On the properties of dynamic value functions in the problem of optimal investment in incomplete markets

Abstract

Abstract We study the analytical properties of a dynamic value function and of an optimal solution to the utility maximization problem in incomplete markets for utility functions defined on the whole real line. It was shown by Kramkov and Sirbu [Ann. Appl. Probab. 16 (2006), no. 3, 1352–1384] that if the relative risk-aversion coefficient of the utility function defined on the half real line is uniformly bounded away from zero and infinity, then the value function at time t = 0 of utility maximization problem is two-times differentiable and the optimal wealth is differentiable in probability with respect to the initial capital. Similar results are true for utility functions defined on the whole real line if instead of relative risk-aversion the same condition on the risk-aversion coefficient is imposed. Besides, assuming the continuity of the filtration F we prove that the derivative of the optimal wealth is strictly positive and that the derivative exists also in the sense of L 1-convergence. This enables us to show the existence of a strictly increasing (with respect to the initial capital) and absolutely continuous modification of the optimal wealth. We also study the differentiability properties of the value function V(t,x) and of the optimal wealth process Xt (x) for any t ∈ [0,T] and give the conditions under which the second derivative of the value function and the derivative of the optimal wealth process are continuous. We need these properties to show that the value function satisfies a certain backward stochastic partial differential equation used to characterize the optimal wealth process.