GKW representation theorem and linear BSDEs under restricted information. An application to risk-minimization

TL;DR

This paper extends the Galtchouk-Kunita-Watanabe representation theorem and linear backward stochastic differential equations to settings with restricted information, enabling new applications in risk-minimization under partial data.

Contribution

It provides the first GKW representation and existence/uniqueness results for linear BSDEs driven by general cadlag martingales under partial information, with an application to risk-minimization.

Findings

Established GKW representation under restricted information

Proved existence and uniqueness of linear BSDEs with partial data

Extended risk-minimization results to partial information framework

Abstract

In this paper we provide Galtchouk-Kunita-Watanabe representation results in the case where there are restrictions on the available information. This allows to prove existence and uniqueness for linear backward stochastic differential equations driven by a general c\`adl\`ag martingale under partial information. Furthermore, we discuss an application to risk-minimization where we extend the results of F\"ollmer and Sondermann (1986) to the partial information framework and we show how our result fits in the approach of Schweizer (1994).