Stochastic processes have been applied to various areas of research to model systems that evolve probabilistically over time.
By the 1930s, Brownian motion (i.e. the random movement of microscopic particles suspended in a liquid or gas, caused by collisions between the said particles and the liquid or gas molecules), as well as a number of continuous time pure jump processes such as the Poisson process (i.e. a process used to explain mathematically the arrival times of objects or individuals entering a queue) had already been extensively studied.
These processes can be considered as the building blocks of processes which are nowadays called Lévy processes. This is because such processes are a superposition of a Brownian motion process and a possibly infinite number of independent Poisson processes.
Lévy processes are a subclass of stochastic processes that are stochastically continuous (i.e. the paths of such a process can experience sudden jumps that occur at random time points) and have increments that are stationary and independent.
During the 1930s, a number of mathematicians, including Lévy, Kolmogorov, Khintchine, Gnedenko and Itô, played an instrumental role in defining and characterising these processes. It must be said that originally such processes were simply referred to as processes with stationary and independent increments.
It was mainly during the 1980s and 1990s that such processes started being called Lévy processes in honour of French mathematician Paul Lévy, who had made major contributions to the area. Tools and techniques from various mathematical areas such as differential geometry, stochastic integration and probability theory in abstract spaces were used to better understand and properly define Lévy processes.
Furthermore, the tremendous power of modern computers also contributed greatly to the understanding of these processes. This is because nowadays researchers can use computers to simulate Lévy processes.
In recent decades, there has been a sharp rise of interest in the study of these processes. This has been fuelled by the fact that their applications are far-reaching. Evidence of this is given by the extensive amount of literature concerning the application of Lévy processes in various fields, which include insurance, queuing theory, telecommunications, quantum theory, extreme value theory and most prominently in finance.
This is because Lévy processes are an excellent tool to model price processes in mathematical finance. In fact, it has been shown that such processes are better suited than Brownian motion to model the increments of logarithmic stock prices. In fact, it has been shown that the log returns of certain financial data follow distributions that are skewed and have heavy tails.
Mark Anthony Caruana is a lecturer at the Department of Statistics and Operations Research within the Faculty of Science of the University of Malta.
Did you know?
• Application of Lévy processes in insurance
Lévy processes have been applied in the insurance sector to estimate the probability of ruin, i.e. the probability that an insurer’s capital is exhausted. In addition, these processes have been used to simulate rare events, sometimes also called extreme events, which include but are not limited to natural disasters and serious accidents which lead to very high insurance claims. These simulations help in understanding how such companies react to these extreme events.
• Application of Lévy processes in finance
Lévy processes provide an extensive tool box to model events that are hard to predict and understand in finance. It is well known that financial markets can exhibit wild fluctuations in the prices of various financial instruments. Such prices might suddenly jump to new heights or crash within a few hours. Lévy processes have the capability of modelling mathematically these wild fluctuations and the sudden jumps.
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Sound bites
• Lévy processes and global warming
The Paris agreement in 2016 marks a global effort to limit global warming. As a result, a number of countries around the world introduced, or are in the process of introducing, a carbon tax to reduce greenhouse gas emissions. The problem, however, lies in the correct approach used to tax such emissions. Lévy processes have been used in this area to model the changes in prices of electricity once this tax is introduced and how this will affect electricity consumption. In such studies, a very specific type of Lévy process was used and is commonly referred to as Normal Inverse Gaussian (NIG).
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