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Kuznetsov, Dmitriy Feliksovich

 Total publications: 125 (118) in MathSciNet: 41 (40) in zbMATH: 40 (40) in Web of Science: 8 (8) in Scopus: 31 (30) Cited articles: 22 Citations in Math-Net.Ru: 23 Citations in Web of Science: 15 Citations in Scopus: 89 Presentations: 1

 Number of views: This page: 25535 Abstract pages: 1774 Full texts: 426 References: 107
Doctor of physico-mathematical sciences (2003)
Speciality: 05.13.18 (Mathematical modelling, calculating methods, and the program systems)
Birth date: 24.04.1970
E-mail:
Website: http://www.sde-kuznetsov.spb.ru
Keywords: iterated stochastic Ito integral, iterated stochastic Stratonovich integral, Wiener process, multidimensional Wiener process, infinite-dimensional $Q$-Wiener process, Ito stochastic differential equation, stochastic differential equation of jump-diffusion type, non-commutative semilinear stochastic partial differential equation (SPDE) with nonlinear multiplicative trace class noise, stochastic Ito-Taylor expansion, stochastic Stratonovich-Taylor expansion, generalized multiple Fourier series, multiple Fourier-Legendre series, multiple trigonometric Fourier series, mean-square approximation of iterated stochastic integrals, approximation with probability $1$ of iterated stochastic integrals, high-order strong numerical methods for Ito stochastic differential equations, numerical modeling of stochastic systems.
UDC: 519.2, 519.21, 519.6, 517.521, 517.521.5, 517.586, 519.85
MSC: 60H10, 60H35, 65C30, 60H05, 42B05, 42C10

Subject:

Fourier method for numerical integration of Ito stochastic differential equations (SDEs)$,$ SDEs of jump-diffusion type as well as for non-commutative semilinear stochastic partial differential equations (SPDEs) with nonlinear multiplicative trace class noise (within the framework of a semigroup approach or an approach based on the so-called mild solution) has been proposed and developed$.$ More precisely$,$ the generalized multiple Fourier series (converging in the sense of norm in Hilbert space $L_2([t,\hspace{0.2mm} T]^k),$ $k\in \mathbb {N}$) in complete orthonormal systems of functions in the space $L_2([t,\hspace{0.2mm} T]^k),$ $k\in \mathbb {N}$ have been applied to expansion and strong approximation (mean-square$,$ mean of degree $p$ $(p>0)$ as well as with probability $1$) of iterated stochastic Ito integrals of the form $$\label{1} \int\limits_t^T\psi_k(t_k)\ \ldots \int\limits_t^{t_{2}} \psi_1(t_1) d{\bf W}_{t_1}^{(i_1)}~ \ldots~ d{\bf W}_{t_k}^{(i_k)},$$ where $k\in \mathbb {N},$ $\psi_l(\tau):\ [t,\hspace{0.2mm} T]\to\mathbb {R},$ $l=1,\ldots,k$ are continuous deterministic functions$,$ ${\bf W}_{\tau} \in \mathbb{R^m}$ is a standard vector Wiener process with independent components ${\bf W}_{\tau}^{(i)},$ $i=1,\ldots,m\$ and $\ {\bf W}_{\tau}^{(0)}:=\tau,\$ $i_1,\ldots,i_k=0,\ 1,\ldots,m.$

Theorem on convergence with probability $1$ for expansions of iterated stochastic Ito integrals $(1)$ of arbitrary multiplicity $k\in\mathbb{N}$ has been fomulated and proved for continuously differentiable functions $\psi_{l}(\tau): [t, T] \to\mathbb {R},$ $l=1,\ldots,k$ as well as for multiple Fourier-Legendre series and multiple trigonometric Fourier series converging in the sense of norm in the space $L_2([t,\hspace{0.2mm} T]^k),$ $k\in \mathbb {N}.$

Generalizations of the Fourier method for complete orthonormal systems of functions with weight $r(t_1) \ldots r(t_k)$ in the space $L_2([t,\hspace{0.2mm} T]^k),$ $k\in \mathbb {N}$ as well as for some other types of iterated stochastic integrals (iterated stochastic integrals with respect to martingale Poisson measures and iterated stochastic integrals with respect to martingales) were obtained$.$

The above results were adapted for iterated stochastic Stratonovich integrals of the form $$\label{2} \int\limits_t^T\psi_k(t_k)\ \ldots \int\limits_t^{t_{2}} \psi_1(t_1)\hspace{0.3mm} \circ d{\bf W}_{t_1}^{(i_1)}\ \ldots\hspace{0.5mm} \circ d{\bf W}_{t_k}^{(i_k)},$$ where $k=\overline{1, 5},\$ $\psi_l(\tau):\ [t,\hspace{0.2mm} T]\to\mathbb {R},\$ $l=1,\ldots,k\$ are smooth nonrandom functions$.$ These results can be interpreted as Wong-Zakai type theorems on the convergence of iterated Riemann-Stieltjes integrals of multiplicities 1-5 to iterated stochastic Stratonovich integrals. The hypothesis on expansion of iterated stochastic Stratonovich integrals of form $(2)$ for the case of arbitrary multiplicity $k\in \mathbb {N}$ has been formulated$.$

We formulated and proved two theorems on expansion of iterated stochastic Stratonovich integrals of form $(2)$ of arbitrary multiplicity $k\in \mathbb {N}$ based on generalized iterated Fourier series (converging pointwise) in complete orthonormal systems of functions in the space $L_2 ([t,\hspace{0.2mm} T]).$

Numerical simulation of iterated stochastic Ito and Stratonovich integrals $(1)$ and $(2)$ is one of the main problems at the stage of numerical realization of high-order strong numerical methods for Ito SDEs and SDEs of jump-diffusion type$.$

Fourier method for iterated stochastic Ito integrals $(1)$ is also applied to the mean-square approximation of iterated stochastic integrals with respect to the infinite-dimensional $Q$-Wiener process$.$ In particular$,$ to the mean-square approximation of integrals of the form $$\int\limits_{t}^{T} \Psi_k(Z) \left( \ldots \left(\hspace{0.2mm} \int\limits_{t}^{t_2} \Psi_1(Z) \psi_1(t_1) d{\bf W}_{t_1}({\bf x})\right) \ldots \right) \psi_k(t_k) d{\bf W}_{t_k}({\bf x}),$$ where $k\in \mathbb {N},\hspace{0.2mm}$ ${\bf W}_{\tau}({\bf x})$ is an $U$-$\hspace{0.2mm}$valued $Q\hspace{0.2mm}$-$\hspace{0.2mm}$Wiener process$,$ $Z:$ $\Omega \rightarrow H$ is an ${\bf F}_t/{\cal B}(H)\hspace{0.2mm}$-$\hspace{0.2mm}$measurable mapping$,$ $\Psi_k(v) (\hspace{0.5mm} \ldots ( \Psi_1(v) ) \ldots )$ is a $k~$-$\hspace{0.2mm}$linear Hilbert-Schmidt operator mapping from $U_0\times\ldots \times U_0$ to $H$ for all $v\in H,~$ the functions $\psi_l(\tau),\$ $l=1,\ldots, k,\$ are the same as in $(1),~$ $Q:$ $U \rightarrow U$ is a trace class operator$,$ $\hspace{0.2mm}$ $U,$ $H$ are separable real-valued Hilbert spaces$,\$ $U_0=Q^{1/2}(U).$

Mean-square approximation of iterated stochastic integrals with respect to the infinite-dimensional $Q$-Wiener process is one of the most difficult problems in numerical implementation of high-order strong approximation schemes (with respect to the temporal discretization) for non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise (approximation schemes based on the so-called mild solution)$.$

Legende polynomials were first applied to the mean-square approximation of iterated stochastic Ito and Stratonovich integrals $(1)$ and $(2)$ with multiplicities $1$ to $6.$ It is shown that the Legendre polynomial system is an optimal system for solving this problem.

Theorems on replacing the order of integration for iterated stochastic Ito integrals and iterated stochastic integrals with respect to martingales were formulated and proved$.$

The so-called unified Ito-Taylor and Stratonovich-Taylor expansions were derived$.$

Strong numerical methods of high-orders of accuracy $\gamma =1.0,$ $1.5,$ $2.0,$ $2.5,$ $3.0, ...$ for Ito SDEs with multidimensional and non-commutative noise were constructed$.$ Among them there are explicit and implicit$,$ one-step and multistep methods as well as the methods of Runge-Kutta type$.$

His research interests also include various types of stochastic integrals and their properties as well as the numerical modeling of linear and nonlinear stochastic dynamical systems$.$

Biography

In 1993 he graduated from Department of Mechanics and Control Processes of Faculty of Physics and Mechanics of Saint-Petersburg State Technical University (Peter the Great Saint-Petersburg Polytechnic University). Ph.D. (1996), doctor of phisico-mathematical sciences (2003), professor of Department of Mathematics of Peter the Great Saint-Petersburg Polytechnic University since 2005, author of monographs on numerical integration of Ito stochastic differential equations and strong approximation of iterated Ito and Stratonovich stochastic integrals.

Main publications:
1. Kuznetsov D. F., “On numerical modeling of the multidimentional dynamic systems under random perturbations with the 2.5 order of strong convergence”, "Automation and Remote Control", 80:5 (2019), 867–881
2. Kuznetsov D. F., “Expansion of iterated Stratonovich stochastic integrals, based on generalized multiple Fourier series”, "Ufa Mathematical Journal", 11:4 (2019), 49–77
3. Kuznetsov D. F., “A comparative analysis of efficiency of using the Legendre polynomials and trigonometric functions for the numerical solution of Ito stochastic differential equations”, "Computational Mathematics and Mathematical Physics", 59:8 (2019), 1236–1250
4. Kuznetsov D. F., “Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations”, "Computational Mathematics and Mathematical Physics", 58:7 (2018), 1058–1070
5. Kuznetsov D. F., “On numerical modeling of the multidimensional dynamic systems under random perturbations with the 1.5 and 2.0 orders of strong convergence”, "Automation and Remote Control", 79:7 (2018), 1240–1254

http://www.mathnet.ru/eng/person34602
https://zbmath.org/authors/?q=ai:kuznetsov.dmitriy-feliksovich
https://mathscinet.ams.org/mathscinet/MRAuthorID/650458
https://elibrary.ru/author_items.asp?spin=6041-1626
ISTINA http://istina.msu.ru/workers/78945709
http://orcid.org/0000-0001-5747-1282
http://www.researcherid.com/rid/R-3032-2017
https://www.scopus.com/authid/detail.url?authorId=7101858544
https://www.researchgate.net/profile/Dmitriy_Kuznetsov
https://arxiv.org/a/kuznetsov_d_1

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