Euler forward-difference scheme

O(At) signifies that in the above approximation the leading term that was neglected is of the order At (we have divided (8-6) by At to get (8-7)). This is the so-called Euler forward-difference scheme. While it is only first-order accurate in At, it has the advantage that it allows for the quantities at timestep n + l being calculated only from those known at timestep n. 

All the terms in the right-hand side of Eq. 12.137 are known, and hence, this makes the forward difference in time rather attractive. However, as in the Euler method (Chapter 7), this forward difference scheme in time suffers the same handicap, that is, it is unstable if the grid size is not properly chosen. Using the stability analysis (von Rosenberg 1969), the criterion for stability (see Problem 12.12) is... 

Assuming a forward difference Euler scheme (Reddy, 1993)... 

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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