On dynamic programming equations for utility indifference pricing under delta constraints

In this paper we study the problem of utility indifference pricing in a constrained financial market, using a utility function defined over the positive real line. We present a convex risk measure − v ( • : y ) satisfying q ( x , F ) = x + v ( F : u 0 ( x ) ) , where u 0 ( x ) is the maximal expected utility of a small investor with the initial wealth x, and q ( x , F ) is a utility indifference buy price for a European contingent claim with a discounted payoff F. We provide a dynamic programming equation associated with the risk measure ( − v ) , and characterize v as a viscosity solution of this equation.