金融随机分析 II Stochastic process.ppt

* 2.4.5 First Passage Time Fix x 0. Define Theorem 1 = * Conclusion 1 Conclusion 2. Brownian motion reaches level x with probability 1. The expected time to reach level x is infinite, that is Remark * 补充 矩母函数（Laplace变换） 为X的矩母函数.（moment generating function) 定义* 对随机变量X及其分布函数F(x),若积分 在某一区间 上存在且有限，则定义区间 上的函数 * 矩母函数有如下性质: 2.设 独立,矩母函数存在, ， 则 4. 若 在包含原点的区间 上存在，则在此区间上的各阶导数都存在，且X的k阶矩可由矩母函数表示： * * * * * * * * * * * * Stochastic calculus for finance 2. Stochastic process 2.1 Stochastic process Def 2.1.1 A stochastic process is a parametrized collection of random variables defined on a probability space (?,F, P) and assuming values in Rn , The parameter space T is usually the halfline ,but it may also be an interval ,the non-negative and even subsets of Rn for n ?1. Which is called a path of Xt. Note that for each t fixed we have a random variable On the other hand, fixing ? ? ? we can consider the function The (finite-dimensional) distributions of the process X={Xt}t ?T are the measures defined on Rnk k=1,2,..., by Here F1, ...,Fk denote Borel sets in Rn. THEOREM (Kolmogorov’s extension theorem). For all measuers on Rnk s.t. let be probability K1 for all permutations ? on {1,2,...k} K2 for all m ? N ,where (of course ) the right hand side has a total of k+m factors. Then there exists a probability space (?,F, P) and a stochastic process {Xt} on ?, s.t. for all ti ?T and all Borel sets Fi. * DEFINTION 2.2.1 An n-dimensional stochastic process {Mt}t≥0 on (?, F ,P) is called a martingale (resp. submartingale, supermartingale) with respect to a filtration {Ft}t≥0 (and with respect to P0) if (Ⅰ) {Mt} is Ft -adapted (Ⅱ) E[| Mt |]∞ for all t, and (III) E[Mt | Fs]= Ms(resp. ≥,≤), a.s. , for all s≤t . (Note: If t∈T={0,1,2,….},then {Mt} is a martingale (resp. submartingale, supermartingale) if and only if E[Mk+1 | Fk]= Mk(resp. ≥,≤), a.s.