Optimal investment and consumption for Ornstein-Uhlenbeck spread financial markets

We develop stochastic optimal control methods for spread financial markets defined by the Ornstein-Uhlenbeck (OU) processes. To this end, we study the Hamilton-Jacobi-Bellman (HJB) equation using the Feynman-Kac (FK) probability representation. We show an existence and uniqueness theorem for the classical solution of the HJB equation, a quasi-linear partial derivative equation of parabolic type. Then we show a special verification theorem and, as a consequence, construct optimal consumption/investment strategies for power utility functions. Moreover, using fixed point tools we study the numeric approximation for the HJB solution and we establish the convergence rate which, as it turns out in this case, is super geometric, i.e., more rapid than any geometric one. Finally, we illustrate numerically the behavior of the obtained strategies. Keywords Stochastic differential equations • Stochastic optimal control • Dynamical programming • Hamilton-Jacobi-Bellman equation • Feynman-Kac mapping • Ornstein-Uhlenbeck processes • Financial Spread markets • Optimal investment/consumption problem • Numerical schemes