Stratonovich stochastic differential equations

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

Before embarking on this discussion one fact must be established. In many applications the autocorrelation function of L(t) is not really a delta function, but merely sharply peaked with a small rc > 0. Accordingly L(t) is a proper stochastic function, not a singular one. Then (4.5) is a well-defined stochastic differential equation (in the sense of chapter XVI) with a well-defined solution. If one now takes the limit rc —> 0 in this solution, it becomes a solution of the Stratonovich form (4.8) of the Fokker-Planck equation. This theorem has been proved officially, but the result can also be seen as follows. 

The Stratonovich interpretation of Eq. (2.238) may also be obtained [31,32] from the white-noise limit of a sequence of stochastic ordinary differential equations (ODEs) of the form... 

Here Cj(f) are solutions of the stochastic differential Ito-Stratonovich equation [26, 34, 99]... 

The method we describe is analogous to the one used by us [52] in order to study dielectric relaxation of polar fluids in the presence of a DC electric field using the Langevin equation. Our calculations are carried out by interpreting the Cartesian components of Gilbert s equation as a set of stochastic nonlinear differential equations of Stratonovich type [12]. 

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