2011/01/27 by Pauline Barrieu, Barrieu, Pauline, Nicole El Karoui +1 · 4 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.48550/arxiv.1101.5282
openalex publication_date 2011/01/27 · arxiv created 2013/06/17 · arxiv updated 2013/06/18 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from ℍ1 to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied in particular to BSDE-like semimartingales. This strong convergence result is then used to prove the existence of solutions of general quadratic BSDEs under minimal exponential integrability assumptions, relying on a regularization in both linear-quadratic growth of the quadratic coefficient itself. On the contrary to most of the existing literature, it does not involve the seminal result of Kobylanski (2000) on bounded solutions.