Anna Karczewska
In the paper stochastic Volterra equations of nonscalar type in Hilbert space are studied. The aim of the paper is to provide some results on stochastic convolution and mild solutions to those Volterra equations. The motivation of the paper comes from a model of aging viscoelastic materials. The pseudo-resolvent approach is used.
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We consider convolution-type stochastic Volterra equations with additive Hilbert-valued fractional Brownian motion, $0<H<1$. We find the weak solution to this stochastic Volterra equation, and study its stochastic integral part, the stochastic convolution, which we show to be mean-zero Gaussian. We develop an Itô isometry for stochastic integrals with respect to a Hilbert-valued fractional Brownian motion, and use it to compute the covariance of the stochastic convolution. This formula, which uses fractional integrals and derivatives, generalizes the well-known formula from the case $H=1/2$.
We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued \Levy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.
In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties of stochastic convolutions are given. The paper provides a sufficient condition for a stochastic convolution to be a strong solution to a stochastic Volterra equation.
Limit theorems for stochastic Volterra processes
本文提出一个抽象的Hilbert空间值框架，用于研究具有算子值Volterra核的随机Volterra方程的马尔可夫提升。主要结果包括极限分布和平稳过程的存在性与特征化，大数定律（含收敛速率）以及在高斯吸引域内过程时间平均的中心极限定理。特别地，作者研究了基于拉普拉斯变换的加权Hilbert空间密度和基于移位半群的Filipović空间的马尔可夫提升，并展示了分数随机Volterra方程在加性或乘性高斯噪声情况下的结果。
In the paper regularity of solutions to stochastic Volterra equations in a separable Hilbert space is studied. Sufficient conditions for the temporal and spatial regularity of stochastic convolutions corresponding to the equations under consideration are provided. The results obtained generalize some well-known regularity results for solutions to stochastic differential equations. The paper is a continuation of previous author's papers concerning stochastic Volterra equations.
The aim of this note is to provide some results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolution of the form $W^Ψ(t):=\int_0^t S(t-τ)Ψ(τ)dW(τ)$, $t\geq 0$, where $S(t), t\geq 0$, is so-called {\em resolvent} for Volterra equation considered,$Ψ$ is an appropriate process and $W$ is a cylindrical Wiener process.
Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations
针对具有完全单调核的随机Volterra积分方程，提出在可分Hilbert空间上的马尔可夫提升新框架，证明提升随机发展方程解的存在性、唯一性与马尔可夫性，并通过渐近耦合方法建立渐近对数Harnack不等式。
Large-time behaviour of solutions to stochastic evolution equations driven by two-sided regular Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions for the existence of limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation driven by the two-sided Rosenblatt process.
Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. In this paper we prove that Picard iterations of BSDEs with globally Lipschitz continuous nonlinearities converge exponentially fast to the solution. Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order $α\in (\frac{1}{2},1)$ under a weaker condition…
The aim of this work is to present, in self-contained form, results concerning fundamental and the most important questions related to linear stochastic Volterra equations of convolution type. The paper is devoted to study the existence and some kind of regularity of solutions to stochastic Volterra equations in Hilbert space and the space of tempered distributions, as well. In recent years the theory of Volterra equations, particularly fractional ones, has undergone a big development. This is an emerging area of research with interesting mathematical questions and various important applicatio…
Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.
Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise
本文在温和正则性假设下，通过将随机Volterra方程转化为Hilbert空间上的等价随机发展方程，利用随机控制论证建立了具有完全单调核和非退化噪声的SVEs的弱适定性，并揭示了奇异核的噪声正则化效应。
Limits of stochastic Volterra equations driven by Gaussian noise
研究高斯噪声驱动的随机Volterra方程，推导温和形式，证明解的存在唯一性、稳定性及极限分布，并刻画其与初始状态无关的条件。
In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of $α$-times resolvent families.
Optimal control of Volterra integral diffusions and application to contract theory
本文研究了一类随机Volterra积分方程的最优控制问题，通过动态规划方法和粘性解理论，在Sobolev空间中建立价值函数的抛物型方程，并应用于时间不一致的委托代理问题。
We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an Itô formula. Moreover, we derive…
We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space.
We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fields can be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation. This stochastic partial differential equation is formulated on a suitable Hilbert space of functions on the positive real line with values in the state s…
In this article we investigate the existence and uniqueness of the stochastic Volterra equation driven by a \levy noise of pure jump type. In particular, we consider the following type of equation $ du(t) = ( A\int_0 ^t b(t-s) u(s)\,ds) \, dt + F(t,u(t))\,dt+ \int_ZG(t,u(t), z) \tilde η(dz,dt) + \int_{Z_L}G_L(t,u(t), z) η_L(dz,dt) ;\, t\in (0,T], $, $u(0)=u_0$, where $Z$ and $Z_L$ are Banach spaces, $\tilde η$ is a time-homogeneous compensated Poisson random measure on $Z$ with \levy measure $ν$ capturing the small jumps, and $η_L$ is a time-homogeneous Poisson random measure on $Z_L$ with fin…
Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators
本文研究了具有完全单调核的随机Volterra积分方程的最优控制问题，通过马尔可夫提升方法将其转化为Hilbert空间上的随机微分方程问题，并证明了新的平滑性质，从而建立了Hamilton-Jacobi-Bellman方程的弱解存在性和唯一性，以及构造了最优反馈控制。
On the Kolmogorov equation associated with Volterra equations and Fractional Brownian Motion
本文研究由Riemann-Liouville型分数布朗运动扰动的Volterra方程所对应的SPDE，通过扩展漂移算子及其Fréchet微分证明Markov随机流的二次可微性，并据此在无穷维微积分意义下求解路径依赖Kolmogorov方程，同时指出温和形式分析中的未解障碍。
Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts
本文研究了随机Volterra积分方程的小时间中心极限定理，并提出了马尔可夫提升框架，用于处理粗糙波动率模型中的定价问题。
Kolmogorov equations for stochastic Volterra processes with singular kernels
本文为具有奇异卷积核的随机Volterra方程建立Kolmogorov方程，通过Hilbert空间和SPDE联系，证明期望满足倒向方程，并给出幂律核下的适定性条件。
This paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating a optimization problem …
In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form as $$ \dd X(t) + \left (\int_0^t b(t-s) A X(s) \, \dd s \right) \, \dd t = \dd W^{_Q}(t), t\in (0,T]; ~ X(0) =X_0\in H, $$ \noindent where $W^Q$ is a $Q$-Wiener process on the Hilbert space $H$ and where the time kernel $b$ is the locally integrable potential $t^{ρ-2}$, $ρ\in (1,2)$, or slightly more general. The operator $A$ is unbounded, li…
Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation
针对带正型记忆项和加性高斯噪声的Hilbert空间值线性随机发展方程，采用隐式Euler时间离散、Laplace变换卷积求积和连续有限元空间离散，证明了全离散解在均方意义下关于时间和空间步长的收敛阶。
Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications
本文研究了在希尔伯特空间中相互作用粒子系统的随机最优控制问题，建立了均场哈密尔顿-雅可比-贝尔曼方程的粘性解理论，并将其应用于经济学中的随机延迟微分方程和随机偏微分方程问题。
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