Prof Ron Doney's homepage, Department of Mathematics, Univ. of Manchester, UK

SCHOOL of MATHEMATICS

Professor Ron Doney

Professor of Probability Theory

Department of Mathematics

Room 13.06

University of Manchester

+44 (0)161 275 5914 (direct line)

+44 (0)161 275 5800 (general office)

+44 (0)161 275 5819 (fax)

rad@maths.man.ac.uk

Research Interests: Stochastic Processes

The stochastic processes I am particularly interested in are random walks, Brownian motion, and Lévy processes (the continuous analogue of random walks). All these processes are relatively simple to describe, and all have been studied intensively. Many of the problems I work on are also simple to state, but not usually simple to solve. Examples include

Which random walks (S_n,n > 0) are such that E(S_N) < infinity, where N = inf {n: S_n >0}?

What is the probability that there exists some t > 0 with t = B_t = sup_{0 < s <
t} B_s, (B is Brownian motion)?

Which Lévy processes have points of increase?

Publications (1996 onwards)

R.A. Doney & J. Bertoin (1996). Some asymptotic results for transient random walks. Adv. App. Prob. 28, 207-226.
R.A. Doney (1996). Increase of Lévy processes. Ann. Prob.24, 961-970.
R.A. Doney & J. Bertoin (1997). Spitzer's condition for random walks and Lévy processes. Ann. Inst. H. Poincare33, 167-178.
R.A. Doney (1997). One-sided local large deviation and renewal theorems in the infinite mean case. Prob. Thy. Rel. Fields107, 451-465.
R.A. Doney (1998). Some calculations for perturbed Brownian motion. Sém. Prob. XXXII, 231-236.
R.A. Doney & M. Yor (1998). On Takac's formula for Brownian motion with drift. J. App. Prob. 35, 272-280.
R.A. Doney, J. Warren & M. Yor (1998). Perturbed Bessel processes. Sém. Prob. XXXII, 237-249.
R.A. Doney (1998). The Martin boundary and ratio limit theorems for killed random walks. J. Lond. Math. Soc.58, 761-768.
R.A. Doney & L. Chaumont (1999). Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Prob. Th. Rel. Fields, 113, 519-534.
R.A. Doney & L. Alili (1999). Wiener-Hopf factorisation revisited and some applications. Stochastics and Stochastics Reports, 66, 87-102.
R.A. Doney (1999). Fluctuation theory for Lévy processes. Bull. I.S.I., LVIII, 455-458.
R.A. Doney & R.A. Maller (2000). Random walks crossing curved boundaries: functional limit theorems, stability and asymptotic distributions for exit times and positions. Adv.Appl.Prob. 32, 1117-1149.
R.A. Doney & L. Chaumont (2000). Some calculations for doubly perturbed Brownian motion. Stoch. Proc. Appl.,85, 61-74.
R.A. Doney, L. Chaumont & Y. Hu. (2000). Upper and lower limits of doubly perturbed Brownian motion. Ann. Inst. H. Poincare., 36, 219-249.
R.A. Doney & L. Alili (2001). Martin boundaries associated with a killed random walk. Ann. Inst. H. Poincare., 37, 313-338..
R.A. Doney (2001). A local limit theorem for moderate deviations. Bull. Lond. Math. Soc. 33, 100-108.
R.A. Doney and Y. Nakhi (2001). Perturbed and non-perturbed Brownian taboo processes. Ann.Inst.H.Poincare, 37, 725-736.
R.A. Doney and R.A. Maller (2002). Stability of the overshoot for Lévy processes. Ann. Prob. 30, 188-212.
R.A. Doney & R.A. Maller (2002). Stability and Attraction to Normality for Lévy processes at zero and infinity. J. Theoretical Probab. 15, 751-792.
R.A. Doney (2002). A note on the strong law. M.C.S.S. report.
R.A. Doney and P.S. Griffin (2003). Overshoots over curved boundaries. Adv. Appl. Prob., 35, 417-448.
R.A. Doney and P. Marchal (2003). A third arc-sine theorem. Bull. Lond. Math. Soc., 35, 536-540..
R.A. Doney and R.A. Maller (2003). Passage times of random walks and Lévy processes across power-law boundaries, to appear in PTRF.
R.A. Doney and P.S. Griffin (2004). Overshoots over curved boundaries II, to appear in Adv. Appl. Probab.
R.A. Doney (2004). Stochastic bounds for Lévy processes, Ann. Prob., 32, 1545-52.
R.A. Doney and R.A. Maller (2004). Moments of passage times for transient. Ann. Inst. H. Poincare, 40, 279-297.
R.A. Doney and A. Bryn-Jones (2004). A functional central limit theorem for random walks conditional to stay non-negative, to appear in Proc. London Math. Soc.
R.A. Doney and A.E. Kyprianou (2004). Overshoots and Undershoots of Lévy processes, to appear in Ann. Appl. Prob. (2005).
R.A. Doney (2004). Some excursion calculations for spectrally one-sided Lévy processes, Sem. de. Probab., XXXVIII, 5-15.
R.A. Doney and R.A. Maller (2005). Cramer's estimate for a reflected Lévy process, to appear in Ann. Appl. Probab. 16, 1445-1451.
R.A. Doney (2004). Small-time behaviour of Lévy processes. Elect. J. Probab., 9, 209-229.
R.A. Doney and T. Zhang (2005). Perturbed Skorokod equations and perturbed reflected diffusion processes in Ann. Inst. H. Poincare, 41, 107-121.
R.A. Doney, L. Alili and L. Chaumont (2005). On a fluctuation identity for random walks and Lévy processes, Bull. London. Math. Soc., 37, 141-148.
R.A. Doney and L. Chaumont (2005). On Lévy processes conditioned to stay positive, to appear in JLMS.
R.A. Doney (2004). Tanaka's construction for random walks and Lévy processes, Sem. de. Probab., XXXVIII, 1-4.
R.A. Doney and R.A. Maller (2005). Almost sure relative stability of the overshoot over power-law boundaries. M.C.S.S. report.
R.A. Doney and R.A. Maller (2005). Curve crossing for the reflected process. M.C.S.S. report.

Grants

In 1996 I was chief investigator on EPSRC VF GR/L/15371, the visiting fellow being L. Chaumont, University Paris VI.
In 96/98 I was the grant holder on EPSRC GR/K/82017, the RA being L. Alili.
In 98/99 I was chief investigator on EPSRC VF GR/L/89594, the visiting fellow being R.A. Maller, University of W. Australia.
In 2000 I was chief investigator on EPSRC GR/N 09046 (V.F. P.Marchal, Paris VI) and on EPSRC GR/N 94939 (V.F. P.Griffin, Syracase).
In 2001 I was chief investigator on EPSRC GR/R53197 (V.F. R.A.Maller, Univ. W. Australia).
In 2002 I was chief investigator on EPSRC GR/R88021 (V.F. V.Vigon, Univ. of Rouen).

Research students

I have recently supervised the following theses.

Angharad Bryn-Jones; A study of random walks conditioned to stay positive. PhD (2003).
Peter Andrew; Small-time behaviour of Lévy processes. PhD (2003).
Franciscus De Weert; Attraction to stable distributions for Lévy processes at zero. M.Phil (2003).

My current students are Neil Farricker, Mladen Savov and Elinor Jones.

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