Dimensional Stochastic Differential Equations Research Articles

The nonlinear dynamics analysis of stochastic delay Jeffcott rotor-seal system with the elastic support

The establishment of stochastic delay Jeffcott rotor-seal system not only considers the track irregularity caused by random noise and the possible influence of stochastic parameter excitation, but also considers the time-delay characteristics of sealing force. The rotation of the rotor shaft causes the gas in the chamber to produce a dynamic effect, resulting in a rotating force, which delays the feedback on the rotating shaft. Firstly, the one-dimensional average Itoˆ differential equation is obtained by simplifying the infinite dimensional stochastic delay differential equation with the perturbation method. Secondly, the global and local stability of the rotor system are obtained by analyzing the singular boundary theory and the maximum Lyapunov exponent. Then, the conditions and types of stochastic bifurcation of rotor system are obtained by analyzing the steady-state joint probability density function. Finally, numerical simulation verifies the accuracy of the theoretical analysis, showing that the time delay affects rotor system to reach the stable critical speed for the first time, and the noise disturbance has a certain stability effect on the rotor system.

Neural network approximation of optimal controls for stochastic reaction-diffusion equations.

We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.

Optimal projection filters with information geometry

We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1–33, 2016) where a direct L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176–213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.

Nonlinear dynamic analysis of a stochastic delay wheelset system

Considering not only the stochastic track irregularity and the possible effect of stochastic parameter excitation, but also the time delay of spring, that is, its response is in place, but the force generated is not in place, a stochastic delay wheelset system is established. The infinite dimensional system is reduced to the finite dimensional stochastic differential equation by using the center manifold, and further reduced to a one-dimensional diffusion process by using the stochastic averaging method. The stability of the wheelset system is obtained by analyzing the singular boundary theory and calculating the maximum Lyapunov exponent. The conditions and types of stochastic bifurcation in wheelset system are obtained by combining probability density function. The numerical simulation verifies the correctness of the theoretical analysis and shows that the time delay affects the critical hunting instability speed of wheelset. The stochastic term affects the lateral displacement of the stochastic delay wheelset system.

Exact Solution and Projection Filters for Open Quantum Systems Subject to Imperfect Measurements

In this letter, we consider an open quantum system undergoing imperfect and indirect measurement. For quantum non-demolition (QND) measurement, we show that the system evolves on an appropriately chosen manifold and we express the exact solution of the quantum filter equation in terms of the solution of a lower dimensional stochastic differential equation. In order to further reduce the dimension of the system under study, we consider the projection on the lower dimensional manifold originally introduced in Gao et al. (2019) for the case of perfect measurements. An error analysis is performed to evaluate the precision of this approximate quantum filter, focusing on the case of QND measurement. Simulations suggest the efficiency of the proposed quantum projection filter, even in presence of a stabilizing feedback control which depends on the projection filter.

Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions

We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum probability: a Grassmann random variable is an homomorphism of an abstract Grassmann algebra into a quantum probability space, i.e. a C^{*}-algebra endowed with a suitable state. We define the notion of Gaussian processes, Brownian motion and stochastic (partial) differential equations taking values in Grassmann algebras. We use them to study the long time behavior of finite and infinite dimensional Langevin Grassmann stochastic differential equations driven by Gaussian space-time white noise and to describe their invariant measures. As an application we give a proof of the stochastic quantization and of the removal of the space cut-off for the Euclidean Yukawa model.

A deep neural network method for solving partial differential equations with complex boundary in groundwater seepage

High dimensional stochastic partial differential equations (SPDEs) attract a lot of attention because of their application in uncertainty quantification (UQ) of ore deposits, petroleum geology and other fields. The groundwater flow in heterogeneous media affects all geological processes, including diagenesis, mineralization and oil accumulation. A point of the deep neural network (DNN) method in machine learning is that it can effectively output the solution of the SPDEs with the complex boundaries such as Newman boundary in a direct way. We propose a novel methodology to construct approximate solutions for SPDEs in groundwater flow by combining two deep convolutional residual networks, which can complete the training without interference in the help of an adaptive functional factor. One of the two network models deals with the inner observed points and the other one is required to satisfy the boundary conditions. This model has great potential for solving more complex boundary problems. Another contribution of this work is that the loss function to train the proposed DNN is obtained by deducing the energy functional based on the variational principles, which integrates the physical meaning into the proposed DNN. In addition, we theoretically prove that the minimization loss function problem has a unique solution, and it always conforms to the weak solution form of the problem no matter how the uncertain parameters change. An unconstrained optimization problem, rather than a constrained problem, can be solved without adding any penalty terms. The practical application shows that our model has low computational cost and strong efficiency for solving high-dimensional SPDEs.

Ising Model on Trees and Factors of IID

We study the ferromagnetic Ising model on the infinite d-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature $$\beta \ge 0$$ satisfies $$\tanh \beta \le (d-1)^{-1}$$ . However, in the reconstruction regime ( $$\tanh \beta > (d-1)^{-\frac{1}{2}}$$ ), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons (Comb Probab Comput 2(2):285–300, 2017). The solution $$\{X_t(v) \}$$ of the SDE is distributed as $$\begin{aligned} X_t(v) = t\tau _v + B_t(v), \end{aligned}$$ where $$\{\tau _v \}$$ is an Ising sample and $$\{B_t(v) \}$$ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever $$\tanh \beta \le c(d-1)^{-\frac{1}{2}}$$ , where $$c>0$$ is an absolute constant.

The dual Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations

We provide the dual result of the Yamada–Watanabe theorem for mild solutions to semilinear stochastic partial differential equations with path-dependent coefficients. An essential tool is the so-called “method of the moving frame”, which allows us to reduce the proof to infinite dimensional stochastic differential equations.

Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients

We provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations in the framework of the semigroup approach with locally monotone coefficients. An important component of the proof is an application of the dilation theorem of Nagy, which allows us to reduce the problem to infinite dimensional stochastic differential equations on a larger Hilbert space. Properties of the solutions like the Markov property are discussed as well.

Disordered high-dimensional optimal control

Mean field optimal control problems are a class of optimization problems that arise from optimal control when applied to the many body setting. In the noisy case one has a set of controllable stochastic processes and a cost function that is a functional of their trajectories. The goal of the optimization is to minimize this cost over the control variables. Here we consider the case in which we have N stochastic processes, or agents, with the associated control variables, which interact in a disordered way so that the resulting cost function is random. The goal is to find the average minimal cost for N → ∞, when a typical realization of the quenched random interactions is considered. We introduce a simple model and show how to perform a dimensional reduction from the infinite dimensional case to a set of one dimensional stochastic partial differential equations of the Hamilton–Jacobi–Bellman and Fokker–Planck type. The statistical properties of the corresponding stochastic terms must be computed self-consistently, as we show explicitly.

Variational approach to nonlinear stochastic differential equations in Hilbert spaces

Here we survey a few functional methods to existence theory for infinite dimensional stochastic differential equations of the form dX+A(t)X(t)=B(t,X(t))dW(t), X(0)=X_0, where A(t) is a non\-linear maximal monotone operator in a variational couple (V,V'). The emphasis is put on a new approach of the classical existence result of N. Krylov and B. Rozovski on existence for the infinite dimensional stochastic differential equations which is given here via the theory of nonlinear maximal monotone operators in Banach spaces. A variational approach to this problem is also developed.

A new discretization scheme for one dimensional stochastic differential equations using time change method

We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of an SDE using a time-changed Brownian motion, which dates back to Doeblin (1940). In cases where the diffusion coefficient is bounded and is β-Holder continuous with 0<β≤1, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat SDEs with no strong solution. Our scheme is the first to achieve strong convergence for the case of 0<β<1∕2.

Asymptotic expansion of the transition density of the semigroup associated to a SDE driven by Lévy noise

In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L ( t ) = L t , t &gt; 0. The transition probability density p t , t &gt; 0 of the semigroup associated to the solution u t , t ⩾ 0 of the SDE is given by a power series expansion. The series expansion of p t can be re-expressed in terms of Feynman graphs and rules. We will also prove that p t , t &gt; 0 has an asymptotic expansion in power of a parameter β &gt; 0, and it can be given by a convergent integral. A remark on some applications will be given in this work.

Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions

We study a stochastic optimal control problem for a two scale system driven by an infinite dimensional stochastic differential equation which consists of “slow” and “fast” components. We use the theory of viscosity solutions in Hilbert spaces to show that as the speed of the fast component goes to infinity, the value function of the optimal control problem converges to the viscosity solution of a reduced effective equation. We consider a rather general case where the evolution is given by an abstract semilinear stochastic differential equation with nonlinear dependence on the controls. The results of this paper generalize to the infinite dimensional case the finite dimensional results of Alvarez and Bardi [SIAM J. Control Optim.40(2001/02) 1159–1188] and complement the results in Hilbert spaces obtained recently in Guatteri and Tessitore [To appear in:Appl. Math. Optim.(2019)https://doi.org/10.1007/s00245-019-09577-y].

A numerical approach to Kolmogorov equation in high dimension based on Gaussian analysis

For Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic Partial Differential Equations (SPDE) and thus contains an underlying Gaussian process which is the key of the algorithm. A series development of the solution in terms of iterated integrals of the Gaussian process is given, it is proved to converge - also in the infinite dimensional limit - and it is numerically tested in a number of examples.

Infinite dimensional affine processes

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada–Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

A variational approach to nonlinear stochastic differential equations with linear multiplicative noise

One introduces a new concept of generalized solution for nonlinear infinite dimensional stochastic differential equations of subgradient type driven by linear multiplicative Wiener processes. This is defined as solution of a stochastic convex optimization problem derived from the Brezis-Ekeland variational principle. Under specific conditions on nonlinearity, one proves the existence and uniqueness of a variational solution which is also a strong solution in some significant situations. Applications to the existence of stochastic total variational flow and to stochastic parabolic equations with mild nonlinearity are given.

An Experimental Implementation for Stochastic Differential Equation Using Euler and Milstein Schemes in One and Two Dimensional Stochastic Differential Equation

An Experimental Implementation for Stochastic Differential Equation Using Euler and Milstein Schemes in One and Two Dimensional Stochastic Differential Equation

Iterated stochastic integrals in infinite dimensions: approximation and error estimates

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.