5. Conclusions
In this paper, we study Lie symmetry analysis for a parabolic Monge-Ampere ` equation in the optimal investment theory . As a result, infinitesimal generator, commutation table of Lie algebra and optimal system of this equation are derived. With the help of optimal systems, symmetry reductions and invariant solutions are obtained. It is shown that the the value function of self-financing optimal investment problem is a smooth and monotonically function of initial wealth y. For n-dimensional case, let r = √ y 2 1 + · · · + y 2 n , Eq.(2.8) becomes Eq.(2.11), so discuss the case n = d = 1 that involve the general case. The solutions which we obtain can be used to self-financing optimal investment problem and to check on the accuracy and reliability of numerical algorithm of Hamilton-Jacobi-Bellman equation (2.8).
