## Economics

LML Fellows **Ole Peters** and **Alex Adamou **spend most of their time with LML’s economics project. They maintain a **blog with background information and latest news**, including a periodically updated set of **lecture notes**.

The basis of the project is summarized as follows:

- Economics was the first discipline to develop the mathematics of randomness (17th century).
- The early conceptualization of randomness has a flaw: it assumes that randomness playing out over time has the same effect as randomness playing out over an ensemble of parallel systems (parallel universes).
- Economics noticed symptoms that arise from this flaw and designed tools, most notably utility theory (18th century), that cure some of them.
- In the 19th century a new conceptualization of randomness emerged in the context of physics, namely in thermodynamics and statistical mechanics. This conceptualization recognizes from the start the central role of time. It thereby resolves the fundamental flaw, rather than treating its symptoms.

Peters used these insights to provide a solution of the leverage optimization problem in finance [1]. He also published a solution of the 300-year-old St. Petersburg paradox [2], a key problem in the foundations of both economics and probability theory.

Peters and Adamou [4] extended the work on leverage optimization [1] to test a prediction it makes about the nature of fluctuations in stock prices. The prediction turned out to be correct. This produces a solution to the Equity Premium Puzzle and suggests an algorithm for setting central bank interest rates.

Following a suggestion from economics Nobel Laureate Ken Arrow, Peters and Adamou [6] published a solution of the insurance puzzle: why do people buy insurance although the transaction reduces their expected wealth? By extension this explains, and helps price, financial derivatives.

Peters and physics Nobel Laureate Murray Gell-Mann [7] published a detailed solution of the gamble-selection problem, which is the foundation of economic decision theory.

In 2016 Adamou and Peters [8] pointed out that the techniques and concepts developed in this context lead to deep insights into the dynamics of economic inequality. Berman, Peters, and Adamou [9] published a detailed study of American wealth distributions.

[9] Y. Berman, O. Peters, and A. Adamou

*Far from equilibrium: Wealth reallocation in the United States.*

**arXiv:1605.05631** (2016).

[8] A. Adamou and O. Peters

*Dynamics of inequality.*

*Significance ***13**, 3, 32–37 (2016).

** doi:10.1111/j.1740-9713.2016.00918.x**

[7] O. Peters and M. Gell-Mann

*Evaluating gambles using dynamics.*

*Chaos ***26**, 023103 (2016).

**arXiv:140585**

** doi:10.1063/1.4940236**

[6] O. Peters and A. Adamou

*Rational insurance with linear utility and perfect information.*

**arXiv:1507.04655** (2015).

[5] O. Peters and W. Klein

*Ergodicity*** b**reaking in geometric Brownian motion.

*Phys. Rev. Lett. *110, 100603 (2013).*
*arXiv:1209.4517

**doi:10.1103/PhysRevLett.110.100603**

[4] O. Peters and A. Adamou

*Stochastic Market Efficiency.
*

*Santa Fe Institute Working Paper*#2013-06-022 (2013).

www.santafe.edu/media/workingpapers/13-06-022.pdf

*arXiv:1101.4548*

www.santafe.edu/media/workingpapers/13-06-022.pdf

[3] O. Peters

*Menger 1934 revisited.*

**arXiv:1110.1578 **(2011).

[2] O. Peters

*The time resolution of the St Petersburg paradox.*

*Phil. Trans. R. Soc. A* 369, 369, 1956, 4913–4931 (2011).

**arXiv:1011.4404**

**doi:10.1098/rsta.2011.0065**

[1] O. Peters

*Optimal leverage from non-ergodicity.*

*Quant. Fin.* **11**, 11, 1593–1602 (2011).

**arXiv:0902.2965v2**

**doi:/10.1080/14697688.2010.513338**