lec5_slide(伊藤积分和随机微分方程)

fg(t)is a continuousrandomprocessXt?h(t) =f(Xt)is also a random process.?h(t) =f(Xt+?Xt)?f(Xt)=f′(Xt)?Xt+O((?Xt)2),?Is(?Xt)2=O(?t2)?Lecture 5: It?o’s Fomula and Stochastic Differential Equations – p. 2/??Functions of process and the chain rule?For differentiableg: [0,+∞)7→IR,f:IR7→IR,?h(t) =f(g(t))is also a function on IR+?h(t) :=h(t+?t)?h(t) =f(g(t+?t))?f(g(t))=f(g(t) +?g)?f(g(t))=f′(g(t))?g(t) +O((?g(t))2)?By(?g(t))2=O(?t2),dh(t) =f′(g(t))dg(t).?Ifg(t)is a continuousrandomprocessXt?h(t) =f(Xt)is also a random process.?h(t) =f(Xt+?Xt)?f(Xt)=f′(Xt)?Xt+O((?Xt)2),?Is(?Xt)2=O(?t2)?(?Wt)2=O(?t).Lecture 5: It?o’s Fomula and Stochastic Differential Equations – p. 2/??It?o’s formula for BM?ForXt=μt+σWt, i.e.,dXt=μdt+σdWtLecture 5: It?o’s Fomula and Stochastic Differential Equations – p. 3/??