BSDEs, Càdlàg Martingale Problems, and Orthogonalization under Basis Risk

Abstract

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(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 öllmer-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.

Special weak Dirichlet processes and BSDEs driven by a random measure

“…In the general case when the forward BSDEs are also driven by random measures, similar results have been established, for instance by [6], for the jump-diffusion case, and by [13], for the purely discontinuous case, in particular when no Brownian noise appears. In the context of martingale driven forward BSDEs, a first approach to the probabilistic representation has been carried on in [30].…”

Martingale driven BSDEs, PDEs and other related deterministic problems

“…A first approach to classical solutions for a general deterministic problem, associated with forward BSDEs with applications to the so called Föllmer-Schweizer decomposition, was performed by [23].…”

The identification problem for BSDEs driven by possibly non quasi-left-continuous random measures

“…In the context of alternative BSDEs with jumps, i.e. the one of martingale driven forward BSDEs of [13], the identification problem was discussed for instance in [26] and [10].…”