In this Note we study a class of BSDEs which admits a particular singularity in their driver. More precisely, we assume that the driver is not integrable and degenerates when approaching to the terminal time of the equation.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded cd
processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs.
We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[Psi(Y_{T})]ge m$, for some (
possibly random) non-decreasing map $Psi$ and some threshold $m$. We name them textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi cite{BoElTo09}. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in F{o}llmer and Leukert cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi cite{BoElTo09}.
In this paper we prove that every random variable of the form $F(M_T)$ with $F:real^d toreal$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $mathcal{F}$ admits an exact integral representati
on with respect to $M$, that is, without any orthogonal component. This representation holds true regardless any regularity assumption on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$ and show they can be solved without an orthogonal component. To this end, we extend first existence results for such BSDEs under a general filtration and then obtain regularity properties such as differentiability for the solution process.
