[英]Milstein algorithm when variance of the noise is different from 1
I want to implement the Milshtein algorithm for an stochastic equation in which the noise is additive. 我想针对一个随机方程(其中的噪声是可加的)实现Milshtein算法。 The equation has the next form.
该方程式具有下一种形式。
dx(t)/dt= q(x(t)) + noise(t)
The noise is a gaussian variable of zero mean and variance 5. The expression of the milsthein algorithm I found in books for this expression reads as follows 噪声是均值为零且方差为5的高斯变量。我在书中为该表达式找到的milsthein算法的表达式如下:
x(t)=x(t) + h q(x(t)) + sqrt(h) u
where "h" is the step of the algorithm, "sqrt" means "square root" and "u" is a gaussian random variable of mean 0 and variance 1. 其中“ h”是算法的步骤,“ sqrt”表示“平方根”,“ u”是均值0和方差1的高斯随机变量。
However, if I want to have a noise of variance 5, should I just make "u" a gaussian variable of variance 5 or something else should be changed? 但是,如果我想使用方差5的噪声,是否应该使“ u”成为方差5的高斯变量,还是应该进行其他更改?
You're correct. 没错 Given your revised stochastic differential equation,
根据您修改后的随机微分方程,
dx(t)/dt = q(x(t)) + √5 noise(t),
we can derive the correct formula using only the linearity of differentiation and formula for variance-1 noise. 我们可以仅使用微分线性和方差1噪声的公式得出正确的公式。 Let
p(z) = q(√5 z)/√5
and y(t) = x(t)/√5
. 设
p(z) = q(√5 z)/√5
和y(t) = x(t)/√5
。
dx(t)/dt = q(x(t)) + √5 noise(t)
= √5 [q(√5 x(t)/√5)/√5 + noise(t)]
= √5 [p(x(t)/√5) + noise(t)] change q -> p where z = x(t)/√5
(dx(t)/dt)/√5 = p(x(t)/√5) + noise(t)
d(x(t)/√5)/dt = p(x(t)/√5) + noise(t) linearity of differentiation
dy(t)/dt = p(y(t)) + noise(t) change x -> y
The update formula for y
is given by Milstein: y
的更新公式由Milstein给出:
y(t) -> y(t) + h p(y(t)) + √h u.
We can derive the update formula for x
. 我们可以推导
x
的更新公式。
x(t)/√5 -> x(t)/√5 + h p(x(t)/√5) + √h u
x(t) -> x(t) + √5 h q(x(t))/√5 + √5 √h u
-> x(t) + h q(x(t)) + √h (√5 u)
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