by
general c{\`a}dl{\`a}g 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 F{\"o}llmer-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.
