BSDEs, Càdlàg martingale problems, and orthogonalization 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 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(XT ; ST ), where S (resp., X) is an underlying price of a traded (resp., nontraded but observable) asset, via the celebrated Follmer{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.