Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility

We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations.

Optimal investment and consumption problems under correlation ambiguity

Consider an economy with $d$ stochastic factors that have an ambiguous variance–covariance matrix. An ambiguity- and risk-averse agent seeks to determine the optimal investment and consumption strategy that is robust to the uncertainty in the covariances. We formulate the robust decision rule as an expected utility maximization over the worst-case scenario with respect to all possible covariances. As this variance–covariance ambiguity leads to robust optimal decisions over a set of non-equivalent probability measures, the $G$-expectation framework is adopted to characterize the problem as a maximin optimization. Our problem formulation can be applied to finite and infinite horizon investment–consumption problems with or without a subsistence consumption constraint. We demonstrate our models using two examples including the defined contribution pension problem and lifetime optimal investment–consumption problems.

OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.