Quadratic BSDEs with convex generators and unbounded terminal conditions

By P. Briand, Ying Hu. 2007

357 citations recorded by the source corpus.

In Briand and Hu (Probab Theory Relat Fields 136(4):604–618, 2006), the authors proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions. However, no uniqueness result was stated in that work. The main goal of this paper is to fill this gap. In order to obtain a comparison theorem for this kind of BSDEs, we assume that the generator is convex with respect to the variable z. Under this assumption of convexity, we are also able to prove a stability result in the spirit of the a priori estimates stated in Karoui et al. (Math Finance 7(1):1–71, 1997). With these tools in hands, we can derive the nonlinear Feynman–Kac formula in this context.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/9d937a74b8ea554ebd0e5d99124989972abbb134

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In Applied Mathematics

In Finance

Citations in the corpus

A financial market with interacting investors: does an equilibrium exist? (2011, result): “Her results were generalized by Briand and Hu [1] and Delbaen et al.”
BSDEs with terminal conditions that have bounded Malliavin derivative (2012, background): “BSDEs with drivers of quadratic growth in z and unbounded terminal conditions have been studied by Briand and Hu [4, 5] as well as Delbaen et al.”
7 – Backward Stochastic Differential Equations (2011, unknown): “BSDEs with drivers of quadratic growth in z and unbounded terminal conditions have been studied by Briand and Hu [12, 13] as well as Delbaen et al.”
Numerical simulation of quadratic BSDEs (2013, unknown): “When the terminal condition is bounded, we refer to [7, 34, 35, 42], and, in the unbounded case, we refer to [3, 8, 9, 19, 20].”
Numerical simulation of BSDEs with drivers of quadratic growth (2010, background): “Many papers deal with existence and uniqueness of solution for such BSDEs: we refe r the reader to [17, 18] when the terminal condition is bounded and [3, 4, 9] for the unbounded case.”
A simple constructive approach to quadratic BSDEs with or without delay (2013, background): “For classical quadratic BSDEs, Briand and Hu [4, 5] proved the existence of a solution for unbounded terminal conditions ξ with exponential moments, and verified that uniqueness holds under an extra convexity condition on the driver F .”
Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs (2011, background): “In the case of an unbounded terminal condition, Briand & Hu [8] work under the additional assumption that the coefficient g is convex with respect to the variable z.”
On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions (2009, methodology): “As in part 5 of [3], this assumption allows us to reduce hypot hesis of [4].”
Optimal consumption and investment in incomplete markets with general constraints (2010, background): “So it can be deduced from Proposition 3 in [3] that (4.”
A class of globally solvable Markovian quadratic BSDE systems and applications (2016, background): “In the one-dimensional case, the situation is fully understood—Markov representatives of solutions to Markovian BSDE in dimension 1 are viscosity solutions to the associated quasilinear PDEs (see [13, 42, 49] or [24]).”
Optimal Stopping for Non-Linear Expectations - Part II (2011, unknown): “Taking fn ≡ g and f = g in Proposition 7 of [4] yields that”