Professor Grigori N Milstein
(1937-2023)
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Professor G.N. Milstein received his undergraduate degree
in mathematics from the Ural State University (UrGU; Sverdlovsk, USSR), which
is now Ural Federal University (Ekaterinburg, Russia). He completed his
PhD studies at the same University. Professor Milstein has been an assistant
professor, associate professor and, after defending his DSc thesis, professor
at the Faculty of Mathematics and Mechanics UrGU (then URFU). He also worked
as Senior Researcher at the Weierstrass Institute for Applied Analysis and
Stochastics (Berlin, Germany) and was a Visiting Professor at University of
Leicester (Leicester, UK) and University of Manchester (Manchester, UK).
Milstein was a world-leading expert in Stochastic
Numerics, Estimation, Control, Stability, Financial Mathematics. Milstein has
a number of pioneering works on Stochastic Numerics.
He published four research monographs and also
contributed to the second edition of R. Khasminskii Stochastic Stability of
Differential Equations, Springer, 2012.
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At Ural State University, 1986
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WIAS, 1996
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New York,
2018
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List of
publications:
Books
1.
Milstein G.N. Numerical Integration of
Stochastic Differential Equations. Ural. State. Univ., Sverdlovsk, 1988, 224
pp.
2.
Milstein G.N.
Numerical Integration of Stochastic Differential Equations. Kluwer Academic
Publishers, Mathematics and its Applications, v. 313, 1995, 170 pp.
3.
Milstein, G.N.
and Tretyakov, M.V. Stochastic Numerics for Mathematical Physics. Series:
Scientific Computation, Springer, 2004, XX+ 596 pp.
4.
Milstein, G.N.
and Tretyakov, M.V. Stochastic Numerics for Mathematical Physics. Series:
Scientific Computation, Springer, 2021, XXV+ 736 pp. Revised and expanded
Second Edition.
Research papers
1.
Milstein G.N., Sholokovich, F.A. Almost
recurrent and uniformly Poisson stable orbits in a linear dynamical system.
Sibirsk. Mat. Z. 2(1961), pp. 567-573.
2.
Milstein G.N. On the
approximate realization of responses using transient curves. Prikl. Mat. Meh.
26(1962), pp. 623--630.
3.
Milstein G.N. An application of successive
approximations for solving an optimal problem. Avtomat. i Telemeh. 25(1964),
pp. 321--329.
4.
Milstein G.N. On the optimal realization of
trajectories. Avtomat. i Telemeh. 26(1965), pp. 621--628.
5.
Milstein G.N. Reduction of
a class of optimum control problems to the simplest variational problem. Prikl.
Mat. Meh. 28(1964), pp. 375--380.
6.
Milstein G.N. On a boundary value problem for a
system of two differential equations. Differencial'nye Uravneniya 1(1965), pp.
1628--1639.
7.
Milstein G.N. Optimal realization of
trajectories with two types of limitations on the controls. In ''Optimal
Systems of Automatic Control'', pp. 45--53, Izdat. ''Nauka'', Moscow, 1967.
8.
Milstein G.N., Ziteneva Ju.A. The minimization
of a convex functional by the free trajectories of a linear system.
Differencial'nye Uravneniya 3(1967), pp. 2081-2093.
9.
Milstein G.N., Repin, Ju.M.
Stability in the mean square of a system of stochastic differential equations.
Priklad. Mat. Meh. 31(1967), pp. 508--510.
10. Milstein
G.N., Repin, Ju.M. The action of a Markov process on a system of differential
equations. Differencial'nye Uravnenija 5(1969), pp. 1371--1384.
11. Milstein
G.N. The stability of a linear system that is subject to the influence of a
Markov chain. Differencial'nye Uravnenija 6(1970), pp. 1982--1993. MR 44
12. Milstein G.N. Interaction of Markov
processes. Theor. Probab. Appl., 17(1972), pp. 36--45.
13. Milstein
G.N. On the problem of analytic construction of regulators for equations of
parabolic and hyperbolic type. Differencial'nye Uravnenija 8(1972), pp.
678--695.
14. Milstein
G.N. Linear Ljapunov functions for equations with positive solutions, and mean
square stability. Dokl. Akad. Nauk SSSR 204(1972), no.3, pp. 550--553.
15. Milstein
G.N. Distribution of the integral of a Markov chain with two states. Ural. Gos.
Univ. Mat. Zap. 8(1972), tetrad' 2, pp. 80--90.
16. Milstein G.N. Mean square
stability of linear systems under the action of a Markov chain. Prikl. Mat.
Meh. 36(1972), pp. 537--545.
17. Milstein G.N. Approximate integration of
stochastic differential equations. Theor. Prob. Appl., 19(1974), no.3, pp.
583--588.
18. Melencova,
Ju.A., Milstein G.N. Optimal estimation of the interval of solvability of a
multipoint boundary value problem. Differencial'nye Uravnenija 10(1974), pp.
1630--1641.
19. Milstein
G.N. Exponential stability of positive semigroups in a linear topological
space. Izv. Vyss. Ucebn. Zaved. Matematika, 1975, no.9, pp. 35--42.
20. Makarov,
Ju.D., Milstein G.N., Poljakov, B.N. Solution of certain optimal problems
having phase coordinate constraints associated with rolling on a blooming mill.
Avtomat. i Telemeh., 1969, no. 8, pp. 178--185.
21. Milstein
G.N. An equivalence principle for stochastic systems in mean--square
minimization problems. Avtomat. i Telemeh., 1974, no.2, pp. 25--30.
22. Milstein
G.N. On dispersion of the integral in linear stochastic systems. Avtomat. i
Telemeh., 1974, no.10, pp. 49--55.
23. Milstein
G.N. Exponential stability of positive semigroups in a topological linear
space. II. Izv. Vyss. Ucebn. Zaved. Matematika, 1975, no.10, pp. 51--61.
24. Marhasin
V.S., Milstein G.N. Statistical analysis of paramecia movements. J. General
Biology, 1975, no.1, pp. 119-125.
25. Milstein
G.N. Linear optimal controllers of a specified structure in systems with
incomplete data. Avtomat. i Telemekh. 1976, no.8, pp. 48--53.
26. Milstein
G.N. An extension of the Ljapunov stability criterion to equations in Hilbert
space with an unbounded operator. Problems of analytical mechanics, stability
and control theories (Proc. Second Cetaev Conf., Kazan, 1973), Vol.2, pp.
242--250, Kazan. Aviacion. Inst., Kazan, 1976.
27. Milstein G.N., Rjashko,
L.B. Optimal stabilization of linear stochastic systems. Prikl. Mat. Meh.,
40(1976), no. 6, pp. 1034--1039.
28. Milstein
G.N. Extension of semigroups of operators to locally convex spaces. Izv. Vyss.
Ucebn. Zaved. Matematika, 1977, no.2, pp. 91--95.
29. Milstein G.N. A method with second order
accuracy for the integration of stochastic differential equations. Theor.
Probab. Appl., 23(1978), no.2, pp. 414--419.
30. Milstein G.N. Stability and
stabilization of periodic motions of autonomous systems. Prikl. Mat. Meh.,
41(1977), no.4, pp. 744--749.
31. Milstein G.N. Probabilistic solution of
linear systems of elliptic and parabolic equations. Theor. Prob. Appl.,
23(1978), no.4, pp. 851--855.
32. Marhasin
V.S., Milstein G.N. The model for rhythm influence on contraction force of
cardiac muscle. Biophysics, 23(1978), no.4, pp. 674-681.
33. Milstein
G.N. Quadratic Ljapunov functionals for systems with aftereffect.
Differencial'nye Uravnenija 17(1981), no.6, pp. 984--993.
34. Melencova,
Ju.A., Milstein G.N. On the optimal estimate of the interval of nonoscillation
for linear differential equations with bounded coefficients. Differencial'nye
Uravnenija 17(1981), no.12, pp. 2160--2175.
35. Milstein G.N. Quadratic Lyapunov
functionals and second moments for stochastic systems with aftereffects. Theor.
Prob. Appl., 26(1981), no.4, pp. 734--744.
36. Milstein
G.N. Design of stabilizing controller with incomplete state data for linear
stochastic system with multiplicative noise. Avtomat. i Telemekh. 1982, no.5,
pp. 98--106.
37. Makarova,
L.A., Milstein G.N., Repin, Ju.M. Krylov and Faddeev methods in the spectrum
control problem. Avtomat. i Telemekh. 1982, no.8, pp. 33--41.
38. Auslaender, E.I., Milstein
G.N. Asymptotic expansions of the Ljapunov index for linear stochastic systems
with small noise. Priklad. Mat. Mekh. 46(1982), no.3, pp. 277--283.
39. Milstein
G.N., Ryashko, L.B. Estimation in controllable stochastic systems with
multiplicative noise. Avtomat. i Telemekh. 1984, no.6, pp. 88--94.
40. Venttsel A.D., Gladyshev S.A., Milstein
G.N. Piecewise constant approximation for Monte Carlo calculation of Wiener
integrals. Theor. Prob. Appl., 29(1984), no.4, pp. 715--722.
41. Gladyshev
S.A., Milstein G.N. The Runge--Kutta method for calculation of Wiener integrals
of functionals of exponential type. Zh. Vychisl. Mat. i Mat. Fiz. 24(1984),
no.8, pp. 1136--1149.
42. Milstein
G.N. Optimal stabilization of linear distributed parameter systems with
controllers of a specified structure. Problems Control Inform. Theory 14(1985),
no.2, pp. 117--129.
43. Milstein G.N. Weak approximation of
solutions of systems of stochastic differential equations. Theor. Prob. Appl.,
30(1985), no.4, pp. 706--721.
44. Milstein
G.N., P'yanzin S.A. Digital modelling of the Kalman-Bucy filter and an optimal
filter in quantized arrival of information. Avtomat. i Telemekh. 1985, no.1,
pp. 59-68.
45. Marhasin
V.S., Milstein G.N., Solov'yeva O.E. On the theory of regulation of contraction
force of the heart muscle. Biophysics, 30(1985), no.2, pp. 322-327.
46. Milstein
G.N., P'yanzin S.A. Regularization and digital modeling of a Kalman--Bucy
filter for systems with degenerate noise in the observations. Avtomat. i
Telemekh. 1987, no.11, pp. 80--92.
47. Milstein
G.N. Strictly positive Lyapunov functionals for linear systems with
aftereffect. Differencial'nye Uravnenija 23(1987), no.12, pp. 2051--2060.
48. Milstein G.N. A theorem on the order of
convergence of mean-square approximations of solutions of systems of stochastic
differential equations. Theor. Prob. Appl., 32(1987), no.4, pp. 809--811.
49. Artem'eva
O.G., Marhasin V.S., Milstein G.N., Solov'yeva O.E., Tsivian P.B. Mechanism of
postextrasystolic potentiation in the heart muscle. Sechenov Physiological J.
USSR, 68(1987), no.10, pp. 1339-1344.
50. Malyuck
V.P., Milstein G.N., Vdovin Yu.A. Certification of standard samples on the
basis of interpolation by multivariate splines. Metrological Service in USSR,
1989, no.1, pp. 24-28. .
51. Milstein
G.N., P'yanzin S.A. A discretization method in the construction of an optimal
filter for systems with delay. Avtomat. i Telemekh. 1990, no.12, pp. 84--93.
52. Milstein G.N., Solov'yeva
O.E. Recurrent estimation and parameter identification in nonlinear deterministic
systems. Prikl. Mat. Mekh. 55(1991), no.1, pp. 39--47
53. Milstein G.N., Ryashko L.B.
Stability and stabilization of the orbits of autonomous systems under random
perturbations. J. Appl. Math. Mechs. 56(1992), no.6, pp. 855-862.
54. Milstein
G.N., Rybkina N.F. An algorithm for random walks over small ellipsoids for
solving the general Dirichlet problem. J. Comput. Math. and Math. Phys.
33(1993), no.5, pp. 631-647.
55. Milstein G.N., Solov'yeva
O.E. The constructing of the filters in the nonlinear deterministic systems. J.
Appl. Math. Mechs. 58(1994), no.6, pp. 30-41.
56. Milstein G.N., Tret'yakov M.V.
Numerical solution of differential equations with colored noise. J. Stat. Phys.
77(1994), no.3/4, pp. 691-715.
57. Milstein G.N. The solving
of boundary value problems by numerical integration of stochastic equations.
Mathematics and Computers in Simulation 38(1995), pp. 77-85.
58. Milstein G.N., Ryashko L.B.
First approximation of quasypotential in problems on the stability of systems
with random nonsingular perturbations. J. Appl. Math. Mech. 59(1995), no.1, pp.
47-56.
59. Milstein G.N. Solution of the first
boundary value problem for equations of parabolic type by means of the
integration of stochastic differential equations. Theor. Probab. Appl.,
40(1995), no.3, pp. 556-563.
60. Milstein G.N.
Application of the numerical integration of stochastic equations for the
solution of boundary value problems with Neumann boundary conditions. Theor.
Prob. Appl., 41(1996), no.1, pp. 170-177.
61. Milstein G.N. The simulation
of phase trajectories of a diffusion process in a bounded domain. Stochastics
and Stochastics Reports, 56(1996), no.1-2, pp. 103-125.
62. Milstein
G.N. Evaluation of moment Lyapunov exponents for second order stochastic
systems. Random&Comput. Dynam., 4(1996), no.4, pp. 301-315.
63. Milstein G.N., Tret'yakov M.V.
Mean-square numerical methods for stochastic differential equations with small
noise. SIAM J. Scien. Comp., 18(1997), no. 4, pp. 1067-1087.
64. Milstein G.N., Tret'yakov M.V.
Numerical methods in the weak sense for stochastic differential equations with
small noise. SIAM J. Numer. Anal., 34(1997), no. 6, pp. 2142-2167.
65. Milstein
G.N. Stability index for invariant manifolds of stochastic systems. Random
& Comput. Dynam., 5(4), 1997, pp. 177-211.
66. Milstein G.N. Weak
approximation of a diffusion process in a bounded domain. Stochastics and
Stochastics Reports, 62(1997), pp. 147-200.
67. Milstein G.N., Platen E.,
Schurz H. Balanced implicit methods for stiff stochastic systems. SIAM J.
Numer. Anal., 35(1998), no. 3, pp. 1010--1019.
68. Milstein G.N. On the
mean-square approximation of a diffusion process in a bounded domain.
Stochastics and Stochastics Reports, 64(1998), pp. 211-233.
69. Milstein G.N., Nussbaum M.
Diffusion approximation for nonparametric autoregression. Prob. Th. Rel.
Fields, 112(1998), no. 4, pp. 535-543.
70. Milstein G.N., Tretyakov M.V.
Simulation of a space-time bounded diffusion. Ann. Appl. Probab., 9(1999), no.
3, pp. 732-779.
71. Milstein G.N., Tretyakov
M.V. Mean velocity of noise-induced transport in the limit of weak periodic
forcing. J. Physics A, 32(1999), no. 31, pp. 5795-5805.
72. Babovsky H., Milstein G.N.
Transport equations with singularity. Transport Theory & Statistical
Physics, 28(1999), no. 6., pp. 575-595.
73. Milstein G.N., Tretyakov
M.V. Numerical algorithms for semilinear parabolic equations with small
parameter based on weak approximation of stochastic differential equations.
Math. Comp., 60(2000), no. 229, pp. 237-267.
74. Milstein G.N., Tretyakov
M.V. Numerical analysis of noise-induced regular oscillations. Physica D,
140(2000), no, 3-4, pp. 244-256.
75. Milstein G.N. Orbital
stability index for stochastic systems. Stochastic Analysis and Applications,
18(2000), no. 5, pp.777-809.
76. Milstein G.N.,
Veretennikov A.Yu. On deterministic and stochastic sliding modes via small
diffusion approximation. Markov Proc. Relat. Fields, 6(2000), no. 3, pp.
371-395.
77. Khasminskii R., Milstein G.N.
On estimation of the linearized drift for nonlinear stochastic differential
equations. Stochastic and Dynamics, 1(2001), no. 1, 23-43.
78. Milstein G.N., Tretyakov M.V.
Noice-induced unidirectional transport. Stochastic and Dynamics, 1(2001), no.
3, 361-375.
79. Milstein G.N., Tretyakov M.V.
Numerical solution of the Dirichlet problem for nonlinear parabolic equations
by a probabilistic approach. IMA J. Numer. Anal., 21(2001), no. 4, pp. 887-917.
80. Milstein G.N., Nussbaum M.
Maximum likelihood estimation of a nonparametric signal in white noise by
optimal control. Stat. Probab. Letters, vol. 55/2(2001), pp. 193-203.
81. Milstein G.N., Tretyakov M.V.,
Repin Yu.M. Symplectic integration of Hamiltonian systems with additive noise.
SIAM J. Numer. Anal., 39(2002), no. 6, pp. 2066-2088.
82. Milstein G.N., Tretyakov M.V.
The simplest random walks for the Dirichlet problem. Theor. Prob. Appl.,
47(2002), no. 1, pp. 39-58.
83. Imkeller P., Milstein G.N.
Moment Lyapunov exponent for conservative systems with small periodic and
random perturbations. Stochastic and Dynamics, 2(2002), no. 1, pp. 25-48.
84. Milstein G.N. The probability approach
to numerical solution of nonlinear parabolic equations. Numer. Methods PDEs,
18(2002), no. 4, pp. 490-522.
85. Milstein G.N., Tretyakov M.V. A
probabilistic approach to the solution of the Neumann problem for nonlinear
parabolic equations. IMA J. Numer. Anal., 22(2002), no. 4, pp. 599-622.
86. Milstein G.N., Schoenmakers
J.G.M. Numerical construction of a hedging strategy against the multi-asset
European claim. Stochastics and Stochastics Reports, 73(2002), no. 1-2, pp.
125-157.
87. Milstein G.N. The asymptotic
behavior of semi-invariants for linear stochastic systems. Stochastic and
Dynamics, 2(2002), no. 2, pp. 281-294.
88. Milstein G.N., Repin Yu.M.,
Tretyakov M.V. Numerical methods for stochastic systems preserving symplectic
structure. SIAM J. Numer. Anal., 40(2003), no. 4, pp. 1583-1604.
89. Milstein G.N., Tretyakov M.V.
Quasi-symplectic methods for Langevin type equations. IMA J. Numer. Anal.,
23(2003), pp. 593-626.
90. Belopolskaya Ya., Milstein
G.N. An approximation method for Navier-Stokes equations by probabilistic
approach. Stat. Probab. Letters, 64(2003), no. 2, pp. 201-211.
91. Milstein G.N., Schoenmakers
J.G.M., Spokoiny V. Transition density estimation for stochastic differential
equations via forward-reverse representations. Bernoulli, 10(2004), No. 2,
281-312.
92. Milstein G.N., Tretyakov M.V.
Evaluation of conditional Wiener integrals by numerical integration of
stochastic differential equations. J. Comput. Physics, 197(2004) Issue 1,
275-298.
93. Khasminskii R., Milstein G.N.
Stability of gyroscopic systems under small random excitations. Stochastics and
Dynamics, 4(2004), no. 1, 107-133.
94. Milstein G.N., Reiss O.,
Schoenmakers J. A new Monte Carlo method for American options. Inter. J.
Theoret. Appl. Finance, 7(2004), no. 5, 591-614.
95. Milstein G.N., Tretyakov M.V.
Numerical analysis of Monte Carlo evaluation of Greeks by finite differences.
J. Comput. Finance 8(2005), 1-33.
96. Spivakovskaya D.,
Heemink A.W., Milstein G.N., Schoenmakers J.G.M. Simulation of particle
concentrations in coastal waters using forward and reverse time diffusion. Adv.
Water Resources 28(2005), 927-938.
97. Milstein G.N., Tretyakov M.V.
Numerical integration of stochastic differential equations with nonglobally
Lipschitz coefficients. SIAM J Num. Anal. 43(2005), no. 3, 1134-1154.
98. Milstein G.N., Tretyakov M.V.
Numerical algorithms for forward-backward stochastic differential equations.
SIAM J Sci. Comp. 28(2006), no. 2, 561-582.
99. Belomestny D., Milstein G.N.
Monte Carlo evaluation of American options using consumption processes. Intern.
J. Theor. Appl. Finance 9(2006), no. 4, 1-27.
100. Milstein G.N., Tretyakov M.V.
Discretization of forward-backward stochastic differential equations and
related quasi-linear parabolic equations. IMA J. Numer. Anal., 27(2007), 24-44.
101. Milstein G.N., Tretyakov
M.V. Computing ergodic limits for Langevin equations. Physica D, 229 (2007),
no. 1, 81-95.
102. Fedotov S., Milstein G.N.,
Tretyakov M.V. Superdiffusion of a random walk driven by ergodic Markov process
with switching. J. Phys. A: Math. Theor. 40 (2007), 5769-5782.
103. Milstein G.N., Schoenmakers
J., Spokoiny V. Forward and reverse representations for Markov chains. Stoch.
Processes Appl., 117 (2007), 1052-1075.
104. Belomestny
D., Milstein G.N. Simulation based option pricing. In: W. Haerdle, N. Hautch, L.
Overbeck (Eds). Applied Quantitative Finance, Springer, 2008, pp. 363-378.
105. Belomestny D., Milstein G.N.,
Spokoiny V. Regression methods in pricing American and Bermudan options using
consumption processes. Quantitative Finance, 9(2009), no. 3, 315--327.
106. Milstein G.N., Tretyakov
M.V. Solving parabolic stochastic partial differential equations via averaging
over characteristics. Math. Comp., 78 (2009), no. 268, 2075--2106.
107. Milstein G.N., Tretyakov M.V.
Monte Carlo algorithms for backward equations in nonlinear filtering. Adv.
Appl. Prob., 41 (2009), no. 1, 63-100.
108. Milstein G.N., Tretyakov M.V.
Practical variance reduction via regression for simulating diffusions. SIAM J.
Numer. Anal., 47 (2009), no. 2, 887-910.
109. Belomestny D., Milstein G.N.,
Schoenmakers J. Sensitivities for Bermudan options by regression methods.
Decisions Econ. Fin., 33 (2010), no. 2, 117-138.
110. Milstein
G.N., Tretyakov M.V. Averaging over characteristics with innovation approach in
nonlinear filtering. In Handbook
of Nonlinear Filtering (Eds. D. Crisan and B. Rozovskii). Oxford University
Press, 2011, pp. 892--919.
111. Milstein G.N., Tretyakov M.V.
Solving the Dirichlet problem for Navier-Stokes equations by probabilistic
approach. BIT Numer. Math., 52(2012), 141--153.
112. Khasminskii
R., Milstein G.N. Moment Lyapunov exponents and stability index. Appendix B in
the book by R. Khasminskii
Stochastic Stability of Differential Equations, Springer, 2012, pp.
281--322.
113. Milstein G.N., Tretyakov M.V.
Probabilistic methods for the incompressible Navier-Stokes equations with space
periodic conditions. Adv. Appl. Probab., 45(2013), no. 3, 742-772.
114. Milstein G.N., Spokoiny V.
Construction of mean-self-financing strategies for European options under
regime-switching. SIAM J. Fin. Math., 5 (2014), no. 1, 532-556.
115. Milstein G.N., Schoenmakers J.
Uniform approximation of the Cox-Ingersoll-Ross process. Adv. in Appl. Probab.
47 (2015), no. 4, 1132-1156.
116. Milstein G.N., Tretyakov M.V.
Layer methods for stochastic Navier-Stokes equations using simplest
characteristics. J. Comput. Appl. Math. 302 (2016), 1-23.
117. Milstein G.N., Schoenmakers J.
Uniform approximation of the Cox-Ingersoll-Ross process via exact simulation at
random times. Adv. Appl. Prob. 48 (2016), no. 4, 1095-1116.
118. Milstein G.N., Tretyakov M.V.
Mean-square approximation of Navier-Stokes equations with additive noise in
vorticity-velocity formulation. Numer. Math. Theory Methods Appl. 14 (2021),
no. 1, 1-30.
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Urals, 1987
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Hamburg, 1998
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Edinburgh,
2007
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