On Stochastic Differential Equations in the Ito and in the Stratonovich Sense
Freidlin, Mark I
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In this paper I looked into some modifications of the standard diffusion equation. First I added ``look back'' in the differential equation and proved that the solution of the new equation converged to the solution of the diffusion equation in the Ito sense. Then I proved that if we use an approximation to the Weiner process as well as ``look back'' our solution will depend on the order in which we take the limits. Specifically if we first let the look back go to zero then let our approximation to the Weiner process converge to the Weiner process we will converge to the diffusion equation understood in the Stratonovich sense and if we first let our approximation converge to the Weiner process then let our ``look back'' go to zero we will converge to the Ito integral.