Brownian Motion and the Stochastic Behaviour of Stocks

Published in Journal of Mathematical Finance, 2021

In this paper, we test the effectiveness of predicting the behavior of stocks utilizing stochastic calculus. We begin by exploring the intuition of Brownian motion by explaining its birth through the observations of Robert Brown and later through Bachelier’s work on its applications to the financial market and finally its rigorous and concretized form proposed by Norbert Wiener. The aforementioned motivates a stochastic differential equation to model the future price fluctuations of a stock wherein It\hat{o} integration is prominent and consequently expanded upon. The final part of this paper focuses on the accuracy of the model by back testing it with Apple stock and deriving a correlation coefficient.

Recommended citation: Tassopoulos, Pantelis, and Yorgos Protonotarios. "Brownian Motion & the Stochastic Behavior of Stocks." Journal of Mathematical Finance 12.1 (2021): 138-149.
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The KPZ Fixed Point and the Directed Landscape

Published in arXiv, 2024

The term “KPZ” stands for the initials of three physicists, namely Kardar, Parisi and Zhang, which, in 1986 conjectured the existence of universal scaling behaviours for many random growth processes in the plane.

A process is said to belong to the KPZ universality class if one can associate to it an appropriate “height function” and show that its 3:2:1 (time : space: fluctuation) scaling limit, see 1.2, converges to a universal random process, the KPZ fixed point. Alternatively, membership is loosely characterised by having: 1. Local dynamics; 2. A smoothing mechanism; 3. Slope-dependent growth rate (lateral growth); 4. Space-time random forcing with the rapid decay of correlations.

The central object that we will study is the so-called KPZ fixed point, which belongs to the KPZ universality class. Many strides have been made in the last couple of decades in this field, with constructions of the KPZ fixed point from certain processes such as the totally asymmetric simple exclusion process (with arbitrary initial condition) and Brownian last passage percolation.

In this essay, we: 1. delineate the origins of KPZ universality; 2. describe and motivate canonical models; 3. give an overview of recent developments, especially those in the 2018 Dauvergne, Ortmann and Virag (DOV) paper; 4. present the strategy of and key points in the proof of the absolute continuity result of the KPZ fixed point by Sarkar and Virag; 5. conclude with remarks for future directions. The presentation is such that the content is displayed in a way that is as self-contained as possible and aimed at a motivated audience that has mastered the fundamentals of the theory of probability.

Recommended citation: Pantelis Tassopoulos. (2024). The KPZ fixed point and the Directed landscape.
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New outlook on profitability of rogue mining strategies

Published in Preprint, 2022

Many of the recent works on the profitability of rogue mining strategies hinge on a parameter called gamma (γ) that measures the proportion of the honest network attracted by the attacker to mine on top of his fork. Cyril Grunspan and Ricardo P´erez-Marco, in two papers released in 2018, have surmised conclusions based on premises that erroneously treat γ to be constant. In this paper, we treat γ as a stochastic process and attempt to find its distribution through a Markov analysis. We begin by making strong assumptions on gamma’s behaviour and proceed to translate them mathematically in order to apply them in a Markov setting. The aforementioned is executed in two separate occasions for two different models. Furthermore, we model the Bitcoin network and numerically derive a limiting distribution whereby the relative accuracy of our models is tested through a likelihood analysis. Finally, we conclude that even with control of 20% of the total hashrate, honest mining is the strongly dominant strategy.

Recommended citation: Pantelis Tassopoulos and Yorgos Protonotarios. (2022). New outlook on profitability of rogue mining strategies.
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