Utility Maximization in Models with Conditionally Independent Increments

We consider the problem of maximizing expected utility from terminal wealth in models with stochastic factors. Using martingale methods and a conditioning argument, we determine the optimal strategy for power utility under the assumption that the increments of the asset price are independent conditionally on the factor process. 1. Introduction. A classical problem in Mathematical Finance is to maximize expected utility from terminal wealth in a securities market (cf. [20, 22] for an overview). This is often called the Merton problem, since it was first solved in a continuous-time setting by Merton [26, 27]. In particular, he explicitly determined the optimal strategy and the corresponding value function for power and exponential utility functions and asset prices modeled as geometric Brownian motions. Since then, these results have been extended to other models of various kinds. For Lévy processes (cf. [3, 7, 8, 15]), the value function can still be determined explicitly, whereas the optimal strategy is determined by the root of a real-valued function. For some affine stochastic volatility models (cf. [19, 21, 23, 25]), the value function can also be computed in closed form by solving some ordinary differential equations, while the optimal strategy can again be characterized by the root of a real-valued function. For more general Markovian models, one faces more involved partial (integro-)differential equations that typically do not lead to explicit solutions and require a substantially more complicated verification procedure to ensure the opti-mality of a given candidate strategy (cf., e.g., [35] for power and [31] for exponential utility). A notable exception is given by models where the stochastic volatility is independent of the other drivers of the asset price process. In this case, it has been shown that the optimal strategy is myopic, that is, only depends on the local dynamics of the asset price (cf., e.g., [11] for exponential and [4, 6, 24] for power utility). In particular, it can be computed without having to solve any differential equations. In the present study, we establish that this generally holds for power utility, provided that the asset price has independent increments conditional on some arbitrary factor process. As in [11], the key idea is to condition on this process,