Anna Karczewska, Carlos Lizama
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.
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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.
The purpose of this paper is to establish the multivariate normal convergence for the average of certain Volterra processes constructed from a fractional Brownian motion with Hurst parameter H>1/2. Some applications to parameter estimation are then discussed.
In this note we prove an existence and uniqueness result of solution for stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2, showing also that the solution has finite moments. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
Based on the recent development of the framework of Volterra rough paths, we consider here the probabilistic construction of the Volterra rough path associated to the fractional Brownian motion with $H>\frac{1}{2}$ and for the standard Brownian motion. The Volterra kernel $k(t,s)$ is allowed to be singular, and behaving similar to $|t-s|^{-γ}$ for some $γ\geq 0$. The construction is done in both the Stratonovich and Itô sense. It is based on a modified Garsia-Rodemich-Romsey lemma which has an interest in its own right, as well as tools from Malliavin calculus. A discussion of challenges and p…
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 consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu-Nualart and Nualart-Saussereau where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space.
Aim of this work is to extend the results of Clément, Da Prato & Prüss on the fractional white noise perturbation with Hurst parameter 0<H<1. We will obtain similar results and it will turn out that the regularity of the solution u(t) of the stochastic Volterra equation increases with Hurst parameter H.
Central limit theorem of Multilevel Monte Carlo Euler estimators for Stochastic Volterra equations with fractional kernels
本文证明了带分数核的随机Volterra方程在Euler离散化下多层Monte Carlo估计量满足Lindeberg-Feller型中心极限定理。
This note is devoted to construct a rough path above a multidimensional fractional Brownian motion $B$ with any Hurst parameter $H\in(0,1)$, by means of its representation as a Volterra Gaussian process. This approach yields some algebraic and computational simplifications with respect to [Stochastic Process. Appl. 120 (2010) 1444--1472], where the construction of a rough path over $B$ was first introduced.
We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in…
This paper is devoted to study a class of stochastic Volterra equations associated with fractional Brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the …
We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an $N$-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equ…
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.
Quasihelix properties of selected Volterra Gaussian processes
本文研究了几种具有温控、幂加权和对数核的高斯Volterra过程的局部准螺旋和广义准螺旋性质，包括温控分数布朗运动和广义分数布朗运动型过程，探讨了参数值对性质的影响。
Strong solutions for singular SDEs driven by long-range dependent fractional Brownian motion and other Volterra processes
该研究针对长程依赖分数布朗运动等Volterra过程驱动的奇异随机微分方程，突破漂移项需Hölder连续的限制，建立了奇异漂移项下的强解存在唯一性结果，采用概率论证方法。
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.
We study the existence and uniqueness of solutions to stochastic differential equations with Volterra processes driven by Lévy noise. For this purpose, we study in detail smoothness properties of these processes. Special attention is given to two kinds of Volterra-Gaussian processes that generalize the compact interval representation of fractional Brownian motion and to stochastic equations with such processes.
Limit distribution of errors in discretization of stochastic Volterra equations with multidimensional kernel
本文研究了具有一般多维核结构的随机Volterra方程离散化误差的极限分布，将其推广至包含非分数型形式的对角矩阵核，并证明了重缩放误差过程的依律稳定收敛且在放宽假设下刻画了极限过程。
This work concerns stochastic Volterra equations with singular kernels. Under the suitable conditions, we prove the central limit theorem for them. Moreover, we apply our result to stochastic Volterra equations with the kernels of fractional Brownian motions with the Hurst parameter $H\in(0, 1)$.
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。
Our study aims to specify the asymptotic error distribution in the discretization of a stochastic Volterra equation with a fractional kernel. It is well-known that for a standard stochastic differential equation, the discretization error, normalized with its rate of convergence $1/\sqrt{n}$, converges in law to the solution of a certain linear equation. Similarly to this, we show that a suitably normalized discretization error of the Volterra equation converges in law to the solution of a certain linear Volterra equation with the same fractional kernel.
Path-dependent Fractional Volterra Equations and the Microstructure of Rough Volatility Models driven by Poisson Random Measures
本文研究了由Poisson随机测度驱动的粗糙波动率模型的微观结构，通过具有幂律衰减的Hawkes核刻画订单影响，证明波动率过程经缩放后收敛于分数Heston模型，并基于路径依赖Volterra方程解的存在唯一性得出结论。
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。
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 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
Volterra processes appear in several applications ranging from turbulence to energy finance where they are used in the modelling of e.g. temperatures and wind and the related financial derivatives. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. In this work we suggest to construct an approximating sequence of Lévy driven Volterra processes, by perturbation of the kernel function. In this way, one can obtain an approximating sequence of semimartingales. Then we consider fractional integration with respect…
By using a simple method based on the fractional integration by parts, we prove the existence and the Besov regularity of the density for solutions to stochastic differential equations driven by an additive Gaussian Volterra process. We assume weak regularity conditions on the drift. Several examples of Gaussian Volterra noises are discussed.
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.
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…
In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional…
We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This formula has a unifying character in the sense that it contains the classical Itō formula for Lévy processes as well as recent change-of-variable formulas for Gaussian processes such as fractional Brownian motion as special cases. Our result also covers fractional Lévy processes (with Mandelbrot-Van Ness kernel) and a wide class of related processes for which such a generalised Itō formula has not yet been available in the literature.
We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a time-dependent kernel with respect to a standard Brownian motion. For these processes which are natural generalization of fractional Brownian motion, we construct a stochastic integral and show some of its main properties: regularity with respect to time and kernel, transformation under an absolutely continuous change of probability, possible approximation schemes and Ito formula.
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。
Fast simulation of Volterra processes using random Fourier features with application to the log-stationary fractional Brownian motion
提出基于随机傅里叶特征（RFF）的Volterra过程快速模拟框架，通过核的谱表示和哈密顿蒙特卡洛采样实现高效近似，并应用于平稳分数布朗运动（S-fBM）核的Volterra过程。
Gaussian fluctuations for stochastic Volterra equations with small noise
本文研究小噪声下随机Volterra方程解围绕确定性极限的波动，利用 Malliavin 计算术证明波动过程满足中心极限定理，并提供收敛速率的最优估计，同时应用分数布朗运动核的随机Volterra方程进行示例说明。
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.
We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with Hölder exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with Hölder exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.
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。
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。
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…
Motivated by the potential applications to the fractional Brownianmotion, we study Volterra stochasticdifferential of the form~:\begin{equation}X\_t = x+ \int\_0^tK(t,s)b(s,X\_s)ds + \int\_0^tK(t,s) σ(s,X\_s)\,dB\_s ,\tag{E} \label{eq:sdefbm}\end{equation}where $(B\_s, \, s\in [0,1])$ is a one-dimensional standard Brownianmotion and $(K(t,s), \, t,s \in [0,1])$ is a deterministic kernelwhose properties will be precised below but for which we don't assumeany boundedness property.
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.
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…
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。
In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted $L^2$-processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as …
Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels
本文证明奇异核随机Volterra方程的Milstein格式收敛速率$n^{-2H}$最优，并揭示归一化误差的极限分布为随机系数线性Volterra方程的非零解。
On (fake) Stationarity in Stochastic Volterra Equations with Affine Drift and Regular Kernels
研究带仿射漂移与正则核的随机沃尔泰拉方程的伪平稳性，通过两种方法诱导伪平稳 regime，证明时移解弱收敛至L²平稳过程，并将结果应用于α≥1的指数-分数SVIE。
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.
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…
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。
The research presented in this article provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, even simple non-linear financial derivatives as vanilla European options are typically priced by means of Monte-Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this…
In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean space, second order SPDEs on complete Riemannian manifolds,…
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。
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。
Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian
本文提出基于 Inverse Gaussian 分布的 iVi 模拟方案，用于处理积分 Volterra 平方根过程和 Volterra Heston 模型，通过重构方程证明其弱收敛性，且在极少时间步长下即可收敛。
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。
Affine Volterra covariance processes and application to commodity markets
本文研究对称正半定矩阵锥上的仿射随机Volterra方程，建立弱存在性并推导Fourier-Laplace变换公式，进而扩展Gibson-Schwartz商品模型为Volterra-Wishart过程，使方差具有记忆性和随机瞬时相关性，同时保持仿射可处理性。
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 present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider
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