Anna Karczewska, Carlos Lizama
In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a $d$-dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.
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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…
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.
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.
We investigate the space-time regularity of the local time associated to Volterra-Lévy processes, including Volterra processes driven by $α$-stable processes for $α\in(0,2]$. We show that the spatial regularity of the local time for Volterra-Lévy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturba\Ption of ODEs by a Volterra-Lévy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differ…
This note aims to give an explicit solution for backward stochastic Volterra integral equations with linear time delayed generators. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ is expressed by Hida-Malliavin derivatives involving $Y$.
Infinite Horizon Optimal Control of Forward-Backward Stochastic Volterra Equations with Delay
研究无限时间 horizon 系统的最优控制问题，利用Hida-Malliavin微积分证明最大原理，并建立无限时间 horizon 向后随机Volterra积分方程的存在性和唯一性结果。
Well-posedness and regularity of mean-field backward doubly stochastic Volterra integral equations and applications to dynamic risk measures
本文研究平均场倒向双重随机Volterra积分方程，证明其M-解的适定性与比较定理，借助Malliavin微积分建立正则性，并应用于动态风险度量的性质分析。
In this paper, we study backward stochastic Volterra integral equations of type-I with time delayed generators. Under some condition (small time horizon or a Lipschitz constant), we derive an existence and uniqueness results. Next, with the help of two examples of this BSVIEs we provide this condition is necessary. However, for special class of delayed generators, the previous existence and uniqueness result hold for an arbitrary time horizon and Lipschitz constant. We end this paper by establishing under an additive assumptions, a path continuity property.
New approach to optimal control of delayed stochastic Volterra integral equations
本文利用Hida-Malliavin calculus研究延迟随机Volterra积分方程的最优控制，推导出逆向随机Volterra积分方程，并建立随机最大原理，为延迟随机系统的最优控制提供了严谨框架。
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 consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-$p$ Hölder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-Hölder) regularity of the solution together with a maximal type bou…
The theory of affine processes has been recently extended to the framework of stochastic Volterra equations with continuous trajectories. These so-called affine Volterra processes overcome modeling shortcomings of affine processes because they can have trajectories whose regularity is different from the regularity of the paths of Brownian motion. More specifically, singular kernels yield rough affine processes. This paper extends the theory by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure together with possible sin…
We consider the regularity of sample paths of Volterra processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a semimartingale and $F$ is a deterministic real-valued function. We derive the information on the modulus of continuity for these processes under regularity assumptions on the function $F$ and show that $M(t)$ has "worst" regularity properties at times of jumps of $X(t)$. We apply our results to obtain the optimal Hölder exponent for fractional Lévy processes.
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…
We consider the regularity of sample paths of Volterra-Lévy processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a Lévy process and $F$ is a deterministic real-valued function. We derive the spectrum of singularities and a result on the 2-microlocal frontier of $\{M(t)\}_{t\in [0,1]}$, under regularity assumptions on the function $F$.
Hölder regularity for backward stochastic Volterra integral equations and applications to numerical schemes
研究倒向随机Volterra积分方程（BSVIE）鞅被积函数的Hölder型正则性估计。通过用BSDE系统逼近BSVIE，利用Malliavin微积分建立鞅分量的一致正则性估计并取极限，还给出显式Euler格式及收敛速率。
In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient conditi…
Regular occupation measures of Volterra processes
本文研究Volterra过程的正则occupation测度，通过结合随机缝合引理和Euler近似，证明了其occupation测度、自交测度和时间边缘的时空正则性，并构建了非高斯Volterra过程类。
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.
Complementary regularity between the integrand and integrator is a well known condition for the integral $\int_0^T f(r) \, \mathrm{d} g(r)$ to exist in the Riemann-Stieltjes sense. This condition also applies to the multi-dimensional case, in particular the 2D integral $\int_{[0, T]^2} f(s,t) \, \mathrm{d} g(s,t)$. In the paper, we give a new condition for the existence of the integral under the assumption that the integrator $g$ is a Volterra covariance function. We introduce the notion of strong Hölder bi-continuity, and show that if the integrand possess this property, the assumption on com…
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 this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the It{ô} setting, which is also new in this jump setting.
Stochastic Volterra equations with Hölder diffusion coefficients
本文研究一维随机Volterra方程，在扩散系数局部Hölder连续且核足够正则的条件下证明了强解的存在性与逐路径唯一性，并分析了样本路径正则性、可积性和半鞅性质。
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…
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。
Weak solutions to distribution-dependent stochastic Volterra equations
本文在系数线性增长和连续性条件及核的 mild 正则性假设下，证明了分布依赖随机Volterra方程弱解的存在性，并研究了解的连续性和可积性。
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.
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.
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。
This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G* of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.
For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward-backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation parti…
In this paper, we establish the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. We first introduce the extended backward stochastic Volterra integral equations (EBSVIEs, for short). Under some mild conditions, we establish the well-posedness of EBSVIEs and obtain some regularity results of the adapted solution to the EBSVIEs via Malliavin calculus. We show that a given function expressed in terms of the solution to the EBSVIEs solves a certain system of non-local paraboli…
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…
We propose a novel class of tempo-spatial Ornstein-Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of ver…
We prove strong existence and uniqueness, and Hölder regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised kernels. We apply these results to the rough Heston model, with square-root diffusion coefficient, recently proposed in Mathematical Finance to model the volatility of asset prices.
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。
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, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the $L^2$-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a bac…
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…
Stochastic Volterra Equations for the Local Times of Spectrally Positive Stable Processes
本文研究谱正稳定过程局部时的空间演化动力学，证明其正半直线局部时满足由Poisson随机测度驱动的随机Volterra方程，借此得到正则性、矩界等结果，并拓展对偶方法建立Laplace泛函的指数-仿射表示。
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。
We extend the work of Delong and Imkeller (2010a,b) concerning Backward stochastic differential equations with time delayed generators (delay BSDE). We give moment and a priori estimates in general $L^p$-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in $L^p$. We introduce decoupled systems of SDE and delay BSDE (delay FBSDE) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDE via the usual representation formulas. We conclude with several path regularity…
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。
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…
In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.
Motivated by the traditional Lotka-Volterra competitive models, this paper proposes and analyzes a class of stochastic reaction-diffusion partial differential equations. In contrast to the models in the literature, the new formulation enables spatial dependence of the species. In addition, the noise process is allowed to be space-time white noise. In this work, wellposedness, regularity of solutions, existence of density, and existence of an invariant measure for stochastic reaction-diffusion systems with non-Lipschitz and non-linear growth coefficients and multiplicative noise are considered.…
Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games
本文研究了一类更一般的预期反向随机Volterra积分方程，建立了其适定性及比较定理，并首次提出非零和随机延迟Volterra积分方程的极大原理。
Strong approximation for stochastic Volterra equations by compound Poisson processes
本文提出通过复合泊松过程对具有奇异系数的随机Volterra方程进行强逼近，克服了传统Euler-Maruyama方法在时间不规则系数和奇异性下的局限性。
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。
On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics
本文研究路径依赖随机Volterra积分方程的强well-posed性，分析其在$ L^p $空间中的局部解的存在唯一性及路径正则性，并提出基于插值$ K $-积分欧拉-马尔可夫方案的数值方法，证明其在$ L^p $范数下的收敛性及收敛速率。
A Malliavin Calculus Approach to Backward Stochastic Volterra Integral Equations
本文运用 Malliavin 分析证明多维 BSVIEs 在含对角过程的非线性生成元下的适定性，并据此解释半线性 PDE 及刻画动态均值-方差投资组合。
This paper formulates and studies a linear quadratic (LQ for short) game problem governed by linear stochastic Volterra integral equation. Sufficient and necessary condition of the existence of saddle points for this problem are derived. As a consequence we solve the problems left by Chen and Yong in [3]. Firstly, in our framework, the term GX^2(T) is allowed to be appear in the cost functional and the coefficients are allowed to be random. Secondly we study the unique solvability for certain coupled forward-backward stochastic Volterra integral equations (FBSVIEs for short) involved in this g…
Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establish large deviations for enhanced fractional Brownian motion. Second, the q-variation embedding, combined with results of rough path theory, provides a different route to a regularity result for stochastic…
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。
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。
Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation
本文研究受加性高斯噪声扰动的带记忆半线性Volterra方程，证明其存在不变测度，并通过引入适应记忆核的函数空间恢复紧性，进而利用Lyapunov结构证明统计稳态具有更高正则性。
Functional integration by parts formulae for stochastic Volterra processes
本文通过Riemann-Liouville分数阶导数建立了随机Volterra过程方向导数的分数阶分部积分公式，揭示了粗糙性增强平滑效应的现象，并应用于前向和粗糙波动率模型。
In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Hölder space which is separable.
In this article we extend the framework of rough paths to processes of variable Hölder exponent or variable order paths. We show how a class of multiple discrete delay differential equations driven by signals of variable order are especially well suited to be studied in a path wise sense. In fact, under some assumptions on the Hölder regularity functions of the driving signal, the local Hölder regularity of the variable order signal may be close to 0, without more than C_{b}^{2} diffusion coefficients (in contrast to higher regularity requirements known from constant order rough path theory). …
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