Stochastic Convolution Research Articles

A weak solution theory for stochastic Volterra equations of convolution type

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in Lp spaces. The main tools are new a priori estimates on Sobolev–Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

A two-parameter Milstein method for stochastic Volterra integral equations

In this paper, a two-parameter Milstein method for stochastic Volterra integral equations is introduced. First, the method is proved to be strongly convergent with order one in Lp norm (p≥1). Then, we investigate the mean square stability of the exact and numerical solutions of a stochastic convolution test equation. Stability conditions are derived. Based on these conditions, analytical and numerical stability regions are plotted and compared with each other. The results show that additional implicitness offers benefits for numerical stability. Finally, some numerical experiments are carried out to confirm the theoretical results.

Strong solutions of semilinear SPDEs with unbounded diffusion

We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded diffusion operators driven by cylindrical Brownian motion via the mild solution approach. In the case of finite dimensional driving noise, provided sufficient regularity on the coefficients, we establish existence and uniqueness of strong solutions. In the case of “critical unboundedness”, we show how to use the stochastic compactness method to obtain a martingale solution.

Evolution of histopathological breast cancer images classification using stochastic dilated residual ghost model

Breast cancer detection is a complex problem to solve, and it is a topic that is still being studied. Deep learning-based models aid medical science by helping to classify benign and malignant cancers and saving lives. Breast cancer histopathological image classification (BreakHis) and breast cancer histopathological annotation and diagnosis (BreCaHAD) datasets are used in the proposed model. The study led to the resolution of four essential issues: 1) Addresses the color divergence issue caused by strain normalization during image generation 2) Data augmentation uses several factors like as flip, rotation, shift, resize, and gamma value in order to overcome overfitting concerns caused by a lack of histopathology images. 3) Using the proposed stochastic dilated convolution (SDC) model, able to find missing features and improve tiny and low-level features such as edge, contour, and color correctness. This model effectively solves depth issues with stochastic pooling and max pooling by combining three proposed units: spatial dilation convolution, channel dilation convolution, and dilated convolution to detect missing features and obtain more information on images. Stochastic pooling may choose any value since it is classified as a heterogeneous approach that can determine tiny and large values. When the max pooling and stochastic pooling are combined, a decent feature map is produced, processed by the dilatation unit, detecting cancer cells without adding to the complexity. 4) To correctly identify breast cancer and extract depth characteristics, the stochastic dilated residual ghost (SDRG) model uses the proposed SDC model, a ghost unit, stochastic upsampling, and downsampling units. The proposed feature selection unit employs linear transformations to produce feature maps to show information based on intrinsic characteristics to eliminate redundant or similar features in convolutional neural networks. Convolution and identity mapping are used to construct and retain intrinsic feature mappings in this skip unit while upsampling with stochastic pooling is used to minimize feature dimensions. The proposed model?s performance was assessed with 150,271 augmented and original histopathology images. The proposed method?s findings compare favorably to eight popular approaches. In addition, the suggested model outperforms the competition in terms of accuracy, average precision score, precision, sensitivity, and f1 score. With an accuracy of 98.41% with BreakHis and 98.60% with the BreCaHAD dataset with enhanced images, the proposed method exceeds numerous state-of-the-art methods.

On a generalized stochastic Burgers' equation perturbed by Volterra noise

In this article, we investigate the existence and uniqueness of local mild solutions for the one-dimensional generalized stochastic Burgers' equation (GSBE) containing a nonlinearity of polynomial type and perturbed by α-regular cylindrical Volterra process and having Dirichlet boundary conditions. The Banach fixed point theorem (or contraction mapping principle) is used to obtain the local solvability results. The L∞-estimate on both time and space for the stochastic convolution involving the α-regular cylindrical Volterra process is obtained with the help of Garsia-Rodemich-Rumsey inequality. Further, the existence and uniqueness of global mild solution of GSBE up to third order nonlinearity is also shown.

Regularity of fractional stochastic convolution and its application to fractional stochastic chaotic systems

The current paper is devoted to the regularity of Caputo-type fractional Onstein–Ulenbeck processes with fractional Brownian motion, and it is applied to establish the existence and uniqueness of the mild solution for Caputo-type fractional Rössler equation and fractional Chen equation with fractional Brownian motion. Some numerical simulations and examples are provided to support the theoretical results.

Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if (S(t,s))_{0leqslant sle tleqslant T} is a C_0-evolution family of contractions on a 2-smooth Banach space X and (W_t)_{tin [0,T]} is a cylindrical Brownian motion on a probability space (Omega ,{mathbb {P}}) adapted to some given filtration, then for every 0<p<infty there exists a constant C_{p,X} such that for all progressively measurable processes g: [0,T]times Omega rightarrow X the process (int _0^t S(t,s)g_s,mathrm{d} W_s)_{tin [0,T]} has a continuous modification and Esupt∈[0,T]‖∫0tS(t,s)gsdWs‖p⩽Cp,XpE(∫0T‖gt‖γ(H,X)2dt)p/2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {E}}\\sup _{t\\in [0,T]}\\Big \\Vert \\int _0^t S(t,s)g_s\\,\\mathrm{d} W_s \\Big \\Vert ^p\\leqslant C_{p,X}^p {\\mathbb {E}} \\Bigl (\\int _0^T \\Vert g_t\\Vert ^2_{\\gamma (H,X)}\\,\\mathrm{d} t\\Bigr )^{p/2}. \\end{aligned}$$\\end{document}Moreover, for 2leqslant p<infty one may take C_{p,X} = 10 D sqrt{p}, where D is the constant in the definition of 2-smoothness for X. The order O(sqrt{p}) coincides with that of Burkholder’s inequality and is therefore optimal as prightarrow infty . Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale g_t,mathrm{d} W_t is replaced by more general X-valued martingales ,mathrm{d} M_t. Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs dut=A(t)utdt+gtdWt,u0=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\,\\mathrm{d} u_t = A(t)u_t\\,\\mathrm{d} t + g_t\\,\\mathrm{d} W_t, \\quad u_0 = 0, \\end{aligned}$$\\end{document}where the family (A(t))_{tin [0,T]} is assumed to generate a C_0-evolution family (S(t,s))_{0leqslant sleqslant tleqslant T} of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

Decompositions of stochastic convolution driven by a white-fractional Gaussian noise

Decompositions of stochastic convolution driven by a white-fractional Gaussian noise

Discrete variance swap in a rough volatility economy

AbstractThe discrete variance swap is one of the most popular volatility derivatives traded on the over‐the‐counter market. This paper discusses its valuation in a rough volatility economy and the impact of roughness on the term structure of discrete variance swap prices. A semianalytic solution is obtained through stochastic convolution. Our numerical experiments show that the roughness of volatility has a significant impact on the concavity of the variance swap term structure.

The stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise

We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schrödinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schrödinger equation in L2(Rn) with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.

Limit theorems for singular Skorohod integrals

In this paper we prove the convergence in distribution of sequences of It\^o and Skorohod integrals with integrands having singular asymptotic behavior. These sequences include stochastic convolutions and generalize the example $\sqrt n\int _0^1 t^n B_tdB_t$ first studied by Peccati and Yor in 2004.

Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.

WELL-POSEDNESS AND CONVERGENCE FOR TIME-SPACE FRACTIONAL STOCHASTIC SCHRÖGER-BBM EQUATION

<abstract> In this paper, the Banach fixed point theorem combined with Mittag-Leffler functions has been used to obtain the existence and uniqueness of global mild solution for a kind of time-space fractional stochastic Schr򤩮ger-BBM equation driven by Gaussian noise. The spatial-temporal regularity of the nonlocal stochastic convolution is established. Furthermore the convergence and simulation is provided by the Galerkin finite element method as well. </abstract>

Kernel estimation for Lévy driven stochastic convolutions

Abstract We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X ⁢ ( t ) = ∫ 0 t a ⁢ ( t - s ) ⁢ d Z ⁢ ( s ) X(t)=\int_{0}^{t}a(t-s)\,dZ(s) , where 𝑍 is a Lévy martingale and the kernel a ( . ) a(\,{.}\,) a deterministic function square integrable on R + \mathbb{R}^{+} . Given 𝑁 i.i.d. continuous time observations ( X i ⁢ ( t ) ) t ∈ [ 0 , T ] (X_{i}(t))_{t\in[0,T]} , i = 1 , … , N i=1,\dots,N , distributed like ( X ⁢ ( t ) ) t ∈ [ 0 , T ] (X(t))_{t\in[0,T]} , we propose two types of nonparametric projection estimators of a 2 a^{2} under different sets of assumptions. We bound the L 2 \mathbb{L}^{2} -risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions

In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform H\"older continuity in time, and strong convergence rates for a class of fully discrete exponential Euler-type numerical approximations of infinite dimensional stochastic convolution processes. The considered approximations involve specific taming and truncation terms and are therefore well suited to be used in the context of SPDEs with non-globally Lipschitz continuous nonlinearities.

A stochastic convolution integral inequality

We prove a stochastic Gronwall lemma of the convolution type. Our results extend that of Scheutzow [A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350019], and the related results established in the non-convolution case. The proofs of the present results are essentially based on the Métivier–Pellaumail inequality for semimartingales.

Joint UAV Access and GEO Satellite Backhaul in IoRT Networks: Performance Analysis and Optimization

With the growing demand for communications in remote and dispersed areas, Internet-of-Remote Things (IoRT) networks with joint unmanned aerial vehicle (UAV) access and geostationary orbit (GEO) satellite backhaul hold great promise to provide sufficient access services to Internet-of-Things (IoT) users and devices. As the fundamental of the performance optimization of IoRT networks, the performance analysis sheds light on the relationship between the network performance (i.e., backlog, delay, and throughput) and access scale (i.e., the numbers of UAVs and UAV users). Aiming at the challenges brought by the complex network structure (i.e., two-level queuing network along with the converged traffic), we introduce the stochastic network calculus-based min-plus convolution and the leftover service to mathematically describe the complex structure. For the analytical challenges of the continuous-time arrival process and heterogeneous two-level link capacities, we innovatively prove their supermartingale features and further derive the closed-form expressions of the network backlog and delay bounds based on the martingale theory. To pursue higher throughput while guaranteeing delay performance, we formulate a mixed-integer optimization problem of the access scale that contains a nondifferentiable variable derived from a transcendental equation. For the tractability, we propose a three-directional iterative (TDI) algorithm to search the optimal solution of the optimization problem. Simulation results verify the tightness of our performance bounds in contrast to the standard bound and the effectiveness of the proposed algorithm.

CARMA Approximations and Estimation

CARMA($p,q$) processes are compactly defined through a stochastic differential equation (SDE) involving $q+1$ derivatives of the Levy process driving the noise, despite this latter one has in general no differentiable paths. We replace the Levy noise with a continuously differentiable process obtained by stochastic convolution. The solution of the new SDE is then a stochastic process which converges to the original CARMA in an $L^2$ sense. CARMA processes are largely applied in finance, and this article mathematically justifies their representation via SDE. Consequently, the use of numerical schemes in simulation and estimation by approximating derivatives with discrete increments, is also justified for every time step. This provides a link between discrete-time ARMA models and CARMA models in continuous-time. We then analyse an Euler method and find a rate of convergence proportional to the square of the time step for discretization and to the inverse of the convolution parameter. These must then be adjusted to get accurate approximations.

Maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations.

This paper presents a survey of maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'.

Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.