## Ben Hambly's research page

### Recent Papers

• B.M. Hambly, R. Xu and H. Yang, Policy gradient methods find the Nash equilibrium in N-player general-sum linear-quadratic games.
• I. Chevyrev, B.M. Hambly and A. Mayorcas, A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity.
• B.M. Hambly and W. Yang, Continuous random field solutions to parabolic SPDEs on p.c.f. fractals.
• P. Cesana and B.M. Hambly, A probabilistic model for interfaces in a martensitic phase transition.
• 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, Policy gradient methods for the noisy linear quadratic regulator over a finite horizon. SIAM J. Control Optim., 59, 3359--3391, 2021.
• T. Ehnes and B.M. Hambly, An Approximation to solutions to heat equations defined by generalized measure theoretic Laplacians. J. Evol. Equ., 21, 805--830, 2021.
• 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 reverting asymptotics for large portfolios of stochastic volatility models. Finance and Stochastics, 24, 757--794, 2020.
• B.M. Hambly and J. Kalsi, Stefan problems for reflected 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.
• 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.
• 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.