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| Kuznetsov Dmitriy Feliksovich |
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| Doctor of physico-mathematical sciences (2003) |
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 $\psi_1(\tau),\ldots,\psi_k(\tau)\in C^1[t,T]$ $(k=1,\ldots,5),$ $\psi_1(\tau),\ldots,\psi_6(\tau)\equiv 1$ $(k=6).$ 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 $\psi_1(\tau), \psi_2(\tau)\in L_2[t,T]$ $(k=1, 2),$ $\psi_1(\tau),\ldots, \psi_k(\tau)\in C[t, T]$ $(k=3,\ 4,\ 5).$
$3.$ 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]\ $ $(k\in\mathbb{N})$ (https://arxiv.org/pdf/2003.14184v57, Sect. 2.31, Theorem 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 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$.$
ISTINA
