Publications
Research monographs
1. Milstein,
G.N. and Tretyakov, M.V. (2004) Stochastic Numerics
for Mathematical Physics. Series: Scientific Computation, Springer, XX+ 596 p., 48 illus. ISBN: 3-540-21110-1.
2. Milstein,
G.N. and Tretyakov, M.V. (2021) Stochastic Numerics
for Mathematical Physics. Series: Scientific Computation, Springer, XXV+ 736 p Revised and expanded Second Edition.
Textbook
Tretyakov,
M.V. (2013) Introductory Course on Financial Mathematics, Imperial
College Press, X+ 266 p. ISBN: 978-1-908977-38-0.
Journal Articles - published
1. Fedotov, S.P. and Tret'yakov,
M.V. (1988): Stationary regimes of heterogeneous chemical reaction at existence
of external noises. Khimicheskay Fizika,
V. 7, No. 11, pp. 1533-1537 (in Russian).
2. Tret'yakov, M.V. and Fedotov, S.P.
(1990): Stationary regimes of heterogeneous chemical reaction at existence of
white Poissonian noise. Khimicheskay
Fizika, V. 9, No. 2, pp. 252-257 (in Russian).
3. Fedotov, S.P. and Tret'yakov, M.V.
(1991): About stochastic ignition of a particle. Khimicheskay
Fizika, V. 10, No. 2, pp. 238-241 (in Russian).
4. Fedotov, S.P.
and Tret'yakov, M.V. (1991): Stochastic criteria for
ignition of single particles. Combust. Sci. and Tech., V. 78,
No. 1-3, pp. 1-6.
5. Buyevich, Yu.A., Tret'yakov,
M.V., and Fedotov, S.P. (1991): Model of stochastic growth of new phase islands
on solid surface. Dokl. Akad.
Nauk SSSR, V. 321, No. 5, pp. 1005-1009 (in Russian);
translation in Dokl. Phys. Chem., V. 6, pp. 852-854
(1991).
6. Buyevich, Yu.A.
and Tret'yakov, M.V. (1992):
Covering of solid surface by a new phase under conditions of metastability reduction by growing nuclei. Dokl. Akad. Nauk,
V. 323, No. 2, pp. 306-310 (in Russian).
7. Tret'yakov, M.V. (1992): Influence of colored noise on combustion of a particle. Ingenerno-Fizicheskii Journal, V. 63, No. 2, pp. 156-160
(in Russian). Translated
in English: Journal of Engineering Physics and Thermophysics,
pp 771-774.
8. Buyevich, Yu.A., Fedotov, S.P., and Tret'yakov,
M.V. (1993): Heterogeneous reaction affected by external noise. Physica A, V. 198, Nos. 3/4,
pp. 354-367.
9. Buyevich,
Yu.A. and Tret'yakov, M.V. (1994): The initiating rate of a
heterogeneous reaction. Journal de Physique II, V. 4, pp.
1605-1616.
10. Milshtein,
G.N. and Tret'yakov,
M.V. (1994): Numerical solution of differential equations with colored noise. J. Stat. Phys., V. 77, Nos. 3/4, pp.
691-715.
11. Tret'yakov, M.V. and Tret'jakov,
S.V. (1994): Numerical integration of Hamiltonian systems with external noise.
Phys. Lett. A, V. 194, pp. 371-374.
12. Milstein, G.N.
and Tret'yakov, M.V. (1997): Mean-square numerical
methods for stochastic differential equations with small noises. SIAM J. Sci. Comput., V. 18, No. 4, pp. 1067-1087.
13. Milstein,
G.N. and Tret'yakov, M.V. (1997): Numerical methods
in the weak sense for stochastic differential equations with small noise. SIAM
J. Numer. Anal., V. 34, No. 6, pp. 2142-2167.
14. Tretyakov, M.V.
(1998): Numerical technique for studying stochastic resonance. Physical Review
E, V. 57, No. 4, pp. 4789-4794.
15. Milstein, G.N. and
Tretyakov, M.V. (1999): Simulation of a space-time diffusion. The Annals of
Applied Probability, V. 9, No. 3, pp. 732-779.
16. Milstein,
G.N. and Tretyakov, M.V. (1999): Mean velocity
of noise-induced transport in the limit of weak periodic forcing. Journal of
Physics A: Mathematical and General, V. 32, No. 31, pp. 5795-5805.
17. Milstein,
G.N. and Tretyakov, M.V. (2000): Numerical algorithms for semilinear
parabolic equations with small parameter based on approximation of stochastic
equations. Mathematics of Computation, V. 69,
No. 229, pp. 237-267.
18. Milstein, G.N.
and Tretyakov, M.V. (2000): Numerical analysis of noise-induced regular
oscillations. Physica D: Nonlinear Phenomena, V.140,
Nos. 3-4, pp. 244-256.
19.
Milstein, G.N. and Tretyakov, M.V. (2001): Noise-induced unidirectional
transport. Stochastics and Dynamics, V. 1, No 1, pp. 361-375.
20. Milstein, G.N.
and Tretyakov, M.V. (2001): Numerical solution of Dirichlet
problems for nonlinear parabolic equations by a probabilistic approach. IMA J.
Num. Anal., V. 21, No 4, pp. 887-917.
21. Tretyakov, M.V.
and Fedotov S. (2001): On FKPP equation with Gaussian shear advection. Physica D: Nonlinear Phenomena, V. 159, Nos. 3-4, pp.
191-202.
22. Milstein,
G.N. and Tretyakov, M.V. (2002): The simplest random walks for the Dirichlet problem. Theory of Probability and its
Applications V. 47, No 1, pp. 39-58 (Russian);
pp. 53-68 (English).
23. Milstein,
G.N., Repin, Yu.M., and Tretyakov, M.V. (2002): Symplectic
integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal. V. 39, No 6, pp. 2066-2088.
24.
Milstein, G.N. and Tretyakov, M.V. (2002): A probabilistic approach to solution
of the Neumann problem for nonlinear parabolic equations. IMA J. Numer. Anal., V. 22, No 4, pp. 599-622.
25. Milstein,
G.N., Repin, Yu.M., and Tretyakov, M.V. (2002): Numerical methods for
stochastic systems preserving symplectic structure.
SIAM J. Numer. Anal., V. 40, No 4, pp. 1583-1604.
26.
Milstein, G.N. and Tretyakov, M.V. (2003): Quasi-symplectic
methods for Langevin-type equations. IMA J. Numer. Anal., V. 23, No 4, pp. 593-626.
27. Milstein, G.N. and
Tretyakov, M.V. (2004): Evaluation of conditional Wiener integrals by
numerical integration of stochastic differential equations. Journal of
Computational Physics, V. 197, No 1, pp. 275-298.
28.
Milstein, G.N. and Tretyakov, M.V. (2005): Numerical integration of stochastic
differential equations with nonglobally Lipschitz
coefficients. SIAM J. Numer. Anal., V. 43, No3,
1139-1154.
29. Milstein,
G.N. and Tretyakov, M.V. (2005): Numerical analysis of Monte Carlo evaluation
of Greeks by finite differences. Journal of
Computational Finance, V. 8, No 3, pp. 1-33.
30. Milstein,
G.N. and Tretyakov, M.V. (2006): Numerical algorithms for forward-backward
stochastic differential equations. SIAM J. Sci. Comp., V. 28, No 2, pp. 561-582.
31. Milstein, G.N. and
Tretyakov, M.V. (2007): Discretization of forward-backward stochastic
differential equations and related quasilinear parabolic equations. IMA J
Numer. Anal. V. 27, No 1, pp. 24-44.
32. Milstein, G.N.
and Tretyakov, M.V. (2007): Computing ergodic limits for Langevin
equations. Physica D: Nonlinear Phenomena, V. 129, No
1, pp. 81-95.
33.
Fedotov S., Milstein, G.N. and Tretyakov, M.V. (2007): Superdiffusion
of a random walk driven by ergodic Markov process with switching. Journal of
Physics A: Mathematical and Theoretical, V. 40, pp. 5769-5782.
34. Milstein,
G.N. and Tretyakov, M.V. (2009): Practical variance reduction via regression
for simulating diffusions. SIAM J. Numer. Anal. V.
47, No. 2, pp. 887-910.
35.
Milstein, G.N. and Tretyakov, M.V. (2009): Monte Carlo algorithms for backward
equations in nonlinear filtering. Advances in Applied Probability, V. 41, No.
1, pp. 63-100.
36.
Milstein, G.N. and Tretyakov, M.V. (2009): Solving parabolic stochastic partial
differential equations via averaging over characteristics. Math. Comp. V. 78,
No 268, pp. 2075-2106.
37. Ruslan L. Davidchack,
Richard Handel, and M.V. Tretyakov (2009): Langevin
thermostat for rigid body dynamics. J. Chem. Phys. V. 130, Issue 23, pp.
234101-14.
38. Robert J. Hardwick,
Michael V. Tretyakov, and Yuri E. Dubrova (2009): Age-related accumulation of
mutations supports a replication-dependent mechanism of spontaneous mutation at
tandem repeat DNA loci in mice. Molecular Biology and Evolution. V. 26, No 11,
pp. 2647-2654.
39.
Jonathan C. Mattingly, Andrew M. Stuart, and M.V. Tretyakov (2010): Convergence
of numerical time-averaging and stationary measures via Poisson equations. SIAM
J. Numer. Anal. V. 48, No 2, pp. 552-577.
40. J.H.
Mentink, M.V. Tretyakov, A. Fasolino, M.I. Katsnelson, Th. Rasing (2010):
Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation. J. Phys.: Condens.
Matter V. 22, Number 17, 176001 (12pp). [included in Highlights of 2010 of
the journal] [SIA and SIB integrators are included in UppASD library]
41.
Dumas, W.M. and Tretyakov, M.V. (2011): Computing conditional Wiener integrals
of functionals of a general form. IMA J. Numer. Anal. V. 31, No 3, pp. 1217-1251.
42. Milstein G.N. and
Tretyakov M.V. (2012): Solving the Dirichlet problem
for Navier-Stokes equations by probabilistic
approach. BIT Numer. Math. V. 52 ,
No 1, pp. 141-153.
43. Z. Zhang, B. Rozovskii, M.V. Tretyakov and G. Karniadakis (2012): A
multi-stage Wiener chaos expansion method for stochastic
advection-diffusion-reaction equations. SIAM J. Sci. Comp. V. 34, No 2, pp.
A914-A936.
44. Bayliss C., Bidmos F., Anjum A., Manchev V., Richards R., Grossier
J-P.; Wooldridge K., Ketley J., Barrow P., Jones M.,
Tretyakov M.V. (2012): Phase variable genes of Campylobacter jejuni exhibit high
mutation rates and specific mutational patterns but mutability is not the major
determinant of population structure during host colonisation. Nucleic Acids
Research V 40, No. 13, pp. 5876-5889.
45. Milstein
G.N. and Tretyakov M.V. (2013): Probabilistic methods for the incompressible Navier‒Stokes equations with space periodic
conditions. Advances in Applied Probability, V. 45, No. 3, pp. 742-772.
46.
Tretyakov M.V. and Zhang Z. (2013): A fundamental mean-square convergence
theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM
J. Numer. Anal. V. 51, No 6, pp. 3135-3162.
47. Krivko
M. and Tretyakov M.V. (2014): Numerical integration of Heath-Jarrow-Morton
model of interest rates. IMA J. Numer. Anal. V. 34,
pp. 147-196.
48. Zhang Z., Tretyakov M.V., Rozovskii
B.L. and Karniadakis G. (2014): A recursive sparse
grid collocation method for differential equations with white noise. SIAM J.
Sci. Comp. V. 36, No 4, pp. A1652-A1677.
49.
Leimkuhler B., Matthews C., and Tretyakov M.V. (2014): On the
long-time integration of stochastic gradient systems. Proc. R. Soc. A V. 470, No 2170, 20140120, 16 pp.
50. Zhang Z., Tretyakov M.V., Rozovskii B.L. and Karniadakis G.
(2015): Wiener chaos vs stochastic collocation methods for linear
advection-diffusion-reaction equations with multiplicative white noise. SIAM J.
Numer. Anal. V. 53, No 1, pp. 153-183.
51. Davidchack
R.L., Ouldridge T.E., and Tretyakov M.V. (2015): New Langevin
and gradient thermostats for rigid body dynamics. J. Chem. Phys. V. 142, No 14,
144114, 13 pp.
[Langevin
C integrator is implemented in LAMMPS]
52.
Park M. and Tretyakov M.V. (2015): A block circulant
embedding method for simulation of stationary Gaussian random fields on
block-regular grids. Int. J. Uncertainty Quantification, V. 5, No. 6, pp.
527-544.
53. Milstein
G.N. and Tretyakov M.V. (2016): Layer methods for Navier-Stokes
equations with additive noise using simplest characteristics. J. Comp Appl.
Maths. V. 302, pp. 1-23.
54. Park M. and Tretyakov M.V.
(2017): Stochastic resin transfer molding process.
SIAM/ASA Journal on Uncertainty Quantification. Vol. 5, No. 1, pp. 1110-1135.
[also downloadable
from arxiv]
55. Davidchack R.L.,
Ouldridge T.E., and Tretyakov M.V. (2017): Geometric integrator for Langevin systems with quaternion-based rotational degrees
of freedom and hydrodynamic interactions. J. Chem. Phys. Vol.147, Issue 22,
224103, 12pp+5pp of Suppl Mat. +videos [also downloadable from arxiv]
56. Matveev M.Y., Ball
F., Jones I.A., Long A.C., Schubel P.J., Tretyakov
M.V. (2018): Uncertainty in geometry of fibre preforms manufactured with
Automated Dry Fibre Placement (ADFP) and its effects on permeability. Journal
of Composite Materials V.52, Issue 16, pp. 2255-2269.
57. Bailey N.Y, Hibberd
S., Power H., Tretyakov M.V. (2018): Evaluation of the
minimum face clearance of a high speed gas lubricated bearing with Navier slip boundary conditions under random excitations.
Journal of Engineering Mathematics Vol. 112, Issue 1, pp. 17-35.
58. Iglesias M., Park M.,
Tretyakov M.V. (2018): Bayesian inversion in resin transfer molding.
Inverse Problems V. 34, No 10, 1005002.
59. Bayliss C.D.,
Fallaize C., Howitt R., Tretyakov M.V. (2019): Mutation and selection in
bacteria: modelling and calibration. Bulletin of Mathematical Biology V. 81, No
3, pp. 639-675.
60. Bailey N.Y, Hibberd S.,
Power H., Tretyakov M.V. (2019): Effect of random forcing on fluid lubricated
bearing. IMA J. Appl. Math. V. 84, Issue 3, pp. 632-649.
61. Milstein G.N. and
Tretyakov M.V. (2020): Mean-square approximation of Navier-Stokes
equations with additive noise in vorticity-velocity formulation. Numerical
Mathematics: Theory, Methods and Applications V. 14, Issue 1, pp. 1-30
62. Matveev,
M.Y., Endruweit, A., Long, A.C., Iglesias, M.A, Tretyakov, M.V. (2021):
Bayesian inversion algorithm for estimating local variations in permeability
and porosity of reinforcements using experimental data. Composites Part A V. 143,
106323.
63. Deligiannidis, D.,
Maurer, S., Tretyakov, M.V. (2021): Random walk algorithm for the Dirichlet problem for parabolic integro-differential
equation. BIT Numerical Mathematics V. 61, pp. 1223-1269.
64. Endruweit, A., Matveev, M.Y., Tretyakov, M.V. (2022): Controlling resin flow in liquid
composite moulding processes through localised irradiation with ultraviolet
light. Polymer Composites V. 43, Issue 11, pp. 8308-8321.
65. Iglesias, M.,
McGrath, D. M., Tretyakov, M. V., Francis, S.T. (2022): Ensemble Kalman inversion for magnetic resonance elastography.
Physics in Medicine & Biology V. 67, Number 23, 235003.
66. Kalise, D., Sharma,
A., Tretyakov, M.V. (2023): Consensus based
optimization via jump-diffusion stochastic differential equations. Mathematical
Models and Methods in Applied Sciences (M3AS) V. 33, No. 2, pp. 289-339 [earlier
version downloadable from arxiv]
67. Leimkuhler, B., Sharma, A., Tretyakov,
M.V. (2023): Simplest random walk for approximating Robin boundary value
problems and ergodic limits of reflected diffusions. The Annals of Applied
Probability V. 33, No. 3, 1904-1960
68. Hinds, P.D.,
Tretyakov, M.V. (2023): Neural variance reduction for stochastic differential
equations. J. Comp. Finance V. 27, No. 3, 1-41 [a version is downloadable from arxiv]
69. Su, H., Tretyakov, M.V., Newton, D.P. (2024): Deep learning of
transition probability densities for stochastic asset models with applications
in option pricing. Management Science (accepted) (an earlier version is
available at [downloadable from arxiv])
Refereed contributions to books
70. Milstein,
G.N. and Tretyakov, M.V. (2011): Averaging over characteristics with innovation
approach in nonlinear filtering. In Handbook of Nonlinear Filtering (Eds. D.
Crisan and B. Rozovskii) Oxford University Press,
ISBN 978-0-19-953290-2, pp. 892-919.
71.
Stanciulescu,
V.N. and Tretyakov, M.V. (2011): Numerical solution of the Dirichlet
problem for linear parabolic SPDEs based on averaging over characteristics. In
Stochastic Analysis 2010 (Ed. D. Crisan), Springer, ISBN 978-3-642-15357-0, pp.
191-212.
72.
Krivko M. and Tretyakov M.V. (2013): Application of simplest random walk
algorithms for pricing barrier options. In: Recent Developments in
Computational Finance (Eds. T. Gerstner and P.E. Kloeden), World Scientific, ISBN 978-981-4436-42-7, pp. 407-428. [pre-publication
is at arxiv]
73. Bailey N.Y, Hibberd S.,
Power H., Tretyakov M.V. (2019): Effect of uncertainty
in external forcing on a fluid lubricated bearing. In: Cavalca
K., Weber H. (eds)
Proceedings of the 10th International Conference on Rotor Dynamics - IFToMM. IFToMM 2018. Mechanisms
and Machine Science, vol 63. Springer.
Recent preprints
74. Maurer, S., Sharp, T.E., Tretyakov, M.V. (2021): Pricing FX options
under intermediate currency. [downloadable from arxiv]
75. Bharath, K., Lewis, A., Sharma, A., Tretyakov, M.V. (2023): Sampling and
estimation on manifolds using the Langevin diffusion.
[downloadable
from arxiv]
76. Garcia, J.A., Gimeno, R., Hinds, P.D., Su, H.,
Tretyakov, M.V. (2024): Understanding inflation:
insights from the term structure of inflation risks.
77. Garcia, J.A., Gimeno, R., Hinds, P.D., Su, H.,
Tretyakov, M.V. (2024): The return of inflation and inflation risks.
Book Editor
Numerical
Mathematics and Advanced Applications. Proceedings of ENUMATH 2011, the 9th
European Conference on Numerical Mathematics and Advanced Applications,
Leicester, September 2011. Cangiani, A.; Davidchack, R.L.;
Georgoulis, E.; Gorban, A.N.; Levesley, J.; Tretyakov, M.V. (Eds.), Springer
2013; VIII, 854 pp. ISBN 978-3-642-33133-6.
Other
Tretyakov
M.V. (2015): Stochastic filtering. In Encyclopedia of
Applied and Computational Mathematics, Editor-in-Chief Bjorn Engquist, Springer, ISBN 978-3-540-70530-7.
Milstein G.N. and Tretyakov M.V.
(2012): Layer methods for Navier-Stokes equations
with additive noise. Research Report No. MA-12-001, Department of Mathematics,
University of Leicester, Leicester (UK), 2012, 35pp. [downloadable from arxiv]