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This course is an introduction to the theory of stochastic processes. Informally, astochastic process is a random quantity that evolves over time, like a gambler’s netfortune and the price fluctuations of a stock on any stock exchange, for instance. Themain aims of this course are: 1) to provide a thorough account of basic probabilitytheory; 2) to introduce the ideas and tools of the theory of stochastic processes; and3) to discuss in depth important classes of stochastic processes, including MarkovChains (both in discrete and continuous time), Poisson processes, the Brownianmotion and Martingales. The course will also cover other important but less routinetopics, like Markov decision processes and some elements of queueing theory.

这是一份unsw新南威尔士大学MATH3801 的成功案例

概率和随机过程|MATH3801 Probability and Stochastic Processes

问题 1.

Suppose phi phi is convex in (a, b)(a, b) and let u in(a, b)u in(a, b). Define begin{aligned} &f(s)=f_{ phi, u}(s)= frac{ phi(u)- phi(s)}{u-s}, quad s in(a, u) &g(t)=g_{ phi, u}(t)= frac{ phi(t)- phi(u)}{t-u}, quad t in(u, b) end{aligned} begin{aligned} &f(s)=f_{ phi, u}(s)= frac{ phi(u)- phi(s)}{u-s}, quad s in(a, u) &g(t)=g_{ phi, u}(t)= frac{ phi(t)- phi(u)}{t-u}, quad t in(u, b) end{aligned} Then (a) ff and gg are non-decreasing, and (b) f(s) leq g(t)f(s) leq g(t) for any ss and tt from the domains of ff and gg, respectively.

证明 .

Proof To prove the statement for ff, we take ats>t (if s0s0, and sufficiently small h>0h>0, the function psi_{t, e, h}(u)=(q+h)(u-t)+ phi(t)- epsilon= psi_{t}(u)+h(u-t)- epsilon psi_{t, e, h}(u)=(q+h)(u-t)+ phi(t)- epsilon= psi_{t}(u)+h(u-t)- epsilon belongs to SS. We take h< min left( frac{ varepsilon}{s-t}, frac{ phi(s)- phi(t)}{s-t}-q right)h< min left( frac{ varepsilon}{s-t}, frac{ phi(s)- phi(t)}{s-t}-q right). Note that frac{ phi(s)- phi(t)}{s-t}> frac{ phi(s)- phi(t)}{s-t}> qq, by 1.5 .41.5 .4

For u leq s, psi_{t, e, h}(u) leq psi_{t}(u) leq phi(u)u leq s, psi_{t, e, h}(u) leq psi_{t}(u) leq phi(u) since h(u-t)- epsilon leq h(s-t)- epsilon leq 0h(u-t)- epsilon leq h(s-t)- epsilon leq 0.For u>tu>t, by 1.5 .4, frac{ phi(u)- phi(t)}{u-t} geq frac{ phi(s)- phi(t)}{s-t}>q+h1.5 .4, frac{ phi(u)- phi(t)}{u-t} geq frac{ phi(s)- phi(t)}{s-t}>q+h. Thus, psi_{t, e, h}(u) leq frac{ phi(u)- phi(t)}{u-t}(u-t)+ phi(t)- epsilon leq phi(u) psi_{t, e, h}(u) leq frac{ phi(u)- phi(t)}{u-t}(u-t)+ phi(t)- epsilon leq phi(u)

本课程是对随机过程理论的介绍。非正式地讲，一个随机过程是一个随时间变化的随机量，比如说赌徒的净财富和任何证券交易所的股票价格波动。财富和任何证券交易所的股票价格波动，例如。本课程的主要目的是本课程的主要目的是 1）提供基本概率理论的全面说明理论；2）介绍随机过程理论的思想和工具；以及3）深入讨论随机过程的重要类别，包括马尔科夫链（包括离散和连续时间）、泊松过程、布朗运动和马廷格。运动和马丁格尔。该课程还将涵盖其他重要的但不太常规的课题，如马尔可夫决策过程和排队理论的一些要素。