Applications of Ito's formula to stochastic mechanics

The Ito formula is applied to find the local solution of the Schrodinger equation and develop partial differential equations for the characteristic function of the state X of a dynamic system subjected to semimartingale noise. It is shown that the solution of the Schrodinger equation at an arbitrary point a: of the domain of definition of this equation is equal to an expectation depending on the boundary conditions of the Schrodinger equation and sample properties of a Brownian motion starting at a. This expectation cannot be obtained analytically but can be estimated by Monte Carlo simulation. The partial differential equation for the characteristic function of the state X provides similar information as the Fokker-Planck equation but it is simpler to solve when dealing with some dynamic systems driven by non-Gaussian white noise. Numerical examples are used to demonstrate the proposed methods.