BSDEs, c{\`a}dl{\`a}g martingale problems and orthogonalisation under basis risk

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 $g(X\mathrm{\_}T,S\mathrm{\_}T)$, 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.