Abstract: The author considered mathematical problem of a class of optimal investment. By variable substitution, the author transformed this mathematical problem into the initial boundary value problem of a class of parabolic MongeAmpère equation. Firstly, the author established a series of a priori estimates, and then using some analytical techniques including the continuity method, the author obtained the existence and uniqueness of the smooth solution of the
 initial boundary value problem.

Key words: parabolic MongeAmpère equation, optimal investment, initial boundary value problem