John Appleby

School of Mathematical Sciences

Research Overview

Overview

My research divides roughly into three overlapping areas: asymptotic analysis of deterministic Volterra equations, the theory of random Volterra equations,
and applications of the above theories in Mathematical Finance.

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Mathematical Finance

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.

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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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Asymptotic analysis of deterministic Volterra equations

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. 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.

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One 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 rate.

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 system.

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Volterra Equations

These 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.

The 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.

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Refereed Publications

Heterogeneity 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.

(with 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

Almost sure stability of linear Ito-Volterra equations with damped stochastic perturbations, Elect. Comm. in Probab. 7, Paper no. 22, 2002, 213-224.
(with D. W. Reynolds) Non-exponential Stability of Scalar Stochastic Volterra Equations equations, to appear Stat. Probab. Lett., 2003
Stabilisation of Functional Differential Equations by Noise, to appear Systems Control Lett., 2003

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Conference Proceedings

A complete market model with feedback. Proceedings 16th IMACS World Congress 2000.

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Preprints

(with Alan Freeman) Exponential Asymptotic Stability of Linear Ito-Volterra Equations with Damped Stochastic Perturbations, submitted Electron. J. Probab., 2003.
p-th mean integrability and almost sure asymptotic stability of solutions of Ito-Volterra equations, submitted J. Integral Equations Appl., 2003.

Destabilisation of functional differential equations by noise, submitted Appl. Math. Lett., 2003.

Almost sure asymptotic stability of Ito-Volterra equations, submitted
Stoch. Anal. Appl., 2003 .
(with Evelyn Buckwar) Noise induced oscillation in solutions of stochastic delay
differential equations
, submitted Dynam. Sys. Appl., 2003 (postscript file only)
(with Aoife Flynn) Stabilisation of Volterra equations by noise, 2003.
(with D. W. Reynolds) Decay Rates of Solutions of Linear Stochastic Volterra Equations, 2002.
(with M. Fabrizio, B. Lazzari, D. W. Reynolds) On Exponential Stability in Linear Viscoelasticity, 2002

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