I will present results on the existence and uniqueness of a self-similar solution of a fully nonlinear, uniformly parabolic equation (an example of which include the Bellman-Isaacs equation arising in the theory of stochastic optimal control and stochastic differential game theory). Using the self-similar solution, we describe the long-time behavior of solutions to the Cauchy problem with nonnegative initial data, and derive a conservation law which generalizes the conservation of mass in the case of the heat equation. The scaling invariance property of the self-similar solution depends on the nonlinear operator, and is in general different from that of the heat kernel. We will see that this difference has a stochastic interpretation. This work is joint with M. Trokhimtchouk