Utility maximisation and utility indifference price for exponential semi-martingale models and HARA utilities

We consider the utility maximisation problem for semi-martingale models and HARA (hyperbolic absolute risk aversion) utilities. Using specific properties of HARA utilities, we reduce the initial maximisation problem to the conditional one, which we solve by applying a dual approach. Then we express the solution of the conditional maximisation problem via conditional information quantities related to HARA utilities, like the Kullback-Leibler information and Hellinger-type integrals. In turn, we express the information quantities in terms of information processes, which is helpful in indifference price calculus. Finally, we give equations for indifference prices. We show that the indifference price for a seller and the minus indifference price for a buyer are risk measures. We apply the results to Black-Scholes models with correlated Brownian motions. Using the identity-in-law technique, we give an explicit expression for information quantities. Then the previous formulas for the indifference price can be applied.