维纳过程和伊藤引理.pptVIP

Chapter 2: Brownian Motions and Stochastic Integrals Wiener Processes: Brownian Motions, Ito Processes W(tk+1)= W(tk) + e(tk) ??t, where ?t = tk+1 – tk, k=0,…,N, t0 = 0, and e(tk) iid N(0,1). For jk, W(tk) - W(tj) = ?i=jk-1 e(ti) ??t. The right-hand side is normally distributed, so is the left-hand side. Clearly, E(W(tk) - W(tj) ) = 0. Var (W(tk) - W(tj)) = E [?i=jk-1 e(ti) ??t]2 = (k-j) ?t = tk – tj. For t1 t2 ? t3 t4, W(t4) - W(t3) is uncorrelated with W(t2) - W(t1). Simulation of Brownian Motion Partition [0,1] into n subintervals each with length 1/n. For each t in [0,1], let [nt] denote the greatest integer part of the number nt. For example, if n=10 and t=1/3, then [nt] =[10/3]=3. For each t in [0,1], define a stochastic process S[nt] = ?i=1[nt] e(i)/?n, e(i) iid N(0,1). Clearly, S[nt] = S[nt]-1 + e([nt])/ ?n, a special form of the additive model defined at the beginning with ?t =1/n and W(t)= S[nt]. At time t=1, S[nt] = Sn = ?i=1n e(i)/?n, which has a standard normal distribution. Even if e(i)s are not normally distributed, CLT shows that Sn will still be normally distributed as long as n is big. This is the key idea in constructing a Brownian motion. By letting n goes to infinity (?t goes to 0), Donsker (1953) proved that the stochastic process S[nt] constructed in this way tends to a Brownian motion. This result is known as the Functional Central Limit Theorem or the Invariance Principle. Therefore, the above equation provides a means to simulate the path of a Brownian motion. All we have to do is to iterate the equation S[nt] = S[nt]-1 + e([nt])/ ?n by taking n bigger and bigger and in the limit, we have a Brownian motion. When ?t goes to 0, the discrete-time random walk equation S[nt] = S[nt]-1+e([nt])/?n, can be approximated by the continuous-time equation dW(t) = e(t)?dt . In advanced courses in Probability, it will be shown that this limiting operation is well-defined and the limitin