We consider an incomplete market with a non-tradable stochastic factor and an investment problem with  optimality criterion based on a functional which is a modification of a monotone mean-variance preferences. We formulate it as a stochastic differential game problem and use Hamilton Jacobi Bellman Isaacs equations to derive the optimal investment strategy and the value function. Finally , we show that our solution coincides with the solution to classical mean-variance problem with risk aversion coefficient which is dependent on stochastic factor .
We consider an incomplete market with a nontradable stochastic factor and a continuous time investment problem with an optimality criterion based on monotone mean-variance preferences. We formulate it as a stochastic differential game problem and use Hamilton-Jacobi-Bellman-Isaacs equations to find an optimal investment strategy and the value function. What is more , we show that our solution is also optimal for the classical Markowitz problem and every optimal solution for the classical Markowitz problem is optimal also for the monotone mean-variance preferences. These results are interesting because the original Markowitz functional is not monotone, and it was observed that in the case of a static one-period optimization problem the solutions for those two functionals are different. In addition, we determine explicit Markowitz strategies in the square root factor models .
R
non-tradable
nontradable
R
an
a continuous time
A
an
D
a functional which is a modification of a
R
Hamilton Jacobi Bellman Isaacs equations to derive the
Hamilton-Jacobi-Bellman-Isaacs equations to find an
R
Finally
What is more
R
coincides with the solution to classical mean-variance problem with risk aversion coefficient which is dependent on stochastic factor
is also optimal for the classical Markowitz problem and every optimal solution for the classical Markowitz problem is optimal also for the monotone mean-variance preferences. These results are interesting because the original Markowitz functional is not monotone, and it was observed that in the case of a static one-period optimization problem the solutions for those two functionals are different. In addition, we determine explicit Markowitz strategies in the square root factor models