The Archetypal Equation with Rescaling and Related Topics

Summary

Archetypal equation is a functional equation involving averaging (expectation) over a random affine transformation of the argument. This equation is related to many important topics such as the Choquet Deny theorem, Bernoulli convolutions, fractals, subdivision schemes in approximation theory, etc. Investigation of the archetypal equation will involving asymptotic analysis of the corresponding Markov chains with affine jumps. The project may have applications to financial modelling based on random processes with multiplicative jumps.

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) (0p1), which describes e.g. the ruin probability for a gambler who spends 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; this analytic result can be obtained using 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 and 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).

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]).

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.

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