Stratonovich calculus

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 -... 

Stratonovich SDEs, unlike Ito SDEs, may thus be manipulated using the familiar calculus of differentiable functions, rather than the Ito calculus. This property of a Stratonovich SDE may be shown to follow from the Ito transformation rule for the equivalent Ito SDE. It also follows immediately from the definition of the Stratonovich SDE as the white-noise limit of an ordinary differential equation, since the coefficients in the underlying ODE may be legitimately manipulated by the usual rules of calculus. 

Using a Langevin dynamics approach, the stochastic LLG equation [Eq. (3.46)] can be integrated numerically, in the context of the Stratonovich stochastic calculus, by choosing an appropriate numerical integration scheme [51]. This method was first applied to the dynamics of noninteracting particles [51] and later also to interacting particle systems [13] (see Fig. 3.5). 

As long as L(t) is not singular one is free to apply any transformation of y using the familiar rules of calculus in particular one may transform (4.5) into the quasilinear equation (4.3). In the latter the limit rc — 0 gives no problem and the result is (4.4). Transforming back to the original variables gives (4.8), as we found before. Hence, whenever the delta function stands for a sharp, but not infinitely sharp peak the Stratonovich interpretation is appropriate. The Ito interpretation cannot even be formulated unless tc is strictly zero. 

Ito [51] obtains different expressions for the Kramers-Moyal coefficients in which the spurious drift term is absent. However use of Ito coefficients involves new rules for calculus and so Stratonovich s method will be used here since it is also in agreement with the original method of Brown [8] and is the correct definition to use in the case of a physical noise which always has a finite correlation time [58] (see B.2). 

Historically, the two major interpretations are from Ito and Stratonovich. In both these formulations, (x) = 5(x) l However, (x) = A(x) in Ito interpretation while (x) = A(x)-jd B(x) in the Stratonovich interpretation. From the practical perspective, Ito interpretation allows one to simulate the SDE using the usual forward Euler scheme. However, special differentiation and integration rules are required for analytical calculations. On the other hand, Stratonovich interpretation allows using the regular rules of calculus but has to be simulated using implicit schemes. We emphasize that the FPE does not suffer from such ambiguity of interpretation SDEs corresponding to different interpretations of the same FPE lead to the same physical results [3, 7]. 

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