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

TL;DR

This paper develops a new formalism for analyzing backward stochastic differential equations driven by cadlag martingales, with applications to hedging under basis risk and explicit solutions for exponential additive processes.

Contribution

It introduces a novel deterministic analysis framework for BSDEs driven by cadlag martingales, extending classical PDE methods to more general stochastic processes.

Findings

New formalism for BSDEs with cadlag martingales

Explicit solutions for exponential additive processes

Application to hedging 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 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.