research divides roughly into three overlapping areas: asymptotic
analysis of deterministic Volterra equations,
the theory of random Volterra
and applications of the above theories in Mathematical
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
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.
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.
Asymptotic analysis of deterministic Volterra equations
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. This phenomena can be exhibited in problems
in the linear theory of viscoelasticity.
These results can be extended to scalar nonlinear problems, and similar
work on systems of equations is in progress. Of particular interest
here is the hierarchy of the equations - influential higher subsystems
can determine a sluggish convergence to equilibrium; the lower subsystems
import the slow decay of the dominant part of the system.
can employ these ideas to study the adjustment of defensive expenditure
in a model in which the participants react to the expenditure of their
competitors - as in an arms race.
In this instance a slow decay in memory - or sluggish adjustment to
current trends - leads to a safe equilibrium being reached very slowly.
However, an explosion in expenditure can still occur at an exponential
In each case above, the slow decay in memory leads to a slow convergence
to equilibrium, thereby providing a model of inertia in an economic
results immediately motivate the following question: does non-exponential
stability still persist in the presence of random perturbations? The
short answer is that the slow convergence property is essentially unaltered
by the addition of randomness, whether the noise arises from an external
source, or is intrinsic to the system. However, this property merits
further investigation, as it poses a sharp question about the modelling
of a "stable subsystem'' of the economy by a stochastic functional
differential equation: if
the convergence is exponential to equilibrium, the system has a finite
memory, and is not subject to inertia; if the convergence is slower,
memory is unbounded, and the slow convergence is caused by an inability
to react sufficiently quickly to new data.
fact that convergence speed need not be exponential to an equilibrium
level means that one should also examine the conditions under which
convergence happens, without regard to the convergence speed - a question
which does not arise with a finite memory. One can show for scalar problems
that under conditions broadly analogous to those which guarantee convergence
for deterministic systems, that convergence occurs. In the above cases,
the effects of inertia in a system are the dominant ones; but what additional
effects does noise have in the presence of inertia? It transpires that
noise can have a stabilising, destabilising, or oscillation-inducing
effect on delay-dependent systems, whether there is a bounded or unbounded
memory of the past.
and Seasonality in Financial Markets, International Journal
of Theoretical and Applied Finance, 3(3), 491-2, 2000.
||(with D. W.
Reynolds) On the non-exponential
convergence of asymptotically stable solutions of linear scalar
Volterra integro-differential equations,
J. Integral Equations Appl., 14(2), 109--118, 2002.
D. W. Reynolds) Subexponential
solutions of linear Volterra integro-differential equations and
transient renewal equations. Proc. Roy. Soc. Edinburgh. Sect.
A, 132:521--543, 2002
sure stability of linear Ito-Volterra equations with damped stochastic
perturbations, Elect. Comm. in Probab. 7, Paper no. 22, 2002,
||(with D. W.
Reynolds) Non-exponential Stability
of Scalar Stochastic Volterra Equations equations, to appear
Stat. Probab. Lett., 2003
of Functional Differential Equations by Noise, to appear Systems
Control Lett., 2003
Freeman) Exponential Asymptotic
Stability of Linear Ito-Volterra Equations with Damped Stochastic
Perturbations, submitted Electron. J. Probab., 2003.
mean integrability and almost sure asymptotic stability of solutions
of Ito-Volterra equations, submitted J. Integral Equations Appl.,
of functional differential equations by noise, submitted Appl.
Math. Lett., 2003.
sure asymptotic stability of Ito-Volterra equations, submitted
Stoch. Anal. Appl., 2003 .
Buckwar) Noise induced oscillation
in solutions of stochastic delay
differential equations, submitted Dynam. Sys. Appl., 2003 (postscript
Flynn) Stabilisation of Volterra
equations by noise, 2003.
||(with D. W.
Reynolds) Decay Rates of Solutions of Linear Stochastic Volterra
||(with M. Fabrizio,
B. Lazzari, D. W. Reynolds) On Exponential Stability in Linear Viscoelasticity,