A First Look at Stochastic Processes (Paperback)

This textbook introduces the theory of stochastic processes, that is, randomness which proceeds in time. Using concrete examples like repeated gambling and jumping frogs, it presents fundamental mathematical results through simple, clear, logical theorems and examples. It covers in detail such essential material as Markov chain recurrence criteria, the Markov chain convergence theorem, and optional stopping theorems for martingales. The final chapter provides a brief introduction to Brownian motion, Markov processes in continuous time and space, Poisson processes, and renewal theory.

Interspersed throughout are applications to such topics as gambler's ruin probabilities, random walks on graphs, sequence waiting times, branching processes, stock option pricing, and Markov Chain Monte Carlo (MCMC) algorithms.

The focus is always on making the theory as well-motivated and accessible as possible, to allow students and readers to learn this fascinating subject as easily and painlessly as possible.

這本教科書介紹了隨機過程的理論，即隨機性隨時間推移。通過具體的例子，如重複賭博和跳躍青蛙，它通過簡單、清晰、邏輯的定理和例子呈現了基本的數學結果。它詳細介紹了馬爾可夫鏈的遞歸條件、馬爾可夫鏈收斂定理以及對於鞅的可選停止定理等重要材料。最後一章簡要介紹了布朗運動、連續時間和空間中的馬爾可夫過程、泊松過程和更新理論。

散佈在全書中的應用包括賭徒破產概率、圖上的隨機遊走、序列等待時間、分支過程、股票期權定價以及馬爾可夫鏈蒙特卡羅（MCMC）算法。

重點始終是使理論盡可能具有動機和易於理解，以便讓學生和讀者能夠盡可能輕鬆無痛地學習這個迷人的主題。