Monotone stability of quadratic semimartingales with applications to general quadratic BSDEs and unbounded existence result

Pauline Barrieu (1) , Nicole El Karoui (2)

1 Department of Statistics
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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 inequalities on the absolute value of the terminal condition. Then, a general stability result, including the strong convergence of the martingale parts, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale.\\ This strong convergence result is then applied to the study of general quadratic BSDEs, which does not involve the usual exponential transformation but relies on a regularization with both linear-quadratic growth of the quadratic coefficient it-self through inf-convolution. Strong convergence results for BSDEs are then obtained in a general framework using the stability results previously obtained using a forward point of view and considering the quadratic BSDEs as a particular type of quadratic semimartingales.

Domaines: Probabilités [math.PR]

Pauline Barrieu, Nicole El Karoui. Monotone stability of quadratic semimartingales with applications to general quadratic BSDEs and unbounded existence result. 2011. ⟨hal-00560153⟩