Stochastic Processes

Let the stochastic process Y(t)=F(X(t),t) for every t, where X is a Wiener process. Then, Ito's Lemma states that the stochastic differential equation for Y can be written

\(dY(t)=∂F∂tdt+∂F∂XdX+12∂2F∂X2(dX)2 Now, suppose that Y(t)=F(X(t),t)=eσX(t)−σ2t/2. Then, the partial derivatives of F are

∂Y∂t∂Y∂X∂2Y∂X2=−12σ2Y=σY=σ2Y Ito's Lemma then gives

dY=(−12σ2)dt+(σ)dX+(12σ2)(dX)2=(−12σ2)dt+(σ)dX+(12σ2)(dt)=σYdX\)