Hannah Geiss: Existence, uniqueness and Malliavin differentiability of quadratic forward-backward SDEs with unbounded terminal conditions

Abstract

The aim of the talk is to present the analysis of forward backward stochastic differential equations (FBSDEs) driven by a Brownian motion or by a Lévy process. The generator and the terminal condition can be path-dependent and satisfy local Lipschitz conditions. The idea of the proof it to start first with a truncated version of the FBSDE. Then Malliavin derivatives provide a representation for the two solution processes which are only implicitly given by the (truncated) FBSDE. Here Malliavin differentiability follows from a recent characterisation for the Wiener Malliavin Sobolev space by S. Geiss and X. Zhou in arxiv.org/pdf/2412.10836.

In the end, by an iteration trick introduced by A. Richou in Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition, Stoch. Proc. Appl. 122, (2012), one can show that there are bounds for the solutions to the truncated version of the FBSDE which do not depend on the truncation level and hence they also hold for the not truncated FBSDE.

This is joint work with Céline Labart, Adrien Richou, and Alexander Steinicke.