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, as well as stochastic Navier-Stokes equations are investigated.
We characterize the Markovian and affine structure of the Volterra Heston model in terms of an infinite-dimensional adjusted forward process and specify its state space. More precisely, we show that it satisfies a stochastic partial differential equation and displays an exponentially-affine characteristic functional. As an application, we deduce an existence and uniqueness result for a Banach-space valued square-root process and provide its state space. This leads to another representation of the Volterra Heston model together with its Fourier-Laplace transform in terms of this possibly infini…
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.
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…
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, we generalize to Gaussian Volterra processes the existence and uniqueness of solutions for a class of non linear backward stochastic differential equations (BSDE) and we establish the relation between the non linear BSDE and the partial differential equation (PDE). A comparison theorem for the solution of the BSDE is proved and the continuity of its law is studied.
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.
In this article, we study the ergodicity of neutral retarded stochastic functional differential equations driven by $α$-regular Volterra process. Based on the equivalence between neutral retarded stochastic functional differential equations and the stochastic evolution equation, we get the ergodicity of neutral retarded stochastic functional differential equations.
We study uniqueness for a class of Volterra-type stochastic integral equations. We focus on the case of non-Lipschitz noise coefficients. The connection of these equations to certain degenerate stochastic partial differential equations plays a key role.
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 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…
We propose a finite difference scheme to simulate solutions to a certain type of hyperbolic stochastic partial differential equation (HSPDE). These solutions can in turn estimate so called volatility modulated Volterra (VMV) processes and Lévy semistationary (LSS) processes, which is a class of processes that have been employed to model turbulence, tumor growth and electricity forward and spot prices. We will see that our finite difference scheme converges to the solution of the HSPDE as we take finer and finer partitions for our finite difference scheme in both time and space. Finally, we dem…
Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional brownian motion and the multifractional Ornstein-Uhlenbeck process. By an Itô formula, proven in the context of Malliavin calculus, the BSDE is associated to a linear second order partial differential equation with terminal condition whose solution is given by a Feynman-Kac type formula. An application to self-financing trading strategies is discussed.
We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.
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…
Using a multiplicative structure (for example that of a Banach algebra) and a partial order we construct a weak version of a Banach space valued stochastic integral with respect to square integrable martingales.
We consider stochastic (partial) differential equations appearing as Markovian lifts of affine Volterra processes with jumps from the point of view of the generalized Feller property which was introduced in e.g.~\cite{doetei:10}. In particular we provide new existence, uniqueness and approximation results for Markovian lifts of affine rough volatility models of general jump diffusion type. We demonstrate that in this Markovian light the theory of stochastic Volterra processes becomes almost classical.
We study a stochastic differential game between two players, controlling a forward stochastic Volterra integral equation (FSVIE). Each player has to optimize his own performance functional which includes a backward stochastic differential equation (BSDE). The dynamics considered are driven by time-changed Lévy noises, with absolutely continuous time-change process. We prove a sufficient maximum principle to characterize Nash equilibria and the related optimal strategies. For this we use techniques of control under partial information, and the non-anticipating stochastic derivative. The zero-su…
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 study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semilinear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the…
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of normalized maxima of partial sums.
Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore classical methods, like dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that by using {\em Malliavin calculus} it is possible to formulate a modified functional type of {\em maximum principle} suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a sufficient and a necessary maximum principle of this type, and then we use the …
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…
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.…
Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces
This paper is concerned with the relationship between forward-backward stochastic Volterra integral equations (FBSVIEs, for short) and a system of (non-local in time) path dependent partial differential equations (PPDEs, for short). Due to the nature of Volterra type equations, the usual flow property (or semigroup property) does not hold. Inspired by Viens-Zhang \cite{Viens-Zhang-2019} and Wang-Yong \cite{Wang-Yong-2019}, auxiliary processes are introduced so that the flow property of adapted solutions to the FBSVIEs is recovered in a suitable sense, and thus the functional Itô's formula is a…
In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation. Applications are given to ergodic control of stochastic partial differential equations.
This paper is devoted to studying stochastic parabolic evolution equations with additive noise in Banach spaces of M-type 2. We construct both strict and mild solutions possessing very strong regularities. First, we consider the linear case. We prove existence and uniqueness of strict and mild solutions and show their maximal regularities. Second, we explore the semilinear case. Existence, uniqueness and regularity of mild and strict solutions are shown. Regular dependence of mild solutions on initial data is also investigated. Finally, some applications to stochastic partial differential equa…
In this article, we show that every centered Gaussian measure on an infinite dimensional separable Fréchet space $X$ over $\mathbb R$ admits some full measure Banach intermediate space between $X$ and its Cameron-Martin space. We provide a way of generating such spaces and, by showing a partial converse, give a characterization of Banach intermediate spaces. Finally, we show an example of constructing an $α$-Hölder intermediate space in the space of continuous functions, $\mathcal C_0[0, 1]$ with the classical Wiener measure.
We obtain limit theorems for a class of nonlinear discrete-time processes $X(n)$ called the $k$-th order Volterra processes of order $k$. These are moving average $k$-th order polynomial forms: \[ X(n)=\sum_{0<i_1,\ldots,i_k<\infty}a(i_1,\ldots,i_k)ε_{n-i_1}\ldotsε_{n-i_k}, \] where $\{ε_i\}$ is i.i.d.\ with $\mathbb{E} ε_i=0$, $\mathbb{E} ε_i^2=1$, where $a(\cdot)$ is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for $X(n)$ to be well-defined in $L^2(Ω)$, and focus on central and non-central limit theorems. We show that normalized partia…
In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $p\in (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic parti…
