[Paper Review] Quadratic BSDEs with jumps and related non-linear expectations: a fixed-point approach ∗

Nabil Kazi-Tani, Dylan Possamaï+1 more|arXiv (Cornell University)|Aug 28, 2012

Stochastic processes and financial applications32 references5 citations

TL;DR

This paper establishes the existence and uniqueness of bounded solutions to quadratic backward SDEs with jumps using a fixed-point approach, proving a non-linear Doob-Meyer decomposition and a converse comparison theorem. It further derives properties of g-expectations and applies the results to dynamic risk measures and their dual representations, including explicit inf-convolution computations.

ABSTRACT

We prove the existence of bounded solutions of quadratic backward SDEs with jumps, using a direct fixed point approach as in Tevzadze [36]. Under an additional standard assumption, we prove a uniqueness result, thanks to a comparison theorem. Then we study the properties of the corresponding g-expectations, we obtain in particular a non linear Doob-Meyer decomposition for g-submartingales and their regularity in time. As a consequence of this results, we obtain a converse comparison theorem for our class of BSDEs. We give applications for dynamic risk measures and their dual representation, and compute their inf-convolution, with some explicit

Motivation & Objective

• To establish the existence of bounded solutions for quadratic backward SDEs with jumps using a direct fixed-point argument.
• To prove uniqueness under a standard integrability assumption via a comparison theorem.
• To study the structural properties of g-expectations associated with such BSDEs.
• To derive a non-linear Doob-Meyer decomposition for g-submartingales and analyze their time regularity.
• To apply the results to dynamic risk measures, including their dual representation and inf-convolution computation.

Proposed method

• A fixed-point approach inspired by Tevzadze [36] is employed to construct solutions to the quadratic BSDEs with jumps.
• The existence of solutions is established under mild integrability conditions on the driver and jump components.
• A comparison theorem is applied to prove uniqueness under an additional standard assumption on the generator.
• The non-linear Doob-Meyer decomposition is derived for g-submartingales using the properties of the solution process.
• The theory of g-expectations is extended to the jump-diffusion setting, revealing structural regularity in time.
• Applications to dynamic risk measures are developed through the dual representation and explicit computation of inf-convolution.

Experimental results

Research questions

• RQ1 Can bounded solutions be proven to exist for quadratic backward SDEs with jumps using a fixed-point method?
• RQ2 Under what conditions is the solution to such BSDEs unique?
• RQ3 What structural properties do g-expectations exhibit in the presence of jumps and quadratic growth?
• RQ4 How does the non-linear Doob-Meyer decomposition behave for g-submartingales in this context?
• RQ5 What are the implications of these results for dynamic risk measures and their dual representations?

Key findings

• Bounded solutions exist for quadratic backward SDEs with jumps using a direct fixed-point argument, extending prior results to the jump-diffusion case.
• Uniqueness of the solution is established under a standard integrability assumption via a comparison theorem.
• A non-linear Doob-Meyer decomposition is derived for g-submartingales, with the decomposition components shown to be regular in time.
• A converse comparison theorem is proven for the class of BSDEs under study, strengthening the link between solution properties and driver behavior.
• The results are applied to dynamic risk measures, yielding their dual representation and explicit computation of the inf-convolution.