Anna Karczewska
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.
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In the paper stochastic Volterra equations with noise terms driven by series of independent scalar Wiener processes are considered. In our study we use the resolvent approach to the equations under consideration. We give sufficient condition for the existence of strong solution to the class of stochastic Volterra equations of convolution type. We provide regularity of stochastic convolution, as well.
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.
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 define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional Brownian motion with Hurst coefficient greater than 1/3
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…
Space-time regularity of linear stochastic partial differential equations is studied. The solution is defined in the mild sense in the state space $L^p$. The corresponding regularity is obtained by showing that the stochastic convolution integrals are Hölder continuous in a suitable function space. In particular cases, this allows to show space-time Hölder continuity of the solution. The main tool used is a hypercontractivity result on Banach-space valued random variables in a finite Wiener chaos.
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 $L^p$ 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 u…
We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and i…
We establish existence and uniqueness for infinite dimensional Riccati equations taking values in the Banach space L 1 ($μ$ $\otimes$ $μ$) for certain signed matrix measures $μ$ which are not necessarily finite. Such equations can be seen as the infinite dimensional analogue of matrix Riccati equations and they appear in the Linear-Quadratic control theory of stochastic Volterra equations.
In the paper we study stochastic convolution appearing in Volterra equation driven by so called Lévy process. By Lévy process we mean a process with homogeneous independent increments, continuous in probability and cadlag.
Weak solutions of Stochastic Volterra Equations in convex domains with general kernels
本文研究了具有连续系数和可能奇异非卷积核的随机Volterra方程的弱存在性，通过引入近似方案并证明其收敛性，强调了解在闭凸集内的随机不变性，并推导了非卷积Riccati方程的Laplace变换表示。
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.
We define a class of functions which have a known decay rate coupled with a periodic fluctuation. We identify conditions on the kernel of a linear summation convolution Volterra equation which give the equivalence of the kernel lying in this class of functions and the solution lying in this class of functions. Some specific examples are examined. In particular this theory is used to provide a counter--example to a result regarding the rate of decay of the auto--covariance function of an ARCH($\infty$) process.
On the existence of weak solutions to stochastic Volterra equations
本文通过新制定的局部鞅问题，证明了具有一般核漂移和卷积或有界核扩散的随机Volterra方程弱解的存在性。
This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product be…
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.
Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation
本文刻画保持非负性的卷积核，据此研究随机 Volterra 方程的闭凸集不变性与一维比较，并为指数组合核提出保持域内的二阶弱误差逼近，应用于粗糙 Heston 模型。
Feller's test for explosions of stochastic Volterra equations
本文针对一维卷积型连续随机Volterra过程的爆炸问题提出Feller检验，推导非奇异核下过程区间内存续的充要条件，通过三类过程验证结果，并讨论Volterra CIR过程的分数核指数和近似方法。
We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel $K(t,s)$ and inhomogeneous drift and diffusion coefficients $b(s,X_s)$ and $σ(s,X_s)$. In the case of affine $b$ and $σσ^T$ we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a kernel of convolution type $K(t,s)=\overline{K}(t-s)$ we establish existence of a solution to the stochastic inhomogeneous Vo…
Failure of the Markov property for stochastic Volterra equations
本文证明了具有Hölder连续系数的随机Volterra方程在广泛Volterra核下不具有马尔可夫性质，并指出其马尔可夫提升通常为无限维。
We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from \cite{HarangTindel} to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of…
A BDG inequality for stochastic Volterra integrals
本文为随机Volterra积分建立了Burkholder-Davis-Gundy型不等式，给出有限与无限区间上的上界，并用于量化不同核的方程间路径距离及均值回归方程的均匀时间界。
The Volterra square-root process on $\mathbb{R}_+^m$ is an affine Volterra process with continuous sample paths. Under a suitable integrability condition on the resolvent of the second kind associated with the Volterra convolution kernel, we establish the existence of limiting distributions. In contrast to the classical square-root diffusion process, here the limiting distributions may depend on the initial state of the process. Our result shows that the non-uniqueness of limiting distributions is closely related to the integrability of the Volterra convolution kernel. Using an extension of th…
Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional rough paths. The existence of such rough paths above a wide class of stochastic processes including the fractional Brownian motion is shown. As applications we consider various types of rough and stochastic (partial) differential equations such as rough differential equations with delay, stochastic Volterra equations driven by Gaussian processes and moving ave…
We extend the new approach introduced in arXiv:1912.02064v2 [math.PR] and arXiv:2102.10119v1 [math.PR] for dealing with stochastic Volterra equations using the ideas of Rough Path theory and prove global existence and uniqueness results. The main idea of this approach is simple: Instead of the iterated integrals of a path comprising the data necessary to solve any equation driven by that path, now iterated integral convolutions with the Volterra kernel comprise said data. This leads to the corresponding abstract objects called Volterra-type Rough Paths, as well as the notion of the convolution…
In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.
Linear-quadratic stochastic Volterra controls II: Optimal strategies and Riccati--Volterra equations
本文研究了具有奇异和非卷积型系数的随机Volterra积分方程线性二次控制问题，从新视角构建因果反馈策略框架，并通过Riccati-Volterra方程刻画最优策略的存在唯一性。
We study a two-dimensional process $(X, Y)$ arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes $X$ and $Y$ can be identified as continuous-state nonlinear branching processes where the evolution of $Y$ is negatively affected by $X$. Assuming that process $X$ extinguishes, i.e. it converges to $0$ but never reaches $0$ in finite time, and process $Y$ converges to $0$, we identify rather sharp conditions under which the process $Y$ exhibits, respe…
Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases
本文研究扰动线性随机Volterra方程的解空间特征，探讨$p$-可和性与可积性条件，并在对角噪声下分析解的收敛性。
We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" d…
We provide a unified treatment of pathwise Large and Moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhijara, Dupuis and Ellis. We show in particular how this framework encompasses most rough volatility models used in mathematical finance and generalises many recent results in the literature.
Scaling Limit Theorems for Multivariate Hawkes Processes and Stochastic Volterra Equations with Measure Kernel
本文建立了多元Hawkes过程的完全尺度极限定理，证明其弱收敛至具有位势测度核的多维随机Volterra方程的解，并给出尺度极限的指数仿射表示与正则性。
The note is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are proved by the fractional method of infinite dimensional stochastic calculus. The paper extends on non-semigroup case some results obtained earlier for semigroups.
Linear-quadratic stochastic Volterra controls I: Causal feedback strategies
本文研究了奇异非卷积系数SVIEs线性二次控制中的因果反馈策略，证明闭环系统解的唯一性，并引入两种新方程建立二次泛函的对偶原理与表示公式。
On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients
本文研究带跳跃和非Lipschitz系数的随机Volterra方程，建立非负解强存在与路径唯一性条件，并应用于Lévy驱动方程，定义alpha稳定CIR过程的Volterra扩展。
Exponential ergodicity and finite-dimensional approximation for Markovian lifts of stochastic Volterra equations
本文通过广义Harris定理建立了随机Volterra方程马尔可夫提升的指数ergodic性，并证明了其平稳解可由有限维SDEs弱近似。
In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on $C_0$-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates t…
Singular backward stochastic Volterra integral equations in infinite dimensional spaces
本文研究无限维奇异 BSVIEs，提出涵盖分数阶核等情形的奇异性条件并建立适定性，进而用于随机 Volterra 方程最优控制及时滞发展方程最大值原理。
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。
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 provide an exhaustive treatment of Linear-Quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than $1/2$ as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state varia…
Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations
本文研究随机Volterra方程马尔可夫近似的收敛性，在Hölder范数下给出误差界，并证明基于sinc求积的近似误差以指数速率衰减，数值实验验证了理论结果。
Stochastic Volterra Equations for Local Times of Spectrally Positive Lévy Processes with Gaussian Components
本文用由高斯白噪声与 Poisson 随机测度驱动的随机 Volterra 方程刻画谱正 Lévy 过程局部时，证明强存在唯一性，并建立比较原理、正则性及 Laplace 泛函表示。
Efficient simulation of a new class of Volterra-type SDEs
提出利用卷积核将Volterra型非马尔可夫过程转化为马尔可夫扩散过程的理论框架，并针对分数核情形设计高效模拟方案，实现1/2阶强收敛。
The Laplace transform of the integrated Volterra Wishart process
本文推导了积分Volterra Wishart过程条件拉普拉斯变换的显式表达式，利用Fredholm行列式与预解式及无限维矩阵Riccati方程给出两种表示，并通过有限维近似实现快速定价。
Moments in Rough Bergomi and Boundary Attainment in Rough Heston
本文证明了粗糙Bergomi价格过程在负相关下整个次临界范围内的有限正矩，并证明了粗糙Heston方差过程在任意正时间以正概率到达零边界。
Utility maximization in multivariate Volterra models
研究粗糙随机环境下多资产市场中幂效用投资者的最优投资组合问题，提出新的Volterra-Wishart随机波动率模型，利用卷积与预解式微积分验证最优策略，扩展至多变量情形并给出存在唯一性定理。
Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations
该研究针对描述带记忆随机系统的随机Volterra积分方程，建立Wong--Zakai收敛性，将镜像填充傅里叶神经算子的解算子学习框架扩展至该类方程，经数值实验验证其预测精度与泛化能力。
We study stochastic convolutions providing by fundamental solutions of a class of integrodifferential equations which interpolate the heat and the wave equations. We give sufficient condition for the existence of function--valued convolutions in terms of the covariance kernel of a noise given by spatially homogeneous Wiener process.
Extinction behaviour for mutually enhancing continuous-state population dynamics
本文研究了由两个独立布朗运动和谱正α-稳定随机测度驱动的随机微分方程系统，分析了不同参数下种群灭绝性质。
Markovian lifting and optimal control for integral stochastic Volterra equations with completely monotone kernels
本文提出利用完全单调核和马尔可夫提升技术，解决有限维积分随机Volterra方程的最优控制问题，并证明最优控制的存在性。
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。
Damage detection in an uncertain nonlinear beam based on stochastic Volterra series
本文提出基于随机 Volterra 级数的方法，通过高阶 Volterra 核的非线性分析，在存在系统不确定性的条件下成功检测出非线性振动梁中的呼吸裂纹。
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。
The Laplace transform of the first passage time density of the Ornstein--Uhlenbeck process for a constant threshold contains a ratio of two parabolic cylinder functions for which no analytical inversion formula is available. Recently derived inverse Laplace transforms for the product of two parabolic cylinder functions together with the convolution theorem of the Laplace transform then allow to derive a new Volterra integral equation for this first passage time density. The kernel of this integral equation contains a parabolic cylinder function and the Fortet renewal equation for the Ornstein-…
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