stochastic logarithm

上一篇 / 下一篇  2011-12-11 00:03:55 / 個人分類:隨機微積分

   IF U=epsilon(X) then X is called the stochastic logarithm of U,denoted L(U)
let U has a SDE and not take value 0
then stochastic logarithm of U stasify the SDE

dX(t)=dU(t)/U(t)
X(0)=0
X(t)=L(U)(t)=ln(U(t)/U(0))+{d[U,U](t)/2U^2 is integred on (0,t)}


U(t)=exp(Bt)
so dU(t)=exp(Bt)+1/2*exp(Bt)dt
hence dXt=dL(U)=dUt/Ut=dBt+(1/2)*dt thus Xt=L(U)=Bt+(1/2)*t

L(U)=Bt+1/2*t


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