On a free boundary problem for an optimal investment problem with different interest rates

Continuous-time mean-variance hedging under different loan and deposit rates

“…Using standard PDE tools including the truncation method, the Leray-Schauder fixed point theorem, the embedding theorem and the Schauder estimation, we get a solution to the fully nonlinear parabolic PDE, from which we eventually construct a C 3,2 solution to the original HJB equation. But different from [7] and [14], we will show the solution is indeed the value function to our mean-variance problem by a verification theorem. An optimal feedback strategy is also obtained during this process.…”

“…There is a very limited number of works that have taken the gap between deposit and loan rates into consideration in the literature. For instance, Fleming and Zariphopoulou [9] considered optimal investment and consumption problems; Bergman [1], Korn [20], and Cvitanic and Karatzas [4] studied option pricing problems; Xu and Chen [38] investigated an optimal consumption-investment problem; Guan [14] studied a utility maximization problem by PDE method, but no verification theorem was provided in the paper so that whether the solution of the HJB equation is the value function of the problem is not known.…”

“…In this paper, we study a mean-variance hedging problem under different deposit and loan rates, whose value function is not quadratic form. Similar to Dai and Yi [7] and Guan [14], we solve the problem by PDE method. We first transform the HJB equation into a fully nonlinear parabolic PDE by intuitive argument.…”

Free boundary problem for an optimal investment problem with a borrowing constraint

“…Li and Xu [13] consider a continuous-time Markowitz's model with bankruptcy prohibition and convex cone portfolio constraints. Guan [8] discusses an investment problem with different interest rates. In this paper, we incorporate the borrowing constraint into our model under the expected utility framework.…”