I completed my PhD at UNSW in 1992 under the supervision of Professor Tony Dooley. I worked for four years in the School of Mathematics at the University of Sydney, before joining UTS in 1998 in the school of Finance and Mathematics. I began full time in the School of Mathematics in 2002. I was honours coordinator in the BMathFin program from 2005 to 2011.
Can supervise: YES
My research is largely devoted to the study of the connection between Lie symmetries, partial differential equations and harmonic analysis. I have a particular interest in the construction of fundamental solutions. I have also dabbled in Clifford analysis and have an active interest in mathematical finance.
I teach measure differential equations, stochastic calculus and advanced analysis. I have also taught advanced calculus, stochastic processes and first year mathematics. I supervise honours students, (more than a dozen) and PhD students. Kelly Lennox completed her PhD on Lie symmetry methods for multi-dimensional parabolic equations under my supervision in 2010. Lars Hunziker completed his PhD on Lie symmetries for non local equations in 2011. I currently have two students working on their PhDs with me.
© 2019 Elsevier B.V. We investigate PDEs of the form ut= [Formula presented] σ2(t,x)uxx−g(x)u which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can be used to obtain Fourier transforms of fundamental solutions of the PDE. We detail explicit computations in the separable volatility case when σ(t,x)=h(t)(α+βx+γx2), g=0, corresponding to the so called Quadratic Normal Volatility Model. We give financial applications and also show how symmetries can be used to compute first hitting distributions.
Craddock, M 2017, 'Fundamental solutions for the two dimensional affine group and Hn+1', Journal of Mathematical Analysis and Applications, vol. 445, no. 1, pp. 953-970.View/Download from: Publisher's site
© 2016 Elsevier Inc. We derive the wave and heat kernels on the ax+b group, as well as the fundamental solution of the group Laplacian. We make particular use of the Kontorovich–Lebedev transform and a recent result of the author to produce new expressions for these kernels. Our results easily extend to the hyperbolic space H n+1 for any n and the explicit formulas are given in n dimensions.
Craddock, M & Yakubovich, S 2017, 'New classes of non-convolution integral equations arising from Lie symmetry analysis of hyperbolic PDEs', Journal of Differential Equations, vol. 263, no. 11, pp. 7412-7447.View/Download from: Publisher's site
© 2017 Elsevier Inc. In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial differential equations. Our analysis leads to new integral equations of non-convolution type, which can be solved by classical methods. We derive solutions of these integral equations, which in turn lead to solutions of the associated Cauchy problems.
In this paper, we construct operators on a Lie symmetry group which may be regarded
as Fourier transforms. Essentially, we integrate solutions generated by Lie
symmetries against suitable test functions. We show that this idea leads to a powerful
method for solving Cauchy problems for parabolic and hyperbolic equations
in two and higher dimensions. We also discuss applications to the elliptic case.
The Clifford-Fourier transform was introduced by Brackx, De Schepper and Sommen who subsequently computed its kernel in dimension d=2. Here we compute the kernel of a fractional version of the transform when d=2 and 4. In doing so we solve appropriate wa
Craddock, MJ & Lennox, KA 2012, 'Lie Symmetry Methods For Multi-dimensional Parabolic PDEs And Diffusions', Journal of Differential Equations, vol. 252, no. 1, pp. 56-90.View/Download from: Publisher's site
In this paper we introduce new methods based upon integrating Lie symmetries for the construction of explicit fundamental solutions of multi-dimensional second order parabolic PDEs. We present applications to the problem of finding transition probability
We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2, R). We then show th
Craddock, MJ 2009, 'Fundamental solutions, transition densities and the integration of Lie symmetries', Journal Of Differential Equations, vol. 246, no. 6, pp. 2538-2560.View/Download from: Publisher's site
In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.
Craddock, MJ & Lennox, KA 2009, 'The Calculation Of Expectations For Classes Of Diffusion Processes By Lie Symmetry Methods', Annals Of Applied Probability, vol. 19, no. 1, pp. 127-157.View/Download from: Publisher's site
This paper uses Lie symmetry methods to calculate certain expectations for a large class of Ito diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form E-x(e(-lambda Xt-f0tg(Xs)ds)) can be reduced to evaluating a single integral of known functions. Given a drift f we determine the functions g for which the corresponding functional can be calculated by symmetry. Conversely, given g, we can determine precisely those drifts f for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.
Craddock, MJ & Lennox, KA 2007, 'Lie group symmetries as integral transforms of fundamental solutions', Journal Of Differential Equations, vol. 232, no. 2, pp. 652-674.View/Download from: Publisher's site
We obtain fundamental solutions for PDEs of the form u(t) = sigma x(gamma)u(xx) + f(x)u(x) - mu x(r)u by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case the problem of finding a fundamental solution can be reduced to inverting a Laplace rtansform or some other classical transform.
Brown, D, Craddock, M, Culshaw, R, Fendel, D & Wu, HH 2006, 'Serge lang and the HIV consensus ', Notices of the American Mathematical Society, vol. 53, no. 9, p. 1006.
Chiarella, C, Craddock, MJ & El-Hassan, N 2003, 'An implementation of Bouchouev's method for a short time calibration of option pricing models', Computational Economics, vol. 22, no. 2-3, pp. 113-138.View/Download from: Publisher's site
Craddock, MJ 2000, 'Symmetry groups of linear partial differential equations and representation theory: the Laplace and Axially symmetric wave equations', Journal of Differential Equations, vol. 166, pp. 107-131.View/Download from: Publisher's site
We examine the Lie point symmetry groups of two important equations of mathematics and mathematical physics. We establish that the action of the symmetry groups are in fact equivalent to principal series representation sof the underlying group. Some application sare given.
Craddock, MJ, Heath, DP & Platen, E 2000, 'Numerical inversion of Laplace transforms: a survey of techniques with applications to derivative pricing', Journal of Computational Finance, vol. 4, no. 1, pp. 57-81.
Craddock, MJ 1995, 'The Symmetry Groups of Linear Partial Differential Equations and Representation Theory, I', Journal of Differential Equations, vol. 116, no. 1, pp. 202-247.View/Download from: Publisher's site
We study the symmetry groups of three closely related PDEs. It is shown that the symmetry groups for these equations are actually global Lie groups and that the symmetry operations arise from standard group representation theory via inter-twinning operators dericed from the fundamental solutions of the equations.
Craddock, MJ 1994, 'Symmetry groups of partial differential equations, separation of variables and direct integral theory', Journal of Functional Analysis, vol. 125, no. 2, pp. 452-479.View/Download from: Publisher's site
We study unitary symmetries of linear differential equations. by applying representations theory we show how to decompose a Hilbert space of solution to certain classes of equations as direct integrals.
Craddock, MJ 2014, 'On an integral arising in mathematical Finance' in Dieci, R, He, XZ & Hommes, C (eds), Nonlinear Economic Dynamics and Financial Modelling: Essays in Honour of Carl Chiarella, Springer, Germany, pp. 355-370.
We consider an integral which arises in several problems in analysis and financial mathematics. We significantly simplify the integral, yielding a tractable form which is more useful for explicit calculations.
Craddock, MJ, Konstandatos, O & Lennox, KA 2009, 'Some recent developments in the theory of Lie group symmetries for PDEs' in Baswell, AR (ed), Advances in Mathematics Research, Nova Science Publishers, United States of America, pp. 1-40.
Lie group symmetry methods provide a powerful tool for the analysis of PDEs. Over the last thirty years, considerable progress has been made in the development of this field. In this article, we provide a brief introduction to the method developed by Lie for the systematic computation of symmetries, then move on to a survey of some of the more recent developments. Our focus is on the use of Lie symmetry methods to construct fundamental solutions of partial differential equations of parabolic type. We will show how recent work has uncovered an intriguing connection between Lie symmetry analysis and the theory of integral transforms. Fundamental solutions of families of PDEs which arise in various applications, can be obtained by exploiting this connection. The major applications we give will be in financial mathematics. We will illustrate our results with the problem of pricing a so called zero coupon bond, as well as giving some applications to option pricing. We also discuss some results on group invariant solutions and show how an important PDE in nilpotent harmonic analysis can be studied via its group invariant solutions.
Craddock, MJ & Lennox, KA Quantitative Finance Research Centre, UTS 2006, Lie Group Symmetries as Integral Transforms of Fundamental Solutions, pp. 1-23, Broadway, NSW.
Research Paper Number: 246 Abstract: This paper uses Lie symmetry group methods to obtain transition probability densities for scalar diffusions, where the diffusion coefficient is given by a power law. We will show that if the drift of the diffusion satisfies a certain family of Riccati equations, then it is possible to compute a generalized Laplace transform of the transition density for the process. Various explicit examples are provided. We also obtain fundamental solutions of the Kolmogorov forward equation for diffusions, which do not correspond to transition probability densities.
Craddock, MJ & Platen, E 2003, 'Symmetry group methods for fundamental solutions and characteristic functions', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
Research Paper Number: 90 Abstract: This paper uses Lie symmetry group methods to analyse a class of partial differential equations of he form It is shown that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find the characteristic functions and transition densities of the corresponding diffusion processes.
Research Paper Number: 60 Abstract: This paper makes use of an integrated benchmark modelling framework that allows us to model credit risk. We demonstrate how to price contingent claims by taking expectations under the real world probability measure in a benchmarked world. Furthermore, put and call options on an index are studied that measure the credit worthiness of a firm.
Chiarella, C, Craddock, MJ & El-Hassan, N 2000, 'The calibration of stock option pricing models using inverse problem methodology', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
Research Paper Number: 39 Abstract: We analyse the procedure for determining volatility presented by Lagnado and Osher, and explain in some detail where the scheme comes from. We present an alternative scheme which avoids some of the technical complications arising in Lagnado and Osher's approach. An algorithm for solving the resulting equations is given, along with a selection of numerical examples.
Craddock, MJ, Heath, DP & Platen, E 1999, 'Numerical inversion of Laplace transforms: A survey of techniques with applications to derivative pricing', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
Research Paper Number: 27 Abstract: We consider different approaches to the problem of numerically inverting Laplace transforms in finance. In particular, we discuss numerical inversion techniques in the context of Asian option pricing.