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 arbitrary 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 approximation$,$ approximation in the mean of degree $p$ $(p>0)$ as well as approximation with probability $1$) of iterated Ito stochastic integrals of the form \begin{equation} \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)}, \end{equation} where $k\in \mathbb {N},\ $ $\psi_{1}(\tau),\ldots,\psi_k(\tau)\in L_2[t, T],\ $ ${\bf W}_{\tau} \in \mathbb{R^m}$ is a standard multidimensional Wiener process with independent components ${\bf W}_{\tau}^{(i)}$ $(i=1,\ldots,m)\ $ and $\ {\bf W}_{\tau}^{(0)}:=\tau,\ $ $i_1,\ldots,i_k$ $=0,\ 1,\ldots,m.$

The relationship of the mentioned expansion with the multiple Wiener stochastic integrals with respect to components of a multidimensional Wiener process and Hermite polynomials of the vector random argument is established. The mean-square approximation error for iterated Ito stochastic integrals of form $(1)$ of arbitrary multiplicity $k,\ $ $k\in\mathbb{N}$ for all possible combinations of indices $i_1,\ldots, i_k \in\{1,\ldots, m\}$ in the framework of this approach has been calculated exactly.

Theorem on convergence with probability $1$ for expansions of iterated Ito stochastic integrals $(1)$ of arbitrary multiplicity $k\in\mathbb{N}$ has been fomulated and proved for the case of multiple Fourier-Legendre series and multiple trigonometric Fourier series converging in the sense of norm in the space $L_2([t,\hspace{0.2mm} T]^k),$ $k\in \mathbb {N}$ as well as for $\psi_{1}(\tau),\ldots,\psi_k(\tau)\in C^1[t,T].$ The rate of convergence in this theorem is found.

Generalizations of the Fourier method for complete orthonormal with weight $r(t_1) \ldots r(t_k)$ systems of functions 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 under a special condition on trace series for iterated Stratonovich stochastic integrals of the form \begin{equation} \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)}, \end{equation} where $k\in \mathbb {N},\ $ $\psi_{1}(\tau),\ldots,\psi_k(\tau)\in L_2[t,T],\ $ ${\bf W}_{\tau} \in \mathbb{R^m}$ is a standard multidimensional Wiener process with independent components ${\bf W}_{\tau}^{(i)}$ $(i=1,\ldots,m)\ $ and $\ {\bf W}_{\tau}^{(0)}:=\tau,\ $ $i_1,\ldots,i_k$ $=0,\ 1,\ldots,m.$ The above condition on trace series was removed for the following three cases.

$1.$ The case of multiple Fourier series in complete orthonormal systems of Legendre polynomials and trigonometric functions (Fourier basis) in $L_2([t,\hspace{0.2mm} T]^k)$ as well as $\psi_1(\tau),\ldots,\psi_5(\tau)\in C^1[t,T]$ $(k=1,\ldots,5),$ $\psi_1(\tau),\ldots,\psi_8(\tau)\equiv 1$ $(k=6, 7, 8).$ The rate of mean-square convergence of expansions of iterated Stratonovich stochastic integrals is found for this case $(k=1,\ldots,5).$

$2.$ The case of multiple Fourier series in arbitrary complete orthonormal systems in $L_2([t,T]^k)$ and $\psi_1(\tau), \ \psi_2(\tau)\in L_2[t,T]$ $(k=1,\ 2),\ $ $\psi_1(\tau)=(\tau-t)^{p_1},$ $ \ldots,\ $ $\psi_4(\tau)=(\tau-t)^{p_4},\ $ $p_1,\ldots,p_4$ $=$ $0,\ 1,\ 2,\ldots\ $ $(k=3,\ 4),\ $ $\psi_1(\tau),\ldots,\psi_6(\tau)\equiv 1\ $ $(k=5,$ $ 6).$

$3.$ The case of multiple Fourier series in arbitrary complete orthonormal systems in $L_2([t,T]^k)$ and $\psi_1(\tau), \ldots, \psi_k(\tau)$ $\in$ $C[t, T]\ $ $(k\in\mathbb{N})$ but under one additional condition (https://arxiv.org/pdf/2003.14184v67, Theorems 2.59, 2.61).

These results can be interpreted as Wong-Zakai type theorems on the convergence of iterated Riemann-Stieltjes integrals to iterated Stratonovich stochastic integrals. The hypothesis on expansion of iterated Stratonovich stochastic integrals of form $(2)$ has been formulated for the case of an arbitrary multiplicity $k\in \mathbb {N}.$

We formulated and proved two theorems on expansion of iterated Stratonovich stochastic integrals of form $(2)$ of arbitrary multiplicity $k\in \mathbb {N}$ based on iterated Fourier series converging pointwise.

Numerical simulation of iterated Ito and Stratonovich stochastic 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 Ito stochastic 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 $v\in H,~$ $\psi_1(\tau),\ldots,\psi_k(\tau)\in L_2[t,T],\ $ $Q:~U \rightarrow U$ is a linear symmetric non-negative 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 Ito and Stratonovich stochastic integrals $(1)$ and $(2)$ with multiplicities $1$ to $6.$ It is shown that the Legendre polynomial system is the optimal system for solving this problem for $k\ge 3.$

Theorems on replacing the order of integration for iterated Ito stochastic 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$.$

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 2006, author of monographs on numerical integration of Ito stochastic differential equations and strong approximation of iterated Ito and Stratonovich stochastic integrals.

1. Kuznetsov D. F., Kuznetsov M. D., “Mean-square approximation of iterated stochastic integrals from strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs based on multiple Fourier-Legendre series”, Recent Developments in Stochastic Methods and Applications, ICSM-5 2020, Springer Proceedings in Mathematics & Statistics, 371, eds. Shiryaev A.N., Samouylov K.E, Kozyrev D.V., Springer, Cham, 2021, 17–32        
2. Kuznetsov D. F., “Explicit one-step numerical method with the strong convergence order of 2.5 for Ito stochastic differential equations with a multi-dimensional nonadditive noise based on the Taylor–Stratonovich expansion”, "Computational Mathematics and Mathematical Physics", 60:3 (2020), 379–389                
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., “Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs”, 2025, 1–1238, arXiv: 2003.14184        

• Peter the Great St. Petersburg Polytechnic University