Multidimensional Wiener Process Research Articles

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

An analysis of approximation algorithms for iterated stochastic integrals and a Julia and Matlab simulation toolbox

For the approximation and simulation of twofold iterated stochastic integrals and the corresponding Lévy areas w.r.t. a multi-dimensional Wiener process, we review four algorithms based on a Fourier series approach. Especially, the very efficient algorithm due to Wiktorsson and a newly proposed algorithm due to Mrongowius and Rößler are considered. To put recent advances into context, we analyse the four Fourier-based algorithms in a unified framework to highlight differences and similarities in their derivation. A comparison of theoretical properties is complemented by a numerical simulation that reveals the order of convergence for each algorithm. Further, concrete instructions for the choice of the optimal algorithm and parameters for the simulation of solutions for stochastic (partial) differential equations are given. Additionally, we provide advice for an efficient implementation of the considered algorithms and incorporated these insights into an open source toolbox that is freely available for both Julia and Matlab programming languages. The performance of this toolbox is analysed by comparing it to some existing implementations, where we observe a significant speed-up.

Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations

For T > 0, we prove theorems concerning sharp asymptotics of the probabilities$$ \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_j^2(t)>{u}^2\right\},\kern1em \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_{j0,T}^2(t)>{u}^2\right\}, $$as u → ∞, where wj(t), j = 1, . . . , n, are independent Wiener processes and wj0,T (t), j = 1, . . . , n, are independent Brownian bridges on the segment [0, T]. Our research method is the double sum method for the Gaussian processes and fields. We also give an application of the obtained results to the statistical tests for the homogeneity hypothesis of k one-dimensional samples.

Sequential convex programming for non-linear stochastic optimal control

This work introduces a sequential convex programming framework for non-linear, finitedimensional stochastic optimal control, where uncertainties are modeled by a multidimensional Wiener process. We prove that any accumulation point of the sequence of iterates generated by sequential convex programming is a candidate locally-optimal solution for the original problem in the sense of the stochastic Pontryagin Maximum Principle. Moreover, we provide sufficient conditions for the existence of at least one such accumulation point. We then leverage these properties to design a practical numerical method for solving non-linear stochastic optimal control problems based on a deterministic transcription of stochastic sequential convex programming.

Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

The article is devoted to approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 3 by the method of multiple Fourier–Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence order 1.5 for Ito stochastic differential equations with multidimensional noncommutative noise. These numerical methods are based on the so-called Taylor–Ito and Taylor–Stratonovich expansions. We calculate the exact lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals. Thus, the computational cost for the implementation of numerical methods can be significantly reduced.

Optimal sampling policy of multi‐dimensional wiener processes with random delivery time

A sampling problem for remote monitoring and estimation of the multi-dimensional Wiener process through a channel with random delivery time under the constraint of maximum sampling frequency is considered. The authors study the optimal sampling policy that minimizes the mean square error. It is showed that the threshold policy is optimal and the near closed-form expressions of the optimal threshold and the corresponding mean square error are given. A bisection method is proposed to search the numerical solutions. The results provide a general framework for remote estimation and sampling, including the special cases that the dimension of Wiener source is one and that the delivery time of the channel tends to zero. Simulation results show that even though there is no constraint on the maximum sampling frequency, age-optimal policy, periodic sampling policy and zero-wait policy are far from optimal.

On Sets of Laws of Continuous Martingales

We prove that the set of laws of stochastic integrals $H\,{\cdot}\, W$, where $W$ is a multidimensional Wiener process and $H^2$ takes values in a compact convex subset $D$ of the set of symmetric positive semidefinite matrices, is weakly dense in the set of laws of martingales $M$ with $d\langle M \rangle/dt$ taking values in $D$.

Simulation of the FPT for Multi-Dimensional Wiener Process with Dynamic Thresholds

Reliability modeling and analysis has becoming an essential issue in complex system management. Degradation system reliability modeling is an important part of the reliability field due to that the failure data is more and more difficult to obtain. Degradation data can well describe the failure process of a system or product. In the stochastic process, the Wiener process has been widely used in literature due to its good properties. Multidimensional diffusion Wiener process describes the multidimensional degradation process of products. First passage time is the lifetime of a system to reach a threshold value. In this paper, a generalized model is developed for evaluating the reliability of the multidimensional degradation process. Meanwhile, we use the Monte Carlo method for the solution of the FPT with dynamic threshold value. The comparisons show that the results of the proposed model agree well with the Monte Carlo results. The research can support decision making in the reliability improving.

ПОСТРОЕНИЕ НАБЛЮДЕНИЯ В МОДЕЛИ ШЕСТАКОВА–СВИРИДЮКА ПРИ ЕГО ИСКАЖЕНИИ МНОГОМЕРНЫМ «БЕЛЫМ ШУМОМ»

The Shestakov–Sviridyuk model is a mathematical model of a measuring unit used to reconstruct a dynamically distorted signal with the help of experimental data. This model is also called the optimal dynamic measurement problem. The theory of optimal dynamic measurement is based on the problem of minimizing the difference between the values of a virtual observation obtained using a computational model and experimental data, usually distorted by some disturbances. The article describes the Shestakov–Sviridyuk model of optimal dynamic measurement in terms of various types of disturbances. It focuses on the preliminary stage of the study of the optimal dynamic measurement problem, namely, the Pyt’ev–Chulichkov method for constructing observation data, i. e. converting experimental data to clean them from disturbances in the form of “white noise”, which is understood as the Nelson–Glicklich derivative of the multidimensional Wiener process. To use this method, a priori information on the properties of the functions describing the observation, is used.

On the support of solutions to stochastic differential equations with path-dependent coefficients

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus.

On Stability and Comparison Theorems for Systems of Stochastic Differential Equations

We prove comparison theorems for stochastic differential equations (briefly, SDE) with respect to a standard multidimensional Wiener process as well as for components of systems of SDE with respect to a multidimensional Wiener process. The obtained results are applied to the study of the stability with probability 1 of the perturbed solutions to the SDE.

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

On the numerical integration of the undamped harmonic oscillator driven by independent additive gaussian white noises

The stochastic exponential method is applied to solve the undamped linear stochastic oscillator driven by a multidimensional Wiener process. The proposed additive noise model is new since the disturbances are introduced in the equivalent first order two dimensional system and can affect both space and velocity. It is shown that three important properties related to long time behavior of its analytical solution are preserved. In addition, the mean-square error of the numerical solution is analyzed. Numerical experiments confirming the theoretical results are presented.

On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the d-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem in the bounded case. The approach is illustrated through the optimal stopping problem of a d-dimensional Wiener process with a positive definite quadratic form reward function.

Constrained Wiener Processes and Their Financial Applications

The extrema of Wiener processes are relevant to the pricing of so-called exotic options, which have many financial applications. The probability densities of such extrema are well known for one dimensional Wiener processes. We employ elementary methods to derive analytical expressions for the densities for multidimensional Wiener processes, with multiple extrema. These take the form of (possibly infinite) series expansions of Gaussian densities. This is undertaken using the characterization of the Wiener process by the heat equation, a well known connection in mathematical physics.

Extrema of Wiener Processes

The extrema of Wiener processes are relevant to the pricing of so-called exotic options, which have many financial applications. The probability den-sities of such extrema are well known for one dimensional Wiener processes. We employ elementary methods to derive analytical expressions for the den-sities for multidimensional Wiener processes, with multiple extrema. These take the form of (possibly infinite) series expansions of Gaussian densities. This is undertaken using the characterization of the Wiener process by the heat equation, a well known connection in mathematical physics.

Stabilization by Artificial Wiener Processes

In this technical note, we restage the positive effects of Gaussian white noises for continuous-time driftless input-affine nonholonomic systems. Compared with previous studies, we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi-dimensional Wiener processes. Furthermore, we found that the positive effects are artificially restaged by numerical calculations with use of Euler scheme for ordinary differential equations. Our results demonstrate that the positive effects act as “hidden independent control inputs” for nonholonomic systems. This implies that our results provide a constructive procedure for designing control inputs such that the origins become $p$ -th moment exponentially or finite-time stable.

State feedback control of interval type-2 T–S model based uncertain stochastic systems with unmatched premises

This paper is concerned with the state feedback control for interval type-2 (IT2) uncertain Itô stochastic fuzzy systems with a multidimensional Wiener process and unmatched premises. The uncertainties are of linear fractional form, and appear not only in the membership functions but also in the parametric matrices of the systems. The fuzzy basis functions of the controllers to be designed are different from those of the IT2 fuzzy model. Since stochastic perturbations and unmatched premises as well as parametric uncertainties are involved in the underlying systems, the stabilization problem becomes more complicated and challenging than that for deterministic systems. Facilitating by space decomposition, the lower and upper membership functions (LUMFs) can be locally represented in terms of the convex combinations of some local basis functions whose coefficients can be obtained via evaluating them at the boundaries of the subspaces decomposed. So the unmatched fuzzy basis functions can be handled in stability analysis of the resulting closed-loop systems with support of these local representations. Then by employing a matrix decomposition technique which is effective in dealing with linear fractional uncertainties which involve a multidimensional Wiener process, a state feedback controller is developed such that the resulting closed-loop IT2 system is stochastically asymptotically stable. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.

T–S fuzzy affine model based non-synchronized state estimation for nonlinear Itô stochastic systems

This paper deals with the robust H∞ filter design problem for a class of nonlinear stochastic systems which are modeled by uncertain Takagi–Sugeno (T–S) fuzzy affine Itô stochastic models affected by a multidimensional Wiener process. The objective is to design an admissible full-order filter so that the resulting stochastic filtering error system is mean-square asymptotically stable with a prescribed disturbance attenuation level in an H∞ sense. It is assumed that the plant premise variables are not necessarily measurable so that the filter implementation with state space partition may be not synchronized with the state trajectories of the plant. Based on piecewise quadratic Lyapunov functions, some new results are proposed for the filtering design of T–S fuzzy affine Itô stochastic systems. The filter gain of each region can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, two simulation examples are provided to illustrate the effectiveness of the proposed filtering design approach.