Overview
My research has
principally been concerned with understanding the qualitative
behaviour of stochastic functional differential equations (SFDEs).
Such equations are dynamical systems which evolve in a random
environment, which are subject to time delays, or in which the
system has a memory of its past behaviour. These properties make
SFDEs useful for modeling the evolution of systems in many branches
of science - control engineering, continuum physics, mathematical
biology and finance, but the study of equations which combine both
random and memory features is still at a relatively early stage.
My particular motivation for studying these equations arises from
mathematical finance; if one drops the assumption that markets are
efficient, and take into account that many traders make use of
extant trends in returns to make their decisions, price dynamics are
then described by SFDEs rather than by stochastic differential
equations (SDEs). Issues in SFDEs closely related to mathematical
finance include
- long memory properties (modeling term structure
of asset returns' autocorrelation function);
- large deviation,
stabilization and explosion theory in highly nonlinear problems
(modeling bubbles and crashes in asset markets);
- oscillation and
periodicity of solutions (arising from agent-based models of
arbitrage and the bounce between bid and ask prices);
- discrete
dynamical systems, which can be viewed as modelling price evolution
in discrete time;
- considering computer simulations and numerical
methods;
- direct applications to finance.
By Stochastic Differential Equations stochastic systems we mean stochastic differential equations,
stochastic differential delay equations which arise from engineering, economics and other disciplines.
One important question is to investigate what the effect of small departures from deterministic differential equations
when the underlying deterministic equation has known asymptotic behaviour. In particular, it is interesting to examine
precisely the maximal extent to which the system can be perturbed before the behaviour changes, and this question has been addressed
for a wide variety of equations with and without memory and with a wide variety of sensitivities to the state.
The development of analytical techniques which deal with highly nonlinear equations therefore form an important part of our
work. Much of our work is for pathwise (or almost sure) properties of the path, but asymptotic behaviour in probability or in a p--th mean sense
is also studied.
Another question involves the growth of solutions and the growth rate of the fluctuations of solutions when the stochastic input in
the system is so large that stability of the paths can no longer be expected. This analysis is of interest in many real world problems, including the size of epidemics of infectious diseases and the extent of fluctuations in financial markets.
If the stochastic terms start to dominate, the role of these terms in suppressing the growth or explosion of solutions, or of stabilising or destabilising solutions becomes of special interest. These results have particular application in control engineering.
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For autonomous functional differential equations with bounded delay which have asymptotically stable equilibria, exponentially fast convergence eventuates. However, for problems with a long memory - unbounded delay - exponentially fast convergence need not always occur.
It transpires (in rough terms) that a stable solution of a linear Volterra equation should decline exponentially to its equilibrium if and only if its memory of the past fades at an exponential rate.
The questions arising from this problem have lead us to show that the convergence rate to equilibrium is exactly the (non-exponential) decay rate of the memory kernel. These results can be used to investigate the long memory properties of systems in population biology, viscoelasticity in continuum physics, and the persistence of influential events in finance.
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When the state of a system changes only at fixed points in time, its evolution can be described
by a difference equation. Many questions that are relevant for continuous--time dynamical systems
are also germane for such discrete time systems, such as asymptotic behaviour.
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In understanding the behaviour of financial markets, two important factors should be taken into consideration.
The first is noise or randomness, arising, for instance, from the unpredictable external environment in which financial markets exist, and the sometimes unpredictable behaviour of agents participating in that market. This view leads naturally to considering the market as a stochastic dynamical system.
A second factor - the presence of a "memory'' in the market - is one less studied for at least two reasons; first, much financial market theory is predicated on the assumption that the past behaviour of markets is essentially irrelevant in determining future market movements: in other words that markets are efficient at processing information about financial assets. However, the occurrence of market events such as stock market crashes, and the presence of large numbers of traders using investment strategies which actively involve past market information calls this assumption into question.
Under the "memoryless" assumption, the price dynamics of the market are forced to follow a Markov process, often a diffusion process driven by Brownian motion. However, if one lifts the Markovian assumption, and allows some path-dependence into the market dynamics, the appropriate mathematical model is a stochastic functional differential equation, or Volterra equation. Very often, a good understanding of the problems involved in the random theory is afforded by first studying the associated deterministic problem.
An instance of such a mathematical model of financial market prices can arise if one considers the effects of chart trading, and other methods of positive feedback trading. Under appropriate simplifying hypotheses, a simple linear Volterra equation can describe the motion of prices. In this framework, it can be shown that the presence of such investors has a destabilising effect on the market, providing a mechanism for market bubbles and crashes. In addition, the presence of diverse classes of investors with differing time horizons tends to add to volatility and explain the term structure of the autocorrelation of returns. Using a more sophisticated and realistic generalisation of this memory dependent model, one can account for "heavy tails'' in the assets returns', as well as providing a model of bubble inflation and deflation.
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My work in numerical analysis concentrates on the property of dynamical consistency
of discretisations of differential systems, both deterministic and stochastic. In the analysis
we seek to recover a property (such as growth, stability, oscillation, stationarity, fluctuation)
and if possible a metric for that property (such as the growth rate, the rate of convergence, the frequency of zero crossings, or the growth rate of fluctuations) when the differential system is discretised. These results have been applied to recover finite-time explosion and growth rates of deterministic delay differential equations, the asymptotic behaviour of stochastic differential equations with Markovian switching in the study of financial markets with regime switching, the preservation of asymptotic behaviour in stochastic dynamical systems modelling simulated annealing, and the persistence of memory in financial markets.
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john.appleby{at}dcu.ie
http://webpages.dcu.ie/~applebyj/research