Existence of solutions to initial value problem for a parabolic Monge-Ampère equation and application

For the initial value problem of the parabolic Monge-Ampère equation VtVxx+rxVxVxx-θVx2=0, (x,t)∈R×[0,T) arising from the optimal investment of mathematical finance theory, we establish the existence of solutions, whose application is also given. Here the initial function is unbounded, and a special property is required for the solution to satisfy.