## Ben Hambly's research page

### Recent Papers

B.M. Hambly, R. Xu and
H. Yang, Linear-quadratic
Gaussian games with asymmetric information: belief corrections
using the opponents actions.

B.M. Hambly, J. Meier and A. Sojmark,
An SPDE
with Robin-type boundary for a system of elastically killed
diffusions on the positive half-line.

B.M. Hambly and
W. Yang, Continuous random field
solutions to parabolic SPDEs on p.c.f. fractals.

G.J. Gibson and B.M. Hambly, From point estimation to Bayesian
inference via dynamical systems (new version January 2018).

### Published Papers since 2000

B.M. Hambly, R. Xu and
H. Yang, Recent advances in
reinforcement learning in finance, Math. Finance. 33, 437-503, 2023.

B.M. Hambly, R. Xu and
H. Yang, Policy gradient
methods find the Nash equilibrium in N-player general-sum
linear-quadratic games, J. Machine Learning Research. 24,
(139), 1-56, 2023.

B.M. Hambly and
P. Koepernik, Dimension
results and local times for superdiffusions on fractals,
Stoch. Proc. Applic. 158, 377-417, 2023.

B.M. Hambly and
J. Meier, McKean-Vlasov
equations with positive feedback through elastic stopping
times, Electron. Comm. Probab. 27, no. 41, 1-13, 2022.

I. Chevyrev, B.M. Hambly and
A. Mayorcas, A stochastic
model of chemorepulsion with additive noise and nonlinear
sensitivity, Stoch. Partial Differ. Equ. Anal. Comput. 11,
730-772, 2023.

P. Cesana and
B.M. Hambly, A probabilistic
model for interfaces in a Martensitic phase transition,
J. Appl. Probab. 59, 1081-1105, 2022.

B.M. Hambly, R. Xu and
H. Yang, Policy gradient
methods for the noisy linear quadratic regulator over a finite
horizon, SIAM J. Control Opt. 59, 3359-3391, 2021.

T. Ehnes and
B.M. Hambly, An
approximation of solutions to heat equations defined by generalized
measure theoretic Laplacians, J. Evol. Equ. 21, 805-830, 2021.

B.M. Hambly and
W. Yang, The damped stochastic wave
equation on p.c.f. fractals, In the proceedings of the
6th Cornell conference on Analysis, Probability and Mathematical
Physics on Fractals. pp521-556, 2020.

B.M. Hambly, J. Kalsi and
J. Newbury, Limit order books,
diffusion approximations and reflected SPDEs: from microscopic to
macroscopic models,
Appl. Math. Finance. 27, 132-170, 2020.

B.M. Hambly and
N. Kolliopoulos, Fast
mean-reversion asymptotics for large portfolios of stochastic
volatility models,
Finance and Stochastics. 24, 757--794, 2020.

B.M. Hambly and J. Kalsi, Stefan
problems for SPDEs driven by space-time white noise,
Stoch. Proc. Applic. 130, 924--961, 2020.

B.M. Hambly and J. Kalsi, A reflected
moving boundary problem driven by space-time white noise,
Stoch. Partial Differ. Equ. Anal. Comput. 7, 746--807, 2019.

D.A. Croydon, B.M. Hambly and T. Kumagai, Heat kernel estimates for FIN
processes associated with resistance forms,
Stoch. Proc. Applic. 129, 2991--3017, 2019.

B.M. Hambly, S. Ledger and A. Sojmark, A McKean-Vlasov equation with positive
feedback and blowups. Ann. Appl. Probab. 29, 2338--2373, 2019.

B.M. Hambly and A. Sojmark, An SPDE model for systemic risk with
endogenous contagion. Finance and Stochastics, 23, 535--594, 2019.

B.M. Hambly and W. Yang, Degenerate limits for one-parameter
families of non-fixed-point diffusions on fractals.
J. Fractal Geom. 6, 1--51, 2019.

F. Ahmad, B.M. Hambly and S. Ledger, A stochastic partial differential
equation model for the pricing of mortgage backed securities.
Stoch. Proc. Applic. 128, 3778--3806, 2018.

M. Chauvent, A.L. Duncan, P. Rassam, O. Birkholz, J. Helie,
T. Reddy, D. Beliaev, B.M. Hambly, J. Piehler, C. Kleanthous and
M.S.P. Samson, How nanoscale protien interactions
determine the mesoscale dynamic organization of bacterial outer
membrane proteins. Nature Communications 9, No. 2846, 1--12, 2018.

B.M. Hambly and W. Yang, Existence and space-time regularity for
stochastic heat equations on p.c.f. fractals.
Elec. J. Probab. 23, no. 22, 1--30, 2018.

B.M. Hambly and N. Kolliopoulos, Stochastic evolution equations for
large portfolios of stochastic volatility models. SIAM
J. Fin. Math. 8, 962--1014, 2017.
Erratum: SIAM J. Fin. Math. 10, 857-876, 2019.

U. Freiberg, B.M. Hambly and J.E. Hutchinson,
Spectral asymptotics for
V-variable Sierpinski gaskets. Ann. Inst. H. Poincare
Probab. Stat. 53, 2162--2213, 2017.

B.M. Hambly and S. Ledger, A stochastic McKean-Vlasov equation for
absorbing diffusions on the half-line. Ann. Appl. Probab. 27,
2698--2752, 2017.

D.A. Croydon, B.M. Hambly and T. Kumagai, Time changes of stochastic
processes associated with resistance
forms. Elec. J. Probab. 22, No 82. 1--41, 2017.

P.A. Charmoy, D.A. Croydon and B.M. Hambly, Central limit theorems for the
spectra of a class of random self-similar fractals. Trans. Amer.
Math. Soc. 369, 8967--9013, 2017.

G. Flint, B.M. Hambly and T.J. Lyons, Discretely sampled signals and the
rough Hoff process. Stoch. Proc. Applic. 126, 2593--2614, 2016.

B.M. Hambly, M. Mariapragassam and C.R. Reisinger, A forward equation for barrier options
under the Brunick-Shreve Markovian projection.
Quantitative Finance, 16, 827--836, 2016.

J.M. Ball, P. Cesana and B.M. Hambly
A probabilistic model for martensitic avalanches. MATEC Web of Conferences Volume: 33 Article Number: 02008, 2015.

B.M. Hambly and J. Vaicenavicius, The 3/2 model as a stochastic
volatility approximation for a large-basket price-weighted
index. Int. Jour. Theoret. Applied Fin. 18, 2015.

L.G. Gyurko, B.M. Hambly and J.H. Witte,
Monte Carlo methods via a dual approach
for some discrete time stochastic control problems. Math. Methods
Oper. Res. 81, 109--135, 2015.

G. Decrouez, B.M. Hambly and O.D. Jones,
The Hausdorff spectrum for a class
of multifractal processes. Stoch. Proc. Applic. 125, 1541--1568,
2015

K. Bujok, B.M. Hambly and C.R. Reisinger,
Multilevel simulation of
functionals of Bernoulli random variables and application to basket credit derivatives.
Meth. Comp. Appl. Probab. 17, 579--604, 2015.

S. Andres, M.T. Barlow, J-D. Deuschel and B.M. Hambly,
Invariance principle for the random
conductance model. Probab. Theory Related Fields 156, 535--580, 2013.

D.W-C. Miao and B.M. Hambly, Recursive formulas for the default probability
distribution of a heterogeneous group of defaultable entities.
J. Credit Risk 8, No 3, 3-40, 2012.

D.A. Croydon, B.M. Hambly and T. Kumagai,
Convergence of mixing times for
sequences of random walks on finite graphs. Elec. J. Probab. 17,
No 3, 1--32, 2012.

N. Bush, B.M. Hambly, H. Haworth, L. Jin and C. Reisinger, Stochastic evolution equations in
portfolio credit modelling. SIAM J. Fin. Math. 2, 627--664, 2011.

J.D. Biggins, B.M. Hambly and O.D. Jones, Multifractal spectra for random
self-similar measures via branching processes.
Adv. Appl. Prob. 43, 1--39, 2011.

B.M. Hambly, Asymptotics for functions associated with
heat flow on the Sierpinski carpet. Canadian J. Math 63, 153--180, 2011.

D.A. Croydon and B.M. Hambly, Spectral asymptotics for
stable trees. Elec. J. Probab. 15, 1772--1801, 2010.

N. Aleksandrov and B.M. Hambly, A dual approach to multiple exercise
option problems under constraints. Math. Methods in
Oper. Res. 71, 503--533, 2010.

B.M. Hambly and T. Kumagai, Diffusion on the scaling limit of
the critical percolation cluster in the diamond hierarchical lattice.
Comm. Math. Phys. 295, 29--69, 2010.

B.M. Hambly and T.J. Lyons,
Uniqueness for the signature of a path of bounded variation and
the reduced path group. Ann. Math. 171, 109--167, 2010.

B.M. Hambly and T.J. Lyons,
Some notes on trees and paths. (an appendix to `Uniqueness for the
signature of a path of bounded variation and the reduced path group').

B.M. Hambly, S. Howison and T. Kluge, Modelling spikes and pricing swing options
in electricity markets. Quant. Fin. 9, 937--949, 2009.

M.T. Barlow and B.M. Hambly, Parabolic Harnack inequality and local
limit theorem for percolation clusters. Elec. J. Probab. 14, 1--26,
2009.

D.A. Croydon and B.M. Hambly, Local limit theorems for sequences of
simple random walks on graphs. Pot. Anal. 29, 351--389, 2008.

D.A. Croydon and B.M. Hambly, Self-similarity and spectral asymptotics for
the continuum random tree. Stoch. Proc. Applic. 118, 730--754, 2008.

B.M. Hambly and L. Jones, Number variance from a
probabilistic perspective: infinite systems of independent Brownian
motions and symmetric $\alpha$-stable
processes. Elec. J. Probab. 12, 862--887, 2007.

B.M. Hambly and L. Jones, Erratum: Number variance from a
probabilistic perspective: infinite systems of independent Brownian
motions and symmetric $\alpha$-stable
processes. Elec. J. Probab. 14, 2009.

B.M. Hambly and J. B. Martin, Heavy tails in last passage
percolation. Probab. Theory and related fields. 137, 227-275, 2007.

B.M. Hambly, V. Metz and A. Teplyaev, Self-similar energies on
p.c.f. self-similar fractals. J. London Math. Soc. 74, 93-112, 2006.

B.M. Hambly and M.L. Lapidus,
Random fractal strings: their zeta functions, complex dimensions and
spectral asymptotics. Trans. Amer. Math. Soc. 358, 285--314, 2006.

B.M. Hambly and T. Kumagai,
Heat kernel estimates for symmetric
random walks on a class of fractal graphs and stability under rough
isometries, in Fractal Geometry and applications: a jubilee of
Benoit Mandelbrot, Proc. Symp. Pure Math. 72, Part 2, 233-259, 2004.

B.M. Hambly and J.H. Jordan, A
random hierarchical lattice: the series-parallel graph and its
properties, Adv. Appl. Prob. 36, 824-838, 2004.

N. Meinshausen and B.M. Hambly, Monte
Carlo methods for the valuation of multiple exercise options.
Math. Finance, 14, 557-583, 2004.

B.M. Hambly and T. Kumagai, Heat
kernel estimates and law of the iterated logarithm
for symmetric random walks on fractal graphs. Proc. of the JAMS
symposium "Discrete Analysis and related topics", Contemporary
Mathematics 347, pp153-172, 2004.

B.M. Hambly and T. Kumagai,
Diffusion processes on fractal fields: heat kernel estimates and large
deviations. Probab. Theory and Related Fields, 127, 305--352, 2003.

B.M. Hambly, G. Kersting and A.E. Kyprianou, Law of the iterated logarithm for
oscillating random walks conditioned to stay positive.
Stoch. Proc. Applic. 108, 327--343, 2003

B.M. Hambly and O.D. Jones, Thick and thin points for random
recursive fractals. Adv. Appl. Probab. 35, 251--277, 2003

B.M. Hambly and S.O. Nyberg, Finitely ramified graph directed
fractals, spectral asymptotics and the multidimensional
renewal theorem. Proc. Edin. Math. Soc. 46, 1--34, 2003.

B.M. Hambly, Fractals and the modelling of
self-similarity. Chapter 10 of Handbook of Statistics, vol 21, pp 371--406, 2003.

B.M. Hambly and T.Kumagai, Asymptotics for the spectral and
walk dimension as fractals approach Euclidean space. Fractals 10, 403--412, 2002.

B.M. Hambly, J. Martin and N. O'Connell, Concentration results for a
Brownian directed percolation problem. Stoch. Proc. Applic. 102, 207--220, 2002.

B.M. Hambly, K. Hattori and T. Hattori,
Self-repelling walk on the Sierpinski gasket. Probab. Theory
Related Fields, 124, 1--25, 2002.

R.F. Bass, B.M. Hambly and T.J. Lyons,
Extending the Wong-Zakai theorem to reversible Markov processes .
J. Euro. Math. Soc. 4, 237--269, 2002.

B.M. Hambly, J. Kigami and T. Kumagai, Multifractal formalisms for the
local spectral and walk dimensions.
Math. Proc. Cambridge Philos. Soc. 132, 555--571, 2002.

B.M. Hambly and O.D. Jones, Asymptotically one dimensional
diffusion on the Sierpinski gasket and multitype branching processes
with varying environment. J. Theor. Prob. 15, 285--322, 2002.

B.M. Hambly, J. Martin and N. O'Connell, Pitman's 2M-X Theorem for skip-free
random walks with Markovian increments. Elec. Comm. Probab. 6, 73--77, 2001.

B.M. Hambly and T. Kumgai, Fluctuation of the transition density
for Brownian motion on random recursive Sierpinski gaskets.
Stoch. Proc. Applic. 92, 61--85, 2001

B.M. Hambly, P. Keevash, N. O'Connell and D. Stark, The
characteristic polynomial of a random permutation matrix.
Stoch. Proc. Applic. 90, 335--346, 2000

A.J. Ganesh, B.M. Hambly, N. O'Connell, D. Stark and P. Upton,
Poissonian behaviour of Ising spin systems in an external field.
J. Stat. Phys. 99, 613--626, 2000

B.M. Hambly, T. Kumagai, S. Kusuoka and X.Y. Zhou, Transition
density estimates for diffusion processes on homogeneous
random Sierpinski carpets.
J. Math. Soc. Japan, 52, 373--408, 2000

B.M. Hambly, Heat kernels and spectral asymptotics for some random Sierpinski
gaskets, in Proceedings of the conference `Fractal geometry
and stochastics II'. Ed: C Bandt, S. Graf and M. Zahle, pp 239--267, 2000

B.M. Hambly
On
the asymptotics of the eigenvalue counting function for random
recursive Sierpinski gaskets. Probab. Theory Related Fields,
117, 221--247, 2000

### Unpublished Papers

N. Aleksandrov and B.M. Hambly, Liquidity modelling and optimal
liquidation in bond markets.