We provide a new method to derive the state price density per unit probabilitybased on option prices and GARCH model. We derive the risk neutraldistribution using the result in Breeden and Litzenberger (1978) and thehistorical density adapting the GARCH model of Barone-Adesi, Engle, andMancini...

We study a quadratic BSDE in a continuous filtration with an unbounded generatorand an infinite time horizon. This equation comes from a stochasticcontrol problem in the context of robust utility maximisation.[...]