stochastic exponential

上一篇 / 下一篇  2011-12-10 23:59:17 / 個人分類:隨機微積分

   X has a stochastic differential and U satisfy dU(t)=U(t)*dX(t)
U(0)=1
U(t)=1+[U(s)對X(s)在(0,t)Rieman-stieltjes積分]
then U is the stochastic exponential of X is denoted epsilon(X)

IF X(t) is finte variation ,then dU(t)=U(t)*dX(t) solution is given by U(t)=exp(X(t))

IF X(t) is Ito process ,the solution is U(t)=epsilon(X)(t)=exp{X(t)-X(0)-(1/2)*[X,X](t)}
ex:S(t) is stock price and its Ito process R(t) is the process of the return dR(t)=dS(t)/S(t)---->dS(t)=dR(t)S(t)

 so stock price is the stochastic exponential of the return




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