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非遍历性财富动力学、凯利准则与波动率税
从几何布朗运动伊藤引理、破产吸收壁到杠铃投资组合的随机控制及《史记·货殖列传》并读
Non-Ergodic Wealth Dynamics, the Kelly Criterion and Volatility Drag: From Itô's Lemma on Geometric Brownian Motion and Ruin Barriers to Barbell Portfolios, Read Beside the Shiji's 'Biographies of the Money-Makers'
DOI10.5281/zenodo.23754709摘要
在乘性的世界里，期望收益为正的选择，未必对一个只能活一次的人或企业长期有利。本书用概率论、随机过程与金融经济学考察这一区别，以及围绕它的流行说法哪些成立。依次讨论：一个每轮期望 +5%、时间平均却约 −5.13% 的抛硬币游戏，集合平均与时间平均、遍历性，彼得斯的遍历性经济学与伯努利对数效用的关系及经济学家的回应；几何布朗运动与伊藤引理，长期增长率 μ − σ²/2 与杠杆下的 r + L(μ − r) − L²σ²/2，两倍凯利杠杆回到无风险利率，每日重置杠杆产品的路径依赖；凯利准则 f* = (bp − q)/b 与 (μ − r)/σ²，半凯利保留约 75% 的增长，参数由有限数据估计时最优倍数 m²/(m² + s²) 小于 1，以及「曾经跌到起点 x 倍」的概率 x^(2/c−1)；带漂移布朗运动的首达时间与吸收壁，现金缓冲按指数降低破产概率，寒冬中削减刚性成本的作用；香农恶魔与再平衡溢价（约为方差的八分之一）及其在交易成本和趋势下的边界、分散化收益；肥尾分布下的极端事件频率，以及一个诚实的杠铃模型：限定最大损失，但高赔率部分定价偏高时长期流血。
八章与司马迁《史记·货殖列传》「无财作力，少有斗智，既饶争时」、计然「知斗则修备」「旱则资舟，水则资车」「贵上极则反贱，贱下极则反贵」、白圭「人弃我取，人取我与」以及无盐氏的十倍利息并读，并指出并读的限度；最后给出一份个人与企业现金流配置的四步生存与复利核查表，以及一项比较算术展示与遍历展示对重复乘性投资任务中所选杠杆影响的预注册随机对照行为实验方案（每组 235 人可分析、招募 277 人，判定规则写成代码）。书中以标注框区分已发表的证据、可运行的 Python 计算（15 段程序，全部核对）、原典与作者的假说。本书不是投资、财务、税务或法律建议。
关键词：遍历性经济学；时间平均；集合平均；几何布朗运动；伊藤引理；波动率拖累；杠杆；凯利准则；分数凯利；参数不确定性；赌徒破产；首达时间；吸收壁；现金缓冲；香农恶魔；再平衡；分散化收益；肥尾；杠铃策略；史记；货殖列传
Abstract
In a multiplicative world, a choice with a positive expected return is not necessarily good in the long run for a person or a firm that lives only once. This book examines that distinction with probability theory, stochastic processes and financial economics, and asks which popular claims about it hold. It covers a coin game with an expected return of +5 % a round and a time-average growth rate of about −5.13 %; ensemble and time averages and ergodicity; Peters' ergodicity economics, its relation to Bernoulli's logarithmic utility, and economists' replies; geometric Brownian motion and Itô's lemma, the long-run growth rate μ − σ²/2 and, with leverage, r + L(μ − r) − L²σ²/2, twice-Kelly leverage falling back to the risk-free rate, and the path dependence of daily-reset leveraged products; the Kelly criterion f* = (bp − q)/b and (μ − r)/σ², half Kelly keeping about 75 % of the growth, the best multiple m²/(m² + s²) below one when the edge is estimated from finite data, and the probability x^(2/c−1) of ever falling to x times the start; first-passage times of Brownian motion with drift and absorbing barriers, cash buffers lowering the probability of ruin exponentially, and the effect of cutting rigid costs in a downturn; Shannon's demon and the rebalancing bonus (about one-eighth of the variance) with its limits under transaction costs and trends, and the diversification return; the frequency of extreme events under fat tails, and an honest barbell model that caps the largest loss but bleeds slowly when its high-payoff part is overpriced.
The eight chapters are read beside Sima Qian's "Biographies of the Money-Makers" in the Shiji: "without capital, labour; with a little, wits; when rich, timing", Jiran's "prepare in advance", "in drought, invest in boats; in floods, carts" and "when the dear reaches its extreme it turns cheap", Bai Gui's "take what others discard, let go what others seek" and the Wuyan family's tenfold interest, noting the limits of such a reading. They end with a four-step survival-and-compounding checklist for household and business cash flow and allocation, and a pre-registered randomised controlled behavioural experiment comparing an arithmetic and an ergodic display of the same asset on the leverage chosen in a repeated multiplicative investment task (235 analysable and 277 recruited per arm, decision rule written as code). Marked boxes separate published evidence, runnable Python calculations (15 programs, all checked), classical sources and the author's hypotheses. This book is not investment, financial, tax or legal advice.
Keywords: ergodicity economics; time average; ensemble average; geometric Brownian motion; Ito's lemma; volatility drag; leverage; Kelly criterion; fractional Kelly; parameter uncertainty; gambler's ruin; first-passage time; absorbing barrier; cash buffer; Shannon's demon; rebalancing; diversification return; fat tails; barbell strategy; Shiji; Huozhi liezhuan
目录 · Contents
- 序：平均数骗了谁English 中文Preface: Whom Does the Average Deceive?
- 第一章 集合平均不等于时间平均English 中文Chapter 1 The Ensemble Average Is Not the Time Average
- 第二章 几何布朗运动、伊藤引理与波动率税English 中文Chapter 2 Geometric Brownian Motion, Itô's Lemma and Volatility Drag
- 第三章 凯利准则与分数凯利English 中文Chapter 3 The Kelly Criterion and Fractional Kelly
- 第四章 破产吸收壁与首达时间English 中文Chapter 4 Absorbing Barriers and First-Passage Times
- 第五章 香农恶魔与再平衡溢价English 中文Chapter 5 Shannon's Demon and the Rebalancing Bonus
- 第六章 尾部风险、肥尾分布与杠铃策略English 中文Chapter 6 Tail Risk, Fat Tails and the Barbell
- 第七章 原典并读：《史记·货殖列传》的治生之道English 中文Chapter 7 Reading the Classics: The "Biographies of the Money-Makers"
- 第八章 实践与预注册试验English 中文Chapter 8 Practice and a Pre-registered Experiment
How to cite · 引用
Fan, Y. (2026). Non-Ergodic Wealth Dynamics, the Kelly Criterion and Volatility Drag: From Itô's Lemma on Geometric Brownian Motion and Ruin Barriers to Barbell Portfolios, Read Beside the Shiji's 'Biographies of the Money-Makers' [非遍历性财富动力学、凯利准则与波动率税] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23754709
BibTeX
@book{fan2026ergodicity,
title = {{Non-Ergodic Wealth Dynamics, the Kelly Criterion and Volatility Drag}: From Itô's Lemma on Geometric Brownian Motion and Ruin Barriers to Barbell Portfolios, Read Beside the Shiji's 'Biographies of the Money-Makers'},
author = {Fan, Yafu},
year = {2026},
publisher = {5D Library},
edition = {Version 1.0.0},
doi = {10.5281/zenodo.23754709},
url = {https://doi.org/10.5281/zenodo.23754709},
note = {Chinese title: 非遍历性财富动力学、凯利准则与波动率税}
}
| DOI | https://doi.org/10.5281/zenodo.23754709 |
|---|---|
| DOI（全部版本 all versions） | https://doi.org/10.5281/zenodo.23754708 |
| ORCID | https://orcid.org/0009-0004-3519-1203 |
| 5D Library ID | 5DL-B-019 |
| 版本 Version | 1.0.0 · 中英双语版 Bilingual edition |
| 出版日期 Date | 2026-10-29 |
| 网址 Online | https://5dlibrary.com/books/ergodicity-kelly-wealth/ |
| SHA-256 · ergodicity-kelly-wealth_bilingual_v1.0.0.pdf | e2aff501c1627b3dd9c20a8c8364012948425e8e54d997a7f33a277b25272aa1 |
| SHA-256 · ergodicity-kelly-wealth_en_v1.0.0.pdf | 740e007b719f65c4e5122356216bc61c0dbb9bd5df1e6a96c34870965a6f0d18 |
| SHA-256 · ergodicity-kelly-wealth_zh_v1.0.0.pdf | 6e5f6fb703fa222378d74dc5aad0bed17e6b0eb330ed2fa29922c4035f24ccaf |
| Licence · 许可 | Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0). https://creativecommons.org/licenses/by-nc-nd/4.0/ |
Disclosure. The author founded and runs Asia Influencer X and Yafu Global, which work in influencer and digital marketing; he holds no financial-services licence and is not a licensed investment adviser. The book links to and recommends no broker, fund, trading software, trading course or financial product; the Chapter 8 experiment is designed to be run and analysed by researchers independent of the author's businesses and recruits no participants from their clients, partners or communities; all the code and the study protocol are published openly for anyone to check. The author also uses this book and his own scripts (for personal market-watching study, not sold or offered to anyone) as a methodological reference for his community courses, the 5D Ascension Cards (Dimensional Shift Card Decks) and the Yafu AI knowledge base; he holds no financial-services licence and provides no investment advice, discretionary account management or trading-signal service.