ABSTRACT
As seen in Chapter 6, the investor’s utility maximization of his terminal wealth can be solved in the continuous-time setting. Using methods of stochastic optimal control, Merton ([385], [386]) proves that the value function is the solution of a non-linear partial differential equation: the Bellman equation. Then, closed-form solutions are available for the HARA utility. However, the dynamic programming method is based on Markovian assumptions. To avoid this hypothesis, another approach has been introduced: the duality portfolio characterization by using the martingale measures, the so-called risk-neutral measures. For complete markets, this set is reduced to one point and the optimal solution is determined from the fundamental result: the terminal wealth of the optimal portfolio is equal (up to a multiplicative constant) to the marginal utility inverse of the density of the martingale measure. This method, illustrated in Chapter 6, has been introduced by Pliska [410], Cox and Huang ([132],[133]), and Karatzas et al. [319]. This is in line with the optimal investment problem for a one-period model with a finite set of random events solved by introducing the Arrow-Debreu state prices.
