I_Probablity_theory-金融随机分析.pdf

Stochastic calculus for Stochastic calculus for finance finance 1 Probablity theory 1 Probablity theory Chen Ping 1 参考书： [1]Steven E Severve,Stochastic calculus for finance 2,Springer,2008 [2] Bernt ksendal ,Stochastic differential equations, An introduction with applications Springer, 2000 2 Introduction PROBLEM Consider the simple stock price model dS t    t S t , S 0 x (1.1)     dt Where S(t) is the stock price at time t, and μ(t) is the relative rate of growth at time t. It might happen that μ(t) is not completely known, but subject to some random environmental effects, so that we have (t) r(t) noise Where we do not know the exact behaviour of the noise term, only its probability distribution. the function r(t)is assumed to be nonrandom. How do we solve (1.1)in this case? 1 Probablity theory  1.1 Probability and measure theory  1.2 Random variable,distribution  1.3. Integration using general probability measure  1.4 conditional Expectation  1.5 Stochastic process  1.6 martingales  1.7 Markov Processes 4 1.1 Probability and measure theory Definition1.1.1: A -algebra is a collection  of subsets of  with the following three properties: 1.   1. If A , then its complement Ac     3. Ak , Ak  k 1  A pair (,