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伊藤引理及推导——应用于随机神经网络等微分方程
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2010-11-09
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伊藤引理及推导——应用于随机神经网络,随机控制系统,社会网络,经济管理方程等微分方程
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Ito’s Lemma and its Derivation
Huiwei Wang
April 22, 2009
Ito’s Lemma is named for its discoverer, the brilliant Japanese mathematician Kiyoshi Ito. The human race lost this extra-
ordinary individual on November 10, 2008. He died at age 93. His work created a field of mathematics that is a calculus of
stochastic variables.
Ito’s lemma (also known as Ito-Doeblin Theorem) is a theorem in stochastic calculus. It tells you that if you have a random
walk, in Y , say, and a function of that randomly walking variable, call it F (Y, t), then you can easily write an expression for
the random walk in F . A function of a random variable is itself random in general.
Changes in a variable such as stock price involve a deterministic component which is a function of time and a stochastic
component which depends upon a random variable. Let S be the stock price at time t and let dS be the infinitesimal change in
S over the infinitesimal interval of time dt. The change in the random variable z over this interval of time is dz. The change in
stock price is given by
dS = adt + bdz, (1)
where a and b may be functions of S and t as well as other variables; i.e., dS = a(S, t, x)dt + b(S, t, x)dz.
The expected value of dz is zero so the expected value of dS is equal to the deterministic component, adt.
The random variable dz represents an accumulation of random influences over the interval dt. The Central Limit Theorem
then implies that dz has a normal distribution and hence is completely characterized by its mean and standard deviation. The
mean or expected value of dz is zero. The variance of a random variable which is the accumulation of independent effects over
an interval of time is proportional to the length of the interval, in this case dt. The standard deviation of dz is thus proportional
to the square root of dt,(dt)
1
2
. All of this means that the random variable dz is equivalent to a random variable w(dt)
1
2
, where
w is a standard normal variable with mean zero and standard deviation equal to unity.
Now consider another variable C, such as the price of a call option, which is a function of S and t, say C = f (S, t). Because
C is a function of the stochastic variable S, C will have a stochastic component as well as a deterministic component. C will
have a representation of the form:
dC = pdt + qdz. (2)
where p and q may be functions of S, t and possibly other variables; i.e., p = p(S , t, x) and q = q(S, t, x).
The crucial problem is how the functions p and q are related to the functions a and b in the equation
dS = adt + bdz. (3)
Ito’s Lemma gives the answer. The deterministic and stochastic components of dC are given by:
p =
∂f
∂t
+ (
∂f
∂S
)a +
1
2
(
∂
2
f
∂S
2
)b
2
q = (
∂f
∂S
)b. (4)
1
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- jk15215532015-07-08谢谢,帮助了我学习。
wanghuiwei_1984
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