随机变量的微分可以表示为dx=a(x)dt+b(x)dw(t)。其中w(t)为Weiner过程，满足(dw)^2=dt。
而对于变量x函数f(x)的微分，Gardiner书上推导f(x)Ito的微分形式的过程为

df(x)=f'(x)dx+(1/2)f''(x)dx^2=f'(x)[a(x)dt+b(x)dw(t)]+(1/2)f''(x)[a(x)dt+b(x)dw(t)]^2。考虑到(dw)^2=dt，且舍去dt的高阶项，有

df(x)=[f'(x)a(x)+(1/2)(b(x))^2]dt+f'(x)b(x)dw(t)。

我想问的是，
1，上式的关于Ito形式的微分推导在哪一部体现了Ito微积分的规则（其中(dw)^2=dt与积分规则无关）；
2，安装这样的程序推导Stratonovich微分形式，要如何推导？
谢谢

Stratonovich calculus 遵循普通微积分的链式法则

[ f(W_T) = f(W_0) + int_0^T f'(W_t),dW_t ]

谢谢季候风的回复。这个问题我还没完全搞懂。等一段时间再仔细考虑一下。我一开始很纳闷，随机积分有明确的定义，但随机函数的微分根本没见到明确的定义，那么它的形式是怎么样，怎么推导都无从谈起。
现在想了一下，这个不同积分规则下的函数微分的要求应该是这样定义的：

1，由dx=a(x)dt+b(x)dw(t)应用给定积分规则可以解出x(t),然后代回f(x)，可以得到一个函数表达，记为f1(x)；
2，由如果已知df（x），那么对int df(x)dx,应用同样积分规则也会得到一个函数，记为f2(x)。
所以df的含义应该是由f1(x)=f2(x)的条件定义。

至于如何得到stratonovich或者一般形式的积分规则下的函数微分形式，还要思考一下。

如果哪位网友能提供某本教材相关的推导，那就非常感谢了。

The rule is about the variance of [int(f'(W)) - f(W_T)].

On each t-subinterval, the problem roughly reduces to the order of the error
[ f(W1)-f(W0) - f'(W)(W1-W0) ], that is, the error of the rectangle approximation of the area under the function f'. The error of first order in (W1-W0)^2 will have finite contribution in the variance of sum (because sum(dW^2) = sum(dt) = T), and any higher order term will vanish in the limit of variance of sum as the length of subinterval goes to 0.

We can do an intuitive argument. If use endpoint value in f'(W), i.e. Ito integral, then the main error comes from the little triangle whose area is 1/2 * f''(W)(W1-W0)^2. But if use midpoint value of f'(W), i.e. Stratonovich integral, then the rectangle approaximation is very accrate and the error comes from the convexity of f'(W), that is, 1/6 * f'''(W)(W1-W0)^3 whose variance is o(t2-t1) and vanish in the limit.

In formula, let Wm be the Brownian motion at the midpoint, and g = f'. then
f'(Wm)(W1-W0) = g(Wm)(W1-W0) = [g(W0) + g'(W0)(Wm-W0) + o(Wm-W0)](W1-Wm+Wm-W0)
= g(W0)(W1-W0) + g'(W0)(Wm-W0)^2 + g'(W0)(Wm-W0)(W1-Wm) + o(Wm-W0)(W1-Wm+Wm-W0)

The first two terms contribut the Ito's forumla after taking variance, the variance of the third term is 0, and the variance the fourth term is of order o(dt) and vanish in the limit. That is, in Stratonovich calculus, one term is sufficient in the integral expansion (mathematically the differential formula is just a brief notion for the integral expansion).

再次谢谢季兄，这么详细的解释。