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?
Do Lévy processes obey analogues of the laws of the iterated logarithm for small times?
Publications
(2000 onwards)
- R. A. Doney & R. A. Maller (2000). Random walks crossing curved boundaries:a functional limit theorem, stability and asymptotic distributions for exit positions. Adv. Appl. Probab., 32, 1117-1142.
- L. Chaumont and R.A. Doney (2000). Some calculations for doubly perturbed Brownian motion. Stoch. Proc. Appl., 85, 61-74.
- L. Chaumont, R. A. Doney, and Y. Hu (2000). Upper and lower limits of doubly perturbed Brownian motion. Ann. Inst. H. Poincare., 36, 219-249.
- L. Alili and R. A. Doney (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 (2001). Fluctuation Theory for Lévy processes. Lévy processes, 57-66, Birkhauser Boston.
- 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 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 (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 Lévy processes. Ann. Inst. H. Poincare, 40, 279-297.
- R. A. Doney (2004). Small-time behaviour of Lévy processes. Elect. J. Probab., 9, 209-229.
- R. A. Doney and T. Zhang (2004). Perturbed Skorokod equations and perturbed reflected diffusion processes. Ann. Inst. H. Poincare, 41, 107-121.
- R. A. Doney (2004). Tanaka's construction for random walks and Lévy processes. Sem. de. Probab., XXXVIII, 1-4.
- R. A. Doney (2004). Some excursion calculations for spectrally one-sided Lévy processes. Sem. de. Probab., XXXVIII, 5-15.
- R. A. Doney and P. S. Griffin (2004). Overshoots over curved boundaries II. Adv. Appl. Probab., 36, 1148-1174.
- R. A. Doney and R. A. Maller (2005). Cramer's estimate for a reflected Lévy process. Ann. Appl. Probab., 16, 1445-1451.
- L. Alili, L. Chaumont and R. A. Doney (2005). On a fluctuation identity for random walks and Lévy processes. Bull. London. Math. Soc., 37, 141-148.
- L. Chaumont and R. A. Doney (2005). On Lévy processes conditioned to stay positive. Electronic J. of Probab. Theory, 10, 948-961.
- R. A. Doney and R. A. Maller (2005). Passage times of random walks and Lévy processes across power law boundaries. Probab. Theory Related Fields, 133, 57-70.
- A. Bryn-Jones and R. A. Doney (2006). A functional central limit theorem for random walks conditioned to stay non-negative. J. London Math.Soc., 74, 244-258.
- R. A. Doney and A. E. Kyprianou (2006) . Overshoots and Undershoots of Lévy processes. Ann. Appl. Probab., 16, 91-106.
- R. A. Doney and R. A. Maller (2007). Almost sure stability of the overshoot over curved boundaries. J. Theoret. Probab, 20, 47-63.
- R. A. Doney. Fluctuation Theory for Lévy processes (2007). Lectures from the 35th Summer School, St Flour, 2005. Lecture Notes in Mathematics 1897.
- R. A. Doney and R. A. Maller. (2007) Curve crossing for the reflected process. Ann. Probab., 35, 1351-1373.
- J. Bertoin, R. A. Doney, and R. A. Maller (2008). Passage of Lévy processes across power law boundaries at small times. Ann. Probab., 36, 160-197.
- L. Chaumont and R. A. Doney (2008). On Lévy processes conditioned to stay positive: correction. Electronic J. of Probab. Theory, 13, 1-4.
- R. A. Doney (2008). A note on the supremum of a stable process. Stochastics, 80,
151-155.
- R. A. Doney , R. A. Maller and M. Savov (2008). Renewal theorems and stability for the reflected process. Stochastic processes and Applications, to appear.
- R. A. Doney and M. Savov (2008). The asymptotic behaviour of densities related to the supremum of a stable process. Ann. Probab., to appear.
- L. Chaumont and R. A. Doney (2009).Invariance principles for local times at the supremum of random walks and Lévy processes, Research report 7, Probability and Statistics group, Manchester University.
- R. A. Doney and M. Savov (2009). Right inverses of Lévy processes, Research report 8, Probability and Statistics group, Manchester University.
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).
- In 2008 I was chief investigator on EPSRC EP/G002827 (V.F. P.Griffin, Syracase).
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).
- Mladen Savov; Asymptotic behaviour of Lévy processes. PhD (2008).
- Elinor Jones; Large deviations of random walks and Lévy processes. PhD (2009).
I am currently available to supervise Ph.D. students.