The theory of functional equations is a growing branch of analysis with many deep results and abundant applications (see [1] for a general introduction). A simple functional-differential equation with rescaling is given by y’(x) + y(x) = p y(2x) + (1-p) y(x/2) (0 < p < 1), which describes e.g. the ruin probability for a gambler who spends his capital at a constant rate (starting with x pounds) but at random time instants decides to bet on the entire current capital and either doubles it (with probability p) or loses a half (with probability 1-p). Clearly, y(x) = const is a solution, and the question is whether or not there are any other bounded, continuous solutions. It turns out that such solutions exist if and only if p < 0.5; remarkably enough, this analytic result is obtained using the martingale techniques of probability theory [2].
The equation above exemplifies the "pantograph equation" introduced by Ockendon & Tayler [6] as a mathematical model of the overhead current collection system on an electric locomotive. In fact, the pantograph equation and its various ramifications have emerged in a striking range of applications including number theory, astrophysics, queues & risk theory, stochastic games, quantum theory, population dynamics, imaging of tumours, etc.
A rich source of functional and functional-differential equations with rescaling is the "archetypal equation" y(x)=E[y(α(x-β))], where α, β are random coefficients and E denotes expectation [3]. Despite its simple appearance, this equation is related to many important topics, such as the Choquet–Deny theorem, Bernoulli convolutions, self-similar measures and fractals, subdivision schemes in approximation theory, chaotic structures in amorphous materials, and many more. The random recursion behind the archetypal equation, defining a Markov chain with jumps of the form x → α(x-β), is known as the “random difference equation”, with numerous applications in control theory, evolution modelling, radioactive storage management, image generation, iterations of functions, investment models, mathematical finance, perpetuities (pensions), environmental modelling, etc. (see [4]).
In brief, the main objective of this PhD project is to continue a deep investigation of the archetypal equation and its generalizations. Research will naturally involve asymptotic analysis of the corresponding Markov chains, including characterization of their harmonic functions [7]. The project may also include applications to financial modelling based on random processes with multiplicative jumps (cf. [5]).
Successful candidates should have an excellent degree in mathematics and/or statistics, with a good background and research interests in one or more of the following areas: probability; random processes; analysis; mathematical statistics.
You will be based within a strong research group in Probability, Stochastic Modelling & Financial Mathematics (http://www.maths.leeds.ac.uk/research/groups/probability-stochastic-modelling-and-financial-mathematics.html). Our research focuses on the study and modelling of systems and processes featured by uncertainty and/or complexity, using advanced theoretical, simulation and numerical methods. It covers a vast variety of modern topics both in probability (including theory of random processes and stochastic analysis) and in a wide range of applications in mathematical and other sciences, spanning from nonlinear dynamical systems and mathematical physics through mathematical biology and complexity theory to mathematical finance and economics.
Informal enquiries: Dr Leonid Bogachev (
[email protected]).
References
[1] Aczél, J. and Dhombres, J. Functional Equations in Several Variables, with Applications to Mathematics, Information Theory and to the Natural and Social Sciences. Cambridge Univ. Press, Cambridge, 1989.
[2] Bogachev, L., Derfel, G., Molchanov, S. and Ockendon, J. On bounded solutions of the balanced generalized pantograph equation. In: Topics in Stochastic Analysis and Nonparametric Estimation (P.-L. Chow et al., eds.), pp. 29–49. Springer, New York, 2008. (doi:10.1007/978-0-387-75111-5_3)
[3] Bogachev, L.V., Derfel, G. and Molchanov, S.A. On bounded continuous solutions of the archetypal equation with rescaling. Proc. Roy. Soc. A, 471 (2015), 20150351, 1–19. (doi:10.1098/rspa.2015.0351)
[4] Diaconis, P. and Freedman, D. Iterated random functions. SIAM Reviews, 41 (1999), 45–76. (doi:10.1137/S0036144598338446)
[5] Kolesnik, A.D. and Ratanov, N. Telegraph Processes and Option Pricing. Springer, Berlin, 2013. (doi:10.1007/978-3-642-40526-6)
[6] Ockendon, J.R. and Tayler, A.B. The dynamics of a current collection system for an electric locomotive. Proc. Roy. Soc. London A, 322 (1971), 447–468. (doi:10.1098/rspa.1971.0078)
[7] Revuz, D. Markov Chains, 2nd edn. North-Holland, Amsterdam, 1984.