Real-valued self-similar Markov processes and their applications

A stochastic process is a mathematical model for the evolution through time of a particle that moves randomly through space. There are many different families of stochastic processes that are, now, well understood with varying degrees of success when building other mathematical models with applications in physics, biology, economics and engineering. Amongst some of the more commonly used stochastic processes are so-called Markov stochastic processes. For these processes, the future random evolution of the particle at any moment in time depends only on its current position and not on its historical path to date. This proposal aims to construct new families of Markov stochastic processes, never dealt with before, which respect a fundamental defining property, namely self-similarity. Roughly speaking, a stochastic process is self-similar when, after an appropriate re-scaling in space and time, the resulting random trajectory is an exact stochastic copy of itself. The importance of this new family of self-similar Markov stochastic processes will also be explored through their application in a number of different probabilistic settings. Specifically we shall:

(a) Lay down the mathematical foundations, showing the existence of the self-similar Markov processes that we are interested in.

(b) Explore some of their unusual properties with reference to the general theory of Markov processes. For example, we shall provide an understanding of the strange phenomenon that can occur with our self-similar family of processes in that their random trajectory "starts from infinity".

(c) Look at stochastic differential equations which are "driven by a self-similar Markov stochastic processes". The former can be considered as a family of stochastic processes whose infinitesimal increments (or arbitrarily small-scale random movements) are determined by the infinitesimal increments of the latter.

(d) Take advantage of the intimate relationship of self-similar Markov processes with other families of stochastic processes, known as Markov additive L'evy processes, to derive new results concerning the latter, which themselves can be fed into other applications.

(e) Take all of the above knowledge and feed it into some concrete probabilistic applications known as optimal stopping problems. The latter have proved to be of prominence in a variety of scenarios which are pertinent to financial modelling.

The research as proposed is relatively theoretical in nature, in the sense that there are no immediate 'users' of the research outside of academia. We should emphasise that this is not untypical of many areas of mathematical research. The development of abstract structures and exploration of their application within other mathematical models is but the first step in the time-honoured 'trickle-down' process by which mathematical knowledge passes through science into society. More often than not, the path from research source to 'user' is unexpected and unpredictable, both in the route as well as the time it will take. It would therefore be unwise to try to make an overenthusiasic impression here that the proposed research has a clear and direct path to beneficiaries outside of the academic community.

Nonetheless, at the very heart of this proposal is the concept of self-similarity. This is a phenomenon which we see in the world all around us. As indicated earlier, there is clearly a vast array of disciplines within the applied sciences, engineering, economics and social-science for which self-similarity is a concept of huge importance. One should therefore feel confident that effects and influences of the research undertaken in this project have excellent chances to find their way through these channels into non-academic environments.

There is plenty of scope within this proposal to explore possibilities to facilitate the aforementioned 'trickle-down' process. There are a number of leads that the PI would like to pursue with regard to both exposing the research in this project to the non-academic community and directing the findings within the project in ways that will optimise future impact. We note that, whilst some of these leads have the potential to develop into full-blown research projects, their exposition here should not be read in the spirit of a formal research proposal, but more in the spirit of an expression of intent to pursue with sincerity for the purpose of non-academic impact, wherever possible. We list three below, but others can easily emerge.

(i) The PI is currently at the opening stages of a research project with Richard Olsen, a hedge-fund manager and the inventor of high-frequency data. Statistical observations by Olsen claim to observe scaling relations between 13 different `events' that occur in financial data. There is a case to show that self-similar Markov models may be used to explain these kinds of phenomena.

(ii) Within the social science literature, there exist mathematical models which claim to describe the evolution of terrorist cells. In the limited literature available, which includes data analysis, there is strong evidence of a both self-similarity as well as a Markovian feature. Here lies an open door which is strongly inviting engagement with the outcomes proposed in this project.

(iii) The PI has contacts within scientific media which he would intent to exploit wherever possible within the context of this project.

In addition to the above, the proposal will host will host a workshop which will invite both mathematicians as well as `users' of self-similarity models.