The Dynamic Programming Equation for Second Order Stochastic Target Problems

Abstract

Motivated by applications in mathematical finance [U. Cetin, H. M. Soner, and N. Touzi, “Options hedging for small investors under liquidity costs,” Finance Stoch., to appear] we continue our study of second order backward stochastic equations. In this paper, we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems. In contrast with previous formulations of similar problems, we restrict control processes to be continuous. This new framework enables us to apply our results to a larger class of models. Also the resulting derivation is more transparent. The main technical tool is the geometric dynamic programming principle in this context, and it is proved by using the framework developed in [H. M. Soner and N. Touzi, J. Eur. Math. Soc. (JEMS), 8 (2002), pp. 201–236].