伊藤の補助定理
この補助定理が金融工学によく使われているそうです。しかし数学的な導出と実施方法を承知していません。
Ito's lemma is based on the Ito diffusion process. I am not sure exactly what this is. In addition, the lemma itself is said to be the stochastic analog of the chain rule, but I am not sure why. Lastly, Ito's lemma is mentioned as being an integral component of the derivation of the Black-Scholes formula, but I am not sure why this is.
The Ito drift diffusion process is:
dX(t) = sigma(t)dB(t) + mu(t)dt
Given a twice differentiable function of t and x,
df(t,X(t)) = *1dt + sigma(t)(df/dx)dB(t)
The chain rule of traditional calculus can be expressed as follows. If z is a function of x and y, and x and y are functions of t, then:
dz/dt = (dz/dx)(dx/dt)+(dz/dy)(dy/dt)
How this is related to the above df differential I am not sure of.
In typical calculus, a function f(t,x) would have:
(df/dt) = (df/dt) + (df/dx)(dx/dt), where the df/dx,dx/dt,df/dt on the right are all partial derivatives and the df/dt on the left is an absolute derivative.
