Generalized stochastic target problems for pricing and partial hedging under loss constraints-application in optimal book liquidation

We consider a singular version with state constraints of the stochastic target problems studied in Soner and Touzi (SIAM J. Control Optim. 41:404-424, 2002; J. Eur. Math. Soc. 4:201-236, 2002) and more recently Bouchard et al. (SIAM J. Control Optim. 48:3123-3150, 2009), among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly fits the market models with proportional transaction costs and the order book liquidation issues. Our main result is a direct PDE characterization of the associated pricing function. As an example application, we discuss the valuation of VWAP-guaranteed-type book liquidation contracts, for a general class of risk functions.