Dawson-Watanabe superprocesses and a singular control problem arising in finance

We consider a class of stochastic control problems with fuel constraint that are closely connected to the problem of finding adaptive mean-variance-optimal portfolio liquidation strategies in the Almgren-Chriss framework. We give a closed-form solution to these control problems in terms of the log-Laplace transforms of certain J-functionals of Dawson-Watanabe superprocesses. This solution can be related heuristically to the superprocess solution of certain quasilinear parabolic PDEs with singular terminal condition as given by Dynkin (1992). It requires us to study in some detail the blow-up behavior of the log-Laplace functionals when approaching the singularity.